=10

THE

SCIENTIFIC PAPERS

OF

JAMES CLEKK MAXWELL

CAMBRIDGE :

PRINTED BY a J. CLAY, M.A. AND SONS,
AT THE UNIVERSITY PRESS.

'

6 i -
A<?/2 in i

TABLE OF CONTENTS.

PAGE

- XXVII. On the Viscosity or Internal Friction of Air and other Gases (The

Bakerian Lecture) 1

- XXVIII. On the Dynamical Theory of Oases 26

XXIX. On the Theory of tlie Maintenance of Electric Currents by Mechanical

Work without the use of Permanent Magnets . . . . . 79

XXX. On the Equilibrium of a Spherical Envelope ...... 86

XXXI. On the best Arrangement for producing a Pure Spectrum on a Screen . 96

XXXII. The Construction of Stereograms of Surfaces . . . . . .101

XXXIII. On Reciprocal Diagrams in Space and their relation to Airy's Function of

Stress . 102

XXXIV. On Governors . . . . . . . . . . . .105

XXXV. "Experiment in Magneto- Electric Induction" (in a letter to W. R. Grove,

F.R.S.) ... 121

XXXVI. On a Method of Making a Direct Comparison of Electrostatic with Electro-

magnetic Force; with a Note on the Electromagnetic Theory of Light . 125

XXXVII. On the Cyclide * 144

XXXVIII. On a Bow seen on the Surface of Ice . . . . . . . .160

XXXIX. On Reciprocal Figures, Frames, and Diagrams of Forces .... 161

XL. On the Displacement in a Case of Fluid Motion ...... 208

XLI. Address to the Mathematical and Physical Sections of the British Associa-
tion (1870) 215 -

XLII. On Colour-Vision at different points of the Retina ..... 230

XLIII. On Hills and Dales 233

XLIV. Introductory Lecture on Experimental Physics ...... 241

XLV. On the Solution of Electrical Problems by the Transformation of Conjugate

Functions ............ 256

XL VI. On the Mathematical Classification of Physical Quantities .... 257

XLVII. On Colour Vision 267

XLVIII. On the Geometrical Mean Distance of Two Figures on a Plane . . . 280
XLIX. On the Induction of Electric Currents in an Infinite Plane Sheet of

uniform Conductivity 286

TABLE OF (OXTKXTS.

L. On the Condition that, in the Transformation of any Figure by Curvilinear

Co-ordinates in Three Dimensions, every angle in the new Figure shall

be equal to the corresponding angle in tiie original Figure . . .

LI. Reprint of Papers on Electrostatics and Magnetism, By Sir W. Thomson.

(Review) ...... .

LIL On the Proof of the Equations of Motion of a Connected System . .

UII. On o Problem in the Calculus of Variations in which the solution is dis-

continuous ............

LIV On Action at a Distance ..........

LV Elements of Natural Philosophy. By Professors Sir W. Thomson and

I'. G. Tait. (Review) ... ......

LVI. On the Tlieory of a System of Electrified Conductors, and other Physical
Theories involving Homogeneous Quadratic Functions ....

LVIL On the Focal Lines of a Refracted Pencil .......

LVI 1 1. An Essay on the Mathematical Principles of Physics. By the Rev. James
ChaUis, M.A., <tc. (Review) .........

LIX. On Loschmidt's Experiments on Diffusion in relation to tiie Kinetic Theory
of Oases . ...........

I A On the Final State of a System of Molecules in motion subject to forces of

any kind ............

LXL Faraday .............

LXII. Molecules (A Lecture) ...........

1A 1 1 1. On Double Refraction in a Viscous Fluid in Motion .....

LXIV. On Hamilton's Characteristic Function for a narrow Beam of Light . .

IAV. On the Relation of Geometrical Optics to other parts of Mathematics and

Physics . ........

I A VI. Plateau on Soap-Bubbles (Review) ........

LXVIL Grove's "Correlation of Physical Forces" (Review) .....

I. XVIII. On tiie application of Kirchhoff's Rules for Electric Circuits to the Solu-
tion of a Geometrical Problem ........

I A I \ Van der Wools on the Continuity of the Gaseous and Liquid States . .

LXX. On the Centre of Motion of the Eye ........

I A \ I . On the Dynamical Evidence of the Molecular Constitution of Bodies (A Lec-
ture) . . .........

LXX 1 1. On tiie Application of Hamilton's Characteristic Function to tfie Theory of
an Optical Instrument symmetrical about its axis .....

LXXIIL Atom ..............

IV. Attraction .............

LXXV. On Bow's method of drawing diagrams in Graphical Statics with illustra-
tions from Peaucelliers Linkage ........

LXXVL On the Equilibrium of Heterogeneous Substances ......

LXXVII. Diffusion of Gases through Absorbing Substances ......

LXXVIIL General considerations concerning Scientific Apparatus .....

PAOE

297

301

308

310
311

329
332

338
343

355
361
379
381

391
393

400

406
407
416

418

439
445

485

492
498
501
505

TABLE OF CONTENTS.

Vll

PAGE

LXXIX. Instruments connected with Fluids 523

LXXX. Whewell's Writings and Correspondence (Review) ..... 528

LXXXI. On Ohm's Law . . ......... 533

LXXXII. On the protection of buildings from lightning ..... 538

LXXXIII. Capillary Action 541

LXXXI V. Hermann Ludwig Ferdinand Helmholtz ... ... 592

LXXXV. On a Paradox in the Theory of Attraction 599

LXXXVI. On Approximate Multiple Integration between Limits by Summation . . 604

LXXX VII. On the Unpublished Electrical Papers of the Hon. Henry Cavendish . . 612

LXXXVIII. Constitution of Bodies 616

LXXXIX. Diffusion 625

XC. Diagrams 647

XCI. Tait's Thermodynamics (Review) 660

XCII. On the Electrical Capacity of a long narrow Cylinder and of a Disk

of sensible thickness . . . . . . . . . .672

XCIII. On Stresses in Ratified Gases arising from Inequalities of Temperature . 681

XCIV. On Boltzmann's Theorem on the average distribution of energy in a system

of material points 713

XCV. The Telephone (Rede Lecture) . .... .742

XCVI. Paradoxical Philosophy (A Review) 756

XCVII. Ether 763

XCVIII. Thomson and Tait's Natural Philosophy (A Revieiv) . 776

XCIX. Faraday 786

C. Reports on Special Branches of Science .... . . 794

CI. Harmonic Analysis 797

ERRATA. VOL. II.

Pfcge 3, line 5, read I. p. 25 instead of xxxi. p. 30.
„ 73, „ 14, „ 60 „ 76.

„ 188, „ 27, „ p. (181) „ p. (189).

cPF
91> " b' " p»~~pdydz " ~V dxdy'

201, „ 10, insert // after the differential operator in equation (18).

In the Bakerian Lecture at page 18, line 21, the value of - r* is given as 11 12-8, where

]

r is the radius of each of the movable discs used iu the experiments on Viscosity. But with
the value of the diameter as given at page 4, line 14, this number should be 1220-8. The
values of the quantity A in the fourth line of the Table given at page 19 should therefore all be
increased by 108. The values of Q, the quantity in the fifth line, increase in the same proportion
as the values of A. Hence according to equations (23) and (24) the values of /*, the coefficient
of Viscosity, will be smaller than they appear in the text and will approximate to those obtained
by more recent experiments. The above inaccuracy in the numerical reductions was pointed out by
Mr Leahy, Pembroke College, Cambridge.

[From the Philosophical Transactions, Vol. CLVI.]

XXVII. THE BAKERIAN LECTURE. — On the Viscosity or Internal Friction

of Air and other Gases.

Received November 23, 1865,— Read February 8, 1866.

THE gaseous form of matter is distinguished by the great simplification
which occurs in the expression of the properties of matter when it passes into
that state from the solid or liquid form. The simplicity of the relations between
density, pressure, and temperature, and between the volume and the number
of molecules, seems to indicate that the molecules of bodies, when in the gaseous
state, are less impeded by any complicated mechanism than when they subside
into the liquid or solid states. The investigation of other properties of matter
is therefore likely to be more simple if we begin our research with matter
in the form of a gas.

The viscosity of a body is the resistance which it offers to a continuous
change of form, depending on the rate at which that change is effected.

All bodies are capable of having their form altered by the action of suf-
ficient forces during a sufficient tune. M. Kohlrausch* has shewn that torsion
applied to glass fibres produces a permanent set which increases with the time
of action of the force, and that when the force of torsion is removed the fibre
slowly untwists, so as to do away with part of the set it had acquired.
Softer solids exhibit the phenomena of plasticity in a greater degree ; but the
investigation of the relations between the forces and their effects is extremely
difficult, as in most cases the state of the solid depends not only on the forces
actually impressed on it, but on all the strains to which it has been subjected
during its previous existence.

* "Ueber die elastische Nachwerkung bei der Torsion," Pogg. Ann. cxix. 1863.
VOL. II. 1

OK THE VISCOSITY OB INTERNAL FRICTION

Professor W. Thomson • has shewn that something corresponding to internal
friction take* place in the torsional vibrations of wires, but that it is much
increased if the wire has been previously subjected to large vibrations. I have
also found that, after heating a steel wire to a temperature below 120°, ita
elasticity was permanently diminished and its internal friction increased.

The viscosity of fluids has been investigated by passing them through
capillary tubes t, by swinging pendulums in them*, and by the torsional vibra-
tions of an immersed disk§, and of a sphere filled with the fluid ||.

The method of transpiration through tubes is very convenient, especially
for comparative measurements, and in the hands of Graham and Poiseuille it
has given good results, but the measurement of the diameter of the tube is
difficult, and on account of the smallness of the bore we cannot be certain that
the action between the molecules of the gas and those of the substance of the
tubes does not affect the result. The pendulum method is capable of great
iiccuracy, and I believe that experiments are in progress by which its merits
us a means of determining the properties of the resisting medium will be tested.
The method of swinging a disk in the fluid is simple and direct. The chief
difficulty is the determination of the motion of the fluid near the edge of the
disk, which introduces very serious mathematical difficulties into the calculation
of the result. The method with the sphere is free from the mathematical
difficulty, but the weight of a properly constructed spherical shell makes it un-
suitable for experiments on gases.

In the experiments on the viscosity of air and other gases which I propose
to describe, I have employed the method of the torsional vibrations of disks,
but instead of placing them in an open space, I have placed them each between
two parallel fixed disks at a small but easily measurable distance, in which
case, when the period of vibration is long, the mathematical difficulties of deter-
mining the motion of the fluid are greatly reduced. I have also used three

* Proceedings of the Royal Society, May 18, 1865.

t Liquids : Poiaeuille, Mem. de Savants Etrangert, 1846. Gases : Graham, Philosophical Transactions,
1846 and 1849.

t B«ly, Phil Trans. 1832 ; Bessel, Berlin Acad. 1826 ; Dubuat, Principes <t HydrauHque, 1786. All
the** are discussed in Professor Stokes's paper " On the Eflect of the Internal Friction of Fluids on the
Motion of Pendulums," Cambridge Phil Trans, vol. IX. pt. 2 (1850).

S Coulomb, Mem. de FlnttittU national, in. p. 246; O. E. Meyer, Pogg. Ann. cxm. (1861) p. 55, and
CrMi» Journal, Bd. 69.

y HelmholU and Pietrowski, SUzungtberichte d«r k. L Akad. April, 1860.

OF AIR AND OTHEE GASES. 3

disks instead of one, so that there are six surfaces exposed to friction, which
may be reduced to two by placing the three disks in contact, without altering
the weight of the whole or the time of vibration. The apparatus was con-
structed by Mr Becker, of Messrs Elliott Brothers, Strand.

Description of the Apparatus.

Plate XXI. p. 30, fig. 1 represents the vacuum apparatus one-eighth of the
actual size. MQRS is a strong three-legged stool supporting the whole. The
top (MM] is in the form of a ring. EE is a brass plate supported by the ring
MM. The under surface is ground truly plane, the upper surface is strengthened
by ribs cast in the same piece with it. The suspension-tube A C is screwed
into the plate EE, and is 4 feet in height. The glass receiver N rests on a
wooden ring PP with three projecting pieces which rest on the three brackets
QQ, of which two only are seen. The upper surfaces of the brackets and the
under surfaces of the projections are so bevilled off, that by slightly turning the
wooden ring in its own plane the receiver can be pressed up against the plate
EE.

F, G, H, K are circular plates of glass of the form represented in fig. 2.
Each has a hole in the centre 2 inches in diameter, and three holes near the
circumference, by which it is supported on the screws LL.

Fig. 6 represents the mode of supporting and adjusting the glass plates.
LL is one of the screws fixed under the plate EE. S is a nut, of which the
upper part fits easily in the hole in the glass plate F, while the under part
is of larger diameter, so as to support the glass plate and afford the means of
turning the nut easily by hand. These nuts occupy little space, and enable
the glass disks to be brought very accurately to their proper position.

ACS, fig. 1, is a siphon barometer, closed at A and communicating with
the interior of the suspension-tube at B. The scale is divided on both sides,
so that the difference of the readings gives the pressure within the apparatus.
T is a thermometer, lying on the upper glass plate. F is a vessel containing
pumice-stone soaked in sulphuric acid, to dry the air. Another vessel, containing
caustic potash, is not shewn. D is a tube with a stopcock, leading to the air-
pump or the gas generator. C is a glass window, giving a view of the suspended
mirror d.

For high and low temperatures the tin vessel (fig. 10) was used. When
the receiver was exhausted, the ring P was removed, and the tin vessel raised

1—2

4 ON THE VISCOSITY OE INTERNAL FRICTION

ao as to envelope the receiver, which then rested on the wooden support YY.
The tin vessel itself rested, by means of projections on the brackets QQ. The
outside of the tin vessel was then well wrapped up in blankets, and the top
the brass plate EE covered with a feather cushion; and cold water, hot
water, or steam was made to flow through the tin vessel till the thermometer T,
seen through the window \V, became stationary.

The moveable parts of the apparatus consist of—

The suspension-piece a, fitting air-tight into the top of the tube and holding
the suspension-wire by a clip, represented in fig. 5.

The axis cdek, suspended to the wire by another clip at C.

The wire was a hard-drawn steel wire, one foot of which weighed 2'6 grains.

The axis carries the plane mirror d, by which its angular position is observed
through the window C, and the three vibrating glass disks /, g, h, represented
in fig. 3. Each disk is 10'5G inches diameter and about '076 thick, and has
a hole in the centre '75 diameter. They are kept in position on the axis by
means of short tubes of accurately known length, which support them on the
axis and separate them from each other.

The whole suspended system weighs three pounds avoirdupoise.

In erecting the apparatus, the lower part of the axis ek is screwed off.
The fixed disks are then screwed on, with a vibrating disk lying between each.
Tubes of the proper lengths are then placed on the lower part of the axis
and between the disks. The axis is then passed up from below through the
disks and tubes, and is screwed to the upper part at e. The vibrating
disks are now hanging by the wire and in their proper places, and the fixed
disks are brought to their proper distances from them by means of the adjusting
nuts.

ns is a small piece of magnetized steel wire attached to the axis.

When it is desired to set the disks in motion, a battery of magnets is
placed under N, and so moved as to bring the initial arc of vibration to the
proper value.

Fig. 4 is a brass ring whose moment of inertia is known. It is placed
centrically on the vibrating disk by means of three radial wires, which keep
it exactly in its place.

Fig. 7 is a tube containing two nearly equal weights, which slide inside
it, and whose position can be read off by verniers.

OF AIR AND OTHEE GASES. 5

The ring and the tube are used in finding the moment of inertia of the
vibrating apparatus.

The extent and duration of the vibrations are observed in the ordinary
way by means of a telescope, which shews the reflexion of a scale in the
mirror d. The scale is on a circular arc of six feet radius, concentric with the
axis of the instrument. The extremities of the scale correspond to an arc of
vibration of 19° 36', and the divisions on the scale to 17. The readings are
usually taken to tenths of a division.

Method of Observation.

When the instrument was properly adjusted, a battery of magnets was
placed on a board below N, and reversed at proper intervals till the arc of
vibration extended slightly beyond the limits of the scale. The magnets were
then removed, and any accidental pendulous oscillations of the suspended disks
were checked by applying the hand to the suspension-tube. The barometer and
thermometer were then read off, and the observer took his seat at the telescope
and wrote down the extreme limits of each vibration as shewn by the numbers
on the scale. At intervals of five complete vibrations, the time of the transits
of the middle point of the scale was observed (see Table I.). When the ampli-
tude decreased rapidly, the observations were continued throughout the experi-
ment ; but when the decrement was small, the observer generally left the room
for an hour, or till the amplitude was so far reduced as to furnish the most
accurate results.

In observing a quantity which decreases in a geometrical ratio in equal
times, the most accurate value of the rate of decrement will be deduced from a
comparison of the initial values with values which are to these in the ratio
of e to 1, where e = 2*71828, the base of the Napierian system of logarithms.
In practice, however, it is best to stop the experiment somewhat before the
vibrations are so much reduced, as the time required would be better spent in
beginning a new experiment.

In reducing the observations, the sum of every five maxima and of the
consecutive five minima was taken, and the differences of these were written as
the terms of the series the decrement of which was to be found.

In experiments where the law of decrement is uncertain, this rough method
is inapplicable, and Gauss's method must be applied ; but the series of amplitudes

6 ox THE vraooemr OR INTERNAL FRICTION

in these experiments is so accurately geometrical, that no appreciable difference
between the results of the two methods would occur.

The logarithm of each term of the series was then taken, and the mean
logarithmic decrement ascertained by taking the difference of the first and last,
of the second and last but one, and so on, multiplying each difference by the
interval of the terms, and dividing the sum of the products by the sum of
the squares of these intervals. Thus, if fifty observations were taken of the
extreme limits of vibration, these were first combined by tens, so as to form
five terms of a decreasing series. The logarithms of these terms were then
taken. Twice the difference of the first and fifth of these logarithms was then
added to the difference of the second and third, and the result divided by
ten for the mean logarithmic decrement in five complete vibrations.

The times were then treated in the same way to get the mean time of
five vibrations. The numbers representing the logarithmic decrement, and the
time for five vibrations, were entered as the result of each experiment*.

The series found from ten different experiments were examined to discover
any departure from uniformity in the logarithmic decrement depending on the
amplitude of vibration. The logarithmic decrement was found to be constant
in each experiment to within the limits of probable error; the deviations from
uniformity were sometimes in one direction and sometimes in the opposite, and
the ten experiments when combined gave no evidence of any law of increase
or diminution of the logarithmic decrement as the amplitudes decrease. The
forces which retard the disks are therefore as the first power of the velocity,
and there is no evidence of any force varying with the square of the velocity,
such as is produced when bodies move rapidly through the air. In these
experiments the maximum velocity of the circumference of the moving disks
was about ^ inch per second. The changes of form in the air between the
disks were therefore effected very slowly, and eddies were not produced t.

The retardation of the motion of the disks is, however, not due entirely
to the action of the air, since the suspension wire has a viscosity of its own,
which must be estimated separately. Professor W. Thomson has observed great
changes in the viscosity of wires after being subjected to torsion and longi-
tudinal strain. The wire used in these experiments had been hanging up for

* See Table II.

t The total moment of the resistances never exceeded that of the weight of ^ grain acting at the edge
of the disks.

OF AIR AND OTHER GASES. 7

some months before, and had been set into torsional vibrations with various
weights attached to it, to determine its moment of torsion. Its moment of
torsion and its viscosity seem to have remained afterwards nearly constant, till
steam was employed to heat the lower part of the apparatus. Its viscosity
then increased, and its moment of torsion diminished permanently, but when
the apparatus was again heated, no further change seems to have taken place.
During each course of experiments, care was taken not to set the disks vibrating
beyond the limits of the scale, so that the viscosity of the wire may be
supposed constant in each set of experiments.

In order to determine how much of the total retardation of the motion is
due to the viscosity of the wire, the moving disks were placed in contact with
each other, and fixed disks were placed at a measured distance above and
below them. The weight and moment of inertia of the system remained as
before, but the part of the retardation of the motion due to the viscosity of
the air was less, as there were only two surfaces exposed to the action of the
air instead of six. Supposing the effect of the viscosity of the wire to remain
as before, the difference of retardation is that due to the action of the four
additional strata of air, and is independent of the value of the viscosity of the
wire.

In the experiments which were used in determining the viscosity of air,
five different arrangements were adopted.

Arrangement 1. Three disks in contact, fixed disks at 1 inch above and below.

2. „ „ „ 0-5 inch.

„ 3. Three disks, each between two fixed disks at distance 0*683.

4. „ „ „ „ „ 0-425.

5. „ „ „ „ „ 0-18475.

By comparing the results of these different arrangements, the coefficient
of viscosity was obtained, and the theory at the same time subjected to a
rigorous test.

Definition of the Coefficient of Viscosity.

The final result of each set of experiments was to determine the value of
the coefficient of viscosity of the gas in the apparatus. This coefficient may be
best defined by considering a stratum of air between two parallel horizontal

g ON THE VISCOSITY OE INTERNAL FRICTION

planes of indefinite extent, at a distance a from one another. Suppose the
upper plane to be set in motion in a horizontal direction with a velocity of
v feet per second, and to continue in motion till the air in the different parts
of the stratum has taken up its final velocity, then the velocity of the air
will increase uniformly as we pass from the lower plane to the upper. If the
air in contact with the planes has the same velocity as the planes themselves,

then the velocity will increase - feet per second for every foot we ascend.

The friction between any two contiguous strata of air will then be equal
to that between either surface and the air in contact with it. Suppose that
this friction is equal to a tangential force / on every square foot, then

f v

/=/v

where p. is the coefficient of viscosity, v the velocity of the upper plane, and
a the distance between them.

If the experiment could be made with the two infinite planes as described,
we should find ft at once, for

fa

P--—'

v

In the actual case the motion of the planes is rotatory instead of recti-
linear, oscillatory instead of constant, and the planes are bounded instead of
infinite.

It will be shewn that the rotatory motion may be calculated on the same
principles as rectilinear motion ; but that the oscillatory character of the motion
introduces the consideration of the inertia of the air in motion, which causes
the middle portions of the stratum to lag behind, as is shewn in fig. 8, where
the curves represent the successive positions of a line of particles of air, which,
if there were no motion, would be a straight line perpendicular to the planes.

The fact that the moving planes are bounded by a circular edge introduces
another difficulty, depending on the motion of the air near the edge being
different from that of the rest of the air.

The lines of equal motion of the air are shewn in fig. 9.

The consideration of these two circumstances introduces certain corrections
into the calculations, as will be shewn hereafter.

In expressing the viscosity of the gas in absolute measure, the measures

OP AIR AND OTHER GASES. 9

of all velocities, forces, &c. must be taken according to some consistent system
of measurement.

If L, M, T represent the units of length, mass, and time, then the dimen-
sions of / (a pressure per unit of surface) are L~1MT~*; a is a length, and v
is a velocity whose dimensions are LT~\ so that the dimensions of /A are

L-IMT~\

Thus if /x be the viscosity of a gas expressed in inch-grain-second measure,
and p.' the same expressed in foot-pound-minute measure, then

p. 1 inch 1 pound 1 second _
(if ' 1 foot ' 1 gram ' 1 minute

According to the experiments of MM. Helmholtz and Pietrowski*, the
velocity of a fluid in contact with a surface is not always equal to that of
the surface itself, but a certain amount of actual slipping takes place in certain
cases between the surface and the fluid in immediate contact with it. In the
case which we have been considering, if va is the velocity of the fluid in
contact with the fixed plane, and f the tangential force per unit of surface, then

f=<rv,,

where tr is the coefficient of superficial friction between the fluid and the par-
ticular surface over which it flows, and depends on the nature of the surface as
well as on that of the fluid. The coefficient cr is of the dimensions L~"MT~l.
If v, be the velocity of the fluid in contact with the plane which is moving
with velocity v, and if a-' be the coefficient of superficial friction for that plane,

f=<r'(v-vl).
The internal friction of the fluid itself is

/•/! , l , a'

Hence v=f (- + -,+

\<r <r

If we make -=/3, and —, = {?, then

* SUzungsberichte der k. k. Akad. AprU 1860.
VOL. II.

10 OK THE VISCOSITY OR INTERNAL FRICTION

or the friction is equal to what it would have been if there had been no
slipping, and if the interval between the planes had been increased by ft + ft".
By changing the interval between the planes, a may be made to vary while
0 + ff remains constant, and thus the value of ft + f? may be determined. In
the case of air, the amount of slipping is so small that it produces no appre-
ciable effect on the results of experiments. In the case of glass surfaces rubbing
on air, the probable value of ft, deduced from the experiments, was /J = '0027
inch. The distance between the moving surfaces cannot be measured so accu-
rately as to give this value of ft the character of an ascertained quantity. The
probability is rather in favour of the theory that there is no slipping between
air and glass, and that the value of ft given above results from accidental
discrepancy in the observations. I have therefore preferred to calculate the
value of /* on the supposition that there is no slipping between the air and
the glass in contact with it.

The value of /t depends on the nature of the gas and on its physical
condition. By making experiments in gas of different densities, it is shewn
that p. remains constant, so that its value is the same for air at 0'5 inch and
at 30 niches pressure, provided the temperature remains the same. This will
be seen by examining Table IV., where the value of L, the logarithm of the
decrement of arc in ten single vibrations, is the same for the same tempei-ature,
though the density is sixty times greater in some cases than in others. In
fact the numbers in the column headed L' were calculated on the hypothesis
that the viscosity is independent of the density, and they agree very well with
the observed values.

It will be seen, however, that the value of L rises and falls with the
temperature, as given in the second column of Table IV. These temperatures
range from 51* to 74° Fahr., and were the natural temperatures of the room
on different days in May 1865. The results agree with the hypothesis that
the viscosity is proportional to (461° + ^), the temperature measured from absolute
zero of the air-thermometer. In order to test this proportionality, the tempe-
rature was raised to 185* Fahr. by a current of steam sent round the space
between the glass receiver and the tin vessel. The temperature was kept up
for several hours, till the thermometer in the receiver became stationary, before
the disks were set in motion. The ratio of the upper temperature (185° F.)
to the lower (51*), measured from — 461° F., was

1-2605.

OF AIR AND OTHER GASES. H

The ratio of the viscosity at the upper temperature to that at the lower was

1-2624,

which shews that the viscosity is proportional to the absolute temperature very
nearly. The simplicity of the other known laws relating to gases warrants
us in concluding that the viscosity is really proportional to the temperature,
measured from the absolute zero of the air-thermometer.

These relations between the viscosity of air and its pressure and temperature
are the more to be depended on, since they agree with the results deduced
by Mr. Graham from experiments on the transpiration of gases through tubes
of small diameter. The constancy of the viscosity for all changes of density
when the temperature is constant is a result of the Dynamical Theory of
Gases'", whatever hypothesis we adopt as to the mode of action between the
molecules when they come near one another. The relation between viscosity
and temperature, however, requires us to make a particular assumption with
respect to the force acting between the molecules. If the molecules act on 'one
another only at a determinate distance by a kind of impact, the viscosity will
be as the square root of the absolute temperature. This, however, is certainly
not the actual law. If, as the experiments of Graham and those of this paper
shew, the viscosity is as the first power of the absolute temperature, then in
the dynamical theory, which is framed to explain the facts, we must assume
that the force between two molecules is proportional inversely to the fifth
power of the distance between them. The present paper, however, does not
profess to give any explanation of the cause of the viscosity of air, but only
to determine its value in different cases.

Experiments were made on a few other gases besides dry air.

Damp air, over water at 70" F. and 4 inches pressure, was found by the
mean of three experiments to be about one-sixtieth part less viscous than dry
air at the same temperature.

Dry hydrogen was found to be much less viscous than air, the ratio of
its viscosity to that of air being '5156.

A small proportion of air mixed with hydrogen was found to produce a
large increase of viscosity, and a mixture of equal parts of air and hydrogen
has a viscosity nearly equal to £fj- of that of air.

The ratio of the viscosity of dry carbonic acid to that of air was found
to be '859.

* " Illustrations of the Dynamical Theory of Gases," Philosophical Magazine, Jan. 1860.

2—2

12 ON THE VISCOSITY OR INTERNAL FRICTION

It appears from the experiments of Mr. Graham that the ratio of the
transpiration time of hydrogen to that of air is '4855, and that of carbonic
acid to air '807. These numbers are both smaller than those of this paper.
I think that the discrepancy arises from the gases being less pure in my
experiments than in those of Graham, owing to the difficulty of preventing air
from leaking into the receiver during the preparation, desiccation, and admission
of the gas, which always occupied at least an hour and a half before the
experiment on the moving disks could be begun.

It appears to me that for comparative estimates of viscosity, the method
of transpiration is the best, although the method here described is better
adapted to determine the absolute value of the viscosity, and is less liable to
the objection that in fine capillary tubes the influence of molecular action
between the gas and the surface of the tube may possibly have some effect.

The actual value of the coefficient of viscosity in inch-grain-second measure,
as determined by these experiments, is

•00001492(461° + 0).
At 62* F. p. = '007802.

Professor Stokes has deduced from the experiments of Baily on pendulums

Vp

which at ordinary pressures and temperatures gives

/* = -00417,

or not much more than half the value as here determined. I have not found
any means of explaining this difference.

In metrical units and Centigrade degrees

^ = '01878(1 + -003650).

M. O. E. Meyer gives as the value of /x in centimetres, grammes, and
seconds, at 18* C.,

•000360.

This, when reduced to metre-gramme-second measure, is

/t = -0360.
I make /*, at 18°C., ='0200.

OF AIR AND OTHER GASES. 13

Hence the value given by Meyer is 1'8 times greater than that adopted in
this paper.

M. Meyer, however, has a different method of taking account of the dis-
turbance of the air near the edge of the disk from that given in this paper.
He supposes that when the disk is very thin the effect due to the edge is
proportional to the thickness, and he has given in Crelle's Journal a vindication
of this supposition. I have not been able to obtain a mathematical solution
of the case of a disk oscillating in a large extent of fluid, but it can easily
be shewn that there will be a finite increase of friction near the edge of the
disk due to the want of continuity, even if the disk were infinitely thin. I
think therefore that the difference between M. Meyer's result and mine is to
be accounted for, at least in part, by his having under-estimated the effect of
the edge of the disk. The effect of the edge will be much less in water than
in air, so that any deficiency in the correction will have less influence on the
results for liquids which are given in M. Meyer's very valuable paper.

Mathematical Theory of the Experiment.

A. disk oscillates in its own plane about a vertical axis between two fixed
horizontal disks, the amplitude of oscillation diminishing in geometrical pro-
gression, to find what part of the retardation is due to the viscosity of the
air between it and the fixed disks.

That part of the surface of the disk which is not near the edge may be
treated as part of an infinite disk, and we may assume that each horizontal
stratum of the fluid oscillates as a whole. In fact, if the motion of every
part of each stratum can be accounted for by the actions of the strata above
and below it, there will be no mutual action between the parts of the stratum,
and therefore no relative motion between its parts.

Let 6 be the angle which defines the angular position of the stratum which
is at the distance y from the fixed disk, and let r be the distance of a point

jn

of that stratum from the axis, then its velocity will be r-r-, and the tangential
force on its lower surface arising from viscosity will be on unit of surface

(P0 /..V

14 ON THE V18O08ITY OR INTERNAL FRICTION

The tangential force on the upper surface will be

<F0

and the mass of the stratum per unit of surface is pdy, so that the equation

of motion of each stratum is

<P0 #0 ..

which is independent of r, shewing that the stratum moves as a whole.

The conditions to be satisfied are, that when y = 0, 8 = 0 ; and that when
y*&, 0=Ce-ucoa(nt + a) .............................. (3).

The disk is suspended by a wire whose elasticity of torsion is such that
the moment of torsion due to a torsion 6 is Itfd, where / is the moment
of inertia of the disks. The viscosity of the wire is such that an angular

tlft dfl

velocity -*- is resisted by a moment 2lk -y-. The equation of motion of the

disks is then

where A=\sLtn*dr = \irt*, the moment of inertia of each surface, and N is the
number of surfaces exposed to friction of air.

The equation for the motion of the air may be satisfied by the solution
0 = e-u{e?vcos (nt + qy) -e'1* cos(nt-qy)} .................. (5),

provided 2pq = <-— ..................................... (6),

and <f~P' = P* .................................. (7);

and in order to fulfil the conditions (3) and (4),

2ln(l - £)(«** + e-*» - 2cos 2qb) = NAp{(pn - lq)(tf* - e~+) + (qn + Ip) 2sm2qb} . . .(8).
Expanding the exponential and circular functions, we find

where c =

F

/ = observed Napierian logarithmic decrement of the amplitude in unit of
time,

£ = the part of the decrement due to the viscosity of the wire.

OF AIR AND OTHER GASES. 15

When the oscillations are slow as in these experiments, when the disks
are near one another, and when the density is small and the viscosity large,
the series on the right-hand side of the equation is rapidly convergent.

When the time from rest to rest was thirty-six seconds, and the interval
between the disks 1 inch, then for air of pressure 29 '9 inches, the successive
terms of the series were

1-0-0-00508 +0-24866 + 0'00072 +0'00386 = 1-24816 ;
but when the pressure was reduced to 1'44 inch, the series became

1-0 - 0-0002448 + -0005768 + '00000008 + '00000002 = 1-0003321.

The series is also made convergent by diminishing the distance between
the disks. When the distance was '1847 inch, the first two terms only were
sensible. When the pressure was 29'29, the series was

1- '000858 +'000278 = 1 -'00058.
At smaller pressures the series became sensibly =1.

The motion of the air between the two disks is represented in fig. 8,
where the upper disk is supposed fixed and the lower one oscillates. A row
of particles of air which when at rest form a straight line perpendicular to the
disks, will when in motion assume in succession the forms of the curves 1, 2,
3, 4, 5, 6. If the ratio of the density to the viscosity of the air is very
small, or if the time of oscillation is very great, or if the interval between
the disks is very small, these curves approach more and more nearly to the
form of straight lines.

The chief mathematical difficulty in treating the case of the moving disks
arises from the necessity of determining the motion of the air in the neigh-
bourhood of the edge of the disk. If the disk were accompanied in its motion
by an indefinite plane ring surrounding it and forming a continuation of its
surface, the motion of the air would be the same as if the disk were of
indefinite extent ; but if the ring were removed, the motion of the air in the
neighbourhood of the edge would be diminished, and therefore the effect of its
viscosity on the parts of the disk near the edge would be increased. The
actual effect of the air on the disk may be considered equal to that on a disk
of greater radius forming part of an infinite plane.

Since the correction we have to consider is confined to the space imme-
diately surrounding the edge of the disk, we may treat the edge as if it were

16

ON THE VISCOSITY OR INTERNAL FRICTION

the straight edge of an infinite plane parallel to xz, oscillating in the direction
of s between two planes infinite in every direction at distance b. Let w be
the velocity of the fluid in the direction of z, then the equation of motion is*

dto d*w <fw\

with the conditions
and

. .

(10>'

w=Q when y=±b .............................. (11),

w = Ccos nt when y = 0, and x is positive. ...(12).

I have not succeeded in finding the solution of the equation as it stands,
but in the actual experiments the time of oscillation is so long, and the space

1 1

between the disks is so small, that we may neglect - - , and the equation
is reduced to

(13)

with the same conditions. For the method of treating these conditions I am
indebted to Professor W. Thomson,, who has shewn me how to transform
these conditions into another set with which we are more familiar, namely,
u> = 0 when x = 0, and w=l when y = 0, and x is greater than +1, and w= — 1
when x is less than — 1. In this case we know that the lines of equal values
of w are hyperbolas, having their foci at the points y = 0, x=±l, and that
the solution of the equation is

u>= -sin"
rr

where r,, r, are the distances from the foci.

If we put

then the lines for which
hyperbolas, and

= - log y(r, + r,)' - 4 + r, + r,}.

TT

(14)

.(15),

is constant will be ellipses orthogonal to the

.(16);

and the resultant of the friction on -any -arc of a curve will be proportional

* Professor Stokes " On the Theories of the Internal Friction of Fluids in Motion, <fec.," Cambridge
Phil. Traru. VoL Tin.

OF AIR AND OTHER GASES. 17

to ^-^o, where <j>0 is the value of <f> at the beginning, and <£, at the end
of the given arc.

In the plane y = 0, when x is very great, <i = -10240;, and when x=l

TT

2

-log 2, so that the whole friction between x=l and a very distant point

is -Iog2x.
ti

Now let w and <£ be expressed in terms of r and 6, the polar co-ordinates
with respect to the origin as the pole ; then the conditions may be stated thus :

When 9=±^, w = 0. When 0 = 0 and r greater than 1, w?=l. When

Zi

6 = TT and r greater than 1, w= — 1.

Now let x', y' be rectangular co-ordinates, and let

2 2

y' = -b& and a;' = -61ogr ........................ (17),

IT TT

and let w and <f> be expressed in terms of x and i/ ; the differential equa-
tions (13) and (16) will still be true; and when y'=±b, w = 0, and when
y* = Q and x' positive, w=l.

x' 2 2

When x' is great, <£ = 7- + -log4, and when x' = 0, <£ = -log2, so that the

0 TT TT

whole friction on the surface is

which is the same as if a portion whose breadth is : - log, 2 had been added

TT

to the surface at its edge.

The curves of equal velocity are represented in fig. 9 at u, v, w, x, y.
They pass round the edge of the moving disk AB, and have a set of asymp-
totes U, V, W, X, Y, arranged at equal distances parallel to the disks.

The curves of equal friction are represented at o, p, q, r, s, t. The form

of these curves approximates to that of straight lines as we pass to the left
of the edge of the disk.

VOL. n. 3

lg ON THE VISCOSITY OB INTERNAL FRICTION

The dotted vertical straight lines O, P, Q, R, S, T represent the position
of the corresponding lines of equal friction if the disk AB had been accom-
panied by an extension of its surface in the direction of B. The total friction
on AB, or on any of the curves w, v, w, Ac., is equal to that on a surface
extending to the point C, on the supposition that the moving surface has an
accompanying surface which completes the infinite plane.

In the actual case the moving disk is not a mere surface, but a plate
of a certain thickness terminated by a slightly rounded edge. Its section may
therefore be compared to the curve uu' rather than to the axis AB.

The total friction on the curve is still equal to that on a straight line
extending to C, but the velocity corresponding to the curve « is less than that
corresponding to the line AB.

If the thickness of the disk is 2$, and the distance between the fixed
disks = 26, so that the distance of the surfaces is 6 — j8, the breadth of the strip
which must be supposed to be added to the surface at the edge will be

a = —

(19).

In calculating the moment of friction on this strip, we must suppose it to
be at the same distance from the axis as the actual edge of the disk. Instead

of A=*-r* in equation (9), we must therefore put A =-r4 + 27rr'a, aud instead
t> ' LI

of b we must put b — /8.

The actual value of - i* for each surface in inches = 1112'8.

£t

The value of / in inches and grains was 175337.

It was determined by comparing the times of oscillation of the axis and
disks without the little magnet, with the times of the brass ring (fig. 4) and
of the tube and weights (fig. 7). Four different suspension wires were used in
these experiments.

The following Table gives the numbers required for the calculation of each
of the five Arrangements of the disks.

* This result is applicable to the calculation of the electrical capacity of a condenser in the form of a
dink between two larger disks at equal distance from it

OF AIB AND OTHER GASES.

19

Arrangement

Case 1

Case 2

CaseS

Case 4

CaseS

.A^— number of surfaces

2

2

6

f.

c

6 - ft = distance of surfaces

1-0

0-5

0-683

0-4.2'j

OOftl7

27T7*3a — effect of ed^e

446-09

235-Q

OQO.QK

1 ftfi-7

ft^-1

A = whole moment of each surface . . .

N A

1558-9
•003815

1347-8
•007318

1405-75

•01 M 1ft

1299-5

.noo/HQ

1198-9

,f\A 7fi ,1 A

2/61og«10 V

If I is the Napierian logarithmic decrement per second, and L the observed
decrement of the common logarithm (to base 10) of the arc in tune T, then

L = lT\og1<se (20).

If n is the coefficient of t in the periodic terms, and T the time of five
complete vibrations,

w7T=107r (21).

Let K=JcT\ogwe (22),

then K is the part of the observed logarithmic decrement due to the viscosity
of the wire, the yielding of the instrument, and the friction of the air on the
axis, and is the same for all experiments as long as the wire is unaltered.

Let /i0 be the value of ft at temperature zero, p. that at any other
temperature 6, then if /i is proportional to the temperature from absolute zero,

/» = (!+ ad) A*. (23),

where a is the coefficient of expansion of air per degree.

Equation (9) may now be written hi the form

frQ(l+x)(l + a6)T+K=L (24),

where l+x is the series in equation (9), x being in most cases small, and may
be calculated from an approximate value of /v

The values of Q are to be taken from the Table according to the arrange-
ment of disks in the experiment.

In this way I have combined the results of forty experiments on dry air
in order to determine the values of fia and K. Seven of these had the first
arrangement, six had the second, six the third, nine the fourth, and twelve

the fifth.

3—2

OS THB VISCOSITY OB INTERNAL FRICTION

The values of Q for the five cases are roughly in the proportions of 1, 2, 4,
6, 12, so that it is easy to eliminate K and find /v I had reason, however, to
U-lieve that the value of K was altered at a certain stage of the experiments
whrn steam was first used to heat the air in the receiver. I therefore intro-
iluced two values of A", A', and A',, into the experiments before and after this
rliange respectively. The values of A', and A", deduced from these experiments
were

In ten lingle vibrations.

A", = -01568
AT, = -01901.

The value of ft in inch-grain-second measure at temperature ff> Fahrenheit

is for air

/x = -0000 1492 (46 1° + 0°).

The value of L was then calculated for each experiment and compared with
the observed value. In this way the error of mean square of a single experi-
ment was found. The probable error of /x, as determined from the equations,
was calculated from this and found to be 0'36 per cent, of its value.

In order to estimate the value of the evidence in favour of there being a
finite amount of slipping between the disks and the air in contact with them,
the value of L for each of the forty experiments was found on ' the supposition
that

£=•0027 inch and ^ = ('000015419) (461°

The error of mean square for each observation was found to be slightly
greater than in the former case ; the probable error of ft was 40 per cent., and
that of /t = 1 '6 per cent.

I have no doubt that the true value of ft is zero, that is, there is no
slipping, and that the original value of ft is the best.

As the actual observations were very numerous, and the reduction of them
would occupy a considerable space in this paper, I have given a specimen of
the actual working of one experiment.

Table I. shews the readings of the scale as taken down at the tune of
observation, with the times of transit of the middle point of the scale after

OF AIR AND OTHER GASES. 21

the fifth and sixth readings, with the sum of ten successive amplitudes deduced
therefrom.

Table II. shews the results of this operation as extended to the rest of
experiment 62, and gives the logarithmic decrement for each successive period
of ten semivibrations, with the mean time and corresponding mean logarithmic
decrement.

Table III. shews the method of combining forty experiments of different
kinds. The observed decrement depends on two unknown quantities, the viscosity
of air and that of the wire.

The experiments are grouped together according to the coefficients of p.
and K that enter into them, and when the final results have been obtained,
the decrements are calculated and compared with the results of observation.
The calculated sums of the decrements are given in the last column.

Table IV. shews the results of the twelve experiments with the fifth arrange-
ment. They are arranged in groups according to the pressure of the air, and
it will be seen that the observed values of L are as independent of the pressure
as the calculated values, in which the pressure is taken into account only in
calculating the value of x in the fifth column. By arranging the values of
L — L' in order .of temperature, it was found that within the range of atmo-
spheric temperature during the course of the experiments the relation between
the viscosity of air and its temperature does not perceptibly differ from that
assumed in the calculation. Finally, the experiments were arranged in order of
time, to determine whether the viscosity of the wire increased during the experi-
ments, as it did when steam was first used to heat the apparatus. There did
not appear any decided indication of any alteration in the wire.

Table V. gives the resultant value of /A in terms of the different units
which are employed in scientific measurements.

Note, added February 6, 1866. — In the calculation of the results of the
experiments, I made use of an erroneous value of the moment of inertia of the
disks and axis =1 '01 2 of the true value, as determined by six series of experi-
ments with four suspension wires and two kinds of auxiliary weights. The

ii

ON THE VISCOSITY OB INTERNAL FKICTION

numbers in the coefficients of m in Table IV. are therefore all too large, and
the value of p la also too large in the same proportion, and should be

/i = -0000 1492 (461'

The same error ran through all the absolute values in other parts of the
paper afl *ent in to the Royal Society, but to save trouble to the reader I
have corrected them where they occur.

TABLE I. — -Experiment 62. Arrangement 5. Dry air at pressure 0'55 inch.

Temperature 68" F. May 9, 1865.

0MM

Male reading

Time
8» +

LOM

scale reading

Time

8309
8071
7852
7650
7460

Sam of greater readings 39342
Sam of less do. 10826

m t

1740
1968
2180
2377
2561

10826

m s
28 18-8

27 42-4

Difference 28516 = sum of 10 amplitudes.

The .observations were continued in the same way till five sets of readings
of this kind were obtained. The following were the results.

TABLE II.

Time*

Stun of ten amplitudes

Logarithm

Log. decrement

b m •

3 28 0-6
34 3-2
40 5-8
46 8-6
52 11-2

28516
19784
13734
9530
6598

4-4550886
4-2963141
4-1377970
3-9790929
3-8194123

0-1587745
0-1585171
0-1587041
0-1596806

Results of experiment 62. Mean time of ten vibrations = 362'G6

Mean log. decrement . . . = O'l 588574.

OF AIE AND OTHER GASES.

23

TABLE III.

Equations from which u. for air was determined ; m = „ — z .

461 + d

Number of
experiments

Arrangement

Result of observation

Result of calculation

3.

1

6-3647 TO + 3^ =-00023167

•00022779

3.

2

11 -2893 m + 3^, = -00028280

•00030214

6.

4

71-2412m + 6.£=-00135467

•00133897

4.

1

8-7221 m+ ±K= -00034562

•00034127

3.

2

ll-6680m + 3^= -00031505

•00033335

12.

5

297 -7880 w + lSJT- -00511708

•00512666

3.

4

36-0551 m + 3K = -00069607

•00070159

6.

3

48-8911 m+ 6A,= -00108215

•00105333

Final result /LI* = '00001510 (46T+0) with probable error 0'36 per cent.
^, = •0000439, JTa= '0000524.

TABLE IV.-~-Experiments with Arrangement 5.

No. of
experi-
ment

Absolute
tempe-
rature
461° + 0

Pressure,
in inches
of mercury

Time of
five double
swings, in
seconds

Correction for
inertia of air
(1+x)
in equation (24)

£ = decrement of logarithm
of arc in ten single vibrations

Diff. Z,-,L'

L' calculated

L observed

62.

529

0-55

362-66

) (

•15719

•15886

+ 167

77.

516

0-50

362-80

J. 1 - -0000157 \

•15378

•15260

-118

80.

527

0-56

364-04

f I

•15648

•15946

+ 279

63.

527-5

5-57

362-72

) (

•15680

•15628

- 52

64.

535

5-97

362-94

V 1- -000157 A

•15875

•15838

- 37

81.

516

5-52

363-80

1 (

•15379

•15389

+ 10

65.

524

25-58

362-64

} (

•15555

•15422

-133

71.

513

19-87

362-50

V 1 - -000486 <

•15299

•15144

-155

72.

514-5

20-31

362-86

\ 1

•15338

•15269 :

- 69

75.

517

29-90

363-8

) (

•15398

•15377

- 21

76.

512-5

29-76

362-89

[ 1- -00058 ^

•15280

•15146

-134

79.

521

28-22

363-9

) I

•15502

•15510

+ 8

* This is the result derived from these equations, which is 1 -2 per cent too large.

24 ON THK VISCOSITY OB INTERNAL FRICTION

TABLE V. — Results.

Coefficient of viscosity in dry air. Units — the inch, grain, and second, and
Fahrenheit temperature,

ft, = -00001 492 (461 + 6) = '006876 + '00001490.

At 60* F." the mean temperature of the experiments, p = '007763. Taking
the foot as unit instead of the inch, /t = '000179 (461 +6). In metrical units
(metre, gramme, second, and Centigrade temperature),

,t = -01878(1 + -003650).

The coefficient of viscosity of other gases is to be found from that of air
by multiplying /t by the ratio of the transpiration time of the gas to that of
air as determined by Graham*.

f POSTSCRIPT. — Received December 7, 1865.

Since the above paper was communicated to the Royal Society, Professor
Stokes has directed my attention to a more recent memoir of M. O. E. Meyer,
"Ueber die innere Reibung der Gase," in Poggendorffs Anncden, cxxv. (1865).
M. Meyer has compared the values of the coefficient of viscosity deduced from
the experiments of Baily by Stokes, with those deduced from the experiments
of Bessel and of Girault. These values are '000104, '000275, and -000384
respectively, the units being the centimetre, the gramme, and the second.
M. Meyer's own experiments were made by swinging three disks on a vertical
axis in an air-tight vessel. The disks were sometimes placed in contact, and
sometimes separate, so as to expose either two or six surfaces to the action
of the air. The difference of the logarithmic decrement of oscillation in these
two arrangements was employed to determine the viscosity of the air.

The effects of the resistance of the air on the axis, mirror, &c., and of
the viscosity of the suspending wires are thus eliminated.

The calculations are made on the supposition that the moving disks are
so far from each other and from the surface of the receiver which contains
them, that the effect of the air upon each is the same as if it were in an
infinite space.

At the distance of 30 millims., and with a period of oscillation of fourteen
seconds, the mutual effect of the disks would be very small in air at the

* Philosophical Transactions, 1849.

VOL. II PLATE IX.

Fig- 1.

[ S R Q P 0

To />«•" pric/r 24

Cambridge University J'ress

OP AIR AND OTHER GASES. 25

ordinary pressure. In November 1863 I made a series of experiments with an
arrangement of three brass disks placed on a vertical axis exactly as in
M. Meyer's experiments, except that I had then no air-tight apparatus, and
the disks were protected from currents of air by a wooden box only.

I attempted to determine the viscosity of air by means of the observed
mutual action between the disks at various distances. I obtained the values
of this mutual action for distances under 2 inches, but I found that the results
were so much involved with the unknown motion of the air near the edge of
the disks, that I could place no dependence on the results unless I had a
complete mathematical theory of the motion near the edge.

In M. Meyer's experiments the time of vibration is shorter than in most
of mine. This will diminish the effect of the edge in comparison with the total
effect, but in rarefied air both the mutual action and the effect of the edge
are much increased. In his calculations, however, the effect of the three edges
of the disks is supposed to be the same, whether they are in contact or sepa-
rated. This, I think, will account for the large value which he has obtained
for the viscosity, and for the fact that with the brass disks which vibrate in
14 seconds, he finds the apparent viscosity diminish as the pressure diminishes,
while with the glass disks which vibrate in 8 seconds it first increases and then
diminishes.

M. Meyer concludes that the viscosity varies much less than the pressure,
and that it increases slightly with increase of temperature. He finds the value
of p, in metrical units (centimetre-gramme-second) at various temperatures,

Temperature. Viscosity.

8°-3 C. '000333

21'-5 C. '000323

34°-4 C. -000366

In my experiments, in which fixed disks are interposed between the moving
ones, the calculation is not involved in so great difficulties ; and the value of /u,
is deduced directly from the observations, whereas the experiments of M. Meyer
give only the value of -J^p, from which \L must be determined. For these
reasons I prefer the results deduced from experiments with fixed disks inter-
posed between the moving ones.

M. Meyer has also given a mathematical theory of the internal friction of
gases, founded on the dynamical theory of gases. I shall not say anything of
this part of his paper, as I wish to confine myself to the results of experiment.

VOL. II. 4

[From the Philosophical Transactions, Vol. CLVH.]

XXVIII. On the Dynamical Theory of Gases.
(Received May 16,— Read May 31, 1866.)

THEORIES of the constitution of bodies suppose them either to be continuous
and homogeneous, or to be composed of a finite number of distinct particles or
molecules.

In certain applications of mathematics to physical questions, it is convenient
to suppose bodies homogeneous in order to make the quantity of matter in each
differential element a function of the co-ordinates, but I am not aware that any
theory of this kind has been proposed to account for the different properties
of bodies. Indeed the properties of a body supposed to be a uniform ^/e/iwm
may be affirmed dogmatically, but cannot be explained mathematically.

Molecular theories suppose that all bodies, even when they appear to our
senses homogeneous, consist of a multitude of particles, or small parts the
mechanical relations of which constitute the properties of the bodies. Those
theories which suppose that the molecules are at rest relative to the body may
be called statical theories, and those which suppose the molecules to be in
motion, even while the body is apparently at rest, may be called dynamical
theories.

If we adopt a statical theory, and suppose the molecules of a body kept at
rest in their positions of equilibrium by the action of forces in the directions
of the lines joining their centres, we may determine the mechanical properties
of a body so constructed, if distorted so that the displacement of each molecule
is a function of its co-ordinates when in equilibrium. It appears from the mathe-
matical theory of bodies of this kind, that the forces called into play by a
small change of form must always bear a fixed proportion to those excited by
a small change of volume.

THE DYNAMICAL THEORY OF GASES. 27

Now we know that in fluids the elasticity of form is evanescent, while
that of volume is considerable. Hence such theories will not apply to fluids.
In solid bodies the elasticity of form appears in many cases to be smaller in
proportion to that of volume than the theory gives*, so that we are forced
to give up the theory of molecules whose displacements are functions of their
co-ordinates when at rest, even in the case of 'solid bodies.

The theory of moving molecules, on the other hand, is not open to these
objections. The mathematical difficulties in applying the theory are considerable,
and till they are surmounted we cannot fully decide on the applicability of the
theory. We are able, however, to explain a great variety of phenomena by the
dynamical theory which have not been hitherto explained otherwise.

The dynamical theory supposes that the molecules of solid bodies oscillate
about their positions of equilibrium, but do not travel from one position to
another in the body. In fluids the molecules are supposed to be constantly
moving into new relative positions, so that the same molecule may travel from
one part of the fluid to any other part. In liquids the molecules are supposed
to be always under the action of the forces due to neighbouring molecules
throughout their course, but in gases the greater part of the path of each
molecule is supposed to be sensibly rectilinear and beyond the sphere of sensible
action of the neighbouring molecules.

I propose in this paper to apply this theory to the explanation of various
properties of gases, and to shew that, besides accounting for the relations of
pressure, density, and temperature in a single gas, it affords a mechanical
explanation of the known chemical relation between the density of a gas and
its equivalent weight, commonly called the Law of Equivalent Volumes. It also
explains the diffusion of one gas through another, the internal friction of a gass
and the conduction of heat through gases.

The opinion that the observed properties of visible bodies apparently at rest
are due to the action of invisible molecules in rapid motion is to be found in
Lucretius. In the exposition which he gives of the theories of Democritus as
modified by Epicurus, he describes the invisible atoms as all moving downwards
with equal velocities, which, at quite uncertain times and places, suffer an
imperceptible change, just enough to allow of occasional collisions taking place

* In glass, according to Dr Everett's second series of experiments (1866), the ratio of the elasticity of
form to that of volume is greater than that given by the theory. In brass and steel it is less.—
March 7, 1867.

4—2

28 THE DYNAMICAL THEORY OF OASES.

between the atoms. These atoms he supposes to set small bodies in motion by
an action of which we may form some conception by looking at the motes in
a sunbeam. The language of Lucretius must of course be interpreted according
to the physical ideas of his age, but we need not wonder that it suggested
to Le Sage the fundamental conception of his theory of gases, as well as his
doctrine of ultramundane corpuscles.

Professor Clausius, to whom we owe the most extensive developments of
the dynamical theory of gases, has given* a list of authors who have adopted
or given countenance to any theory of invisible particles in motion. Of these,
Daniel Bernoulli, in the tenth section of his Hydrodynamics, distinctly explains
the pressure of air by the impact of its particles on the sides of the vessel
containing it.

Clausius also mentions a book entitled Deux Traites de Physique Mfoanique,
publics par Pierre Prevost, comme simple Editeur du premier et comme Auteur
du second, Geneve et Paris, 1818. The first memoir is by G. Le Sage, who
explains gravity by the impact of " ultramundane corpuscles " on bodies. These
corpuscles also set in motion the particles of light and various ethereal media,
which in their turn act on the molecules of gases and keep up their motions.
His theory of impact is faulty, but his explanation of the expansive force of
gases is essentially the same as in the dynamical theory as it now stands.
The second memoir, by Prevost, contains new applications of the principles of
Le Sage to gases and to light. A more extensive application of the theory of
moving molecules was made by Herapathf. His theory of the collisions of
perfectly hard bodies, such as he supposes the molecules to be, is faulty,
inasmuch as it makes the result of impact depend on the absolute motion of
the bodies, so that by experiments on such hard bodies (if we could get them)
we might determine the absolute direction and velocity of the motion of the
earth J. This author, however, has applied his theory to the numerical results
of experiment in many cases, and his speculations are always ingenious, and
often throw much real light on the questions treated. In particular, the theory
of temperature and pressure in gases and the theory of diffusion are clearly
pointed out.

* PoggendorflTs Annalen, Jan. 1862. Translated by G. C. Foster, B.A., Phil. Mag. June, 1862.

t Mathematical Physics, <fea, by John Herapath, Esq. 2 vols. London: Whittaker and Co., and
Herapath'g Railway Journal Office, 1847.

* Mathematical Physics, ttc., p. 134.

THE DYNAMICAL THEORY OF GASES. 29

Dr Joule* has also explained the pressure of gases by the impact of their
molecules, and has calculated the velocity which they must have in order to
produce the pressure observed in particular gases.

It is to Professor Clausius, of Zurich, that we owe the most complete
dynamical theory of gases. His other researches on the general dynamical theory
of heat are well known, and his memoirs On the land of Motion which we call
Heat, are a complete exposition of the molecular theory adopted in this paper.
After reading his investigationf of the distance described by each molecule
between successive collisions, I published some propositions J on the motions and
collisions of perfectly elastic spheres, and deduced several properties of gases,
especially the law of equivalent volumes, and the nature of gaseous friction. I
also gave a theory of diffusion of gases, which I now know to be erroneous,
and there were several errors in my theory of the conduction of heat in gases
which M. Clausius has pointed out in an elaborate memoir on that subject §.

M. O. E. Meyer II has also investigated the theory of internal friction on
the hypothesis of hard elastic molecules.

In the present paper I propose to consider the molecules of a gas, not
as elastic spheres of definite radius, but as small bodies or groups of smaller
molecules repelling one another with a force whose direction always passes very
nearly through the centres of gravity of the molecules, and whose magnitude
is represented very nearly by some function of the distance of the centres of
gravity. I have made this modification of the theory in consequence of the
results of my experiments on the viscosity of air at different temperatures, and
I have deduced from these experiments that the repulsion is inversely as the
fifth power of the distance.

If we suppose an imaginary plane drawn through a vessel containing a great
number of such molecules in motion, then a great many molecules will cross
the plane in either direction. The excess of the mass of those which traverse
the plane in the positive direction over that of those which traverse it in the
negative direction, gives a measure of the flow of gas through the plane in
the positive direction.

* Some Remarks on Heat and the Constitution of Elastic Fluids, Oct. 3, 1848.

t Phil. Mag. Feb. 1859.

J "Illustrations of the Dynamical Theory of Gases," Phil. Mag. I860, January and July.

§ Poggendorff, Jan. 1862; Phil. Mag. June, 1862.

|| "Ueber die innere Reibung der Gase" (Poggendorff, Vol. cxxv. 1865).

30 THE DYNAMICAL THEORY OF OASES.

If the plane be made to move with such a velocity that there is no excess
of flow of molecules in one direction through it, then the velocity of the plane
w the mean velocity of the gas resolved normal to the plane.

There will still be molecules moving in both directions through the plane,
and carrying with them a certain amount of momentum into the portion of
gaa which lies on the other side of the plane.

The quantity of momentum thus communicated to the gas on the other
side of the plane during a unit of time is a measure of the force exerted on
this gas by the rest. This force is called the pressure of the gas.

If the velocities of the molecules moving in different directions were inde-
pendent of one another, then the pressure at any point of the gas need not
be the same in all directions, and the pressure between two portions of gas
separated by a plane need not be perpendicular to that plane. Hence, to
account for the observed equality of pressure in all directions, we must suppose
some cause equalizing the motion in all directions. This we find in the deflection
of the path of one particle by another when they come near one another.
Since, however, this equalization of motion is not instantaneous, the pressures
in all directions are perfectly equalized only in the case of a gas at rest, but
when the gas is in a state of motion, the want of perfect equality in the
pressures gives rise to the phenomena of viscosity or internal friction. The
phenomena of viscosity in all bodies may be described, independently of hypo-
thesis, as follows : —

A distortion or strain of some kind, which we may call S, is produced in
the body by displacement. A state of stress or elastic force which we may
call F is thus excited. The relation between the stress and the strain may
be written F=ES, where E is the coefficient of elasticity for that particular
kind of strain. In a solid body free from viscosity, F will remain = ES, and

_

dt~ J dt '

If, however, the body is viscous, F will not remain constant, but will tend to
disappear at a rate depending on the value of F, and on the nature of the
body. If we suppose this rate proportional to F, the equation may be written

dF_ dS F
eft dt ~ T'

THE DYNAMICAL THEORY OP GASES. 31

which will indicate the actual phenomena in an empirical manner. For if S
be constant,

shewing that F gradually disappears, so that if the body is left to itself it
gradually loses any internal stress, and the pressures are finally distributed as
in a fluid at rest.

If -,— is constant, that is, if there is a steady motion of the body which
continually increases the displacement,

shewing that F tends to a constant value depending on the rate of displace-
ment. The quantity ET, by which the rate of displacement must be multiplied
to get the force, may be called the coefficient of viscosity. It is the product
of a coefficient of elasticity, E, and a time T, which may be called the "time
of relaxation " of the elastic force. In mobile fluids T is a very small fraction
of a second, and E is not easily determined experimentally. In viscous solids
T may be several hours or days, and then E is easily measured. It is possible
that in some bodies T may be a function of F, and this would account for
the gradual untwisting of wires after being twisted beyond the limit of perfect
elasticity. For if T diminishes as F increases, the parts of the wire furthest
from the axis will yield more rapidly than the parts near the axis during the
twisting process, and when the twisting force is removed, the wire will at first
untwist till there is equilibrium between the stresses in the inner and outer
portions. These stresses will then undergo a gradual relaxation ; but since the
actual value of the stress is greater in the outer layers, it will have a more
rapid rate of relaxation, so that the wire will go on gradually untwisting for
some hours or days, owing to the stress on the interior portions maintaining
itself longer than that of the outer parts. This phenomenon was observed by
Weber in silk fibres, by Kohlrausch in glass fibres, and by myself in steel wires.
In the case of a collection of moving molecules such as we suppose a gas
to be, there is also a resistance to change of form, constituting what may be
called the linear elasticity, or "rigidity" of the gas, but this resistance gives
way and diminishes at a rate depending on the amount of the force and on
the nature of the gas.

THE DYNAMICAL THEORY OF GASES.

Suppose the molecules to be confined in a rectangular vessel with perfectly
elastic sides, and that they have no action on one another, so that they never
strike one another, or cause each other to deviate from their rectilinear paths.
Then it can easily be shewn that the pressures on the sides of the vessel due
to the impacts of the molecules are perfectly independent of each other, so that
Ihe mass of moving molecules will behave, not like a fluid, but like an elastic
solid. Now suppose the pressures at first equal in the three directions perpen-
dicular to the sides, and let the dimensions a, b, c of the vessel be altered
by small quantities, Sa, 86, Sc.

Then if the original pressure in the direction of a was p, it will become

/ 08a 86 Sc\

p (1-3 -r- ) ;

*r\ a b c]

or if there is no change of volume,

Sp=_2 8«

p a '

shewing that in this case there is a "longitudinal" elasticity of form of which
the coefficient is 2p. The coefficient of "Rigidity" is therefore =p.

This rigidity, however, cannot be directly observed, because the molecules
continually deflect each other from their rectilinear courses, and so equalize the
pressure in all directions. The rate at which this equalization takes place is
great, but not infinite ; and therefore there remains a certain inequality of
pressure- which constitutes the phenomenon of viscosity.

I have found by experiment that the coefficient of viscosity in a given gas
is independent of the density, and proportional to the absolute temperature, so

that if ET be the viscosity, ET oc £ .

But E=p, therefore T, the time of relaxation, varies inversely as the density
and is independent of the temperature. Hence the number of collisions pro-
ducing a given deflection which take place in unit of time is independent of
the temperature, that is, of the velocity of the molecules, and is proportional
to the number of molecules in unit of volume. If we suppose the molecules
hard elastic bodies, the number of collisions of a given kind will be proportional
to the velocity, but if we suppose them centres of force, the angle of deflection
will be smaller when the velocity is greater ; and if the force is inversely as
the fifth power of the distance, the number of deflections of a given kind will

THE DYNAMICAL THEORY OF GASES. 33

be independent of the velocity. Hence I have adopted this law in making my
calculations.

The effect of the mutual action of the molecules is not only to equalize
the pressure in all directions, but, when molecules of different kinds are present,
to communicate motion from the one kind to the other. I formerly shewed
that the final result in the case of hard elastic bodies is to cause the average
vis viva of a molecule to be the same for all the different kinds of molecules.
Now the pressure due to each molecule is proportional to its vis viva, hence
the whole pressure due to a given number of molecules in a given volume will
be the same whatever the mass of the molecules, provided the molecules of
different kinds are permitted freely to communicate motion to each other.

When the flow of vis viva from the one kind of molecules to the other
is zero, the temperature is said to be the same. Hence equal volumes of
different gases at equal pressures and temperatures contain equal numbers of
molecules.

This result of the dynamical theory affords the explanation of the " law of
equivalent volumes" in gases.

We shall see that this result is true in the case of molecules acting as
centres of force. A law of the same general character is probably to be found
connecting the temperatures of liquid and solid bodies with the energy possessed
by their molecules, although our ignorance of the nature of the connexions
between the molecules renders it difficult to enunciate the precise form of the law.

The molecules of a gas in this theory are those portions of it which move
about as a single body. These molecules may be mere points, or pure centres
of force endowed with inertia, or the capacity of performing work while losing
velocity. They may be systems of several such centres of force, bound together
by their mutual actions, and in this case the different centres may either be
separated, so as to form a group of points, or they may be actually coincident,
so as to form one point.

Finally, if necessary, we may suppose them to be small solid bodies of a
determinate form ; but in this case we must assume a new set of forces binding
the parts of these small bodies together, and so introduce a molecular theory
of the second order. The doctrines that all matter is extended, and that no
two portions of matter can coincide in the same place, being deductions from
our experiments with bodies sensible to us, have no application to the theory
of molecules.

VOL. II. 5

34 THE DYNAMICAL THEORY OF GASES.

The actual energy of a moving body consists of two parts, one due to the
motion of its centre of gravity, and the other due to the motions of its parts
relative to the centre of gravity. If the body is of invariable form, the motions
of its parts relative to the centre of gravity consist entirely of rotation, but
if the parta of the body are not rigidly connected, their motions may consist
tit' oscillations of various kinds, as well as rotation of the whole body.

The mutual interference of the molecules in their courses will cause their
energy of motion to be distributed in a certain ratio between that due to the
motion of the centre of gravity and that due to the rotation, or other internal
motion. If the molecules are pure centres of force, there can be no energy of
rotation, and the whole energy is reduced to that of translation ; but in all
other cases the whole energy of the molecule may be represented by ^Mifft,
where £ is the ratio of the total energy to the energy of translation. The
ratio /8 will be different for every molecule, and will be different for the same
molecule after every encounter with another molecule, but it will have an
average value depending on the nature of the molecules, as has been shown
by Clausius. The value of ft can be determined if we know either of the
specific heats of the gas, or the ratio between them.

The method of investigation which I shall adopt in the following paper,
is to determine the mean values of the following functions of the velocity of
all the molecules of a given kind within an element of volume : —

(a) the mean velocity resolved parallel to each of the coordinate axes;

(£) the mean values of functions of two dimensions of these component
velocities ;

(y) the mean values of functions of three dimensions of these velocities.

The rate of translation of the gas, whether by itself, or by diffusion through
another gas, is given by (a), the pressure of the gas on any plane, whether
normal or tangential to the plane, is given by (/3), and the rate of conduction
of heat through the gas is given by (y).

I propose to determine the variations of these quantities, due, 1st, to the
encounters of the molecules with others of the same system or of a different
system ; 2nd, to the action of external forces such as gravity ; and 3rd, to the
passage of molecules through the boundary of the element of volume.

I shall then apply these calculations to the determination of the statical
cases of the final distribution of two gases under the action of gravity, the

THE DYNAMICAL THEORY OF GASES. 35

equilibrium of temperature between two gases, and the distribution of tempe-
rature in a vertical column. These results are independent of the law of force
between the molecules. I shall also consider the dynamical cases of diffusion,
viscosity, and conduction of heat, which involve the law of force between the
molecules.

On the Mutual Action of Two Molecules.

Let the masses of these molecules be Mlt M3, and let their velocities
resolved in three directions at right angles to each other be £,, TJI} £, and
£„ 17,, £,. The components of the velocity of the centre of gravity of the two
molecules will be

The motion of the centre of gravity will not be altered by the mutual
action of the molecules, of whatever nature that action may be. We may
therefore take the centre of gravity as the origin of a system of coordinates
moving parallel to itself with uniform velocity, and consider the alteration of
the motion of each particle with reference to this point as origin.

If we regard the molecules as simple centres of force, then each molecule
will describe a plane curve about this centre of gravity, and the two curves
will be similar to each other and symmetrical with respect to the line of apses.
If the molecules move with sufficient velocity to carry them out of the sphere
of their mutual action, their orbits will each have a pair of asymptotes inclined

TT

at an angle - — 9 to the line of apses. The asymptotes of the orbit of Ml

I

will be at a distance &, from the centre of gravity, and those of Mt at a
distance 63, where

The distance between two parallel asymptotes, one in each orbit, will be

If, while the two molecules are still beyond each other's action, we draw
a straight line through M1 in the direction of the relative velocity of Jf, to
Mt, and draw from Mt a perpendicular to this line, the length of this perpen-

5—2

36 THE DYNAMICAL THEORY OF OASES.

ilicular will be b, and the plane including b and the direction of relative motion
will bo the plane of the orbits about the centre of gravity.

When, after their mutual action and deflection, the molecules have again
reached a distance such that there is no sensible action between them, each
will be moving with the same velocity relative to the centre of gravity that
it had before the mutual action, but the direction of this relative velocity will
be turned through an angle 20 in the plane of the orbit.

The angle 0 is a function of the relative velocity of the molecules and of
It, the form of the function depending on the nature of the action between
the molecules.

If we suppose the molecules to be bodies, or systems of bodies, capable of
rutation, internal vibration, or any form of energy other than simple motion of
translation, these results will be modified. The value of 6 and the final velocities
<>t' the molecules will depend on the amount of internal energy in each molecule
before the encounter, and on the particular form of that energy at every instant
during the mutual action. We have no means of determining such intricate
actions in the present state of our knowledge of molecules, so that we must
content ourselves with the assumption that the value of 0 is, on an average,
the same as for pure centres of force, and that the final velocities differ from
the initial velocities only by quantities which may in each collision be neglected,
although in a great many encounters the energy of translation and the internal
energy of the molecules arrive, by repeated small exchanges, at a final ratio,
which we shall suppose to be that of 1 to ft— 1.

We may now determine the final velocity of M, after it has passed beyond
the sphere of mutual action between itself and Mt.

Let V be the velocity of J/, relative to Mt, then the components of V are

The plane of the orbit is that containing V and b. Let this plane be
inclined ^ to a plane containing V and parallel to the axis of x ; then, since
the direction of V is turned round an angle 20 in the plane of the orbit,
while its magnitude remains the same, we may find the value of f, after the
encounter. Calling it £',,

~ sn * cos

THE DYNAMICAL THEORY OF GASES. 37

There will be similar expressions for the components of the final velocity
of Ml in the other coordinate directions.

If we know the initial positions and velocities of Ml and Mt we can
determine V, the velocity of Mr relative to Jf2; b the shortest distance between
M! and Mt if they had continued to move with uniform velocity in straight
lines ; and <f> the angle which determines the plane in which V and b lie.
From V and 6 we can determine 6, if we know the law of force, so that the
problem is solved in the case of two molecules.

When we pass from this case to that of two systems of moving molecules,
we shall suppose that the time during which a molecule is beyond the action
of other molecules is so great compared with the time during which it is
deflected by that action, that we may neglect both the time and the distance
described by the molecules during the encounter, as compared with the time
and the distance described while the molecules are free from disturbing force.
We may also neglect those cases in which three or more molecules are within
each other's spheres of action at the same instant.

On the Mutual Action of Two Systems of Moving Molecules.

Let the number of molecules of the first kind in unit of volume be Nlt

the mass of each being M^ The velocities of these molecules will in general

be different both in magnitude and direction. Let us select those molecules
the components of whose velocities lie between

£ and & + d£, 17, and 17, + drjl} £, and £, + d£lt

and let the number of these molecules be dN^. The velocities of these molecules
will be very nearly equal and parallel.

On account of the mutual actions of the molecules, the number of mole-
cules which at a given instant have velocities within given limits will be
definite, so that

dNt*fi(fatt*t&& (a).

We shall consider the form of this function afterwards.

Let the number of molecules of the second kind in unit of volume be
Nt, and let dN3 of these have velocities between £, and & + d£3, 77, and
£, and £, + cZ£.,, where

38 TH» DYNAMICAL THEORY OF OASES.

The Telocity of any of the dNt molecules of the first system relative to
the dtft molecules of the second system is V, and each molecule J/, will in
the time & describe a relative path F& among the molecules of the second
system. Conceive a space bounded by the following surfaces. Let two cylindrical
Mrfew* have the common axis F& and radii b and b + db. Let two planes
be drawn through the extremities of the line V8t perpendicular to it. Finally,
let two planes be drawn through VSt making angles tf> and </> + (/»/> with a plane
through V parallel to the axis of x. Then the volume included between the
four planes and the two cylindric surfaces will be Vbdbd<fit.

If this volume includes one of the molecules Mt, then during the time 8*
there will be an encounter between Ml and J/,, in which b is between b and
h + db, and <f> between <f> and <f> + d<f>.

Since there are <&V, molecules similar to 3f, and cZiV, similar to Mt in unit
of volume, the whole number of encounters of the given kind between the two
systems will be

Now let Q be any property of the molecule Mlt such as its velocity in a
given direction, the square or cube of that velocity or any other property of the
molecule which is altered in a known manner by an encounter of the given
kind, so that Q becomes Qf after the encounter, then during the time 8< a
certain number of the molecules of the first kind have Q changed to Q', while
the remainder retain the original value of Q, so that

8QdN, = (<?-

or -(V-tyVtotod+UfjUf* ..................... (3).

Here - * ' refers to the alteration in the sum of the values of Q for the
, molecules, due to their encounters of the given kind with the dN, molecules

of the second sort. In order to determine the value of ^ ', the rate of

ot

alteration of Q among all the molecules of the first kind, we must perform
the following integrations : —

1st, with respect to <f> from <j> = 0 to ^ = 2rr.

2nd, with respect to b from 6 = 0 to 6 = oo . These operations will give

THE DYNAMICAL THEOUY OF GASES. 39

the results of the encounters of every kind between the eZ-ZV, and dNt mole-
cules.

3rd, with respect to dN^, or /„ ($ai)tQ d&d^dt,,.

4th, with respect to dN1} or £ (fi^i) d^d^d^.

These operations require in general a knowledge of the forms of /, and /,.

1st. Integration with respect to <f>.

Since the action between the molecules is the same in whatever plane it

P*
takes place, we shall first determine the value of (Q' — Q)d<ji in several cases,

) o
making Q some function of £, 77, and £.

(a) Let Q = £ and <? = ?„ then

Bin* 0 .................. (4).

08) Let <? = fraud <? = £?,

Tif

...... (5).

By transformation of coordinates we may derive from this

(V .Vi - ^o d$ = orao* KMif i1?* ~ Jfi^1 + * (Jfi ~ M*} (^2 + ^i)} 87r sinl ^

-SJf.tf.-f,)^-^] ...... (6),

with similar expressions for the other quadratic functions of £ t], £.

(y) Let <? = £(fr + V + £.')> and Q^f^V + '?? + £?); then putting

fr + ^ + tt = V?, £& + ^.1?. + O. = ^
and (£-£)' + fo - ^)2 + & -£,)'=F', we find

+ f ^^V (8ir 8in' 0 - STT sin' 20) 2 (£-£)( Z7- F,2

\M1 + MJ

(M \"
saFJ(87rsi

+ U»-rVr ) (Sir sins 0 - 2ir sin' 20) 2 (£ - £) Fa

40 THE DYNAMICAL THEORY OF OASES.

Theae are the principal functions of (, r), f whose changes we shall have to
consider; we shall indicate them by the symbols a, /8, or y, according as the
function of the velocity is of one, two, or three dimensions.

2nd. Integration with respect to b.

We have next to multiply these expressions by bdb, and to integrate with
respect to 6 from 6 = 0 to b = <x> . We must bear in mind that 6 is a function
of 6 and V, and can only be determined when the law of force is known.
In the expressions which we have to deal with, 6 occurs under two forms
only, namely, sin' 6 and sin1 20. If, therefore, we can find the two values of

B,= \ tirbdbBm*0, and Bt= f° irbdb sin* 26 (8),

Jo Jo

we can integrate all the expressions with respect to 6.

Bl and Bt will be functions of V only, the form of which we can determine
only in particular cases, after we have found 6 as a function of b and V.

Determination of B for certain laws of Force.

Let us assume that the force between the molecules Mt and Mt is repul-
sive and varies inversely as the nth power of the distance between them, the
value of the moving force at distance unity being K, then we find by the
equation of central orbits,

n-l\aj
where x = - , or the ratio of b to the distance of the molecules at a given

time : x is therefore a numerical quantity ; a is also a numerical quantity and
is given by the equation

The limits of integration are x = 0 and x = x', where x' is the least positive
root of the equation

THE DYNAMICAL THEORY OF GASES. 41

It is evident that 6 is a function of a and w, and when n is known 6
may be expressed as a function of a only.

so that if we put

r= I 4:irada sin2 0, A3= I vada sin2 20 ................ (13),

Jo Jo

.4! and A, will be definite numerical quantities which may be ascertained when
n is given, and Bl and S, may be found by multiplying 4, and A3 by

Before integrating further we have to multiply by V, so that the form in
which V will enter into the expressions which have to be integrated with respect
to dNt and dNt will be

It will be shewn that we have reason from experiments on the viscosity
of gases to believe that n = 5. In this case V will disappear from the expres-
sions of the form (3), and they will be capable of immediate integration with
respect to dNl and dNt.

If we assume n = 5 and put a* = 2 cot" 2<^> and x = Jl — tan1 <£ cos «|>,

IT
/"I fijj

^-0Wcos2(f> /, . , .-s-rr

2 r J o v 1 -^ sin <p sin ii }.

(14),

where ,f8ln^ is the complete elliptic function of the first kind and is given in
Legendre's Tables. I have computed the following Table of the distance of the
asymptotes, the distance of the apse, the value of 6, and of the quantities whose
summation leads to A1 and Av

VOL. II. 6

THE DYNAMICAL THEORY OF GASES.

*

1

DUtanee

Of.p-

I

nn<»

on*

nn'90
nn'Sf

5 6

infinite

infinite

6 6

0

0

5 0

180]

2391

0 31

•00270

•01079

10 0

L6M

1684

1 53

•01464

•03689

15 0

1316

1366

4 47

•02781

•11048

20 0

1092

1172

8 45

•05601

•21885

25 0

916

1036

14 15

•10325

•38799

30 0

760

931

21 42

•18228

•62942

35 0

603

845

31 59

•31772

•71433

40 0

420

772

47 20

•55749

102427

41 0

374

758

51 32

•62515

•96763

42 0

324

745

56 26

•70197

•85838

43 0

264

732

62 22

•78872 -67866

44 0

187

719

70 18

•88745

•40338

44 30

132

713

76 1

•94190

•21999

45 0

0

707

90 0

1-00000

•00000

A,

sin' 0 = 2'6595 ........................... (15),

The paths described by molecules about a centre
of force S, repelling inversely as the fifth power of
the distance, are given in the figure.

The molecules are supposed to be originally
moving with equal velocities in parallel paths, and
the way in which their deflections depend on the
distance of the path from S is shewn by the dif-
ferent curves in the figure.

3rd. Integration with respect to dN,.

We have now to integrate expressions involving various functions of £ r,.
£, and V with respect to all the molecules of the second sort. We may
write the expression to be integrated

///«»%«

THE DYNAMICAL THEORY OF GASES. 43

where Q is some function of £ 17, £, &c., already determined, and /, is the
function which indicates the distribution of velocity among the molecules of the
second kind.

In the case in which n = 5, V disappears, and we may write the result
of integration QN3,

where Q is the mean value of Q for all the molecules of the second kind,
and jVj is the number of those molecules.

If, however, n is not equal to 5, so that V does not disappear, we should
require to know the form of the function ft before we could proceed further
with the integration.

The only case in which I have determined the form of this function is
that of one or more kinds of molecules which have by their continual encoun-
ters brought about a distribution of velocity such that the number of molecules
whose velocity lies within given limits remains constant. In the Philosophical
Magazine for January 1860, I have given an investigation of this case, founded
on the assumption that the probability of a molecule having a velocity resolved
parallel to x lying between given limits is not in any way affected by the
knowledge that the molecule has a given velocity resolved parallel to y. As
this assumption may appear precarious, I shall now determine the form of the
function in a different manner.

On the Final Distribution of Velocity among the Molecules of Two Systems acting
on one another according to any Law of Force.

From a given point O let lines be drawn representing in
direction and magnitude the velocities of every molecule of
either kind in unit of volume. The extremities of these lines
will be distributed over space in such a way that if an ele-
ment of volume dV be taken anywhere, the number of such
lines which will terminate within dV will be f(r)dV, where
r is the distance of dV from O.

Let OA = a be the velocity of a molecule of the first kind, and OB = b
that of a molecule of the second kind before they encounter one another, then

6—2

44

THE DYNAMICAL THEORY OF GASES.

BA will be the velocity of A relative to B; and if we divide AB in G
invenelj aa the masses of the molecules, and join OG, OG will be the velocity
of the centre of gravity of the two molecules.

Now let OA' = tf' and 0/T=*5' be the velocities of the two molecules after
the encounter, GA^GA' and GB = GBf, and A'GR is a straight line not
necessarily in the plane of OAB. Also AGA' = 20 is the angle through which
the relative velocity is turned in the encounter in question. The relative motion
of the molecules is completely defined if we know BA the relative velocity
before the encounter, 26 the angle through which BA is turned during the
encounter, and ^ the angle which defines the direction of the plane in which
BA and B*A' lie. All encounters in which the magnitude and direction of BA,
.UK! also B and <^», lie within certain almost contiguous limits, we shall class
as encounters of the given kind. The number of such encounters in unit of
time will be

njifde ....................................... (17),

where n, and n, are the numbers of molecules of each kind under consideration,
and F is a function of the relative velocity and of the angle 6, and de depends
on the limits of variation within which we class encounters as of the same kind.

Now let A describe the boundary of an element of volume dV while AB
and A'B move parallel to themselves, then B, A', and B' will also describe
equal and similar elements of volume.

The number of molecules of the first kind, the lines representing the velo-
cities of which terminate in the element dV at A, will be

«i=/i(a)dr .................................... (18).

The number of molecules of the second kind which have velocities corresponding
to OB will be

(19);

and the number of encounters of the given kind between these two sets of
molecules will be

(20).

The lines representing the velocities of these molecules after encounters of the
given kind will terminate within elements of volume at A' and R, each equal
to dV.

THE DYNAMICAL THEORY OF GASES. 45

In like manner we should find for the number of encounters between
molecules whose original velocities corresponded to elements equal to dV described
about A' and R, and whose subsequent velocities correspond to elements equal
to dV described about A and B,

f1(aU(b')(dVYF'de .............................. (21),

where F' is the same function of RA' and A'GA that F is of BA and AGA\
F is therefore equal to F',

When the number of pairs of molecules which change their velocities from
OA, OB to OA', OR is equal to the number which change from OA', OB'
to OA, OB, then the final distribution of velocity will be obtained, which
will not be altered by subsequent exchanges. This will be the case when

Now the only relation between a, b and a, V is

Mtf + Mp^MjP + MJP, ......................... (23),

rt» «J

whence we obtain /(<*) = ^e ""',/,(&) = (73e"^ .............. . ......... (24),

where Mjt^Mft ................................... (25).

(*+**+{•
By integrating \l\Cf~ ** dgdr) d£, and equating the result to Nv we

obtain the value of Cv If, therefore, the distribution of velocities among N,
molecules is such that the number of molecules whose component velocities are
between £ and £+c/£, rj and rj + d-rj, and £ and £ + d£ is

TV

(2G),
o

then this distribution of velocities will not be altered by the exchange of
velocities among the molecules by their mutual action.

This is therefore a possible form of the final distribution of velocities. It
is also the only form ; for if there were any other, the exchange between
velocities represented by OA and OA' would not be equal. Suppose that the
number of molecules having velocity OA' increases at the expense of OA.
Then since the total number of molecules corresponding to OA' remains constant,
OA' must communicate as many to OA", and so on till they return to OA.

Hence if OA, OA', OA", &c. be a series of velocities, there will be a
tendency of each molecule to assume the velocities OA, OA', OA", &c. in order,
returning to OA. Now it is impossible to assign a reason why the successive

46 THE DYHAMICAL THEORY OF GASES.

velocities of a molecule should be arranged in this cycle, rather than in the
reverse order. If, therefore, the direct exchange between OA and OA' is not
equal, the equality cannot be preserved by exchange in a cycle. Hence the
direct exchange between OA and OA' is equal, and the distribution we have
determined is the only one possible.

This final distribution of velocity is attained only when the molecules have
had a great number of encounters, but the great rapidity with which the
encounters succeed each other is such that in all motions and changes of
the gaseous system except the most violent, the form of the distribution of
velocity is only slightly changed.

When the gas moves in mass, the velocities now determined are com-
pounded with the motion of translation of the gas.

When the differential elements of the gas are changing their figure, being
compressed or extended along certain axes, the values of the mean square of
the velocity will be different in different directions. It is probable that the
form of the function will then be

(27),

where a, /8, y are slightly different. I have not, however, attempted to investi-
gate the exact distribution of velocities in this case, as the theory of motion
of gases does not require it.

When one gas is diffusing through another, or when heat is being conducted
through a gas, the distribution of velocities will be different in the positive
and negative directions, instead of being symmetrical, as in the case we have
considered. The want of symmetry, however, may be treated as very small in
most actual cases.

The principal conclusions which we may draw from this investigation are
as follows. Calling a the modulus of velocity,

1st. The mean velocity is v = -/=a ..................... (28).

•Jir

n

2nd. The mean square of the velocity is v'^a1 ....................... (29).

2t

3rd. The mean value of (• is fi = -a> ....................... (30).

THE DYNAMICAL THEORY OF GASES. 47

4th. The mean value of f4 is ^4 = |a4 ................................. (31).

5th. The mean value of £y is |y = ia4 ................................. (32).

6th. When there are two systems of molecules

Mjaf =M& .................................. (33),

whence Mj)? = Mtv; .................................. (34),

or the mean vis viva of a molecule will be the same in each system. This
is a very important result in the theory of gases, and it is independent of the
nature of the action between the molecules, as are all the other results relating
to the final distribution of velocities. We shall find that it leads to the law
of gases known as that of Equivalent Volumes.

Variation of Functions of the Velocity due to encounters between the Molecules.

80
We may now proceed to write down the values of -~— in the different

cases. We shall indicate the mean value of any quantity for all the molecules
of one kind by placing a bar over the symbol which represents that quantity
for any particular molecule, but in expressions where all such quantities are to
be taken at their mean values, we shall, for convenience, omit the bar. We
shall use the symbols 81 and 8., to indicate the effect produced by molecules of
the first kind and second kind respectively, and S3 to indicate the effect of
external forces. We shall also confine ourselves to the case in which « = 5,
since it is not only free from mathematical difficulty, but is the only case which
is consistent with the laws of viscosity of gases.

In this case V disappears, and we have for the effect of the second system
on the first,

where the functions of £ rj, £ in j(O/-Q)d<f> must be put equal to their mean
values for all the molecules, and A1 or A^ must be put for A according as
sin* Q or sin" 20 occurs in the expressions in equations (4), (5), (6), (7). We
thus obtain

w "'-'*^'*-6' ................................. (36);

4g

THE DYNAMICAL THEORY OP OASES.

+ AtMt(rh-rll + {.-£, -2£^?,

(37);

( Jf, - Jf.) (f ,17, + &»,)} - 3AJI . (£ -

(39),

using the symbol 8, to indicate variations arising from the action of molecules
of the second system.

These are the values of the rate of variation of the mean values of
£• f'i (itn ^d £|P?> for the molecules of the first kind due to their encounters
with molecules of the second kind. In all of them we must multiply up all
functions of £, 17, £, and take the mean values of the products so found. As
this has to be done for all such functions, I have omitted the bar over each
function in these expressions.

To find the rate of variation due to the encounters among the particles of
the same system, we have only to alter the suffix (t) into (J) throughout, and to
change K, the coefficient of the force between Ml and Mt into Klt that of the
force between two molecules of the first system. We thus find

(42);

THE DYNAMICAL THEORY OF GASES. 49

S£F2 f K \
i \ Pigi^ i / Jxi \ MW A Q /£ p-2_£]7

»*' 80 \2J.f7 ^ Zi'i

These quantities must be added to those in equations (36) to (39) in order
to get the rate of variation in the molecules of the first kind due to their
encounters with molecules of both systems. When there is only one kind of
molecules, the latter equations give the rates of variation at once.

On the Action of External Forces on a System of Moving Molecules.

We shall suppose the external force to be like the force of gravity, pro-
ducing equal acceleration on all the molecules. Let the components of the force
in the three coordinate directions be X, Y, Z. Then we have by dynamics for
the variations of £ £2, and £F2 due to this cause,

.(44);
.(45);

^ = r,X+tY (46);

(47);

v" St
where S3 refers to variations due to the action of external forces.

On the Total rate of change of the different functions of the velocity of the mole-
cules of the first system arising from their encounters with molecules of both
systems and from the action of external forces.

To find the total rate of change arising from these causes, we must add

S,<? Sa<? W

8* ' & ' ' 8t '

the quantities already found. We shall find it, however, most convenient in
the remainder of this investigation to introduce a change in the notation, and
to substitute for

£ 17, and £, « + £ v + rj, and w + £ (48),

VOL. II. 7

THE DYNAMICAL THEORY OP OASES.

where u, v, and w are so chosen that they are the mean values of the com-
ponents of the velocity of all molecules of the same system in the immediate
neighbourhood of a given point. We shall also write

p ............................... (49),

where p, and p, are the densities of the two systems of molecules, that is, the
maw in unit of volume. We shall also write

•*• (

• MM .w

Pi. Pit * *t an quantities the absolute values of which can be deduced

from experiment. We have not as yet experimental data for determining
M, N, or K.

We thus find for the rate of change of the various functions of the velocity,

(a)

- 2£'} + fa

..(52);

also °^= -

Mt

,-u,)(t>,~i

{2Al

.(53).

(y) As the expressions for the variation of functions of three dimensions
in mixed media are complicated, and as we shall not have occasion to use
them, I shall give the case of a single medium,

X

(54).

THE DYNAMICAL THEORY OF GASES. 51

Theory of a Medium composed of Moving Molecules.

We shall suppose the position of every moving molecule referred to three
rectangular axes, and that the component velocities of any one of them, resolved
in the directions of x, y, z, are

where u, v, w are the components of the mean velocity of all the molecules
which are at a given instant in a given element of volume, and £ 77, £ are
the components of the relative velocity of one of these molecules with respect
to the mean velocity.

The quantities u, v, w may be treated as functions of x, y, z, and t, in
which case differentiation will be expressed by the symbol d. The quantities
£ rj, £, being different for every molecule, must be regarded as functions of t
for each molecule. Their variation with respect to t will be indicated by the
symbol 8.

The mean values of £* and other functions of f, 77, £ for all the molecules
in the element of volume may, however, be treated as functions of x, y, z,
and t.

If we consider an element of volume which always moves with the velocities
u, v, w, we shall find that it does not always consist of the same molecules,
because molecules are continually passing through its boundary. We cannot
therefore treat it as a mass moving with the velocity u, v, w, as is done in
hydrodynamics, but we must consider separately the motion of each molecule.
When we have occasion to consider the variation of the properties of this element
during its motion as a function of the time we shall use the symbol 9.

We shall call the velocities u, v, w the velocities of translation of the medium,
and f, 77, £ the velocities of agitation of the molecules.

Let the number of molecules in the element dx dy dz be JV dx dy dz, then
we may call N the number of molecules in unit of volume. If M is the mass
of each molecule, and p the density of the element, then

p ....................................... (55).

Transference of Quantities across a Plane Area,

We must next consider the molecules which pass through a given plane
of unit area in unit of time, and determine the quantity of matter, of momentum,

7—2

J3 THE DYNAMICAL THEORY OF

of heat, Ac. which is transferred from the negative to the positive side of this
plane in unit of time.

We shall first divide the N molecules in unit of volume into classes
according to the value of (, 17, and £ for each, and we shall suppose that the
number of molecules in unit of volume whose velocity in the direction of x lies
between £ and (+<l(, i) and y + drj, £ and C + d£ is dN, dN will then be a
function of the component velocities, the sum of which being taken for all the
molecules will give N the total number of molecules. The most probable form
of this function for a medium in its state of equilibrium is

(56).

In the present investigation we do not require to know the form of this
function.

Now let us consider a plane of unit area perpendicular to x moving with
a velocity of which the part resolved parallel to x is «'. The velocity of the
plane relative to the molecules we have been considering is u' — (u + g), and since
there are dN of these molecules in unit of volume it will overtake

such molecules in unit of time, and the number of such molecules passing
from the negative to the positive side of the plane, will be

Now let Q be any property belonging to the molecule, such as its mass,
momentum, vis viva, &c., which it carries with it across the plane, Q being
supposed a function of £ or of £, 77, and £, or to vary in any way from one
molecule to another, provided it be the same for the selected molecules whose
number is dN, then the quantity of Q transferred across the plane in the
positive direction in unit of time is

or u-u'QdN+^QdN ........................... (57).

If we put QN for fQdN, and ~£QN for tfQdN, then we may call Q the
mean value of Q, and fQ the mean value of £Q, for all the particles in the
element of volume, and we may write the expression for the quantity of Q
which crosses the plane in unit of time

(u-u')QN+JQN .............................. (58).

THE DYNAMICAL THEORY OF GASES. 53

(a) Tramference of Matter across a Plane — Velocity of the Fluid.

To determine the quantity of matter which crosses the plane, make Q equal
to M the mass of each molecule; then, since M is the same for all molecules
of the same kind, M=M; and since the mean value of £ is zero, the expres-
sion is reduced to

(u-u')MN=(u-u')p (59).

If u — u', or if the plane moves with velocity u, the whole excess of matter
transferred across the plane is zero ; the velocity of the fluid may therefore be
defined as the velocity whose components are u, v, w.

(ft) Transference of Momentum across a Plane — System of Pressures at am/

point of the Fluid.

The momentum of any one molecule in the direction of x is M(u + £).
Substituting this for Q, we get for the quantity of momentum transferred
across the plane in the positive direction

(u-u')up + ?p (60).

If the plane moves with the velocity u, this expression is reduced to £'p.
where fl represents the mean value of f .

This is the whole momentum in the direction of x of the molecules projected
from the negative to the positive side of the plane in unit of time. The
mechanical action between the parts of the medium on opposite sides of the
plane consists partly of the momentum thus transferred, and partly of the
direct attractions or repulsions between molecules on opposite sides of the plane.
The latter part of the action must be very small in gases, so that we may
consider the pressure between the parts of the medium on opposite sides of the
plane as entirely due to the constant bombardment kept up between them.
There will also be a transference of momentum in the directions of y and z
across the same plane,

(u-u')vp + fr)p (61),

and (u-u')wp + i%p (62),

where £77 and f£ represent the mean values of these products.

54 THE DYNAMICAL THEORY OF OASES.

If the plane moves with the mean velocity u of the fluid, the total force
exerted on the medium on the positive side by the projection of molecules into
it from the negative side will be

a normal pressure £*p in the direction of x,

a tangential pressure £rjp in the direction of y,

and a tangential pressure g£p in the direction of z.

If X, Y, Z are the components of the pressure on unit of area of a plane
whose direction cosines are /, m, n,

(63).
Z=

When a gas is not in a state of violent motion the pressures in all direc-
tions are nearly equal, in which case, if we put

ep + ^P + Cp = 3p .............................. (64), ^

the quantity p will represent the mean pressure at a given point, and ^p, tfp,
and f p will differ from p only by small quantities ; i)£p, ££p, and £r)p will then
be also small quantities with respect to p,

Energy in the Medium — Actual Heat.

The actual energy of any molecule depends partly on the velocity of its
centre of gravity, and partly on its rotation or other internal motion with respect
to the centre of gravity. It may be written

lM{(« + ff + (v + tf + (w + Q} + lEM .................. (65),

where \EM is the internal part of the energy of the molecule, the form of
which is at present unknown. Summing for all the molecules in unit of volume,
the energy is

$(u' + v' + iv')p + t(e + r)> + £)p + tEp ..................... (66).

The first term gives the energy due to the motion of translation of the
medium in mass, the second that due to the agitation of the centres of gravity
of the molecules, and the third that due to the internal motion of the parts of
each molecule.

THE DYNAMICAL THEORY OP GASES. 55

If we assume with Clausius that the ratio of the mean energy of internal
motion to that of agitation tends continually towards a definite value (ft — I),
we may conclude that, except in very violent disturbances, this ratio is always
preserved, so that

E = (p-l)(?+if + ?) ........................ (67).

The total energy of the invisible agitation in unit of volume will then be

WP + tf + P) ................................. (68),

or f£p ............................... „'..; ......... (69).

This energy being in the form of invisible agitation, may be called the
total heat in the unit of volume of the medium.

(y) Transference of Energy across a Plane — Conduction of Heat*
Putting C = i/8(p + V + p)Jf, and u = u' ........................ (70),

we find for the quantity of heat carried over the unit of area by conduction
in unit of time

where f , &c. indicate the mean values of £*> &c> They are always small
quantities.

On the Rate of Variation of Q in an Element of Volume, Q being any property

of the Molecules in that Element.

Let Q be the value of the quantity for any particular molecule, and Q
the mean value of Q for all the molecules of the same kind within the element.

The quantity Q may vary from two causes. The molecules within the
element may by their mutual action or by the action of external forces produce
an alteration of Q, or molecules may pass into the element and out of it, and
so cause an increase or diminution of the value of Q within it. If we employ
the symbol 8 to denote the variation of Q due to actions of the first kind on
the individual molecules, and the symbol 3 to denote the actual variation of Q
in an element moving with the mean velocity of the system of molecules under

56 TOE DYNAMICAL THEORY OF OASES.

uderation, then by the ordinary investigation of the increase or diminution
or matter in an element of volume as contained in treatises on Hydrodynamics,

(72),

where the last three terms are derived from equation (59) and two similar
equations, and denote the quantity of Q which flows out of an element of volume,
that element moving with the velocities u', v't w'. If we perform the differen-
tiations and then make u' = «, v' = v, and w' = w, then the variation will be that
in an element which moves with the actual mean velocity of the system of
molecules, and the equation becomes

Equation of Continuity.

Put Q = M the mass of a molecule ; M is unalterable, and we have,
putting MN=p,

(du dv dw\

<74>>

which is the ordinary equation of continuity in hydrodynamics, the element
being supposed to move with the velocity of the fluid. Combining this equa-
tion wi£h that from which it was obtained, we find

a more convenient form of the general equation.

Equations of Motion (a).

To obtain the Equation of Motion in the direction of x, put Q = MJ(ul + (l),
the momentum of a molecule in the direction of x.

80
We obtain the value of ^ from equation (51), and the equation may be

written

~ dz (P® = ^V-P. K ~ «0 + Xfr. -(76).

THE DYNAMICAL THEORY OF GASES. 57

In this equation the first term denotes the efficient force per unit of volume,
the second the variation of normal pressure, the third and fourth the variations
of tangential pressure, the fifth the resistance due to the molecules of a different
system, and the sixth the external force acting on the system.

The investigation of the values of the second, third, and fourth terms
must be deferred till we consider the variations of the second degree.

Condition of Equilibrium of a Mixture of Gases.

In a state of equilibrium u^ and Ma vanish, p^* becomes plt and the tan-
gential pressures vanish, so that the equation becomes

<">•

which is the equation of equilibrium in ordinary hydrostatics.

This equation, being true of the system of molecules forming the first medium
independently of the presence of the molecules of the second system, shews
that if several kinds of molecules are mixed together, placed in a vessel and
acted on by gravity, the final distribution of the molecules of each kind will
be the same as if none of the other kinds had been present. This is the same
mode of distribution as that which Dalton considered to exist in a mixed
atmosphere in equilibrium, the law of diminution of density of each constituent
gas being the same as if no other gases were present.

This result, however, can only take place after the gases have been left
for a considerable time perfectly undisturbed. If currents arise so as to mix
the strata, the composition of the gas will be made more uniform throughout.

The result at which we have arrived as to the final distribution of gases,
when left to themselves, is independent of the law of force between the
molecules.

Diffusion of Gases.

If the motion of the gases is slow, we may still neglect the tangential
pressures. The equation then becomes for the first system of molecules .

p^ + ^kA^p^-u^ + Xp, ................... (78),

VOL. II. 8

58 THE DYNAMICAL THEORY OF OASES.

and for the second,

(79).

In all cases of quiet diffusion we may neglect the first term of each
equation. If we then put Pt+p,=p, and /», + />, = /», we find by adding,

(80).

If we also put pltil +ptut=pu, then the volumes transferred in opposite direc-
tions across a plane moving with velocity u will be equal, so that

Here pt (u, — u) is the volume of the first gas transferred in unit of time
across unit of area of the plane reduced to pressure unity, and at the actual
temperature; and p,(u — u,) is the equal volume of the second gas transferred
across the same area in the opposite direction.

The external force X has very little effect on the quiet diffusion of gases
in vessels of moderate size. We may therefore leave it out in our definition
of the coefficient of diffusion of two gases.

When two gases not acted on by gravity are placed in different parts of
a vessel at equal pressures and temperatures, there will be mechanical equi-
librium from the first, and u will always be zero. This will also be approxi-
mately true of heavy gases, provided the denser gas is placed below the lighter.
Mr Graham has described in his paper on the Mobility of Gases*, experiments
which were made under these conditions. A vertical tube had its lower tenth
part filled with a heavy gas, and the remaining nine-tenths with a lighter gas.
After the lapse of a known time the upper tenth part of the tube was shut
off, and the gas in it analyzed, so as to determine the quantity of the heavier
gas which had ascended into the upper tenth of the tube during the given time.

In this case we have « = 0 .................................. (82),

P>P* 1 fa /Q0\

plpJcA>pdx-

* Philosophical Transactions, 1863.

THE DYNAMICAL THEORY OF GASES. 59

and by the equation of continuity,

dp, d ,

3?+sfc*>-° (84)>

whence 1

dt p1pJcAl p da?

or if we put D = ,\ .

p

dt '

The solution of this equation is

If the length of the tube is a, and if it is closed at both ends,

-— t TTX — t irx

** a "a -\ )'

where Clt Cv (7, are to be determined by the condition that when t = Q, pl=p,
from x = 0 to x = -foci, and p1 = 0 from x = -faa to x = a. The general expression
for the case in which the first gas originally extends from x = 0 to x = b, and
in which after a time t the gas from x = 0 to x = c is collected, is

», 6 2a f -*?t . irb . ire 1 -t'-^t . 2nb . 2irc }

-*-=-+- \e °' sin— sin— +—.e a" sm — sin- +&C.V (89),

p a ir*c { a a 2' aa j

tn

where ' • is the proportion of the first gas to the whole in the portion from
x = Q to x = c.

In Mr Graham's experiments, in which one-tenth of the tube was filled
with the first gas, and the proportion of the first gas in the tenth of the tube
at the other end ascertained after a time t, this proportion will be

We find for a series of values of — taken at equal intervals of time T,

P

m log, 10 a"
where 7 = • . . . •• -^ .

8—2

(0 THE DYNAMICAL THEORY OF OASES.

Time. $•

0 0

T -01193

2T '02305

3J -03376

4T -04366

5T -05267

671 '06072

-07321
-08227
-08845
oo -10000

Mr Graham's experiments on carbonic acid and air, when compared with this
Table, give T=500 seconds nearly for a tube 0'57 metre long. Now

log, 10 a' ,.

1 "

T

whence D = -0235

for carbonic acid and air, in inch-grain-second measure

Definition of the Coefficient of Diffusion.

D is the volume of gas reduced to unit of pressure which passes in unit
of time through unit of area when the total pressure is uniform and equal to p,
and the pressure of either gas increases or diminishes by unity in unit of
distance. D may be called the coefficient of diffusion. It varies directly as the
square of the absolute temperature, and inversely as the total pressure p.

The dimensions of D are evidently DT~l, where L and T are the standards
of length and time.

In considering this experiment of the interdiffusion of carbonic acid and
air, we have assumed that air is a simple gas. Now it is well known that
the constituents of air can be separated by mechanical means, such as passing
them through a porous diaphragm, as in Mr Graham's experiments on Atmolysis.

THE DYNAMICAL THEORY OP GASES. 61

The discussion of the interdiffusion of three or more gases leads to a much
more complicated equation than that which we have found for two gases, and
it is not easy to deduce the coefficients of interdiffusion of the separate gases.
It is therefore to be desired that experiments should be made on the inter-
diffusion of every pair of the more important pure gases which do not act
chemically on each other, the temperature and pressure of the mixture being
noted at the time of experiment.

Mr Graham has also published in Brande's Journal for 1829, pt. 2, p. 74,
the results of experiments on the diffusion of various gases out of a vessel
through a tube into air. The coefficients of diffusion deduced from these ex-
periments are —

Air and Hydrogen '026216

Air and Marsh-gas , '010240

Air and Ammonia... '00962

Air and Olefiant gas '00771

Air and Carbonic acid '00682

Air and Sulphurous acid '00582

Air and Chlorine '00486

The value for carbonic acid is only one third of that deduced from the
experiment with the vertical column. The inequality of composition of the
mixed gas in different parts of the vessel is, however, neglected ; and the dia^
meter of the tube at the middle part, where it was bent, was probably less
than that given.

Those experiments on diffusion which lasted ten hours, all give smaller
values of D than those which lasted four hours, and this would also result
from the mixture of the gases in the vessel being imperfect.

Interdiffusion through a small hole.

When two vessels containing different gases are connected by a small hole;
the mixture of gases in each vessel will be nearly uniform except near the
hole ; and the inequality of the pressure of each gas will extend to a distance
from the hole depending on the diameter of the hole, and nearly proportional
to that diameter.

n

THE DYNAMICAL THEORY OF O

Henoe in the equation

.(92)

the term j* will vary inversely as the diameter of the hole, while «, and u,
will not vary considerably with the diameter.

Hence when the hole is very small the right-hand side of the equation
may be neglected, and the flow of either gas through the hole will be inde-
pendent of the flow of the other gas, as the term kAp1pt(ut — ut) becomes com-
paratively insignificant.

One gas therefore will escape through a very fine hole into another nearly
as fast as into a vacuum ; and if the pressures are equal on both sides, the
volumes diffused will be as the square roots of the specific gravities inversely,
which is the law of diffusion of gases established by Graham*.

Variation of the invisible agitation (y8).
By putting for Q in equation (75)

(93),

and eliminating by means of equations (76) and (52), we find

fdv,

iT.

i du\ .. fdu, dv\ _ f d

.(94).

In this equation the first term represents the variation of invisible agitation
or heat ; the second, third, and fourth represent the cooling by expansion ; the

* Trans. Royal Society of Edinburgh, Vol. XH. p. 222.

THE DYNAMICAL THEORY OF GASES. 63

fifth, sixth, and seventh the heating effect of fluid friction or viscosity ; and

the last the loss of heat by conduction. The quantities on the other side of

the equation represent the thermal effects of diffusion, and the communication
of heat from one gas to the other.

The equation may be simplified in various cases, which we shall take in
order.

1st. Equilibrium of Temperature between two Gases. — Law of Equivalent Volumes.

We shall suppose that there is no motion of translation, and no transfer
of heat by conduction through either gas. The equation (94) is then reduced
to the following form,

If we put

W-g, ....... (%),

we find (Qt-Ql)=--(M^, + MlPA}(Qt-Ql} ......... (97),

1

Q,-Q>=Ce-°<, where n = (MtpA + M^,) - ....... (98).

--

If, therefore, the gases are in contact and undisturbed, Q1 and Q, will
rapidly become equal. Now the state into which two bodies come by exchange
of invisible agitation is called equilibrium of heat or equality of temperature-
Hence when two gases are at the same temperature,

Q1 = Q, .................................... (99),

64 THE DYNAMICAL THEORY OF OASES.

Hence if the pressures as well as the temperatures be the same in two

— = — (100),

/>. P.

or the mnnnnn of the individual molecules are proportional to the density of
the gas.

This result, by which the relative masses of the molecules can be deduced
from the relative densities of the gases, was first arrived at by Gay-Lussac
from chemical considerations. It is here shewn to be a necessary result of the
Dynamical Theory of Gases ; and it is so, whatever theory we adopt as to
the nature of the action between the individual molecules, as may be seen by
equation (34), which is deduced from perfectly general assumptions as to the
nature of the law of force.

We may therefore henceforth put - ! for -^ , where su s, are the specific

Oj JU. j

gravities of the gases referred to a standard gas.

If we use 6 to denote the temperature reckoned from absolute zero of a
gas thermometer, M, the mass of a molecule of hydrogen, F0' its mean square
of velocity at temperature unity, s the specific gravity of any other gas referred
to hydrogen, then the mass of a molecule of the other gas is

M=M0s (101).

Its mean square of velocity, V* = - V*6 (102).

Pressure of the gas, P = $-0Vt* (103).

We may next determine the amount of cooling by expansion.

Cooling by Expansion.

Let the expansion be equal in all directions, then

du_dv_dw_ 1 dp

and -r- and all terms of unsymmetrical form will be zero.

THE DYNAMICAL THEORY OF GASES. §5

If the mass of gas is of the same temperature throughout there will be
no conduction of heat, and the equation (94) will become

2-£ = 3/3-=3/3 ........................ (106),

which gives the relation between the density and the temperature in a gas
expanding without exchange of heat with other bodies. We also find

dp _ 8p 80

P " P " e

_2 + 3^9p

~W~ 7' }>

which gives the relation between the pressure and the density.

Specific Heat of Unit of Mass at Constant Volume.
The total energy of agitation of unit of mass is £/JF3 = £Z?, or

*-¥* <«>•>•

If, now, additional energy in the form of heat be communicated to it without
changing its density,

a*. ?£ &»?££* (no).

2/3 2 p 6

Hence the specific heat of unit of mass of constant volume is in dynamical
measure

**.-*££. (in)

~a0 " 2 P0

VOL. II.

THB DYNAMICAL THEORY OF OASBS.

Sj*cijic Heat of Unit of Mass at Constant Pressure.

By the addition of the heat dE the temperature was raised 30 and the
pressure 9p. Now, let the gas expand without communication of heat till the
pressure sinks to its former value, and let the final temperature be 0 + V0.
The temperature will thus sink by a quantity dd-d'0, such that

80-3*0 __ 2_ 3p 2 W_

~~e 2+3/s y = 2+30 e •

and the specific heat of unit of mass at constant pressure is

8ff_2 + 3£ p , .

3-0- ~2~~ P0-

The ratio of the specific heat at constant pressure to that of constant volume
is known in several cases from experiment. We shall denote this ratio by

whence 0 = f ]- ................................. (115).

The specific heat of unit of volume in ordinary measure is at constant volume

and at constant pressure

y

_

T-l JQ"
where J is the mechanical equivalent of unit of heat.

From these expressions Dr Rankine* has calculated the specific heat of air,
and has found the result to agree with the value afterwards determined ex-
perimentally by M. Regnaultt.

* Transactions oflhe Royal Society of Edinburgh, Vol. XX. (1850).
f CompUt Rendtu, 1853.

THE DYNAMICAL THEORY OF GASES.

67

Thermal Effects of Diffusion.

If two gases are diffusing into one another, then, omitting the terms rela-
ting to heat generated by friction and to conduction of heat, the equation (94)
gives

-...(118).

By comparison with equations (78), and (79), the right-hand side of this equa-
tion becomes

X (p,u, + pM) + Y (p^ + Piv,) + Z (Plw, + p,wt)

/dpl dpt dp, \ /dp, dp., dp,,

- \-±&+-£ **+-£: Vi] ~ \^r u*+ j !v, + -4-'
\dx ay dz J \dx dy dz

- M % (u' + v? + <) - ip, - « + v; + wfi.

The equation (118) may now be written

d.pW\

--

...(119).

The whole increase of energy is therefore that due to the action of the
external forces minus the cooling due to the expansion of the mixed gases.
If the diffusion takes place without alteration of the volume of the mixture,
the heat due to the mutual action of the gases in diffusion will be exactly
neutralized by the cooling of each gas as it expands in passing from places
where it is dense to places where it is rare.

9—2

68

THE DYNAMICAL THKOKY OF OASES.

Determination of the Inequality of Pressure in different directions due to (/,<•

Motion of the Medium.

Let iis put />,£,'-.?. + ?• and P&=P* + <I> ...................... (120).

Then by equation (52),

(2M>A> + S ^OM. - *

"''-*^^-^-^) J

the last terra depending on diffusion; and if we omit in equation (75) terms
of three dimensions in f, rj, £, which relate to conduction of heat, and neglect
quantities of the form £rjp and pi?—p, when not multiplied by the large coeffi-
cients k, kt, and i,, we get

du , dv , dw\
+ dy + Tz)

If the motion is not subject to any very rapid changes, as in all cases

except that of the propagation of sound, we may neglect ~ . In a single

ot

system of molecules

,; .-. ... JjU

2p (du . (du dv
whence ?= __^_ |__^_ + _ + _ .................. {124).

If we make ^ rj*- =/*

fj. will be the coefficient of viscosity, and we shall have by equation (120),

du (du dv

m-*(&+3j

dv . (du dv
Ty-*(d-X + dy

dw . (du dv

THE DYNAMICAL THEORY OF GASES. 69

and by transformation of co-ordinates we obtain

idv dw

dw . du\ (127)_

fdu dv^
\dy dxj

These are the values of the normal and tangential stresses in a simple gas
when the variation of motion is not very rapid, and when p, the coefficient
of viscosity, is so small that its square may be neglected.

Equations of Motion corrected for Viscosity.

Substituting these values in the equation of motion (76), we find
du dp (d*u d*u d*ii\ ^ d (du dv dw\

with two other equations which may be written down with symmetry. The
form of these equations is identical with that of those deduced by Poisson""
from the theory of elasticity, by supposing the strain to be continually relaxed
at a rate proportional to its amount. The ratio of the third and fourth terms
agrees with that given by Professor Stokesf.

If we suppose the inequality of pressure which we have denoted by q to
exist in the medium at any instant, and not to be maintained by the motion
of the medium, we find, from equation (123),

q^Ce-*** ..................................... (129)

= <7<f? if r= -=!- = £ .................. (130);

p

the stress q is therefore relaxed at a rate proportional to itself, so that

| = | ................................. (131).

We may call T the modulus of the time of relaxation.

* Journal de I'Ecole Polylechnigue, 1829, Tom. xm. Cah. xx. p. 139.

t " On the Friction of Fluids in Motion and the Equilibrium and Motion of Elastic Solids," Cambridge
Phil. Trans. Vol. VHI. (1845), p. 297, equation (2).

70 THK DYNAMICAL THBOBY OF GASES.

If we next make i-3, so that the stress q does not become relaxed, the
medium will be an elastic solid, and the equation

T T \

= 0 (132)

mav

where o, /?, y are the displacements of an element of the medium, and pa
is the normal pressure in the direction of z. If we suppose the initial value
of this quantity zero, and pm originally equal to p, then, after a small dis-
placement,

da

tP <134>;

and by transformation of co-ordinates the tangential pressure

(135).

The medium has now the mechanical properties of an elastic solid, the
rigidity of which is p, while the cubical elasticity is $p*.

The same result and the same ratio of the elasticities would be obtained
if we supposed the molecules to be at rest, and to act on one another with
forces depending on the distance, as in the statical molecular theory of elas-
ticity. The coincidence of the properties of a medium in which the molecules
are held, in equilibrium by attractions and repulsions, and those of a medium
in which the molecules move in straight lines without acting on each other
at all, deserve notice from those who speculate on theories of physics.

The fluidity of our medium is therefore due to the mutual action of the
molecules, causing them to be deflected from their paths.

The coefficient of instantaneous rigidity of a gas is therefore p •>

The modulus of the time of relaxation is T 1 .... (136).

The coefficient of viscosity is /* =pT

Now p varies as the density and temperature conjointly, while T varies
inversely as the density.

Hence p. varies as the absolute temperature, and is independent of the
density.

« Camb. PhU. Trans. VoL Tin. (1845), p. 311, equation (29).

THE DYNAMICAL THEORY OF GASES. 71

This result is confirmed by the experiments of Mr Graham on the Tran-
spiration of Gases*, and by my own experiments on the Viscosity or Internal
Friction of Air and other Gases f.

The result that the viscosity is independent of the density, follows from
the Dynamical Theory of Gases, whatever be the law of force between the
molecules. It was deduced by myself J from the hypothesis of hard elastic
molecules, and M. 0. E. Meyer § has given a more complete investigation on
the same hypothesis.

The experimental result, that the viscosity is proportional to the absolute
temperature, requires us to abandon this hypothesis, which would make it vary
as the square root of the absolute temperature, and to adopt the hypothesis
of a repulsive force inversely as the fifth power of the distance between the
molecules, which is the only law of force which gives the observed result.

Using the foot, the grain, and the second as units, my experiments give
for the temperature of 62* Fahrenheit, and in dry air,

/* = 0-0936.

If the pressure is 30 inches of mercury, we find, using the same units,

p = 477360000.

Since pT=fi, we find that the modulus of the time of relaxation of
rigidity in air of this pressure and temperature is

f a second.

This time is exceedingly small, even when compared with the period of
vibration of the most acute audible sounds ; so that even in the theory of
sound we may consider the motion as steady during this very short time, and
use the equations we have already found, as has been done by Professor Stokes ||.

* Philosophical Transactions, 1846 and 1849.

t Proceedings of the Royal Society, February 8, 1866 ; Philosophical Transactions, 1866, p. 249.
J Philosophical Magazine, January 1860. [Vol. I. xx.]
§ Poggendorff'a Annalen, 1865.

|| " On the effect of the Internal Friction of Fluids on the motion of Pendulums," Cambridge Transac-
tions, Vol. ix. (1850), art. 79.

THE DYNAMICAL THEORY OF OASES.

Viscosity of a Mixture of GOKS.

In a complete mixture of gases, in which there is no diffusion going on,
tin- velocity at any point is the same for all the gases.

dv dw\ TT

.>.|iiiition (122) becomes

Similarly,
p, U- -

, ...... (139).

, + 3 M^pfr -k(3At- 2At)

where p and q refer to the mixture, we

Since p=pi+p, and
shall have

where ft is the coefficient of viscosity of the mixture.

If we put *, and *, for the specific gravities of the two gases, referred
to a standard gas, in which the values of p and p at temperature 0, and pt
and p.,

'

where /t is the coefficient of viscosity of the mixture, and

*!+*,

(141).

This expression is reduced to /*, when p, = 0, and to /a,, when ^, = 0. For
other values of p, and p, we require to know the value of k, the coefficient

THE DYNAMICAL THEORY OF GASES. 73

of mutual Interference of the molecules of the two gases. This might be
deduced from the observed values of p, for mixtures, but a better method is
by making experiments on the interdiffusion of the two gases. The experi-
ments of Graham on the transpiration of gases, combined with my experiments
on the viscosity of air, give as values of kt for air, hydrogen, and carbonic acid,

Air ............... &1= 4-81 xlO10,

Hydrogen ....... &, = 1 42'8 X 1 010,

Carbonic acid . . . &, = 3 • 9 x 1 010.

The experiments of Graham in 1863, referred to at page 58, on the inter-
diffusion of air and carbonic acid, give the coefficient of mutual interference
of these gases,

Air and carbonic acid ..... ,& = 5'2xl010;

and by taking this as the absolute value of k, and assuming that the ratios
of the coefficients of interdiffusion given at page 76 are correct, we find

Air and hydrogen ...... & = 29'8 x 101

10

These numbers are to be regarded as doubtful, as we have supposed air to
be a simple gas in our calculations, and we do not know the value of k between
oxygen and nitrogen. It is also doubtful whether our method of calculation
applies to experiments such as the earlier observations of Mr Graham.

I have also examined the transpiration-tunes determined by Graham for
mixtures of hydrogen and carbonic acid, and hydrogen and air, assuming a
value of k roughly, to satisfy the experimental results about the middle of
the scale. It will be seen that the calculated numbers for hydrogen and car-
bonic acid exhibit the peculiarity observed in the experiments, that a small
addition of hydrogen increases the transpiration-time of carbonic acid, and that
in both series the tunes of mixtures depend more on the slower than on the
quicker gas.

The assumed values of k in these calculations were —

For hydrogen and carbonic acid k = l2'5 x 1010,
For hydrogen and air ............ k= 18'8 x 1010;

and the results of observation and calculation are, for the times of transpiration
of mixtures of —

VOL. II. 10

74

THE DYNAMICAL THEORY OF OASES.

Hyiravn tad Carbonic »eid

OtMOTWl

r»lrnilft*rl

Hydrogen and Air

ObMTOd

Calculated

100 0

•4321

•4375

100 0

•4434

•4375

»?•$ a-fi

•4714

•4750

95 5

•5282

•6300

95 5

•5157

•5089

90 10

•5880

•6028

90 10

•5722

•6678

75 25

•7488

•7438

75 25

•6786

•6822

50 50

•8179

•8488

50 50

•7339

•7652

25 75

•8790 -8946

25 75

•7535

•7468

10 90

•8880

•8983

10 90

•7521

•7361

5 95

•8960 -8996

0 100

•7470

7272

0 100

•9000

•9010

The numbers given are the ratios of the transpiration-times of mixtures to
tliat of oxygen as determined by Mr Graham, compared with those given by the
equation (140) deduced from our theory.

Conduction of Heat in a Single Medium (y).
The rate of conduction depends on the value of the quantity

where f, £rf, and ££* denote the mean values of those functions of £ 17, £ for
all the molecules in a given element of volume.

As the expressions for the variations of this quantity are somewhat compli-
cated in a mixture of media, and as the experimental investigation of the con-
duction of heat in gases is attended with great difficulty, I shall confine myself
here to the discussion of a single medium.

Putting

Q = M(u + t){tf + * + vf+Zu£+taHi + tool + $(? + if + ?)} ......... (142),

and neglecting terms of the forms £7 and f1 and £rf when not multiplied by
the large coefficient &,, we find by equations (75), (77), and (54),

(143).

The first term of this equation may be neglected, as the rate of conduction
will rapidly establish itself. The second term contains quantities of four dimen-

THE DYNAMICAL THEORY OF GASES. 75

sions in £ 77, £, whose values will depend on the distribution of velocity among
the molecules. If the distribution of velocity is that which we have proved to
exist when the system has no external force acting on it and has arrived at
its final state, we shall have by equations (29), (31), (32),

and the equation of conduction may be written

(144),
(145),
(146);

[Addition made December 17, 1866.]

[Final Equilibrium of Temperature.~\

[The left-hand side of equation (147), as sent to the Royal Society, con-
tained a term 2(/3 — 1)--^-, the result of which was to indicate that a column

' p dx

of air, when left to itself, would assume a temperature varying with the height,
and greater above than below. The mistake arose from an error* in equation
(143). Equation (147), as now corrected, shews that the flow of heat depends
on the variation of temperature only, and not on the direction of the variation
of pressure. A vertical column would therefore, when in thermal equilibrium,
have the same temperature throughout.

When I first attempted this investigation I overlooked the fact that £4 is
not the same as f . f ", and so obtained as a result that the temperature
diminishes as the height increases at a greater rate than it does by expansion
when air is carried up in mass. This leads at once to a condition of instability,

* The last term on the left-hand side was not multiplied by ft.

10—2

7.-, THE DYNAMICAL THEORY OF OASES.

which is inconsistent with the second law of thermodynamics. I wrote to
Profewor Sir W. Thomson about this result, and the difficulty I had met with,
but presently discovered one of my mistakes, and arrived at the conclusion
that the temperature would increase with the height. This does not lead to
mechanical instability, or to any self-acting currents of air, and I was in some
degree satisfied with it. But it is equally inconsistent with the second law of
thermodynamics. In fact, if the temperature of any substance, when in thermic
equilibrium, is a function of the height, that of any other substance must be
the same function of the height. For if not, let equal columns of the two
substances be enclosed in cylinders impermeable to heat, and put in thermal
communication at the bottom. If, when in thermal equilibrium, the tops of
the two columns are at different temperatures, an engine might be worked
by taking heat from the hotter and giving it up to the cooler, and the
refuse heat would circulate round the system till it was all converted into
mechanical energy, which is in contradiction to the second law of thermo-
dynamics.

The result as now given is, that temperature in gases, when in thermal
equilibrium, is independent of height, and it follows from what has been said
that temperature is independent of height in all other substances.

If we accept this law of temperature as the actual one, and examine our
assumptions, we shall find that unless f = 3£* . £*, we should have obtained a
different result. Now this equation is derived from the law of distribution of
velocities to which we were led by independent considerations. We may there-
fore regard this law of temperature, if true, as in some measure a confirmation
of the law of distribution of velocities.]

Coefficient of Conductivity.

If C is the coefficient of conductivity of the gas for heat, then the quantity
of heat which passes through unit of area in unit of time measured as me-
chanical energy, is

by equation (147).

THE DYNAMICAL THEORY OP GASES. 77

Substituting for /J its value in terms of y by equation (115), and for &,
its value in terms of p. by equation (125), and calling pa, />„, and 00 the simul-
taneous pressure, density, and temperature of the standard gas, and s the spe-
cific gravity of the gas in question, we find

°-lTFD iSt 5 <->• ,

For air we have y= 1*409, and at the temperature of melting ice, or
274°'6C. above absolute zero, /*- = 918*6 feet per second, and at 160>6 C.,

Hi = 0*0936 in foot-grain-second measure. Hence for air at 16°*6C. the conduc-
tivity for heat is

(7=1172 (150).

That is to say, a horizontal stratum of air one foot thick, of which the upper
surface is kept at 17"C., and the lower at 16° C., would in one second transmit
through every square foot of horizontal surface a quantity of heat the mechanical
energy of which is equal to that of 2344 grains moving at the rate of one foot
per second.

Principal Forbes* has deduced from his experiments on the conduction of
heat in bars, that a plate of wrought iron one foot thick, with its opposite
surfaces kept 1°C. different in temperature, would, when the mean temperature
is 25°C., transmit in one minute through every square foot of surface as much
heat as would raise one cubic foot of water 0°*0127 C.

Now the dynamical equivalent in foot-grauvsecond measure of the heat
required to raise a cubic foot of water 1°C. is 1*9157 x 10'°.

It appears from this that iron at 25°C. conducts heat 3525 times better
than air at 16°' 6 C.

M. Clausius, from a different form of the theory, and from a different value
of fi, found that lead should couduct heat 1400 times better than air. Now
iron is twice as good a conductor of heat as lead, so that this estimate is not
far different from that of M. Clausius in actual value;

In reducing the value of the conductivity from one kind of measure to
another, we must remember that its dimensions are MLT"3, when expressed in
absolute dynamical measure.

* " Experimental Inquiry into the Laws of the Conduction of Heat in Bars," Edinburgh Transactions,
1861—62.

78 THE DYNAMICAL THEORY OF GASES.

Since all the quantities which enter into the expression for C are constant
except p., the conductivity is subject to the same laws as the viscosity, that is,
it is independent of the pressure, and varies directly as the absolute tempera-
ture. The conductivity of iron diminishes as the temperature increases.

Also, since y is nearly the same for air, oxygen, hydrogen, and carbonic
oxide, the conductivity of these gases will vary as the ratio of the viscosity
to the specific gravity. Oxygen, nitrogen, carbonic oxide, and air will have
equal conductivity, while that of hydrogen will be about seven times as great.

The value of y for carbonic acid is 1'27, its specific gravity is JJ- of
oxygen, and its viscosity ^ of that of oxygen. The conductivity of carbonic
acid for heat is therefore about $ of that of oxygen or of air.

[From the Proceedings of the Royal Society, No. 91, 1867.]

XXIX. On the Theory of the Maintenance of Electric Currents by Mechanical
Work without the use of Permanent Magnets.

THE machines lately brought before the Royal Society by Mr Siemens
and Professor Wheatstone consist essentially of a fixed and a moveable electro-
magnet, the coils of which are put in connexion by means of a commutator.

The electromagnets in the actual machines have cores of soft iron, which
greatly increase the magnetic effects due to the coils ; but, in order to simplify
the expression of the theory as much as possible, I shall begin by supposing
the coils to have no cores, and, to fix our ideas, we may suppose them in
the form of rings, the smaller revolving within the larger on a common
diameter.

The equations of the currents in two neighbouring circuits are given in
my paper " On the Electromagnetic Field*," and are there numbered (4) and (5),

-^t

~

where x and y are the currents, £ and 77 the electromotive forces, and R and S
the resistances in the two circuits respectively. L and N are the coefficients
of self-induction of the two circuits, that is, their potentials on themselves
when the current is unity, and M is their coefficient of mutual induction,
which depends on their relative position. In the electromagnetic system of
measurement, L, M, and N are of the nature of lines, and R and S are
velocities. L may be metaphorically called the " electric inertia " of the first
circuit, N that of the second, and L + 2M+ N that of the combined circuit.

* Philosophical Transactions, 1865, p. 469.

gO MAINTENANCE OF ELECTRIC CURRENTS BY MECHANICAL WORK.

Let us first take the case of the two circuits thrown into one, and the
two coils relatively at rest, so that M is constant. Then

= Q (1),

whence a-a*"™**' ................................... (2),

where x. is the initial value of the current. This expression shews that the
current, if left to itaelf in a closed circuit, will gradually decay.

L+2M+N
If we put ~ ~r .............................. ' ''

then x = x.e"; ..................................... (4).

The value of the time r depends on the nature of the coils. In coils of
similar outward form, T varies as the square of the linear dimension, and
inversely as the resistance of unit of length of a wire whose section is the
sum of the sections of the wires passing through unit of section of the coil.

In the large experimental coil used in determining the B.A. unit of re-
sistance in 1864, T was about '01 second. In the coils of electromagnets T is
much greater, and when an iron core is inserted there is a still greater in-
crease.

Let us next ascertain the effect of a sudden change of position in the
secondary coil, which alters the value of M from M1 to Ms in a time tt — <,,
during which the current changes from x, to x,. Integrating equation (1) with
respect to t, we get

' l = 0 ............ (5).

(R + S) f'

J ti

If we suppose the time so short that we may neglect the first term in com-
parison with the others, we find, as the effect of a sudden change of position,

(L + 2Mt + N)xt^(L+2Ml + N)xl ................... (6).

This equation may be interpreted in the language of the dynamical theory,
by saying that the electromagnetic momentum of the circuit remains the same
after a sudden change of position. To ascertain the effect of the commutator,
let us suppose that, at a given instant, currents x and y exist in the two
coils, that the two coils are then made into one circuit, and that x' is the

MAINTENANCE OF ELECTRIC CURRENTS BY MECHANICAL WORK. 81

current in the circuit the instant after completion; then the same equation (1)
gives

(L + 2M+N)x' = (L+M)x + (N+H)y (7).

The equation shews that the electromagnetic momentum of the completed
circuit is equal to the sum of the electromagnetic momenta of the separate coils
just before completion.

The commutator may belong to one of four different varieties, according to
the order in which the contacts are made and broken. If A, B be the ends
of the first coil, and C, D those of the second, and if we enclose in brackets
the parts in electric connexion, we may express the four varieties as in the
following Table : —

(1) (2) (3) (4)

(AC) (BD) (AC) (BD) (AC) (BD) (AC) (BD)
(ABCD) (ABC) (D) (A) (BCD) (A) (B) (C) (D)

(AD) (BC) (ABCD) (ABCD) (AD) (BC)

(AD) (BC) (AD) (BC)

In the first kind the circuit of both coils remains uninterrupted ; and when
the operation is complete, two equal currents in opposite directions are com-
bined into one. In this case, therefore, y= — x, and

(L+ZM+N}x' = (L-N)x (8).

When there are iron cores in the coils, or metallic circuits in which inde-
pendent currents can be excited, the electrical equations are much more com-
plicated, and contain as many independent variables as there can be independent
electromagnetic quantities. I shall therefore, for the sake of preserving simplicity,
avoid the consideration of the iron cores, except in so far as they simply in-
crease the values of L, M, and N.

I shall also suppose that the secondary coil is at first in a position in
which M=0, and that it turns into a position in which M= —M, which will
increase the current in the ratio of L + N to L — 2M+N.

The commutator is then reversed. This will diminish the current in a ratio
depending on the kind of commutator.

The secondary coil is then moved so that M changes from M to 0, which
will increase the current in the ratio of L + 2M+N to L + N.

VOL. II. ll

MAINTENANCE OF ELECTRIC CURRENTS BY MECHANICAL WORK.

During the whole motion the current has also been decaying at a rate
which varies according to the value of L + 2M+N; but since M varies from
+ .V to -J/, we may, in a rough theory, suppose that in the expression for
the decay of the current J/=0.

If the secondary coil makes a semirevolution in time T, then the ratio of
the current XH after a semirevolution, to the current x, before the semirevolution,

will be

x -?

'=« 'r,

a;

L + N
where Tat R+~S ....................................... ^ ''

and r is a ratio depending on the kind of commutator.
For the first kind,

<"»•

By increasing the speed, T may be indefinitely diminished, so that the
question of the maintenance of the current depends ultimately on whether r is
greater or less than unity. When r is greater than 1 or less than — 1, the
current may be maintained by giving a sufficient speed to the machine ; it will
be always in one direction in the first case, and it will be a reciprocating current
in the second.

When r lies between +1 and — 1, no current can be maintained.

Let there be p windings of wire in the first coil and q windings in the
second, then we may write

L = lp', M=mpq, N=n<f ........................ (11),

where I, m, n are quantities depending on the shape and relative position of
the coils. Since L — 2M+N must always be a positive quantity, being the
coefficient of self-induction of the whole circuit, ln — m3, and therefore LN—M*
must be positive. When the commutator is of the first kind, the ratio r is
greater than unity, provided pm is greater than qn ; and when

r-(l"fc) <12)'

which is the maximum value of r.

MAINTENANCE OF ELECTRIC CURRENTS BY MECHANICAL, WORK. 83

When the ratio of p to q lies between that of n to m and that of m to I,
r lies between + 1 and — 1, and the current must decay ; but when pi is less
than qm, a reciprocating current may be kept up, and will increase most rapidly
when

I .,\

}-—\

V In] '
and ^-(l.-S'1 (13).

When the commutator is of the second kind, the first step is to close both
circuits, so as to render the currents in them independent. The second circuit
is then broken, and the current in it is thus stopped. This produces an effect
on the first circuit by induction determined by the equation

Lx + My = Lx' + My' .............................. (14).

In this case M= — MM y = x, and y' = Q, so that

(L-M]x = Lx ................................. (15);

where x is the original, and x' the new value of the current.

The next step is to throw the circuits into one, M being now positive.
If x" be the current after this operation,

" ....................... (16).

The whole effect of this commutator is therefore to multiply the current by
the ratio

The whole effect of the semirotation is to multiply the current by the ratio

L + ZM+N
L-2M+N'

The total effect of a semi re volution supposed instantaneous is to multiply the
current by the ratio

D-M*
~L(L-2M+N)'

84 MAINTENANCE OF BLBCTBIC CUBBBNT8 BY MECHANICAL WORK.

If p and q be the number of windings in the first and second coils respec-
tively, the ratio r becomes

"

which is greater than 1, provided 2lmp is greater than (ln + m')q. When

we have for the maximum value of r,

m

r=l

_

2T

n\in -i- + n -m

Vm' + ^7 *^+m /

In the experiment of Professor Wheatetone, in which the ends of the
primary coil were put in permanent connexion by a short wire, the equations
are more complicated, as we have three currents instead of two to consider.
The equations are

*-Kz ............ (17),

(18),

where Q, K, and z are the resistance, self-induction, and current in the short
wire. The resultant equations are of the second degree ; but as they are only
true when the magnetism of the cores is considered rigidly connected with the
currents in the coils, an elaborate discussion of them would be out of place
in what professes to be only a rough explanation of the theory of the experi-
ments.

Such a rough explanation appears to me to be as follows : —

Without the shunt, the current in the secondary coil is always in rigid
connexion with that in the primary coil, except when the commutator is
changing. With the shunt, the two currents are in some degree independent;
and the secondary coil, whose electric inertia is small compared with that of
the primary, can have its current reversed and varied without being clogged
by the sluggish primary coil.

MAINTENANCE OF ELECTRIC CURRENTS BY MECHANICAL WORK. 85

On the other hand, the primary coil loses that part of the total current
which passes through the shunt; but we know that an iron core, when highly
magnetized, requires a great increase of current to increase its magnetism,
whereas its magnetism can be maintained at a considerable value by a current
much less powerful. In this way the diminution in resistance and self-induction
due to the shunt may more than counterbalance the diminution of strength in
the primary magnet.

Also, since the self-induction of the shunt is very small, all instantaneous
currents will run through it rather than through the electromagnetic coils, and
therefore it will receive more of the heating effect of variable currents than
a comparison of the resistances alone would lead us to expect.

[Extracted from The Quarterly Journal of Pure and Applied Mathematics, No. 32, 1867.]

XXX. On the Equilibrium of a Spherical Envelope.

I PROPOSE to determine the distribution of stress in an indefinitely thin
und inextensible spherical sheet, arising from the action of external forces applied
to it at any number of points on its surface.

Notation. Let two systems of lines, cutting each other at right angles, be
drawn upon any surface, and let their equations be

G and <>txz = H,

where each curve is found by putting G or H equal to a constant, and com-
bining it with the equation of the surface itself, which we may denote by

Now let G be made constant, and let H vary, and let dS1 be the element
of length of the curve (G = constant) intercepted between the two curves for

which // varies by dH, then -TTT will be a function of H and G.

ao,

In the same way, making dSt an element of the curve (//= constant) we
may determine -j-^ as a function of H and G.

Now let the element dSt experience a stress, consisting of a force X in
the direction in which G increases, and Y in the direction in which H in-
creases, acting on the positive side of the linear element dSu and equal and
opposite forces acting on the negative side. These will constitute a longitudinal
tension normal to dSlt which we shall denote by

X

EQUILIBRIUM OF A SPHERICAL ENVELOPE. 87

and a shearing force on the element, which we shall call

T

In like manner, if the element dS3 is acted on by forces Y' and X', it
will experience on its positive side a tension and a shearing force, the values
of which will be

=Jr and =x*_

That the moments of these forces on the elements of area dS1} dSt may
vanish,

or the shearing force on dS^ must be equal to that on dS^.

When there is no shearing force, then pn and p^ are called the principal
stresses at the point, and if pa vanishes everywhere, the curves, (G = constant)
and (H= constant), are called lines of principal stress. In this case the con-
ditions of equilibrium of the element dS^dS^ are

, , } - = () (2)

dHdG+(P" p"> dGdH

Py+P^^N.-N, (3).

rt rt

The first and second of these equations are the conditions of equilibrium
in the directions of the first and second lines of principal tension respectively.

The third equation is the condition of equilibrium normal to the surface ;
r, and rt are the radii of curvature of normal sections touching the first and
second lines of principal stress. They are not necessarily the principal radii of
curvature. JV, is the normal pressure of any fluid on the surface from the side
on which r, and ra are reckoned positive, and N., is the normal pressure on the
other side.

If the systems of curves G and H, instead of being lines of principal stress,
had been lines of curvature, we should still have had the same equation (3),
but rl and r, would have been the principal radii of curvature, and pn and p^

gg EQUILIBRIUM OF A SPHERICAL ENVELOPE.

would have been the tensions in the principal planes of curvature, and not
necessarily principal tensions.

In the case of a spherical surface not acted on by any fluid pressure,
rt*Tv and JV, «= A"t «= 0, so that the third equation becomes

Pu+P* = 0 (-4),

whence we obtain from the first and second equations

where <7, is a function of H, and Ct of G. If then we draw two lines of
the system (//= constant) at such a distance that pu (dS^ = (dh)* at any point
where (dh)' is constant, this equation will continue true through the whole
length of these lines, that is, the principal stresses will be inversely as the
square of the distance between the consecutive lines of stress. Since this is
true of both sets of lines, we may assume the form of the functions G and
H, so that they not only indicate lines of stress, but give the value of the
stress at any point by the equations

/dH* dG\*

where (//= constant) is a line of principal tension, and (G = constant) a line of
principal pressure.

If we now draw on the spherical surface lines corresponding to values of G
differing ' by unity, and also lines corresponding to values of H differing by
unity, these two systems of lines will intersect everywhere at right angles, and
the distance between two consecutive lines of one system will be equal to the
distance between two consecutive lines of the other, and the principal stresses
will be in the directions of the lines, and inversely as the square of the
intervals between them.

Now if two systems of lines can be drawn on a surface so as to fulfil
these conditions, we know from the theory of electrical conduction in a sheet
of uniform conductivity, that if one set of the curves are taken as equipotential
lines, the other set will be lines of flow, and that the two systems of lines
will give a solution of some problem relating to the flow of electricity through
a conducting sheet. But we know that unless electricity be brought to some
point of the sheet, and carried off at anoUier point, there can be no flow of

EQUILIBRIUM OF A SPHERICAL ENVELOPE. 89

electricity in the sheet. Hence, if such systems of lines exist, there must
be some singular points, at which all the lines of flow meet, and at which

dH ,

is infinite.
ao,

7TT

If -y~- is nowhere infinite, there can be no systems of lines at all,
and if -TOT is infinite at any point, there is an infinite stress at that point,

which can only be maintained by the action of an external force applied at that
point.

Hence a spherical surface, to which no external forces are applied, must
be free from stress, and it can easily be shewn from this that, when the external
forces are given, there can be only one system of stresses ha the surface.

This is not true in the case of a plane surface. In a plane surface,
equation (3) disappears, and we have only two differential equations connecting
the stresses at any point, which are not sufficient to determine the distribution
of stress, unless we have some other conditions, such as the equations of elas-
ticity, by which the question may be rendered determinate.

The simplest case of a spherical surface acted on by external forces is that
in which two equal and opposite forces P are applied at the extremities of a
diameter. There will evidently be a tension along the meridian lines, com-
bined with an equal and opposite pressure along the parallels of latitude, and
the magnitude p of either of these stresses will be

p

where a is the radius of the sphere, P the force at the poles, and 6 the
angular distance of a point of the surface from the pole. If <£ is the longitude,
and if i\ and r, are the rectilineal distances of a point from the two poles
respectively, and if we make

£ = log.^ and H=<f> ........................... (8),

then G and H will give the lines of principal stress, and

<D= -T?7 I = I-TT7

27ra \ao2/ 2va \ao,y

VOL. II. 12

<JU EQUILIBRIUM OF A SPHERICAL ENVELOPE.

To pass from this case to that in which the two forces are applied at
any points, I shall make use of the following property of inverse surfaces.

If a surface of any form is in equilibrium under any system of stresses,
and if lines of principal stress be drawn on it, then if a second surface be the
i morse of the first with respect to a given point, and if lines be drawn on
it which are inverse to the lines of principal stress in the first surface, and
if along these lines stresses are applied which are to those in the corresponding
point of the first surface inversely as the squares of their respective distances
from the point of inversion, then every part of the second surface will either
be in equilibrium, or will be acted on by a resultant force in the direction
of the point of inversion.

For, if we compare corresponding elements of lines of stress in the two
.surfaces, we shall find that the forces acting on them are in the same plane
with the line through the point of inversion, and make equal angles with it.
The moment of the force on either element about the point of inversion is
therefore as the product of the length of the element, into its distance
from the point of inversion into the intensity of the stress. But the length
of the element is as its distance, and the stress is inversely as the square of
the distance, therefore the moments of the stresses about the point of inversion
are equal, and in the same plane. If now any portion of the first surface is
in equilibrium, it will be in equilibrium as regards moments about the point
of inversion. The corresponding portion of. the second surface will also be in
equilibrium as regards moments about the point of inversion. It is therefore
either in equilibrium, or the resultant force acting on it passes through the
point of inversion.

Now let the first surface be a sphere ; we know that the second surface
is also a sphere. In the first surface the condition of equilibrium normal to
the surface is pa+pa = 0- In the second surface the stresses are to those
in the first in the inverse ratio of the squares of the distances. Hence in
the second surface also, pn + pa = 0, or there is equilibrium in the direction of
the normal. But we have seen that the resultant, if any, is in the radius
vector. Therefore, if we except the limiting case in which the radius vector
is perpendicular to the normal, the equilibrium is complete in all directions.

We may now, by inverting the spherical surface, pass from the case of
a sphere acted on by a pair of tensions applied at the extremities of a

EQUILIBRIUM OF A SPHERICAL ENVELOPE. 91

diameter to that in which the forces are applied at the extremities of any
chord of the sphere, subtending at the centre an angle = 2a. The lines of tension
will be circles passing through the extremities of the chord. Let the angle
which one of these circles makes with the great circle through these points
be <j>. The angle (f> is the same as the corresponding angle in the inverse
surface.

The lines of pressure, being orthogonal to these, will be circles whose
planes if produced pass through the polar of the chord. Let r,, r2 be the
distances of any point on the sphere from the extremities of the chord, then

7*

is constant for each of these circles, and has the same value as it has in
?\

the inverse surface. Hence, if we make

=<j> (10),

G and H will give the lines of principal stress, and the absolute value of the
stress at any point will be

P sin a. (dG\* P sin a [dH\*

p = — - — j~o ) = — 7T~ iFcf ) (11)-

Zrra \ctOj/ 2ira \ttOj/

If we draw tangent planes to the sphere at the extremities of the chord,
and if qv qt are the perpendiculars from any point of the spherical surface
on these planes, it is easy to shew that at that point

p = Pa^^.. ..(12).

2irq,q,

If any number of forces, forming a system in equilibrium, be applied at
different points of a spherical envelope, we may proceed as follows. First
decompose the system of forces into a system of pairs of equal and opposite
forces acting along chords of the sphere. To do this, if there are n forces
applied at n points, draw a number of chords, which must be at least 3 (n — 2),
so as to render all the points rigidly connected. Then determine the tension
along each chord due to the external forces. If too many chords have been
drawn, some of these tensions will involve unknown quantities. The n forces
will now be transformed into as many pairs of equal and opposite forces as
chords have been drawn.

12—2

•.}_• EQUILIBRIUM OF A SPHERICAL ENVELOPE.

Next find the distribution of stress in the spherical surface due to each
of these pairs of forces, and combine them at every point by the rules for the
composition of stress*. The result will be the actual distribution of stress, and
if any unknown forces have been introduced in the process, they will disappear
from the result

The calculation of the resultant stress from the component stresses, when
these are given in terms of u asymmetrical spherical co-ordinates of different
systems, would be very difficult; I shall therefore shew how to effect the same
object by a method derived from Mr Airy's valuable paper, "On the strains
in the interior of beamsf."

If we place the point of inversion on the surface of the sphere, the in-
verse surface is a plane, and if p^, p,^ and pn represent the components of
in the plane referred to rectangular axes, we have for equilibrium

<I3>>

These equations are equivalent to the following :

where F is any function whatever of x and y.

The form of the function F cannot, in the case of a plane, be determined
from the equations of equilibrium, as strains may exist independently of ex-
ternal forces. To solve the question we require to know not only the original
strains, but the law of elasticity of the plane sheet, whether it is uniform, or
variable from point to point, and in different directions at the same point.
When, however, we have found two solutions of F corresponding to different
cases, we can combine the results by simple addition, as the expressions (equa-
tion 15) are linear in form.

In the case of two forces acting on a sphere, let A, B (fig. 29) be the
points corresponding to the points of application in the inverse plane ; AP = ?•„

* Rankin's Applied Mechanics, p. 82.

+ Philosophical Transactions, 1863, Part I. p. 49.

EQUILIBRIUM OF A SPHERICAL ENVELOPE. 93

= ru angle APB = XAT=<$>. Bisect AB in C, and draw PD perpendicular
to AB. Then the Hue of tension at P is a circle through A and B, for which
(£ is constant, and the line of pressure is an orthogonal circle for which the
ratio of rt to r3 is constant, and the angle <f> and the logarithms of the ratio
of r, to rt differ from the corresponding quantities in the sphere only by con-
stants. We may therefore put

_ Psina (dG\ _ Psina tdH\ _
Pu~ ~~ '' (dSJ '' ~P*'

where the values of G and H are the same as in equations (8) and (10).
Transforming these principal stresses into their components, we get

dx

dG*\
dy

.(16),

Psina dGdG , }

P*»~ 27ra Z~dxdy (17)>

Psma/dGl" d<Gft\

Puu = I — T- T-

2ira \dy\ dx\J

From the relations between G and H we have

dG dH . dG dH

-j- = — j- and -7— = -j- (19),

dx dy dy dx

#G#G . d*H d*H
whence ^+^ = ° ^d d*+^ = 0 (2°)'

The values of the component stresses, being expressed as functions of the
second degree in G, cannot be compounded by adding together the corresponding
values of G, we must, therefore, in order to express them as linear functions,
find a value of F, such that

Psina (fdG\*

dy1'

We shall then have, by differentiation with respect to x, and integration
with respect to y,

(22),

> (23).

dxdy 2ira J dx dy
d'F Psina

94 EQUILIBRIUM OF A SPHERICAL ENVELOPE.

Since ^' + ^«0 in this case, the arbitrary functions must be zero, and

djf dtf
we have to find the value of F from that of G by ordinary integration of (21).

The result is

..r if the co-ordinates of A and B are (au 6,) and (o» 6,),
Pain a f. (2x-al-a,)(a,-a1) + (2y-6.-6,)(6.-6l) .
- - ' - « g

x-Oi/J

If we obtain the values of F for all the different pairs of forces acting
on the sphere, and add them together, we shall find a new value of F, the
second differential coefficients of which, with respect to x and y will give a
system of components of stress in the plane, which, being transferred to the
sphere by the process of inversion, will give the complete solution of the
problem in the case of the sphere.

We have now solved the problem in the case of any number of forces
applied to points of the spherical surface, and all other cases may be reduced
to this, but it is worth while to notice certain special cases.

If two equal and opposite twists be applied at any two points of the
sphere, >we can determine the distribution of stress. For if we put

and JT=log-1-^ ..................... (26),

the equations of equilibrium will still hold, and the principal stresses at any
point will be inclined 45° to those in the case already considered.

If M be the moment of the couple in a plane perpendicular to the chord,
the absolute value of the principal stresses at any point is

In figure 30 are represented the stereographic projections of the
lines of stress in the cases which we have considered. When a tension is
applied along the chord AB, the lines of tension are the circles through AB,

VOL. 11. PLATE X.

fe

Cambridge UmveriHy Prcsf

EQUILIBRIUM OF A SPHERICAL ENVELOPE. 95

and the lines of pressure are the circles orthogonal to them. These circles are
so drawn in the figure that the differences of the values of G and H are ^$ir.

The spiral lines which pass through the intersections of these circles are
the principal lines of stress in the case of twists applied to the spherical
surface at the extremities of the chord AS.

In the case of a sphere acted on by fluid pressure, if the pressure is a
function of the distance from a given point, we may take the line joining
that point with the centre of the sphere as an axis, and then the lines of
equal fluid pressure will be circles, and the fluid pressure N will be a function
of 6, the angular distance from the pole. If we suppose the total effect of
pressure balanced ,by a single force at the opposite pole, then we shall have
for the equilibrium of the segment, whose radius is 6,

= a f JVsin20cZ0 ........................... (28),

Jo

and pa

to determine pn the tension in the meridian, and pa that hi the parallels of
latitude.

[From the ProcMdings of the Royal Society of Edinburgh, Session 1867-68.]

YXXT. On the best Arrangement for producing a Pure Spectrum on a Scr-

IN experiments on the spectrum, it is usual to employ a slit through
which the light is admitted, a prism to analyse the light, and one or more
lenses to bring the rays of each distinct kind to a distinct focus on the screen.
The most perfect arrangement is that adopted by M. Kirchhoff, in which two
achromatic lenses are used, one before and the other after the passage of the
light through the prism, so that every pencil consists of parallel rays while
passing through the prism.

But when the observer has not achromatic lenses at his command, or when,
as in the case of the highly refrangible rays, or the rays of heat, he is re-
stricted in the use of materials, it may still be useful to be able to place
the lenses and prism in such a way as to bring the rays of all colours to
their foci at approximately the same distance from the prism.

We shall first examine the effect of the prism in changing the conver-
gency or divergency of the pencils passing through it, and then that of the
lens, so that by combining the prism and the lens we may cause their defects
to disappear.

When a pencil of light is refracted through a plane surface, its convergency
or divergency is less in that medium which has the greatest refractive index ;
and this change is greater as the angle of incidence is greater, and also as
the index of refraction is greater.

When the pencil passes through a prism its convergency or divergency
will be diminished as it enters, and will be increased when it emerges, and
the emergent pencil will be more or less convergent or divergent than the
incident one, according as the angle of emergence is greater or less than that of

THE BEST ARRANGEMENT FOR PRODUCING A PURE SPECTRUM ON A SCREEN. 97

incidence. This effect will increase with the difference of these angles and with
the refractive index.

When the angle of incidence is equal to that of emergence the convergence
of the pencil is unaltered, but since the more refrangible rays have the greatest
angle of emergence, their convergency or divergency will be greater than that
of the less refrangible rays.

This effect will be increased by making the angle of incidence less, and
that of emergence greater ; and it will be diminished by increasing the angle
of incidence, that is, by turning the prism round its edge towards the slit. If
the angle of the prism is not too great, the convergency or divergency of all
the pencils may be made the same (approximately) by turning the prism in this
way.

This correction, however, diminishes the separation of the colours. It is
inapplicable to a prism of large angle, and it takes no account of the chromatic
aberration of the lens. By altering the arrangement, the lens may be made to
correct the prism. The effect of a convex lens is to increase the convergency, and
to diminish the divergency, of every pencil ; but the change is greatest on the
most refrangible rays. The prism, except when its base is very much turned
towards the slit, makes the highly refrangible rays more convergent or more
divergent than the less refrangible rays, according as they were convergent or
divergent originally. If the rays pass through the prism before they reach the
lens, the pencils will be divergent at incidence, and the more refrangible will
be most divergent at emergence. If they then fall on the lens, they will be
more converged than the rest ; so that by a proper arrangement all may be
brought to their foci at approximately the same distance. If the violet rays
come to their focus first, we must turn the base of the prism more towards the
light, and vice versa.

We proceed to the numerical calculation of the proper arrangement.

To find the variation of position of the focus of light passed through a prism'
as dependent on the nature of the light.

VOL. n. 13

98 THE BEST ARRANGEMENT FOR PRODUCING

Let p. be the index of refraction of the prism, a its angle, <j>, and fa the
angles of incidence and emergence, 0, and 0, the angles of the ray within the
prism with the normals to the first and second surfaces, 8 the difference of these
angles — then by geometry

and by the law of refraction,

.sin^, = fxsin 0,, sin fa = p. sin 0S,

fa is constant, being the angle of incidence for all kinds of light, but the other
angles vary with p., so that

i i 6, sin 0, d0, _ sin 0, dfa sin a

lip. p. COS 0, ' dp. p. COS 0^ ' <//X COS 0, COS «£, '

The last expression gives the dispersion, or breadth of the spectrum, and
shews that it increases as the base of the prism is turned from the light.

As the slit is parallel to the edge of the prism, we have only to consider
the primary foci of the pencils when we wish to render the image distinct.

Let r, be the distance of the focus of incident light from the prism,
that of the emergent light, and u that within the prism, all measured to the
right, then by the ordinary formula,

v, u u v,

cos* fa ~ p, cos2 0, ' p, cos8 0, ~~ cos* fa '

r, cos5 0, cos* fa = r, cos' 0, cos* fat
Taking the differential coefficient of the logarithms of these quantities,

1. <h\ _ 2sm 0, d0, _ 2sin fa dfa _ !_ c/v, _ 2sin0., d0t
v, dp - 0, dp. cos fa dp, v, dp. cos 0, dp. '

1 dvi _^ 1 dvs _ 2 sin fa sin a 2 sin' 0, 2 sin 0, sin 0,

/', (//X Vj rf/t COS 0, COS* fa p. COS* 0, /H. COS 0, COS 0, '

Substituting for these angles their values in terms of a and S we find

1 di\ 1 dv, _ 4 sin a (/r- l) sin a — sinSjl +/TCOS (a — 8)}
v, dp. vt dp. (cos a + cos 8)' ' 1 — ?>/r { 1 — cos (a — 8)}

The quantity on the right of this equation is always positive, unless the
value of 8 exceeds that given by the equation

(/i* — 1 ) sin a = sin 8 {1 + /*' cos (a — 8)}.

A PURE SPECTRUM ON A SCREEN. 99

If fj. = 1-5 and a = 60°, then 8 = 22° 5 2', that is, the ray within the prism
makes an angle of 11° 26' with the base, which corresponds to an angle of
incidence 82250', or the incident ray is inclined 7° 10' to the face of the prism.

If two lenses'"" are used, of the same material with the prism, we may
correct the defects of the prism without turning it so far from its position of
least deviation.

Let a be the distance of the slit from the prism, and b that of the screen
from the prism. Let /, be the focal length of the lens between the prism
and slit, and f, that of the lens between the prism and the screen, then the
condition of a flat image is

1 di\ 1 dv, _ 1 /i\ vt\

7 """ "™ ~7 — " ? ( ~~f ' ? ) •

V.dfJ. V3rf/A jll-1 V/ fj

Let us first find the conditions of a flat spectrum S = o.
When 8 = 0, vl = v,a and we obtain the conditions

1 1 1 _ 1 _ /I 1\ ( 1 + cos a)' {1 - fa* (1 - cos a)} _ A 1\

from which ft and /, may be found in terms of known quantities.

When a, the angle of the prism, is 60°, then

C- — •* ~ _ / i \ q 7 ~, -i \

If we now make /x = 1'5 we find c = '5025, and

1 _ -4975 _ ^025
f=~^~ b
1 _ 1-5025 ^025

• — 7 •

/, b a

If a = ^b, the first lens may be dispensed with, and the second lens will
correct the defects of the prism.

* [These lenses are supposed to be close to the edge of the prism.]

13—2

100 THE BEST ARRANGEMENT FOR PRODUCING A PURE SPECTRUM ON A SCREEN.

If a is greater in proportion to b, a concave lens must be placed in front
of the prism. The most convenient arrangement will be that in which tl it-
prism ia placed in the position of least deviation, and the lens placed between
the prism and the screen, while the distance from the slit to the prism is to
that between the prism and the screen as 1 — c is to c. For quartz, in which

p. = 1*58 4 for the ordinary ray, - = 2'53, so that the best arrangement is

C

«i=l '53 6, or the lens should be placed on the side next the screen, and the
distance from the slit which admits the light to the prism should be about
one and a half times the distance from the prism to the screen.

[From the Proceedings of the London Mathematical- Society, Vol. n.]

XXXII. The Construction of Stereograms of Surfaces.

To make a surface visible, lines must be drawn upon it ; and to exhibit
the nature of the surface, these lines ought to be traced on .the surface ac-
cording to some principle, as for instance the contour lines and lines of greatest
slope on a surface may be drawn. Monge represents surfaces to the eye by
their two systems of lines of curvature, which have the advantage of being
independent of the direction assumed for the co-ordinate axes. For stereoscopic
representation it is necessary to choose curves which are easily followed by the
eye, and which are sufficiently different in form to prevent a curve of the
one figure from being visually united with any other than the corresponding
curve in the other figure. I have found the best way in practice to be as
follows : — First determine how many curves are to be drawn on the surface,
and at what intervals, and find the numerical values of the co-ordinates of
points on these curves. A convenient method of drawing the figures according
to Cartesian co-ordinates is to draw an equilateral triangle, find its centre of
gravity, and take a point about ^ of the side of the triangle distant hori-
zontally from the centre of gravity as the origin, and the lines joining this
point with the angles as unit axes. For the other figure, the point must be
taken on the opposite side of the centre of gravity. I have found that this
rule gives a convenient amount of relief to the figures. When the co-ordinates
of a point are easily expressed in terms of tetrahedral co-ordinates x, y, z, u\
I draw the two lines (x = 0, y = 0) and (2 = 0, w = 0) in both figures. By means
of a sector I. divide the first line in P in the ratio of z to w, and the second
in Q in the ratio of x to y ; I then lay a rule along the line PQ (without
drawing the line) and divide PQ in the ratio of x + y to z + if>, in order to find
the point R in the figure. By drawing the two lines once for all in each
figure, and performing the same process of finding ratios in each figure, we get
each point without making any marks on the paper, which come to be very
troublesome in complicated figures. In this way I have drawn the figures of
cyclides, &c.

[Proceedings of the Aonrfon Mathematical Society, Vol. II.]

XXXIII. On Reciprocal Diagrams in Space, and their relation to

Function of Stiress.

LET F be any function of the co-ordinates x, y, z of a point in space, and
l«'t £ *?• £ be another system of co-ordinates which we may suppose referred
t«> axes parallel to the axes of x, y, z, but at such a distance (in thought)
that figures referred to z, y, z do not interfere with the figures referred to f, 17, £.

We shall call the figure or figures referred to x, y, z the First Diagram,
and those referred to f, 77, £ the Second Diagram.

Let the connection between the two diagrams be expressed thus—

(IF _dF (IF

*~dx' V'lly' ^~dz'

When the form of F is known, £ rj and £ may be found for every value
of x, y, z, and the form of the second diagram fixed.

To complete the second diagram, let a function <£ of £ 77, £ be found from
the equation

<f>=

then it is easily shewn that

(t<f>

r — tt

" J

' —

-

Hence the first diagram is determined from the second by the same process
that the second is determined from the first. They are therefore Reciprocal
I>i;igrams both as regards their form and their functions.

But reciprocal diagrams have a mechanical significance which is capable
<•!' extensive applications, from the most elementary graphic methods for calcu-
lating the stresses of a roof to the most intricate questions about the internal

RECIPROCAL DIAGRAMS IN SPACE. 103

molecular forces in solid bodies. I shall indicate two independent methods of
representing internal stress by means of reciprocal diagrams.

First Method. Let a, b be any two contiguous points in the first diagram,
and a, y8 the corresponding points in the second. Let an element of area be
described about ab perpendicular to it. Then if the stress per unit of area on

this surface is compounded of a tension parallel to a/8 and equal to P~ and

CltV

11 7 D ,

a, pressure parallel to cut and equal to P ( +~r^ + -r , P being a constant

introduced for the sake of homogeneity, then a state of internal stress denned
in this way will keep every point of the first figure in equilibrium.

The components of stress, as thus defined, will be

p p

~ ' Pzx ~ ' Pxy~

dydz' zx ~ dzdx ' xy~ dxdy '

and these are easily shewn to fulfil the conditions of equilibrium.

If any number of states of stress can be represented in this way, they
can be combined by adding the values of their functions (F), since the quan-
tities are linear. This method, however, is applicable only to certain states of
stress ; but if we write

_d3B &C _<fC #A _d\A d?B

Pzx~ dz* + dy3 ' Pvy~dx* + dz* ' Pzz~ dif + dx* '

(PA d"B &C

Pyz~ dydz' Pzx~ dzdx' Pxy~ dxdy'

we get a general method at the expense of using three functions A, B, C,
and of giving up the diagram of stress.

Second Method. Let a be any element of area in the first diagram, and
a the corresponding area in the second. Let a uniform normal pressure equal
to P per unit of area act on the area a, and let a force equal and parallel
to the resultant of this pressure act on the area a, then a state of internal
stress in the first figure defined in this way will keep every point of it in
equilibrium.

1,,J RECIPROCAL DIAGRAMS IN SPACE.

The com|»>m>nt8 of stress as thus defined will be

/•' ~&F\ d'FJF '_ <fF

(d'FJF '_ <fF A
~ '/•' '/:</*!/'

\

/
\

lf'F d*F d'F \
jE2x1xdjl dJdydz)' Pa~ \dxdy ,ly,lz dfdzdx)'

y

This is a more complete, though not a perfectly general, representation of
a state of stress ; but as the functions are not linear, it is difficult of application.

If however we confine ourselves to problems in two dimensions, either
method leads to the expression of the three components of stress in terms of
the function inti-oduced by the Astronomer- Royal*.

d>F

dF •//•'

Here *=dx' y= dy '

and if ab and a^8 be corresponding lines, the whole stress across the lin«
is perpendicular to aft and equal to Paf3.

* "On Strains in the Interior of Beams." Phil. Tram. 1863.

[From the Proceedings of the Royal Society, No. 100, 1868.]

XXXIV. On Governors.

A GOVERNOR is a part of a machine by means of which the velocity of
the machine is kept nearly uniform, notwithstanding variations in the driving-
power or the resistance.

Most governors depend on the centrifugal force of a piece connected with
a shaft of the machine. When the velocity increases, this force increases, and
either increases the pressure of the piece against a surface or moves the piece,
and so acts on a break or a valve.

In one class of regulators of machinery, which we may call moderators*,
the resistance is increased by a quantity depending on the velocity. Thus in
some pieces of clockwork the moderator consists of a conical pendulum revolving
within a circular case. When the velocity increases, the ball of the pendulum
presses against the inside of the case, and the friction checks the increase of
velocity.

In Watt's governor for steam-engines the arms open outwards, and so
contract the aperture of the steam-valve.

In a water-break invented by Professor J. Thomson, when the velocity is
increased, water is centrifugally pumped up, and overflows with a great velocity,
and the work is spent in lifting and communicating this velocity to the water.

In all these contrivances an increase of driving-power produces an increase
of velocity, though a much smaller increase than would be produced without
the moderator.

But if the part acted on by centrifugal force, instead of acting directly
on the machine, sets in motion a contrivance which continually increases the

* See Mr C. W. Siemens "On Uniform Rotation," Phil. Trans. 1866, p. 657.
VOL. II. 14

1(XJ GOVERNORS.

resistance M long ss the velocity is above its normal value, and reverses its
action when the velocity is below that value, the governor will bring the
velocity to the same normal value whatever variation (within the working limits
of the machine) be made in the driving-power or the resistance.

I propose at present, without entering into any details of mechanism, to
direct the attention of engineers and mathematicians to the dynamical theory
of such governors.

It will be seen that the motion of a machine with its governor consists
in general of a uniform motion, combined with a disturbance which may be
expressed as the sum of several component motions. These components may be
of four different kinds : —

1. The disturbance may continually increase.

2. It may continually diminish.

8. It may be an oscillation of continually increasing amplitude.

4. It may be an oscillation of continually decreasing amplitude.

The first and third cases are evidently inconsistent with the stability of
the motion ; and the second and fourth alone are admissible in a good governor.
This condition is mathematically equivalent to the condition that all the possible
roots, and all the possible parts of the impossible roots, of a certain equation
shall be negative.

I have not been able completely to determine these conditions for equations
•i' a higher degree than the third; but I hope that the subject will obtain
the attention of mathematicians.

The actual motions corresponding to these impossible roots are not generally
taken notice of by the inventors of such machines, who naturally confine their
attention to the way in which it is designed to act; and this is generally
expressed by the real root of the equation. If, by altering the adjustments of
the machine, its governing power is continually increased, there is generally ;i
limit at which the disturbance, instead of subsiding more rapidly, becomes an
oscillating and jerking motion, increasing in violence till it reaches the limit of
action of the governor. This takes place when the possible part of one of tin-
impossible roots becomes positive. The mathematical investigation of the motion
may be rendered practically useful by pointing out the remedy for these distur-
Itances.

Tli is has been actually done in the case of a governor constructed by Mr
Fleeming Jenkin, with adjustments, by which the regulating power of the

GOVERNORS.

107

governor could be altered. By altering these adjustments the regulation could
be made more and more rapid, till at last a dancing motion of the governor,
accompanied with a jerking motion of the main shaft, shewed that an alteration
had taken place among the impossible roots of the equation.

I shall consider three kinds of governors, corresponding to the three kinds
of moderators already referred to.

In the first kind, the centrifugal piece has a constant distance from the
axis of motion, but its pressure on a surface on which it rubs varies when the
velocity varies. In the moderator this friction is itself the retarding force. In
the governor this surface is made moveable about the axis, and the friction
tends to move it ; and this motion is made to act on a break to retard the
machine. A constant force acts on the moveable wheel in the opposite direction
to that of the friction, which takes off the break when the friction is less than
a given quantity.

Mr Jenkin's governor is on this principle. It has the advantage that the
centrifugal piece does not change its position, and that its pressure is always
the same function of the velocity. It has the disadvantage that the normal
velocity depends in some degree on the coefficient of sliding friction between
two surfaces which cannot be kept always in the same condition.

In the second kind of governor, the centrifugal piece is free to move further
from the axis, but is restrained by a force the intensity of which varies with
the position of the centrifugal piece in such a way that, if the velocity of
rotation has the normal value, the centrifugal piece will be in equilibrium in
every position. If the velocity is greater or less than the normal velocity, the
centrifugal piece will fly out or fall in without any limit except the limits of
motion of the piece. But a break is arranged so that it is made more or less
powerful according to the distance of the centrifugal piece from the axis, and
thus the oscillations of the centrifugal piece are restrained within narrow limits.

Governors have been constructed on this principle by Sir W. Thomson and
by M. Foucault. In the first, the force restraining the centrifugal piece is that of
a spring acting between a point of the centrifugal piece and a fixed point at
a considerable distance, and the break is a friction-break worked by the reaction
of the spring on the fixed point.

In M. Foucault's arrangement, the force acting on the centrifugal piece is
the weight of the balls acting downward, and an upward force produced by

14—2

108 GOVERNORS.

weights acting on a combination of levers and tending to raise the balls. The
resultant vertical force on the balls is proportional to their depth below the
centre of motion, which ensures a constant normal velocity. The break is:—
in the first place, the variable friction between the combination of levers and
the ring on the shaft on which the force is made to act ; and, in the second
place, a centrifugal air-fan through which more or less air is allowed to pass,
according to the position of the levers. Both these causes tend to regulate the
velocity according to the same law.

The governors designed by the Astronomer-Royal on Mr Siemens's principle
for the chronograph and equatorial of Greenwich Observatory depend on nearly
similar conditions. The centrifugal piece is here a long conical pendulum, not
far removed from the vertical, and it is prevented from deviating much from a
fixed angle by the driving- force being rendered nearly constant by means of a
differential system. The break of the pendulum consists of a fan which dips
into a liquid more or less, according to the angle of the pendulum with the
vertical. The break of the principal shaft is worked by the differential apparatus ;
and the smoothness of motion of the principal shaft is ensured by connecting it
with a fly-wheel.

In the third kind of governor a liquid is pumped up and thrown out over
the sides of a revolving cup. In the governor on this principle, described by
Mr C. W. Siemens, the cup is connected with its axis by a screw and a spring,
in such a way that if the axis gets ahead of the cup the cup is lowered and
more liquid is pumped up. If this adjustment can be made perfect, the normal
velocity of the cup will remain the same through a considerable range of
driving-power.

It appears from the investigations that the oscillations in the motion must
be checked by some force resisting the motion of oscillation. This may be done
in some cases by connecting the oscillating body with a body hanging in a
viscous liquid, so that the oscillations cause the body to rise and fall in the liquid.

To check the variations of motion in a revolving shaft, a vessel filled with
viscous liquid may be attached to the shaft. It will have no effect on uniform
rotation, but will check periodic alterations of speed.

Similar effects are produced by the viscosity of the lubricating matter in
the sliding parts of the machine, and by other unavoidable resistances ; so that
it is not always necessary to introduce special contrivances to check oscillations.

GOVERNORS. \Q<J

I shall call all such resistances, if approximately proportional to the velocity,
by the name of " viscosity," whatever be their true origin.

In several contrivances a differential system of wheel-work is introduced
between the machine and the governor, so that the driving-power acting on the
governor is nearly constant.

I have pointed out that, under certain conditions, the sudden disturbances
of the machine do not act through the differential system on the governor, or
vice versa. When these conditions are fulfilled, the equations of motion are not
only simple, but the motion itself is not liable to disturbances depending on
the mutual action of the machine and the governor.

Distinction between Moderators and Governors,

In regulators of the first kind, let P be the driving-power and R the
resistance, both estimated as if applied to a given axis of the machine. Let V

dx

be the normal velocity, estimated for the same axis, and -*- the actual velocity,

and let M be the moment of inertia of the whole machine reduced to the
given axis.

Let the governor be so arranged as to increase the resistance or diminish
the driving- power by a quantity f \-ji~ V\, then the equation of motion will be

dt \ dtt

When the machine has obtained its final rate the first term vanishes, and

Hence, if P is increased or R diminished, the velocity will be permanently
increased. Regulators of this kind, as Mr Siemens* has observed, should be
called moderators rather than governors.

* " On Uniform Rotation," Phil. Trans. 1866, p. 657.

J 10 GOVERNORS.

In the second kind of regulator, the force F (^ - V\ , instead of being

applied directly to the machine, is applied to an independent moving piece, B,
which continually increases the resistance, or diminishes the driving-power, by
a quantity depending on the whole motion of B.

If y represents the whole motion of B, the equation of motion of B is

and that of M

where G is the resistance applied by B when B moves through one unit of space.
We can integrate the first of these equations at once, and we find

(5);

so that if the governor B has come to rest x= Vt, and not only is the velocity
of the machine equal to the normal velocity, but the position of the machine
is the same as if no disturbance of the driving-power or resistance had taken
place.

Jenkin's Governor. — In a governor of this kind, invented by Mr Fleeming
Jenkin, and used in electrical experiments, a centrifugal piece revolves on the
principal axis, and is kept always at a constant angle by an appendage which
slides on the edge of a loose wheel, B, which works on the same axis. The
pressure on the edge of this wheel would be propoi-tional to the square of the
velocity ; but a constant portion of this pressure is taken off by a spring which
acts on the centrifugal piece. The force acting on B to turn it round is therefore

jp**"_/7'.

dt

and if we remember that the velocity varies within very narrow limits, we may
write the expression

where F is a new constant, and F, is the lowest limit of velocity within which
the governor will act.

GOVERNORS. HI

Since this force necessarily acts on B in the positive direction, and since
it is necessary that the break should be taken off as well as put on, a weight
W is applied to B, tending to turn it in the negative direction ; and, for a
reason to be afterwards explained, this weight is made to hang in a viscous
liquid, so as to bring it to rest quickly.

The equation of motion of B may then be written

where Y is a coefficient depending on the viscosity of the liquid and on other
resistances varying with the velocity, and W is the constant weight.

Integrating this equation with respect to t, we find

Yy-Wt ........................ (7).

If B has come to rest, we have

f .............................. (8),

or the position of the machine is affected by that of the governor, but the
final velocity is constant, and

W
V> + ^ = V. ................................... (9),

where V is the normal velocity.

The equation of motion of the machine itself is

0, ............... (10).

This must be combined with equation (7) to determine the motion of the
whole apparatus. The solution is of the form

x = A1e**t + A1e'* + Aten*t+ Vt ........................ (11),

where nlt nn n, are the roots of the cubic equation

MBnt + (l£Y+FB)n' + FYn + FG = Q .................. (12).

If n be a pair of roots of this equation of the form a±J—lb, then the
part of x corresponding to these roots will be of the form

112 GOVERNORS.

If a is a negative quantity, this will indicate an oscillation the amplitude
of which continually decreases. If a is zero, the amplitude will remain constant,
and if a is positive, the amplitude will continually increase.

One root of the equation (12) is evidently a real negative quantity. The
condition that the real part of the other roots should be negative is

Y G

<iuantity-

This ia the condition of stability of the motion. If it is not fulfilled there
will be a dancing motion of the governor, which will increase till it is as great
as the limits of motion of the governor. To ensure this stability, the value of
)" must be made sufficiently great, as compared with G, by placing the weight
H' in a viscous liquid if the viscosity of the lubricating materials at the axle
is not sufficient.

To determine the value of F, put the break out of gear, and fix the
moveable wheel ; then, if V and V be the velocities when the driving-power
is P and P',

P-P

F=

V-V"

To determine G, let the governor act, and let y and y be the positions
oi' the break when the driving-power is P and P", then

P—Pf
G = - A-.

y-y

General Theory of Chronometric Centrifugal Pieces.

Sir W. Thomson's and M. Foucault's Governors. — Let A be the moment of
inertia of a revolving apparatus, and 6 the angle of revolution. The equation
of motion is

where L is the moment of the applied force round the axis.

GOVERNORS.

11;;

Now, let A be a function of another variable <£ (the divergence of the
centrifugal piece), and let the kinetic energy of the whole be

where B may also be a function of <£, if the centrifugal piece is complex.

If we also assume that P, the potential energy of the apparatus, is a
function of <f>, then the force tending to diminish (f>, arising from the action of

dP
gravity, springs, &c., will be -jj .

The whole energy, kinetic and potential, is

, ............. (2).

Differentiating with respect to tt we find

d^LdAdfT dBdj* dP\ d0tW d^_d^-
dt \*^ dt + *d<l>& + cfy) 4 l Tt dt3 4 dt dt>

TdQ_M/dAd0d$ A
dt ~ dt \ddt dt+ dt"

whence we have, by eliminating L,

*/»^\ i^^fo-X^®-^

dt ( dt) "* 34 dt +* (ty dt d<j> "

The first two terms on the right-hand side indicate a force tending to
increase <J>, depending on the squares of the velocities of the main shaft and
of the centrifugal piece. The fca-ce indicated by these terms may be called the
centrifugal force.

If the apparatus is so arranged that

P = ^o)2 + const ............................... (5),

where w is a constant velocity, the equation becomes
d

VOL. II.

15

I i 4 GOVERNORS.

In this case the value of <j> cannot remain constant unless the angular
velocity is equal to to.

A abaft with a centrifugal piece arranged on this principle has only one
velocity of rotation without disturbance. If there be a small disturbance, the
equations for the disturbance 0 and <f> may be written

d>B dA < ,

dA d$_

-

dA
The period of such small disturbances is -/v (AB)~* revolutions of the shaft.

They will neither increase nor diminish if there are no other terms in the
equations.

To convert this apparatus into a governor, let us assume viscosities X
and Y in the motions of the main shaft and the centrifugal piece, and a

dA
resistance G<f> applied to the main shaft. Putting -,, <a = K, the equations be-

come

The condition of stability of the motion indicated by these equations is
that all the possible roots, or parts of roots, of the cubic equation

ABnt + (AY+SX)nt+(XY+Kt)n + GK=0 ............... (11)

shall be negative ; and this condition is

(12).

Combination of Governors. — If the break of Thomson's governor is applied
to a moveable wheel, as in Jenkins governor, and if this wheel works a
steam-valve, or a more powerful break, we have to consider the motion of
three pieces. Without entering into the calculation of the general equations of

GOVERNORS.

115

motion of these pieces, we may confine ourselves to the case of small distur-
bances, and write the equations

dW

dt

64
lli

dt

~dl

= 0

.(13),

where 6, <f>, $ are the angles of disturbance of the main shaft, the centrifugal
arm, and the moveable wheel respectively, A, B, C their moments of inertia,
X, Y, Z the viscosity of their connexions, K is what was formerly denoted by

dA

j-f- w, and T and J are the powers of Thomson's and Jenkin's breaks respectively.

The resulting equation in n is of the form

An' + Xn Kn+T J
-K Bn+Y 0

0 -T Cri>

= 0

(14),

or

n' + n'f-

Y Z\ ,rXYZ/A B (7\.£*1 ]

h B + cr l L^^c U"1" r+ z) + AB]

JXYZ+KTC+K3^ KTZ KTJ

f\ ABC )^nABCl ABC

(15).

I have not succeeded in determining completely the conditions of stability
of the motion from this equation ; but I have found two necessary conditions,
which are in fact the conditions of stability of the two governors taken
separately. If we write the equation

= 0 ..................... (16),

then, in order that the possible parts of all the roots shall be negative, it is
necessary that

pq>r and ps>t ............................ (17).

I am not able to shew that these conditions are sufficient. This compound
governor has been constructed and used.

15—2

GOVERNORS.

On the Motion of a Liquid in a Tube revolving about a Vertical Axis.

C. ir. Sitmenst Liquid Governor. — Let p be the density of the fluid, k
the section of the tube at a point whose distance from the origin measured
iilong the tube is *, r, 0, z the co-ordinates of this point referred to axes fixed
with respect to the tube, Q the volume of liquid which passes through any
s<»ction in unit of time. Also let the following integrals, taken over the whole
tube, be

C ...................... (1),

the lower end of the tube being in the axis of motion.

Let ^ be the angle of position of the tube about the vertical axis, then
the moment of momentum of the liquid in the tube is

(2).

The moment of momentum of the liquid thrown out of the tube in unit of
time is

~*s*~

where r is the radius at .the orifice, k its section, and a the angle between the
direction of the tube there and the direction of motion.
The energy of motion of the fluid in the tube is

(4).

The energy of the fluid which escapes in unit of time is

The work done by the prime mover in turning the shaft in unit of time is

7 I 1 I t 1 "Tf i •»•*•*.

_ dH

dt-dt(d

The work spent on the liquid in unit of time is

<I\V d

dt •

GOVEBNORS, 117

Equating this to the work done, we obtain the equations of motion

These equations apply to a tube of given section throughout. If the fluid
is in open channels, the values of A and C will depend on the depth to
which the channels are filled at each point, and that of k will depend on the
depth at the overflow.

In the governor described by Mr C. W. Siemens in the paper already
referred to, the discharge is practically limited by the depth of the fluid at
the brim of the cup.

The resultant force at the brim is /= -Jg* + wV.

If the brim is perfectly horizontal, the overflow will be proportional to a;*
(where x is the depth at the brim), and the mean square of the velocity relative
to the brim will be proportional to x, or to Q*.

If the breadth of overflow at the surface is proportional to xn, where x is the
height above the lowest point of overflow, then Q will Vary as xn+*, and the mean

square of the velocity of overflow relative to the cup as a; or as -^—^ .

v

If n = — -J, then the overflow and the mean square of the velocity are both
proportional to x.

From the second equation we find for the mean square of velocity

If the velocity of rotation and of overflow is constant, this becomes

'-2g(h + z) (10),

From the first equation, supposing, as in Mr Siemens's construction, that
cosa=0 and .6 = 0, we find

] I g GOVERNORS.

In Mr Siemens's governor there is an arrangement by which a fixed rela-
tion is established between L and z,

L=-Sz .................................... (12),

„

„.„« --**+••« ........................ <13>-

If the conditions of overflow can be so arranged that the mean square of
velocity, represented by w , is proportional to
the spring which determines S is also arranged so that

the velocity, represented by w , is proportional to Q, and if the strength of

the equation will become, if 2gh = a>V,

. ,«-(£ '-')+% -«(3 -) ................... <>*>•

which shews that the velocity of rotation and of overflow cannot be constant
unless the velocity of rotation is w.

The condition about the overflow is probably difficult to obtain accurately
in practice ; but very good results have been obtained within a considerable range
of driving-power by a proper adjustment of the spring. If the rim is uniform,
there will be a maximum velocity for a certain driving-power. This seems to
be verified by the results given at p. 667 of Mr Siemens's paper.

If the flow of the fluid were limited by a hole, there would be a minimum
velocity instead of a maximum.

The differential equation which determines the nature of small disturbances
is in general of the fourth order, but may be reduced to the third by a
proper choice of the value of the mean overflow.

Theory of Differential Gearing.

In some contrivances the main shaft is connected with the governor by a
wheel or system of wheels which are capable of rotation round an axis, which
is itself also capable of rotation about the axis of the main shaft. These two
axes may be at right angles, as in the ordinary system of differential bevel
wheels; or they may be parallel, as in several contrivances adapted to clockwork.

GOVERNORS. 119

Let £ and 77 represent the angular position about each of these axes respec-
tively, 0 that of the main shaft, and <£ that of the governor; then 6 and <f>
are linear functions of £ and 77, and the motion of any point of the system can
be expressed in terms either of £ and 77 or of 6 and <f>.

Let the velocity of a particle whose mass is m resolved in the direction of

x be

dx d

with similar expressions for the other co-ordinate directions, putting suffixes
2 and 3 to denote the values of p and q for these directions. Then Lagrange's
equation of motion becomes

where H and H are the forces tending to increase f and 77 respectively, no
force being supposed to be applied at any other point.

Now putting Sx=p18t; + q1§T), ................................. (3),

and

the equation becomes

and since 8£ and 877 are independent, the coefficient of each must be zero. •

If we now put

2 (rap5) = L, 2 (inpq) = M, S (rag"2) = N (6),

the equations of motion will be

If the apparatus is so arranged that M=0, then the two motions will be
independent of each other ; and the motions indicated by f and -r\ will be about

120

conjugate axes— that is, about axes such that the rotation round one of them
does not t«nd to produce a force about the other.

Now let 8 be the driving-power of the shaft on the differential system,
and * that of the differential system on the governor; then the equation of
motion becomes

-**n-<> ...... (9);

r, =
and if we put U =

+ NS1 \

(11),
+ NS1

the equations of motion in 9 and <f> will be

+3f-§'

(12).

If M' = 0, then the motions in 6 and <f> will be independent of each oilier.
If If is also 0, then we have the relation

LPQ + MRS=0 ................................. (13);

and if this is fulfilled, the disturbances of the motion in 6 will have no effect
on the motion in tf>. The teeth of the differential system in gear with the
main shaft and the governor respectively will then correspond to the centres of
percussion and rotation of a simple body, and this relation will be mutual.

In such differential systems a constant force, H, sufficient to keep the
governor in a proper state of efficiency, is applied to the axis rj, and tin-
motion of this axis is made to work a valve or a break on the main shaft
of the machine. H in this case is merely the friction about the axis of £
If the moments of inertia of the different parts of the system are so arranged
that J/' = 0, then the disturbance produced by a blow or a jerk on the machine
will act instantaneously on the valve, but will not communicate any impulse to
the governor.

[From the Philosophical Magazine, for May, 1868.]

XXXV. "Experiment in Magneto- Electric Induction."
IN A LETTER TO W. R. GROVE, F.R.S.*

8, PALACE GARDENS TERRACE, W.
March 27, 1868.

DEAR SIR,

Since our conversation yesterday on your experiment on magneto-
electric inductionf, I have considered it mathematically, and now send you the
result. I have left out of the question the secondary coil, as the peculiar effect
you observed depends essentially on the strength of the current in the primary
coil, and the secondary sparks merely indicate a strong alternating primary
current. The phenomenon depends on the magneto-electric machine, the electro-
magnet, and the condenser.

The machine produces in the primary wire an alternating electromagnetic
force, which we may compare to a mechanical force alternately pushing and
pulling at a body.

The resistance of the primary wire we may compare to the effect of a
viscous fluid in which the body is made to move backwards and forwards.

The electromagnetic coil, on account of its self-induction, resists the starting
and stopping of the current, just as the mass of a large boat resists the efforts
of a man trying to move it backwards and forwards.

The condenser resists the accumulation of electricity on its surface, just as
a railway-buffer resists the motion of a carriage towards a fixed obstacle.

* Communicated by Mr W. R. Grove, F.R.S.
t See Phil. Mag. S. 4. March 1868, p. 184.

VOL. II. 16

EXPERIMENT IN MAGNETO-ELECTRIC INDUCTION.

Now let us suppose a boat floating in a viscous fluid, and kept in its
place by buffers fore and oft abutting against fixed obstacles, or by elastic
ropes attached to fixed moorings before and behind. If the buffers were away,
the mass of the boat would not prevent a man from pulling the boat along
with a long-continued pull ; but if the man were to push and pull in alter-
nate seconds of time, he would produce very little motion of the boat. The
buffers will effectually prevent the man from moving the boat far from its
position by a steady pull ; but if he pushes and pulls alternately, the period
of alternation being not very different from that in which the buffers would
cause the boat to vibrate about its position of equilibrium, then the force which
acts in each vibration is due, partly to the efforts of the man, but chiefly to
the resilience of the .buffers, and the man will be able to move the boat much
further from its mean position than he would if he had pushed and pulled at
tin- same rate at the same boat perfectly free.

Thus, when an alternating force acts on a massive body, the extent of the
displacements may be much greater when the body is attracted towards a
position of equilibrium by a force depending on the displacement than when
the body is perfectly free.

The electricity in the primary coil when it is closed corresponds to a free
body resisted only on account of its motion ; and in this case the current
produced by an alternating force is small. When the primaiy coil is interrupted
by a condenser, the electricity is resisted with a force proportional to the
accumulation, and corresponds to a body whose motion is restrained by a spring;
and in this case the motion produced by a force which alternates with sufficient
rapidity may be much greater than in the former case. I enclose the mathe-
matical theory of the experiment, and remain,

Yours truly,

J. CLERK MAXWELL.

Mathematical Theory of the Experiment.

Let M be the revolving armature of the magneto-electric machine, N, IS
the poles of the magnets, x the current led through the coil of the electro-
magnet R, and interrupted by the condenser C. Let the plates of the condenser
be connected by the additional conductor y.

EXPERIMENT IN MAGNETO-ELECTRIC INDUCTION.

123

Let Msmd be the value of the potential of the magnets on the coil of
the armature ; then if the armature revolves with the angular velocity n, the
electromotive force due to the machine is Mncosnt.

Let R be the resistance of the wire which forms the coil of the armature
M and that of the fixed electromagnet.

Let L be the coefficient of self-induction, or the " electromagnetic mass " of
these two coils taken together.

Let x be the value of the current in this wire at any instant, then Lx
will be its " electromagnetic momentum."

Let (7 be the capacity of the condenser, and P the excess of potential of
the upper plate at any instant, then the quantity of electricity on the upper
plate is CP.

Let p be the resistance of the additional conductor, and y the current in
it. We shall neglect the self induction of this current.

We have then for this conductor,

(i).

For the charge of the condenser,

dt

.(2).

For the current x,

If we assume

we find

dx

Mncosnt + Rx + L -57 + P =
at

x — A cos (nt + a),

(3).

a = cot

-~
Cpn

16—2

\-24 EXPERIMENT IN MAGNETO-ELECTRIC INDUCTION.

The quantity of the alternating current is determined by A ; and the value
of a only affects the epoch of the maximum current. If we make p = 0, the
effect is that of closing the circuit of x, and we find

,.
"

This expression shews that the condenser has no effect when the current
is closed.

If we make p = <», the effect is that of removing the conductor y, and
thus breaking the circuit. In this case

JfV

Tliis expression gives a greater value of A than when the circuit is closed,
provided ZCLn* is greater than unity, which may be ensured by increasing the
capacity of the condenser, the self-induction of the electromagnetic coil, or the
velocity of rotation.

If CL?R = \, the expression is reduced to

_Mn
'•~R-

This is the greatest effect which can be produced with a given velocity, and
is the same as if the current in the coil had no "electromagnetic momentum.'

If the electromagnet has a secondary coil outside the primary coil so as
to form an ordinary induction-coil, the intensity of the secondary current will
depend essentially on that of the primary which has just been found. Although
the reaction of the secondary current on the primary coil will introduce a
greater complication in the mathematical expressions, the remarkable phenomenon
described by Mr Grove does not require us to enter into this calculation, as
the secondary sparks observed by him are a mere indication of what takes
place in the primary coil.

[From the Philosophical Transactions, Vol. CLVin.]

XXXVI. On a Method of Making a Direct Comparison of Electrostatic with
Electromagnetic Force; with a Note on the Electromagnetic Theory of Light.

Received June 10,— Read June 18, 1868.

THERE are two distinct and independent methods of measuring electrical
quantities -with reference to received standards of length, time, and mass.

The electrostatic method is founded on the attractions and repulsions be-,
tween electrified bodies separated by a fluid dielectric medium, such as air ;
and the electrical units are determined so that the repulsion between two small
electrified bodies at a considerable distance may be represented numerically by
the product of the quantities of electricity, divided by the square of the distance.

The electromagnetic method is founded on the attractions and repulsions
observed between conductors carrying electric currents, and separated by air ;
and the electrical units are determined so that if two equal straight conductors
are placed parallel to each other, and at a very small distance compared with
their length, the attraction between them may be represented numerically by
the product of the currents multiplied by the sum of the lengths of the
conductors, and divided by the distance between them.

These two methods lead to two different units by which the quantity of
electricity is to be measured. The ratio of the two units is an important
physical quantity, which we propose to measure. Let us consider the relation
of these units to those of space, time, and force (that of force being a function
of space, time, and mass).

In the electrostatic system we have a force equal to the product of two
quantities of electricity divided by the square of the distance. The unit of
electricity will therefore vary directly as the unit of length, and as the square
root of the unit of force.

A DIRECT COMPARISON OF ELECTROSTATIC

In the electromagnetic system we have a force equal to the product of two
currents multiplied by the ratio of two lines. The unit of current in this
system therefore varies as the square root of the unit of force; and the unit
of electrical quantity, which is that which is transmitted by the unit current
in unit of time, varies as the unit of time and as the square root of the unit
«-f force.

The ratio of the electromagnetic unit to the electrostatic unit is therefore
that of a certain distance to a certain time, or, in other words, this ratio is a
velocity; and this velocity will be of the same absolute magnitude, whatever
standards of length, time, and mass we adopt.

The electromagnetic value of the resistance of a conductor is also a quantity
of the nature of a velocity, and therefore we may express the ratio of the two
electrical units in terms of the resistance of a known standard coil; and this
expression will be independent of the magnitude of our standards of length,
time and mass.

In the experiments here described no absolute measurements were made,
either of length, time, or mass, the ratios only of these quantities being in-
volved ; and the velocity determined is expressed in terms of the British Asso-
ciation Unit of Resistance, so that whatever corrections may be discovered to
be applicable to the absolute value of that unit must be also applied to the
velocity here determined.

A resistance-coil whose resistance is equal to about 28 '8 B. A. units would
represent the velocity derived from the present experiments in a manner inde-
|>endent of all particular standards of measure.

The importance of the determination of this ratio in all cases in which
electrostatic and electromagnetic actions are combined is obvious. Such cases
occur in the ordinary working of all submarine telegraph-cables, in induction-
coils, and in many other artificial arrangements. But a knowledge of this ratio
is, I think, of still greater scientific importance when we consider that the
velocity of propagation of electromagnetic disturbances through a dielectric medium
depends on this ratio, and, according to my calculations*, is expressed by the
very same number.

The first numerical determination of this quantity is that of Weber and
Kohlrauscht, who measured the capacity of a condenser electrostatically by

* "A Dynamical Theory of the Electromagnetic Field," Philosophical Transactions, 1865.
t Pogg, Ann. Aug. 1856, Bd. xcix. p. 10.

WITH ELECTEOMAGNETIC FORCE. 127

comparison with the capacity of a sphere of known radius, and electromag-
netically by passing the discharge from the condenser through a galvanometer.

The Electrical Committee of the British Association have turned their
attention to the means of obtaining an accurate measurement of this velocity,
and for this purpose have devised new forms of condensers and contact-breakers ;
and Sir William Thomson has obtained numerical values of continually increasing
accuracy by the constant improvement of his own methods.

A velocity which is so great compared with our ordinary units of space
and time is probably most easily measured by steps, and by the use of several
different instruments ; but as it seemed probable that the tune occupied in the
construction and improvement of these instruments would be considerable, I
determined to employ a more direct method of comparing electrostatic with
electromagnetic effects.

I should not, however, have been able to do this, had not Mr Gassiot,
with his usual liberality, placed at my disposal his magnificent battery of 2600
cells charged with corrosive sublimate, with the use of his laboratory to work in.

To Mr Willoughby Smith I am indebted for the use of his resistance-
coils, giving a resistance of more than a million B. A. units, and to Messrs
Forde and Fleeming Jenkin for the use of a galvanometer and resistance-coils,
a bridge and a key for double contacts.

Mr C. Hockin, who has greatly assisted me with suggestions since I first
devised the experiment, undertook the whole work of the comparison of the
currents by means of the galvanometer and shunts. He has also tested the
resistances, and in fact done everything except the actual observation of equi-
librium, which I undertook myself.

The electrical balance itself was made for me by Mr Becker.

The electrostatic force observed was that between two parallel disks, of
which one, six inches diameter, was insulated and maintained at a high
potential, while the other, four inches diameter, was at the same potential as
the case of the instrument.

In order to insure a known quantity of electricity on the surface of this
disk, it was surrounded by the "guard-ring" introduced by Sir W. Thomson,
so that the surface of the disk when in position and that of the guard-ring
were in one plane, at the same potential, and separated by a very narrow
space. In this way the electrical action on the small disk was equal to that
due to a uniform distribution over its front surface, while no electrical action

A DIRECT COMPARISON OF ELECTROSTATIC

•••ultl exist at its sides or back, as these were at the same potential with the
surrounding surfaces.

The large disk was mounted on a slide worked by a micrometer-screw.
The qmftll disk was suspended on one arm of a torsion-balance so that in its
position of equilibrium its surface and that of the guard-ring were in one plane.

If E is the difference of potential between the two disks in electromag-
netic measure, the attraction between them is

where a is the radius of the small disk, b its distance from the large one,
and v is the velocity representing the ratio of the electromagnetic to the elec-
trostatic unit of electricity.

The electromagnetic force observed was the repulsion between two circular
coils, of which one was attached to the back of the suspended disk, and the
other was placed behind the large disk, being separated from it by a plate of
glass and a layer of Hooper's compound. A current was made to pass through
these coils in opposite directions, so as to produce a repulsion

9 A
'

(2),
where n and »' are the number of windings of each coil, y is the current, and

«*

a, and a, are the mean radii of the coils, and &' the mean distance of their
planes, and Ee and Ft are the complete elliptic functions for modulus c = siny.

2 A <2d'

When b' is small compared with a', -jr- becomes very nearly -p- •

If we take into account the fact that the section of each coil is of sensible
area, this formula would require correction; but in these coils the depth v;i*
equal to the breadth of the section, whence it follows, by the differential

WITH ELECTROMAGNETIC FORCE.

129

equation of the potential of two coils, given at p. 508 * of my paper on the
Electromagnetic Field,

d*M d*M 1 dM

_| _ Q /5\

do? db* a da

('2\
1— iV-rJ, where a is the depth
a /

of the coil — a correction which is in this case about 1 — "000926.

A. Suspended disk and coil.

A'. Counterpoise disk and coil.

C. Fixed disk and coil.

B,. Great battery. Ba. Small battery.

G,. Primary coil of galvanometer.

Secondary coil. R. Great resistance.

Torsion head and tangent screw.

One quarter of the micrometer-box, disks, and coils is cut away to shew the interior. The case of the
instrument is not shewn. The galvanometer and shunts were 10 feet from the Electric Balance.

G,
T.

K. Double key. g. Graduated glass scale.
C'. Electrode of fixed disk.

x. Current through R.

x. Current through G,. x — x'. Current through S.

y. Current through the three coils and G2.
S. Shunt. M. Mercury cup.

* [Vol. i. p. 591.]

VOL. II.

17

130 A DIRBCT COMPARISON OF ELECTROSTATIC

The suspended ooU, besides the repulsion due to the fixed coil, experiences
a couple due to the action of terrestrial magnetism. To balance this couple,
a coil exactly similar was attached to the other arm of the torsion-balance,
and the current in the second coil was made to flow in the opposite direction
to that in the first. When the current was made to flow through both coils,
no effect of terrestrial magnetism could be observed.

The torsion-balance consisted of a light brass frame, to which the suspended
coils and disks were attached so that the centre of each coil was about eight
inches from the vertical axis of suspension. This frame was suspended by a
copper wire (No. 20), the upper end of which was attached to the centre <>t
a torsion head, graduated, and provided with a tangent screw for small angular
adjustments. The torsion head was supported by a hollow pillar, the base <>t
which was clamped to the lid of the instrument so as to admit of sniull
adjustments in every direction.

The fixed disk and coil were mounted on a slide worked by a micrometer-
ncrew, and were protected by a cylindrical brass box, the front of which,
forming the guard-ring, 7 inches in diameter, had a circular aperture 4 '26 inches
diameter, within which the suspended disk, 4'13 inches diameter, was free to
move, leaving an interval of '065 of an inch between the disk and the aper-
ture. A glass scale with divisions of -j-J^ of an inch was attached to the
suspended disk on the side which was not electrified, and this was viewed by
a microscope attached to the side of the instrument and provided with cross
wires • at the focus.

The disk worked by the micrometer was carefully adjusted by the maker,
so as to be parallel to the inner surface of the guard-ring, or front face <>t
the micrometer-box. This front face of the micrometer-box, when in position
in the instrument, was made vertical by means of three adjusting screws.
The suspended disk was then pressed against the fixed disk by means of a
slight spring, and the fixed disk was gradually moved forward by the micro-
meter-screw, while at the same time the graduated scale was observed through
the microscope. In this way the graduations on the scale were compared with
the readings of the micrometer. This was continued till the large disk came
into contact with the guard-ring at one point, when the regularity of tin-
motion was interrupted. A very small motion was then sufficient to bring the
whole circumference of the disk into contact with the guard-ring, when the
motion ceased altogether. This motion was not much more than one-thousandth
of an inch.

WITH ELECTROMAGNETIC FORCE. 131

This disk was then brought to the position of first contact, and the
microscope was adjusted so that a known division of the glass scale was
bisected by the cross wires. A small piece of silvered glass was fastened to
the outside of the guard-ring, and another to the back of the suspended disk ;
and these were adjusted so as to be in one plane, and to give a continuous
image of reflected objects when the disks were in contact and the surface of
the suspended disk was therefore in the plane of the surface of the guard-
ring. The fixed disk was then screwed back, and the torsion-balance was
adjusted so that the suspended disk when in equilibrium was in. precisely the
same position as before. This was tested by observing the coincidence of the
zero division of the glass scale with the cross wires of the microscope, and
by examining the reflections from the two pieces of silvered glass. The torsion-
balance could be moved bodily in any horizontal direction by adjusting the
base of the pillar; it could be raised or lowered by a winch, and it could
be turned about any horizontal axis by sliding weights, and round the vertical
axis by a tangent screw of the torsion head. In this way the position of
equilibrium of the suspended disk could be made to coincide with the plane
of the guard-ring to the thousandth of an inch; and the adjustment when
made continued very good from day to day, soft copper wire, stretched straight,
not having the tendency to untwist gradually which I have observed in steel
wire. The weight of the torsion piece was about 1 Ib. 3 oz., and the time of
a double oscillation about fourteen seconds. The oscillations of the suspended
disk, when near its sighted position, were found to subside very rapidly, the
energy of the motion being expended in pumping the air through the narrow
aperture between the guard-plate and the suspended disk.

The electrical arrangements were as follows : —

One electrode of Mr Gassiot's great battery was connected with a key.
When the key was pressed connexion was made to the fixed disk, and thence,
through Mr Willoughby Smith's resistance-coils, to a point where the current
was divided between the principal coil of the galvanometer and a shunt, S,
consisting of Mr Jenkin's resistance- coils. These partial currents reunited at a
point where they were put in connexion with the other electrode of the battery,
with the case of the instrument, and with the earth.

Another battery was employed to send a current through the coils. One
electrode of this battery was connected with a second contact piece of the

17—2

132 A DIRECT COMPARISON OF ELECTROSTATIC

key, so thut. when the key was pressed, the current went first through the
secondary ooil of the galvanometer, consisting of thirty windings of thick wire,
then through the fixed coil, then to the suspension wire, and so through the
two suspended coils to the brass frame of the torsion-balance and the suspended
disk. A stout copper wire, well amalgamated, hanging from the centre of the
toreinii-lnliince into a cup of mercury, made metallic communication to the case,
to earth, and to the other electrode of the battery.

When these arrangements had been made, the observer at the microscope,
when the suspended disk was stationary at zero, made simultaneous contact
with both batteries by means of the key. If the disk was attracted, the
great battery was the more powerful, and the micrometer was worked so as
to increase the distance of the disk. If the disk was repelled, the fixed disk
had to be moved nearer to the suspended disk, till a distance was found at
which, when the scale was at rest and at zero, no effect was produced by
the simultaneous action of the batteries. With the forces actually employed
the equilibrium of the scale at zero was unstable ; so that when the adjust-
ment was nearly perfect the force was always directed from zero, and contacts
had to be made as the scale was approaching zero, in such a way as to
bring it to rest, if possible, at zero.

In the meantime the other observer at the galvanometer was taking
advantage of these contacts to alter the shunt S, till the effects of the two
currents on the galvanometer-needle balanced each other.

When a satisfactory case of equilibrium had been observed simultaneously
at the galvanometer and at the torsion-balance, the micrometer-reading and the
resistance of the shunt were set down as the results of the experiment.

The chief difficulties experienced arose from the want of constancy in the
batteries, the ratio of the currents varying very rapidly after first making
contact. I think that by increasing considerably the resistance of the great
battery-circuit, the current could be made more uniform.

When a sufficient number of experiments on equilibrium had been made,
a current was made to pass through the secondary coil of the galvanometer,
and was then divided between a shunt of 31 units BA and the primary mil
<>f the galvanometer with a resistance & added. S" was then varied till the
needle was in equilibrium. In this way the magnetic effects of the two coils
were compared.

The resistance of the galvanometer and of all the coils were tested by

WITH ELECTROMAGNETIC FORCE. 133

Mr Hockin, who also made all the observations with the galvanometer and its
adjusting shunts.

To determine v from these experiments, we have first, since the attraction
is equal to the repulsion,

,214

If x is the current of the great battery passing through the great resistance
R, and if x of this passes through the galvanometer whose resistance is G,
and x — x' through the shunt S to earth, then

E = Rx + Gx' .................................... (7),

and Gx=S(x-x') ................................... (8).

Also if g, is the magnetic effect of the principal coil of the galvanometer,
and g.2 that of the secondary coil, then when the needle is in equilibrium

9iaf=ff& ....................................... (9).

In the comparison of the coils of the galvanometer, if x^ and y^ are the
currents through each, we have

But yt is divided into two parts, of which x1 passes through the galvan-
ometer G and the shunt 5", and the other, yl — x', passes through the shunt

of 31 Ohms. Hence

x1(G + &)=(y1-x,)3l ........................... (11).

From these equations we obtain as the value of v,

1 a f~B IRQ n\ 31

V= . .—-T—J -»-+#+ G) -fr—q, -- ............ (12),

-V 2^1 \ o -

an equation containing only known quantities on the right-hand side. Of these,
n and ri are the numbers of windings on the two coils, a is the mean of
the radii of the suspended disk and the aperture, b is the distance between

2A
the fixed disk and the suspended disk, -5- is found from a^ and aa, the mean

radii of the coils, and b' their mean distance by equation (3).

R is the great resistance, G that of the galvanometer, S that of the shunt
in the principal experiment, and & that of the additional resistance in the com-
parison of galvanometer-coils.

A DIRKCT COMPARISON OF ELECTROSTATIC

In this expression the only quantities which must be determined in
absolute measure are the resistances. The other quantities which must be
measured are the ratios of the radius of the disk to its distance from the
fixed disk, and the ratio of the radius of the coils to the distance between
them. These ratios and the number of windings in the coils are of course
abstract numbers.

In the experiments,

n = 144 n' = 121

a = 2-0977 inches. a = I '934 inch.

To determine a', the circumference of every layer of the coils was measured
with watch-spring, the thickness of which was '008 inch.

One turn of the micrometer-screw was found by Mr Hockin to be equal
to "0202 inch. If TO is the micrometer-reading in terms of the screw,

b = m -1270, &' = w + 26-31.

In terms of the micrometer measure we have for a and a',

a=103'85 turns, a' = 95 75 turns.

The resistances were determined by Mr Hockin as follows :

R=l 102 000 Ohms.
G= 46 220 „

The experiments were made for two days, using a small battery charged
with bichromate of potash. The current due to this battery was found to
diminish so rapidly that a set of Grove's cells was used on the third day,
which was found to be more constant than the great battery. A proper com-
bination of the two batteries would perhaps produce a current which would
diminish according to the same law as that of the great battery. Another
difficulty arose from the fact that when the connexions were made, but before
the key was pressed, if the micrometer was touched by the hand the disk was
attracted. This I have not been able satisfactorily to account for, except by
leakage of electricity from the great battery through the floor. When the
micrometer was not touched, the disk remained at its proper zero. In certain
experiments I kept my hand always on the micrometer in order to be able
to adjust it more accurately. These experiments gave a value of v much too
small, on account of the additional attraction. When I discovered the attraction,

WITH ELECTROMAGNETIC FORCE.

135

I took care to make the observations without touching the micrometer, and
took advantage of the attraction to check the oscillations of the disk. The
experiments in which these precautions were taken agree together as well as
I could expect, and lead me to think that, with the experience I have acquired,
still better results might be obtained by the same method. It must be borne
in mind that none of the results were calculated till after the conclusion of all
the experiments, and that the rejected experiments were condemned on account
of errors observed while they were being made.

Any leakage arising from want of insulation of the fixed disk would
introduce no error, as the difference of potentials between the two disks is
measured by the current in the galvanometer, through a known resistance,
independently of any leakage.

All that is essential to accuracy is that the position of equilibrium before
making contact should be at true zero, the same as when there is no electrical
action, and that this equilibrium should not be disturbed when simultaneous
contact is made with both batteries.

Experiments on May 8. »S' = 1710 Ohms.

Number of
experiment

Great
battery-cells

Small
battery-cells

Distance of disks
by micrometer

Resistance of
shunt S

Value of v in
Ohms

1

1000

6

12-41

6870

28-591

2

1000

6

12-36

6940

28-430

8*

1800

8

16-99

5074

28-886

9

1800

8

17-02

5110

28-686

10

1800

8

19-91

4430

28-910

11

1800

7

20-07

4410

28-850

12

2600

9

25-08

3700

28-762

13

2600

9

25-12

3690

28-795

14

2600

9

25-29

3680

28-735

15 L'UIM)

9

25-18

3690

28-752

16

2600

9

25-19

3695

28-709

17

1800

7

19-69

4435

29-474

Mean value of v = 28798 Ohms, or B. A. units,

or 288,000,000 metres per second,
or 179,000 statute miles per second.

The " probable error " is about one-sixth per cent.

* In experiment 8 Mr Hockin and I changed places.

136 A DIRECT COMPARISON OF ELECTROSTATIC WITH ELECTROMAGNETIC FORCE.

Experiments 3, 4, 5, 6, 7 were rejected on account of the micrometer
being touched during the observation of equilibrium. These experiments gave
an average value of t> = 27*39.

The value of v derived from these experiments is considerably smaller than
that which was obtained by MM. Weber and Kohlrausch, which was 31 '074
Ohms, or 310,740,000 metres per second.

Their method involved the determination of the electrostatic capacity of a
condenser, the electrostatic determination of its potential when charged, and
the electromagnetic determination of the quantity of electricity discharged through
a galvanometer.

The capacity of the condenser was measured by dividing its charge repeat-
edly with a sphere of known radius. Now, since all condensers made with
solid dielectrics exhibit the phenomena of " electric absorption," this method
would give too large a value for the capacity, as the condenser would become
recharged to a certain extent after each discharge, so that the repeated division
of the charge would have too small an effect on the potential. The capacity
being overestimated, the number of electrostatic units in the discharge would
be overestimated, and the value of v would be too great.

In pointing out this as a probable source of error in the experiments of
MM. Weber and Kohlrausch, I mean to indicate that I have such confidence
in the ability and fidelity with which their investigation was conducted, that
I am obliged to attribute the difference of their result from mine to a phe-
nomenon the nature of which is now much better understood than when their
experiments were made.

On the other hand, the result of present experiments depends on the
accuracy of the experiments of the Committee of the British Association on
Electric ^Resistance. The B. A. unit is about 8 '8 per cent, larger than that
determined by Weber in 1862, and about 1'2 per cent, less than that derived
by Dr Joule from his experiments on the dynamical equivalent of heat by
comparing the heating effects of direct mechanical agitation with those of electric
currents.

I believe that Sir William Thomson's experiments, not yet published, give
a value of v not very different from mine. His method, I believe, also depends
on the value of the B. A. unit.

The lowest estimate of the velocity of light, that of the late M. Foucault, is

298,000,000 metres per second.

A COMPARISON OF THE ELECTRIC UNITS, &C. 137

Note on the Electromagnetic Theory of Light.

In a paper on the Electromagnetic Field* some years ago, I laid before
the Royal Society the reasons which led me to believe that light is an elec-
tromagnetic phenomenon, the laws of which can be deduced from those of
electricity and magnetism, on the theory that all these phenomena are affections
of one and the same medium. Two papers appeared in Poggendorff's Anna-
len, for 1867, bearing on the same subject. The first, by the late eminent
mathematician Bernhardt Eiemann, was presented in 1858 to the Royal Society
of Gottingen, but was withdrawn before publication, and remained unknown
till last year. Riemann shews that if for Laplace's equation we substitute

0 ........................ (13),

V being the electrostatic potential, and a a velocity, the results will agree
with known phenomena in all parts of electrical science. This equation is equi-
valent to a statement that the potential V is propagated through space with
a certain velocity. The author, however, seems to avoid making explicit
mention of any medium through which the propagation takes place, but he
shews that this velocity is nearly, if not absolutely, equal to the known velocity
of light.

The second paper, by M. Lorenz, shews that, on Weber's theory, periodic
electric disturbances would be propagated with a velocity equal to that of light.
The propagation of attraction through space forms part of this hypothesis also,
though the medium is not explicitly recognised.

From the assumptions of both these papers we may draw the conclusions,
first, that action and reaction are not always equal and opposite, and second,
that apparatus may be constructed to generate any amount of work from its
resources.

For let two oppositely electrified bodies A and B travel along the line
joining them with equal velocities in the direction AB, then if either the
potential or the attraction of the bodies at a given time is that due to their
position at some former time (as these authors suppose), B, the foremost body,
will attract A forwards more than A attracts B backwards.

* Philosophical Transactions, 1865, p. 459. [Vol. I. p. 527.]
VOL. II. 18

138 A COMPARISON OF THE ELECTRIC UNITS

Now let A and B be kept asunder by a rigid rod.

The combined system, if set in motion in the direction AB, will pull in
that direction with a force which may either continually augment the velocity,
or may be used as an inexhaustible source of energy.

I think that these remarkable deductions from the latest developments of
Weber and Neumann's theory can only be avoided by recognizing the action of
a medium in electrical phenomena.

The statement of the electromagnetic theory of light in my former paper
was connected with several other electromagnetic investigations, and was there-
fore not easily understood when taken by itself. I propose, therefore, to state
it in what I think the simplest form, deducing it from admitted facts, and
shewing the connexion between the experiments already described and those
which determine the velocity of light.

The connexion of electromagnetic phenomena may be stated in the following
manner.

THEOREM A. — If a closed curve be drawn embracing an electric current,
then the integral of the magnetic intensity taken round the closed curve is
equal to the current multiplied by 4ir.

The integral of the magnetic intensity may be otherwise defined as the
work done on a unit magnetic pole carried completely round the closed curve.

This well-known theorem gives us the means of discovering the position
and magnitude of electric currents, when we can ascertain the distribution of
magnetic force in the field. It follows directly from the discovery of (Ersted.

THEOREM B. — If a conducting circuit embraces a number of lines of mag-
netic force, and if, from any cause whatever, the number of these lines is
diminished, an electromotive force will act round the circuit, the total amount
of which will be equal to the decrement of the number of lines of magnetic
force in unit of time.

The number of lines of magnetic force may be otherwise defined as the
integral of the magnetic intensity resolved perpendicular to a surface, multiplied
by the element of surface, and by the coefficient of magnetic induction, the
integration being extended over any surface bounded by the conducting circuit.

This theorem is due to Faraday, as the discoverer both of the facts and
of this mode of expressing them, which I think the simplest and most com-
prehensive.

AND ON THE ELECTROMAGNETIC THEORY OF LIGHT. 139

THEOREM C. — When a dielectric is acted on by electromotive force it ex-
periences what we may call electric polarization. If the direction of the elec-
tromotive force is called positive, and if we suppose the dielectric bounded by
two conductors, A on the negative, and B on the positive side, then the
surface of the conductor A is positively electrified, and that of B negatively.

If we admit that the energy of the system so electrified resides in the
polarized dielectric, we must also admit that within the dielectric there is a
displacement of electricity in the direction of the electromotive force, the amount
of this displacement being proportional to the electromotive force at each point,
and depending also on the nature of the dielectric.

The energy stored up in any portion of the dielectric is half the product
of the electromotive force and the electric displacement, multiplied by the volume
of that portion.

It may also be shewn that at every point of the dielectric there is a
mechanical tension along the lines of electric force, combined with an equal
pressure in all directions at right angles to these lines, the amount of this
tension on unit of area being equal to the amount of energy in unit of volume.

I think that these statements are an accurate rendering of the ideas of
Faraday, as developed in various parts of his "Experimental Researches."

THEOREM D. — When the electric displacement increases or diminishes, the
effect is equivalent to that of an electric current in the positive or negative
direction.

Thus, if the two conductors in the last case are now joined by a wire,
there will be a current in the wire from A to B.

At the same time, since the electric displacement in the dielectric is
diminishing, there will be an action electromagnetically equivalent to that of an
electric current from B to A through the dielectric.

According to this view, the current produced in discharging a condenser
is a complete circuit, and might be traced within the dielectric itself by a
galvanometer properly constructed. I am not aware that this has been done,
so that this part of the theory, though apparently a natural consequence of
the former, has not been verified by direct experiment. The experiment would
certainly be a very delicate and difficult one.

Let us now apply these four principles to the electromagnetic theory of
light, considered as a disturbance propagated in plane waves.

18—2

140 A COMPARISON OF THE ELECTRIC UNITS

Let the direction of propagation be taken as the axis of z, and let all
the quant ities be functions of z and of t the time; that is, let every portion
of any plane perpendicular to z be in the same condition at the same instant.

Let us also suppose that the magnetic force is in the direction of the
^xyi of y, and let ft be the magnetic intensity in that direction at any point.

Let the closed curve of Theorem A consist of a parallelogram in the
plane yz, two of whose sides are 6 along the axis of y, and z along the axis
of :. The integral of the magnetic intensity taken round this parallelogram is
b(ft, — ft), where ft, is the value of ft at the origin.

Now let p be the quantity of electric current in the direction of x per
unit of area taken at any point, then the whole current through the parallelo-
gram will be

I bpdz,

and we have by (A),

l(ftt-ft) = 47r Plpdz.

If we divide by b and differentiate with respect to z, we find

Let us next consider a parallelogram in the plane of xz, two of whose
sides are a along the axis of x, and z along the axis of z.

If' P is the electromotive force per unit of length in the direction of x,
then the total electromotive force round this parallelogram is a(P — P0).

If p. is the coefficient of magnetic induction, then the number of lines of
force embraced by this parallelogram will be

and since by (B) the total electromotive force is equal to the rate of dimi-
nution of the number of lines in unit of time,

Dividing by a and differentiating with respect to z, we find

dft

AND ON THE ELECTROMAGNETIC THEORY OF LIGHT.

141

Let the nature of the dielectric be such that an electric displacement /
is produced by an electromotive force P,

P = ¥- (16),

where k is a quantity depending on the particular dielectric, which may be
called its "electric elasticity."

Finally, let the current p, already considered, be supposed entirely due
to the variation off, the electric displacement, then

We have now four equations, (14), (15), (16), (17), between the four
quantities /J, p, P, and f. If we eliminate p, P, and f, we find

CLL 47TJLC CtfZr

IfwePut G£-p (19)>

the well-known solution of this equation is

/3 = <f>l(z— Vt) + (j)3(z+ Vt) (20),

shewing that the disturbance is propagated with the velocity V.
The other quantities p, P, and f can be deduced from /8.

Thus, if /8 = c cos -r- (z -

A.

C 27T,

T, ...

-

I have in the next place to shew that the velocity F is the same
quantity as that found from the experiments on electricity.

For this purpose let us consider a stratum of air of thickness b bounded
by two parallel plane conducting surfaces of indefinite extent, the difference
of whose potentials is E.

143 A COMPARISON OF THE ELECTRIC UMTS

The electromotive force per unit of length between the surfaces is P = !•'..

The electric displacement is/=^P.

The energy in unit of volume and the tension along the lines of force
per unit of area is } Pf.

The attraction X on an area ira* of either surface is

.(22).

If this area is separated by a small interval from the rest of the plane
surface, as in the experiment, and if this interval is small compared with the
radius of the disk, the lines of force belonging to the disk will be separated
from those belonging to the rest of the surface by a surface of revolution,
the section of which, at any sensible distance from the surface, will be a circle
whose radius is a mean between those of the disk and the aperture. This
radius must be taken for a in the equation (22)*.

Let us next consider the magnetic force near a long straight conductor
carrying a current y. The magnetic force will be in the direction of a tangent
to a circle whose axis is the current ; and the intensity will be uniform round
this circle. If the radius is 6, and the magnetic intensity yS, the integral
round the circle will be 1irbft = ^iry by (A).

Hence ^ = 2f ....................................... (23)'

Let a wire carrying a current y' be placed parallel to the first at a
distance 6, and let us consider a portion of this wire of length I. This portion
will be urged across the lines of magnetic force, and the electromagnetic force
I' will be equal to the product of the length of the portion, multiplied by

* [Note added Dec. 28, 1868.— I have since found that if a, is the radius of the disk, and a, that of
the aperture of the guard-ring, and 6 the distance from the large fixed disk, then we must substitute for

p the more approximate expression" ' +n£- \, where a is a quantity which cannot exceed -^-(o,-o,).
-J. C. M.]

AND ON THE ELECTROMAGNETIC THEORY OF LIGHT. 143

the current and by the number of lines which it crosses per unit of distance
through which it moves, or, in symbols,

I . (24).

=^~byy

If the two wires instead of being straight are circular, of radius a', and
if V the distance between them is very small compared with the radius, the
attraction will be the same as if they were straight, and will be

C)~rr,'

F_ ^iTTll ,
= 2p-y-yy (25).

When // is not very small compared with a', we must use the equation

2A

(3) to calculate the value of -g- by elliptic integrals.

Making X= Y and comparing with equation (6), we find

but, by (19), P = I^-

Hence v = pV (27),

where v is the electromagnetic ratio and V is the velocity of light.

But since all the experiments are made in air, for which fi is assumed equal
to unity, as the standard medium with which all others are compared, we have
finally

v=V (28),

or the number of electrostatic units in one electromagnetic unit of electricity is
numerically equal to the velocity of light.

[Extracted from The Quarterly Journal of Pure and Applied Ufatfiematics, No. 34, 1867.]

XXXVII. On the Cyclide.

l\ optical treatises, the primary and secondary foci of a small pencil are
sometimes represented by two straight lines cutting the axis of the pencil at
right angles in planes at right angles to each other. Every ray of the pencil
is supposed to pass through these two lines, thus forming what M. Pliiker*
has called a congruence of the first order.

The system of rays, as thus defined, does not fulfil the essential condi-
tion of all optical pencils, that the rays shall have a common wave-sur;
for no surface can be drawn which shall cut all the rays of such a pencil
at right angles.

Sir W. R. Hamilton has shewn, that the primary and secondary foci are
in general the points of contact of the ray with the surface of centres of
the wave-surface, which forms a double caustic surface. If we select a pencil
of rays corresponding to a given small area on the wave-surface, their points
of contact will lie on two small areas on the two sheets of the caustic
surface. The sections of the pencil by planes perpendicular to its axis will
appear, when the pencil is small enough, as two short straight lines in pi
perpendicular to each other.

I propose to determine the form of the wave-surface, when one or both
of the so-called focal lines is really a line, and not merely the projection of
a small area of a curved surface.

Let us first determine the condition that all the normals of a surface
may pass through one fixed curve.

Let R be a point on the surface, and RP a normal at P, meeting the
fixed curve at P. Let PT be a tangent to the fixed curve at P, and HI'T
a plane through R and PT.

* Philosophical Transactions, 1864.

THE CYCLIDE. 145

Of the two lines of curvature through R, the first touches the plane
RPT, and the second is perpendicular to it. Hence, if the plane RPT turn
about the tangent PT as an axis, it will always be normal to the second
line of curvature. The second line of curvature is therefore a circle, and PT
passes through its centre perpendicular to its plane.

All the normals belonging to the second line of curvature are of equal
length, and equally inclined to PT, so that they may be considered either
as the generating lines of a right cone whose axis is the tangent to the
fixed curve, or as the radii of a sphere whose centre is at P, and which
touches the surface all along the line of curvature.

The surface may therefore be defined as the envelope of a series of
spheres, whose centres lie on the fixed curve, and whose radii vary according
to any law.

If the normal passes through two fixed curves, the surface must also be
the envelope of a second series of spheres whose centres lie on the second
fixed curve, and each of which touches all the spheres of the first series.

If we take any three spheres of the first series, the surface may be
defined as the envelope of all the spheres which touch the three given spheres
in a continuous manner.

This is the definition given by Dupin, in his Applications de Geometric
(p. 200), of the surface of the fourth order called the Cyclide, because both
series of its lines of curvature are circles.

If the three fixed spheres be given, they may either be all on the same
side (inside or outside) of the touching sphere, or any one of the three may
be on the opposite side from the other two. There are thus four different
series of spheres which may be described touching the same three spheres,
but we cannot pass continuously from one series to another, and the normals
to the four corresponding cyclides pass through different fixed curves.

Let us next consider the nature of the two fixed curves. Since all the
normals pass through both curves, and since all those which pass through a
point P are equally inclined to the tangent at P, the second curve must
lie on a right cone. If now the point P be taken so that its distance from
a point Q in the second curve is a minimum, then PQ will be perpendicular
to PT, and the right cone will become a plane, therefore the second curve is
a plane conic. In the same way we may shew, that the first curve is a plane
conic.

VOL. n. 19

THE CYCLIDE.

The two curves are therefore plane oonics, such that the cone whose base
is one of the conies, and its vertex any point of the other, is a right cone.
The conies are therefore in planes at right angles to each other, and the foci
of one are the vertices of the transverse axis of the other. We shall call tin •>«•
curves the focal conies of the cyclide.

Let the equations to a point on an ellipse be

a; = ccosa, y = (c1 - 6')* sin a, 2 = 0 (1),

where a is the eccentric angle, and let the equations to a point on tin-
hyperbola be

a: = 6sec-B, y = 0, z = (c1 - &')' tan B (2),

where B is an angle, then these two conies fulfil the required conditions. For
uniformity, we shall sometimes make use of the hyperbolic functions

cosfy8 = |(e/l + e-»), and sin/j£ = £ (<f-e~f) (3),

and we shall suppose /? and B so related, that

cos hft = sec B, whence sin hft = tan B } , .

sec hfi = cos B, tan lift = sin B \

The equations to a point in the hyperbola may then be written

x = bcoahp, y = 0, z = (c1 - ft1)* sin lift (5).

Construction of the Cyclide by Points. First Method.

Let P be a point on the ellipse, Q a point on the hyperbola, then

PQ = csecB-bcosa (6 ).

\.i\v take on PQ a point R, so that

/'/,' = ?• -b cos a (7).

or QR = r — c&ecB ( 8 ),

where r is a constant, then if a and B vary, R will give a system of points
on the cyclide (bcr).

For if P be fixed while Q varies, R will describe a circle, and if Q 1>«
fixed while P varies, R will describe another circle, and these circles cut at
right angles, and are both at right angles to PQR at their intersection, and

THE CYCLIDE.

147

since P and Q are any points on the conies, the whole system of circles will
form a cyclide.

The circle corresponding to P a fixed point on the ellipse, is in a plane
which cuts that of xz along the line

br
x = ~, 2/ = 0 (9),

and makes with it an angle

The circle corresponding to Q a fixed point in the hyperbola, is in a
plane which cuts that of xy along the line

* = f • * = 0 (10).

and makes with it an angle

Hence the planes of all the circles of either series pass through one of two
fixed lines, which are at right angles to each other, and at a minimum distance

The line of intersection of the planes of the two circles through the point
R will therefore pass through both the fixed lines at points S and T, where
the co-ordinates of S are

/*1* if — ft" l*

x = cr> y = (c °) iana> z = 0 (11),

b o

and those of T,

x = — , y = Q, z= — — — sinB (12),

c c

and it is easily shewn that

„„ 1 - T sec a

o^t o ,,Q\

TR = —r —P (13)"

1 — cos B
c

Hence, we deduce the following :

19—2

148 THE CYCLIDE.

Second Constntction by Points.

Draw the two fixed lines, and find points S and T given by equations (11)
iuul (12), then draw ST, and cut it in R, so that the ratio of the segments is
that given by equation (13). R will be a point on the cyclide.

This construction is very convenient for drawing any projection of the
cyclide, as the distances are measured along the projections of the fixed lines,
and the line ST can be divided in the required ratio by means of a ruler
and " sector," without making any marks on the paper, except the position of
the required point Jt.

In this way I have drawn stereoscopic diagrams of four varieties of the
cyclide, viewed from a position nearly in the line

x=y = z,
.shewing the circles corresponding to various values of a and B*.

On the. Forms of the Cyclide.

We shall suppose b and c to be given, and trace the effect of giving
different values to 7*. Since the cyclides corresponding to negative values of r
differ from those corresponding to equal positive values merely by having the

* Arote on a Real-Image Stereoscope. In ordinary stereoscopes the virtual images of two pictures are
Huper|>o»ed, and the observer, looking through two lenses, or prisms, or at two mirrors, sees the figure
;t|ij«rviitly behind the optical apparatus. In a stereoscope, which I have had made by Elliott Brothers,
the observer looks at a real image of the pictures, which appears in front of the instrument, and he is not
conscious of using any optical apparatus.

This stereoscope consists of a frame to support the double picture, which may be a common stereoscopic
•lide inverted. One foot from this a frame is placed, containing (tide by side two convex lenses of half a
foot focal length, and having their centres distant one and a quarter inches horizontally. One foot 1"
these U placed a convex lens of two-thirds of a foot focal length and three inches diameter.

The observer stands about two feet from the large lens, so that with the right eye he sees an image of
the left-hand picture, and with the left eye an image of the right-hand picture.

These images are formed by pencils which pass centrically through the two small lenses respectively,
so that they are free from distortion, and they appear to be nearly at the same distance as the large lens,
•o that the observer fixing his eyes on the frame of the large lens sees the combined figures at once.

The figures of the cyclide, though constructed for this stereoscope, may be used with an ordinary
stereoscope, or they may be united by squinting, which is a very effective method.

THE CYCLIDE. 149

positive and negative ends of the axis of x reversed, we need study the positive
values only.

(1) When r lies between zero and b, the section in the plane of the
ellipse consists of two circles, whose centres are the foci, and which intersect at
a point of the ellipse. The section in the plane of the hyperbola consists of
two circles exterior to each other, whose centres are the vertices of the ellipse.
The cyclide itself consists of two lobes exterior to each other, of which the
negative one is the largest, and increases with r, while the positive lobe
decreases. Each lobe terminates in two conical points, where it meets the other
lobe. The cone of contact at these singular points is a right cone, whose axis
is the tangent to the ellipse, and whose semi-vertical angle is

The whole figure resembles two pairs of horns, each pair joined together
by their bases, and the two pairs touching at the tips of the horns. Figure I.*
represents a cyclide of this kind. The 'continuous curves represent the lines
of curvature of both series. The dotted curves represent the ellipse and hyper-
bola through which the normals pass, and the dotted straight lines repre-
sent the axis of x, and the two straight lines through which the planes of
the circles pass.

(2) When r lies between b and c, the cyclide consists of a single sheet
in the form of a ring, the section of which is greatest on the negative side.
Figure II. represents a cyclide of this kind.

(3) When r is greater than c, the cyclide again consists of two sheets,
the one within the other, meeting each other in two Conical points which are'
situated on the positive branch of the hyperbola. The semi-vertical angle at
these points, is

There is also in all forms of the cyclide, a singular tangent plane which

touches the cyclide along a circle corresponding to B = ± - . Figure III. repre-
ia

* [Page 159].

|J(» THE CYCLIDE.

aenta such a cyclide. The outer sheet with its circles of contact and re-entering
conical points, and the inner spindle with its conical points meeting those of
tlu- outer sheet, have a certain resemblance to the outer and inner sheets

• •f FrvsMfl's Wave-Surface ; and, if we bear in mind that the wave-surface
has four such singular points while the cyclide has only two, we may find
Figure III. useful in forming an idea of the singular points of the wave-surface.

If we give to r all values from -f « to — o> , the cyclide assumes the
forms (3), (2), (1), (-1), (-2), (-3) in succession, and every point of space
w traversed four times by the surface. For when r is infinite, any given
point R is within the spindle or inner sheet of (3). As r diminishes, the
spindle contracts, and when r = c it vanishes; so that for a certain value, r,,
greater than c, the surface of the spindle passes through the point R.

At this instant the outer sheet of the cyclide is still beyond R, but as
r diminishes, the surface contracts, and finally vanishes when r = —b, before
which it must have passed through a value ?•,, for which the surface passes
through R. At this instant the surface may have the form either of the outer
sheet of (3), or of the ring cyclide of one sheet (2), or of the negative lobe

"I" (I)-

The positive lobe of (1) begins to appear when r becomes less than b, and

increases as ;• diminishes, till when r= — b it becomes a ring, and when r= — c
it becomes the outer sheet of the cyclide ( — 3).

This surface, therefore, for some value, rt, of r, passes through the point R.
This value r, is necessarily less than ?v

When r = — c the interior sheet of ( — 3) is developed, and increases Ln-

• Infinitely as r diminishes, so that for some value, rv of r, which is less than
<•„ the point R is on the surface of this interior sheet. We thus see that the
cyclide may be said to have four sheets, though not more than two can be
real at once. These four sheets touch at three conical points.

The first sheet, corresponding to r,, is the interior lobe of the cyclide (3),
;md always touches the second sheet at a conical point on the positive branch
of the hyperbola.

The second sheet, corresponding to rt, has three different forms, being either
the outer sheet of (3), the ring cyclide of one sheet (2), or the negative lobe
of (l). When the first sheet exists, it meets it at a conical point on the
|K»itive hyperbola, and when the third sheet exists, it meets it at a conical
point on the ellipse.

THE CYCLIDE.

The third sheet, corresponding to ra, has also three different forms. It
may be either the positive lobe of (l), the ring cyclide ( — 2), or the outer
sheet of ( — 3). In the first case it has a conical point on the ellipse where
it meets the second sheet. In the second case it has no conical point, and
in the third it meets the fourth sheet in a conical point on the negative
hyperbola.

The fourth sheet is the interior spindle of the cyclide ( — 3), and always
meets the third sheet at a conical point on the negative hyperbola.

Parabolic Cyclides.

When the values of b, c, r, and x are each increased by the same quan-
tity, and if this quantity is indefinitely increased, the two conies become in
the limit two parabolas in perpendicular planes, the focus of one being the vertex
of the other, and the cyclide becomes what we may call the parabolic cyclide.

When r lies between b and c, the cyclide consists of one infinite sheet,
lying entirely between the planes a; = 26 — r and x = 2c — r. The portions of space
on the positive and negative side of the sheet are linked together as the earth
and the air are linked together by a bridge, the earth, of which the bridge
forms part, embracing the air from below, and the air embracing the bridge
from above. In fact the earth and bridge form a ring of which one side is
much larger than the other.

A parabolic ring cyclide in which 2r = b + c, is represented in Figure IV.

When r does not lie between b and c, the cyclide consists of a lobe with
two conical points, and an infinite sheet with two conical points meeting those
of the lobe.

Surfaces of Revolution.

When 6 = 0, the cyclide is the surface formed by the revolution of a circle
of radius r about a line in its own plane distant c from the centre. If r is
less than c, the form is that of an anchor ring. If r is greater than c, the
surface consists of an outer and an inner sheet, meeting in two conical points.
When b = c, the cyclide resolves itself into two spheres, which touch externally
if )• is less than b, and internally if r is greater than b. When b = c = 0, the
two spheres become one.

152 THE CYCLIDE.

If the origin be transferred to a conical point, and if the dimensions of
the figure be then indefinitely increased, the cyclide becomes ultimately a right
cone, having the same conical angle as the original cyclide. If b = c, the cone
becomes a plane.

If b remains finite, while c, r, and x are each increased by the same
indefinitely great quantity, the cyclide ultimately becomes a right cylinder, whose
radius is r-c.

Inversion of the Cyclide.

Since every sphere, when inverted by means of the reciprocals of the radii
drawn to a fixed point, becomes another sphere, every cyclide similarly inverted
l»ecomes another cyclide. There is, however, a certain relation between the
parameters of the one cyclide and those of the other, namely

or

(15).

If the point of inversion be taken on either of the circles

c' = °> z = 0 (16),

or a? + zI-2^-&'-r'-»-c1 = 0, y = 0 (17),

c
crx

the cyclide will become a surface of revolution in which 6 = 0, and

r" c'-r5

if the point of inversion be on the first circle, or

if it be on the second.

When r is less than c, the first circle is real; and when r is greater
than 6, the second circle is real In the ring-cyclide r is between b and c,
and the cyclide can be transformed into an anchor ring in two different ways.

THE CYCLIDE. j 53

If the cyclide has conical points, and if one of them be made the point
of inversion, the cyclide becomes a right cone, whose semi-vertical angle is

(52 — •rM /yj_c2\i

-j— — J if r is less than b, or cos'1 (;?— n) if r is greater than 6.

If the point of inversion be at any other point of the surface, the cyclide
becomes a parabolic cyclide.

be
If the point of inversion be x = — , y = Q, z= 0, the cyclide is inverse to

itself.

On the Conjugate Isothermal Functions on the Cyclide.

DEFINITION. If on any surface two systems of curves be drawn, each
individual curve being defined by the value of a parameter corresponding to it,
and if the two systems of curves intersect everywhere at right angles, and if
the intercept of a curve of the second system between two consecutive curves
of the first system has the same ratio to the intercept of a curve of the first
system between two consecutive curves of the second, as the difference of the
parameters of the two curves of the first system has to the difference of the
parameters of the two curves of the second system, then the two systems of
curves are called conjugate isothermal lines, and the two parameters conjugate
isothermal functions. If the surface be now supposed to be a uniform con-
ducting lamina placed between non-conducting media, one set of these lines will
be isothermal for heat or equipotential for electricity, and the other set will
be lines of flow. (See Lame" on Isothermal Functions.)

This property of lines on a surface is not changed by inversion.

In the cyclide, we find the intercept ds of a line of curvature of the first
system is

, r — ccoshfi , „ ?ou , , .

£&»- —TO , (c--b-)*da ..................... (20),

c cos A/3 — o cos a v

and the intercept dst of a line of curvature of the second system is

(21).

c cos —

VOL. II. 20

154 THE CYCLIDE.

If now 8 be a function of a, and <f> of ft, such that

then 6 and <£ will be conjugate isothermal functions.

If 6 = k I- -and<£ = /tf -0 (22),

Jr-fecoso, Jr-ccoshft

the condition will be satisfied. If r is greater than 6, we find

tan . = ± — (23).

(r — b')* b — rcosa

It is more useful to have a expressed in terms of 6, thus :

0J-&*)»sin —
sin a —

rcos

-
C0sa= - V . .............................. (24).

If r is less than 6, we have only to write the hyperbolic functions of

a a

,. , ,,, instead of the circular functions of /•t_j-\\ » and to write (6s — j-3)* for

(,--&')'.

Similarly, we obtain for the relation between ft and <£, when r is greater
than c,

c + r cos h -r~-

r + c cos h

(25).

j-r —$-,
(r1 — c')*

When r is less than c we must substitute (c1 — r4)1 for (j-1 — c1)* and turn
the hyperbolic functions into circular functions.

THE CYCLIDE.

155

Having found these conjugate isothermal -functions we may deduce from
them any number of other pairs, as 01 and <£„ where

(26)-

On Confocal Cyclides.

A system of cyclides in which the focal ellipse and hyperbola remain the
same, while r has various values, may be called a confocal system. This system
of cyclides and the two systems of right cones which have their vertices in
one conic and pass through the other, form three systems of orthogonal surfaces,
and therefore intersect along their lines of curvature. By inversion we may get
three systems of cyclides intersecting orthogonally.

A system of confocal cyclides may also be considered as a system of wave
surfaces in an isotropic medium, corresponding to a pencil of rays, each ray of
which intersects the two focal conies. Each cyclide corresponds to a certain
value of r, which we may call the length of the ray of that cyclide.

Now let us consider the system of confocal conicoids, whose equation is of
the form

By putting p = c, we get the ellipse

I " _1 n O ff)Q\

<?*<? — ty~ * ''

By putting p = b, we get the hyperbola

I-^P"1^-0 (29)'

These two conies therefore belong to the system and may be called its focal
conies. If, with any point R for vertex, we draw cones through the ellipse and
hyperbola, these will be confocal cones, whose three axes are normal to the
three conicoids of the system which pass through R. The two cones will inter-
sect at right angles along four generating lines r1} r2, rt, rt> which are normal
to four cyclides passing through the point R.

20— '2

156 THE CYCLIDE.

The normal to the ellipsoid through R will be the real axis of the cone
which pane* through the ellipse, and will bisect the angle between r, and r,,
and also tliat between r, and r,. If the ellipsoid were reflective, a ray incident
in the direction r, would be reflected in the direction of r, reversed ; hence,
by the wave theory, r, + r, is constant for the ellipsoid. At the point of tin-
ellipsoid (p « constant) where it is cut by the axis of z,

r, = x + c,

So that the equation of the ellipsoid (r = constant) may be expressed in terms
of r, and r, thus :

2p ...................................... (30).

The normal to the ellipsoid also bisects the angle between r, and rit whence we
deduce another form of the equation of the same ellipsoid

r, + rt=-2p ..................................... (31).

Hence, the general relation among the values of r,

0 .............. . .................... (32).

The normal to the hyperboloid of one sheet (/* = constant) bisects the angle
between r, and r,, and also that between r, and rtt whence we obtain the
equations

............................ (33).

The normal to the hyperboloid of two sheets (v = constant) bisects the angles
between r, and r,, and between r, and rt, whence

(34).

These are the equations to the conicoids in terms of the four rays of the pencil.
The equations to the four cyclides in terms of elliptic co-ordinates are easily
deduced from them

(35).

THE CYCLIDE.

Since the quantities p> c, p, b, v, 0

are in descending order of magnitude, it is evident that

ri> P, ^, r3, -p, rt
are also in descending order of magnitude.

The general equation to the cyclide in elliptic co-ordinates is

(r-p-p. + v)(r-p + p.-v)(r + p-fJ,-v)(r + p + iJi + v) = () (36),

which may be expressed in Cartesian co-ordinates thus :

When b = c there are two focal points F and F' and the values of the
four rays are

rl= RF+c
r,= RF-c\

r,= -RF+c\ <38)'

rt=-RF-c.
The equation of the ellipsoid

2p = r1 + r1 = RF+RF' (39)

in this case expresses the property of the prolate spheroid, that the sum of the
distances of any point from the two foci is constant.

In like manner, the equation

rt + r, = 2v = RF' — RF (40)

expresses the property of the hyperboloid of revolution of two sheets, that the
difference of the focal distances is constant.

In order to extend a property analogous to this to the other conicoids, let
us conceive the following mechanical construction :

Suppose the focal ellipse and hyperbola represented by thin smooth wires,
and let an indefinite thin straight rod always rest against the two curves, and
let r be measured along the rod from a point fixed in the rod. Let a string
whose length is b + c be fastened at one end to the negative focus of the ellipse
and at the other to the point ( + b) of the rod, and let the string slide on the
ellipse at the same point as the rod rests on it. To keep the string always
tight let another equal string pass from the positive focus of the ellipse round

158 THE CYCLIDE.

the curre to the point (-&) of the rod. These strings will determine the point
of the rod which rests on any given point of the ellipse.

Let the rod also rest on the hyperbola, so that either the positive portion
of the rod rest* on the positive branch of the hyperbola, or the negative portion
of the rod reete on the negative branch.

Then the point r of the rod lies in the surface of the cyclide whose para-
meter ia r, and as the rod is made to slide on the ellipse and hyperbola, the
point r will explore the whole surface of the cyclide.

If we consider any point of space R, the rod will pass through it in four
different positions corresponding to the four intersections of the cones whose
vertex is R passing through the ellipse and hyperbola.

The first position, r,, corresponds to the first sheet of the cyclide wlm-li
passes through R, If we denote the intersection of the rod with the ellipse by
E, and its intersection with the positive and negative branches of the hyperbola
by +H and — H, then the order of the intersections will be in this case

E, +H, R.

The second position, r,, corresponds to the second sheet, and the order of
intersections is either

E, R, +H or -H, E, R.

The third position, rv corresponds to the third sheet, and the order of the
intersections is either

R, E, +H, or -//, R, E.

The fourth position, rt, corresponds to the fourth sheet, and the order of the
intersections is

R, -H, E,
the letters being always arranged in the order in which r increases.

The complete system of rays is an example of a linear congruence of the
fourth order.

Now if two rods, each fulfilling the above conditions, intersect at R in any
two of these four positions, and if a string of sufficient length be fastened to a
sufficiently distant negative point of the first rod, be passed round the point /.'.
and l)e fastened to a sufficiently distant negative point of the second rod, ami if

THE CYCLIDE. 159

the two rods be then moved always keeping the string tight at the point of
intersection R, then R will trace out a conicoid.

If the rods are in the first and second positions, or in the third and fourth,
the conicoid will be an ellipsoid.

If the rods are in the first and third positions, or in the second and fourth,
the conicoid will be an hyperboloid of one sheet.

If the rods are in the first and fourth positions, or in the second and third,
the conicoid will be an hyperboloid of two sheets.

In the parabolic confocal system, the fourth sheet of the cyclide is a plane,
and rt is parallel to the axis of x. Hence if rays parallel to the axis of a
paraboloid are reflected by the surface, they will all pass through the two focal
parabolas of the system, and the wave surface after reflexion will be a cyclide,
and if the rays are twice reflected, they will become again parallel to the axis.

[From the Edinburgh Royal Society Proceedings, Vol. VH.]

XXXVIII. On a Bow seen on the Surface of Ice.

UN the 26th of January, about noon, I observed the appearance of a
coloured bow on the frozen surface of the ditch which surrounds S. John's
C'/ollege, Cambridge. Its appearance and position seemed to correspond with those
of an ordinary primary rainbow. I at once made a rough measurement of the
angle on the board of a book which I had with me, and then borrowed from
Dr Parkinson, President of S. John's College, a sextant with which I found
that the angle between the bright red and the shadow of the large mirror
was 41*50', and that for bright blue 40° 30'. The angle for the extreme red
of the primary bow, as given in • Parkinson's Optics, is 42° 20', and that for
violet 40* 32'. The bows formed by ice crystals are seen on the same side as
the sun, and not on the opposite side. I suppose the bow which I saw to be
formed by small drops of water lying on the ice. If the lower part of each
drop were flattened, so as to bring the point at which the reflexion takes
place nearer to the points of incidence and emergence, the effect would be of
the same kind as that of a diminution of the index of refraction — that is, the
angle of the bow would be increased. How a drop of water can lie upon ice
without wetting it, and losing its shape altogether, I do not profess to explain.

Only a small part of the ice presented this appearance. It was best seen
when the incident and emergent rays were nearly equally inclined to the
horizontal. The ice was very thin, and I was not able to get near enough
tn the place where the bow appeared to see if the supposed water drops really
existed.

[From the Transactions of the Royal Society of Edinburgh, Vol. xxvi.]

XXXIX. On Reciprocal Figures, Frames, and Diagrams of Forces.
(Received 17th Dec. 1869; read 7th Feb. 1870.)

Two figures are reciprocal when the properties of the first relative to the
second are the same as those of the second relative to the first. Several kinds
of reciprocity are known to mathematicians, and the theories of Inverse Figures
and of Polar Reciprocals have been developed at great length, and have led to
remarkable results. I propose to investigate a different kind of geometrical
reciprocity, which is also capable of considerable development, and can be ap-
plied to the solution of mechanical problems.

A Frame may be defined geometrically as a system of straight lines con-
necting a number of points. In actual structures these lines are material pieces,
beams, rods, or wires, and may be straight or curved ; but the force by which
each piece resists any alteration of the distance between the points which it joins
acts in the straight line joining those points. Hence, in studying the equilibrium
of a frame, we may consider its different points as mutually acting on each
other with forces whose directions are those of the lines joining each pair of
points. When the forces acting between the two points tend to draw them
together, or to prevent them from separating, the action along the joining line
is called a Tension. When the forces tend to separate the points, or to keep
them apart, the action along the joining line is called a Pressure.

If we divide the piece joining the points by any imaginary section, the
resultant of the whole internal force acting between the parts thus divided will
be mechanically equivalent to the tension or pressure of the piece. Hence, in
order to exhibit the mechanical action of the frame in the most elementary
manner, we may draw it as a skeleton, in which the different points are joined

VOL. II. 21

162 RECIPROCAL FIGURES, FRAMES,

by straight lines, and we may indicate by numbers attached to these lines the
tensions or pressures in the corresponding pieces of the frame.

The diagram thus formed indicates the state of the frame in a way which
is geometrical as regards the position and direction of the forces, but arith-
metical as regards their magnitude.

But, by assuming that a line of a certain length shall represent a force
of a certain magnitude, we may represent every force completely by a line.
This is done in Elementary Statics, where we are told to draw a line from
the point of application of the force in the direction in which the force acts,
and to cut off as many units of length from the line as there are units of
force in the force, and finally to mark the end of the line with an arrow-
head, to shew that it is a force and not a piece of the frame, and that it
acts in that direction and not the opposite.

By proceeding in this way, we should get a system of arrow-headed forces
superposed on the skeleton of the frame, two equal and opposite arrows for
every piece of the frame.

To test the equilibrium of these forces at any point of concourse, we
should proceed by the construction of the parallelogram of forces, beginning
with two of the forces acting at the point, completing the parallelogram, and
drawing the diagonal, and combining this with the third force in the same way,
till, when all the forces had been combined, the resultant disappeared. We
should thus have to draw three new lines, one of which is an arrow, in taking
in each force after the first, leaving at last not only a great number of useless
lines, but a number of new arrows, not belonging to the system of forces, and
only confusing to any one wishing to verify the process.

To simplify this process, we are told to construct the Polygon of Forces,
by drawing in succession lines parallel and proportional to the different forces,
each line beginning at the extremity of the last. If the forces acting at the
point are in equilibrium, the polygon formed in this way will be a closed one.

Here we have for the first time a true Diagram of Forces, in which every
force is not only represented in magnitude and direction by a straight line,
but the equilibrium of the forces is manifest by inspection, for we have only
to examine whether the polygon is closed or not. To secure this advantage,
however, we have given up the attempt to indicate the position of the force,
for the sides of the polygon do not pass through one point as the forces do.
We must, therefore, give up the plan of representing the frame and its forces

AND DIAGRAMS OF FORCES. 163

in one diagram, and draw one diagram of the frame and a separate diagram
of the forces. By this method we shall not only avoid confusion, but we shall
greatly simplify mechanical calculations, by reducing them to operations with
the parallel ruler, in which no useless lines are drawn, but every line repre-
sents an actual force.

A Diagram of Forces is a figure, every line of which represents in mag-
nitude and direction the force acting along a piece of the frame.

To express the relation between the diagram of the frame and the dia-
gram of forces, the lines of the frame should each be indicated by a symbol,
and the corresponding lines of the diagram of forces should be indicated by
the same symbol, accented if necessary.

We have supposed the corresponding lines to be parallel, and it is neces-
sary that they should be parallel when the frame is not in one plane ; but
if all the pieces of the frame are parallel to one plane, we may turn one of
the diagrams round a right angle, and then every line will be perpendicular
to the corresponding line.

If any number of lines meet at the same point in the frame, the corre-
sponding lines in the diagram of forces form a closed polygon.

It is possible, in certain cases, to draw the diagram of forces so that if
any number of lines meet in a point in the diagram of forces, the corre-
sponding lines in the frame form a closed polygon.

In such cases, the two diagrams are said to be reciprocal in the sense in
which we use it in this paper. If either diagram be taken as representing the
frame, the lines of the other diagram will represent a system of forces which,
if applied along the corresponding pieces of the frame, will keep it in equi-
librium.

The properties of the " triangle " and " polygon " of forces have been long
known, and a " diagram " of forces has been used in the case of the " funi-
cular polygon," but I am not aware of any more general statement of the
method of drawing diagrams of forces before Professor Rankine applied it to
frames, roofs, &c., in his Applied Mechanics, p. 137, &c. The "polyhedron of
forces," or the proposition that forces acting on a point perpendicular and pro-
portional to the areas of the faces of a polyhedron are in equilibrium, has, I
believe, been enunciated independently at various times, but the application of
this principle to the construction of a diagram of forces in three dimensions
was first made by Professor Rankine in the Philosophical Magazine, Feb. 1864.

21—2

164 RECIPROCAL FIGURES, FRAMES,

In the Philosophical Magazine for April 1864, I stated some of the properties
of reciprocal figures, and the conditions of their existence, and shewed that
any plane rectilinear figure which is a perspective representation of a closed
polyhedron with plane faces has a reciprocal figure. In Sept. 1867, I communi-
cated to the British Association a method of drawing the reciprocal figure,
founded on the theory of reciprocal polars*.

I have since found that the construction of diagrams of forces in which
each force is represented by one line, had been independently discovered by
Mr W. P. Taylor, and had been used by him as a practical method of deter-
mining the forces acting in frames for several years before I had taught it in
King's College, or even studied it myself. I understand that he is preparing
a statement of the application of the method to various kinds of structures in
detail, so that it can be made use of by any one who is able to draw one
line parallel to another.

Professor Fleeming Jenkin, in a paper recently published by the Society,
has fully explained the application of the method to the most important cases
occurring in practice.

In the present paper I propose, first, to consider plane diagrams of frames
and of forces in an elementary way, as a practical method of solving questions
about the stresses in actual frameworks, without the use of long calculations.

I shall then discuss the subject in a theoretical point of view, and give a
method of defining reciprocal diagrams analytically, which is applicable to
figures either of two or of three dimensions.

Lastly, I shall extend the method to the investigation of the state of
stress in a continuous body, and shall point out the nature of the function of
stress first discovered by the Astronomer Royal for stresses in two dimensions,
extending the use of such functions to stresses in three dimensions.

On Reciprocal Plane Rectilinear Figures,

DEFINITION. — Two plane rectilinear figures are reciprocal when they consist
of an equal number of straight lines, so that corresponding lines in the two
figures are at right angles, and corresponding lines which meet in a point in
the one figure form a closed polygon in the other.

• [See pp. 169 and 188].

AND DIAGRAMS OP FORCES. 165

Note. — It is often convenient to turn one of the figures round in its own
plane 90°. Corresponding lines are then parallel to each other, and this is
sometimes more convenient in comparing the diagrams by the eye.

Since every polygon in the one figure has three or more sides, every
point in the other figure must have three or more lines meeting in it. Since
every line in the one figure has two, and only two, extremities, every line in
the other figure must be a side of two, and only two, polygons. If either of
these figures be taken to represent the pieces of a frame, the other will repre-
sent a system of forces such that, these forces being applied as tensions or
pressures along the corresponding pieces of the frame, every point of the frame
will be in equilibrium.

The simplest example is that of a triangular frame without weight, ABC,
jointed at the angles, and acted on by three forces, P, Q, R, applied at the
angles. The directions of these three forces must meet in a point, if the frame
is in equilibrium. We shall denote the lines of the figure by capital letters,
and those of the reciprocal figure by the corresponding small letters ; we shall
denote points by the lines which meet in them, and polygons by the lines
which bound them.

Here, then, are three lines, A, B, C, forming a triangle, and three other
lines, P, Q, R, drawn from the angles and meeting in a point. Of these
forces let that along P be given. Draw the first line p of the reciprocal
diagram parallel to P, and of a length representing, on any convenient scale,
the force along P. The forces along P, Q, R are in equilibrium, therefore, if
from one extremity of p we draw q parallel to Q, and from the other
extremity r parallel to R, so as to form a triangle pqr, then q and r will
represent on the same scale the forces along Q and R.

To determine whether these forces are tensions or pressures, make a point
travel along p in the direction in which the force in P acts on the point of

166 RECIPROCAL FIGURES, FRAMES,

concourse of PQK, and let the point travel in the same direction round the
polygon pqr. Then, the direction in which the point travels along any side
of the polygon will be the direction in which the force acts along the corre-
sponding piece of the frame on the point of concourse. If it acts from the
point of concourse, the force is a tension ; if towards it, it is a pressure.

The other extremity of P meets B and C, and the forces along these
three pieces are in equilibrium. Henoe, if we draw a triangle, having /> for
one side and lines parallel to B and C for the others, the sides of this triangle
will represent the three forces.

Such a triangle may be described on either side of p, the two together
would form a parallelogram of forces ; but the theory of reciprocal figures indi-
cates that only one of these triangles forms part of the diagram of forces.

The rule for such cases is as follows : — Of the two extremities of p, one
corresponds to the closed figure PRB, and the other to the closed figure PQC,
these being the polygons of which P is a side in the first figure.

We must, therefore, draw b parallel to B from the intersection of p and r,
and not from the other extremity, and we must draw c parallel to C from
the intersection of p and q.

We have now a second triangle, pbc, corresponding to the forces acting at
the point of concourse of P, B, C. To determine whether these forces are
tensions or pressures, we must make a point travel round pbc, so that its
course along p is in the opposite direction to its course round pqr, because the
piece P acts on the points PBC and PQR with equal and opposite forces.

If we now consider the equilibrium of the point of concourse of QC and A,
we shall find that we have determined two of these forces by the lines q and c,
and that the third force must be represented by the line a which completes the
triangle '/•••'.

We have now constructed a complete diagram of forces, in which each
force is represented by a single line, and in which the equilibrium of the forces
meeting at any point is expressed visibly by the corresponding lines in the
other figure forming a closed polygon.

There are in this figure six lines, having four points of concourse, and
forming four triangles. To determine the direction of the force along a given
line at any point of concourse, we must make a point travel round the cor-
responding polygon in the other figure in a direction which is positive with
respect to that polygon. For this purpose it is desirable to name the polygons

AND DIAGRAMS OF FOKCES. 167

in a determinate order of their sides, so arranged that, when we arrive at the
same side in naming the two polygons which it divides, we travel along it in
opposite directions. For instance, if pqr be one of the polygons, the others are
pbc, qca, rob.

Note. — It may be observed, that after drawing the lines p, q, r, b, c with
the parallel ruler, the line a was drawn by joining the points of concourse of
q, r and b, c ; but, since it represents the force in A, a is parallel to A.
Hence the following geometrical theorem : —

If the lines PQR, drawn from the angles of the triangle ABC, meet in a
point, then if pqr be a triangle with its corresponding sides parallel to P, Q, R,
and if a, b, c be drawn from its corresponding angles parallel to A, B, C, the
lines a, b, c will meet in a point.

A geometrical proof of this is easily obtained by finding the centres of the
four circles circumscribing the triangles ABC, AQR, BRP, CPQ, and joining
the four centres thus found by six lines.

These lines meet in the four centres, and are perpendicular to the six
lines, A, B, C; P, Q, R; but by turning them round 90° they become parallel
to the corresponding lines in the original figure.

The diagram formed in this way is definite in size and position, but any
figure similar to it is a reciprocal diagram to the original figure. I have
explained the construction of this, the simplest diagram of forces, more at
length, as I wish to shew how, after the first line is drawn and its extremities
fixed on, every other line is drawn in a perfectly definite position by means of
the parallel ruler.

In any complete diagram of forces, those forces which act at a given point
in the frame form a closed polygon. Hence, there will be as many closed
polygons in the diagram as there are points in the frame. Also, since each
piece of the frame acts with equal and opposite forces on the two points which
form its extremities, the force in the diagram will be a side of two different
polygons. These polygons might be drawn in any positions relatively to each
other ; but, in the diagrams here considered, they are placed so that each force
is represented by one line, which forms the boundary between the two polygons
to which it belongs.

If we regard the polygons as surfaces, rather than as mere outlines, every
polygon will be bounded at every point of its outline by other polygons, so

],,.; RECIPROCAL FIGURES, FRAMES,

that the whole assemblage of polygons will form a continuous surface, which
muat either be an infinite surface or a closed surface.

The diagram cannot be infinite, because it is made up of a finite number
of finite lines representing finite forces. It must, therefore, be a closed surface
returning on itself, in such a way that every point in the plane of the diagram
either does not belong to the diagram at all, or belongs to an even number
of sheets of the diagram.

Any system of polygons, which are in contact with each other externally,
may be regarded as a sheet of the diagram. When two polygons are on the
same side of the line, which is common to them, that line forms part of the
common boundary of two sheets of the diagram. If we reckon those areas
positive, the boundary of which is traced in the direction of positive rotation
round the area, then all the polygons in each sheet will be of the same sign as
the sheet, but those sheets which have a common boundary will be of opposite
sign. At every point in the diagram there will be the same number of positive
as of negative sheets, and the whole area of the positive sheets will be equal to
that of the negative sheets.

The diagram, therefore, may be considered as a plane projection of a closed
polyhedron, the faces of the polyhedron being surfaces bounded by rectilinear
polygons, which may or may not, as far as we yet know, lie each in one plane.

Let us next consider the plane projection of a given closed polyhedron.
If any of the faces of this polyhedron are not plane, we may, by drawing
additional lines, substitute for that face a system of triangles, each of which is
necessarily in a plane. We may, therefore, consider the polyhedron as bounded
by plane faces. Every angular point of this polyhedron will be defined by its
projection on the plane and its height above it.

Let us now take a fixed point, which we shall call the origin, and draw
from it a perpendicular to the plane. We shall call this line the axis. If we
then draw from the origin a line perpendicular to one of the faces of the poly-
hedron, it will cut the plane at a point which may be said to correspond to the
projection of that face. From this point draw a line perpendicular to the plane,
and take on this line a point whose distance from the plane is equal to that
of the intersection of the axis with the face of the polyhedron produced, but on
the other side of the plane. This point in space will correspond to the face of
the polyhedron. By repeating this process for every face of the polyhedron, we
shall find for every face a corresponding point with its projection on the plane.

AND DIAGRAMS OF FORCES. 169

To every edge of the polyhedron will correspond the line which joins the
points corresponding to the two faces which meet in that edge. Each of these
lines is perpendicular to the projection of the other; for the perpendiculars
from the origin to the two faces, lie in a plane perpendicular to the edge in
which they meet, and the projection of the line corresponding to the edge is
the intersection of this plane with the plane of projection. Hence, the edge is
perpendicular to the projection of the corresponding line. The projection of the
edge is therefore perpendicular to the projection of the corresponding line, and
therefore to the corresponding line itself. In this way we may draw a diagram
on the plane of projection, every line of which is perpendicular to the corre-
sponding line in the original figure, and so that lines which meet in a point
in the one figure form a closed polygon in the other.

If, in a system of rectangular co-ordinates, we make z = 0 the plane of pro-
jection, and x = 0, y = Q, z = c the fixed point, then if the equation of a plane be

z = Ax + By+C,
the co-ordinates of the corresponding point will be

£=cA, v = cB, £=-C,

and we may write the equation

c(z + £)=xt; + yr}.

If we suppose £ 17, £ given as the co-ordinates of a point, then this equa-
tion, considering x, y, z as variable, is the equation of a plane corresponding
to the point.

If we suppose x, y, z the co-ordinates of a point, and £ 77, £ as variable,
the equation will be that of a plane corresponding to that point.

Hence, if a plane passes through the point xyz, the point corresponding
to this plane lies in the plane corresponding to the point xyz.

These points and planes are reciprocally polar in the ordinary sense with
respect to the paraboloid of revolution

We have thus arrived at a construction for reciprocal diagrams by con-
sidering each as a plane projection of a plane-sided polyhedron, these polyhedra
being reciprocal to one another, in the geometrical sense, with respect to a cer-
tain paraboloid of revolution.

VOL. II. 22

170 RECIPROCAL FIGURES, FRAMES,

Each of the diagrams must fulfil the conditions of being a plane projection
of a plane-sided polyhedron, for if any of the sides of the polyhedron of which
it is the projection are not plane, there will be as many points corresponding
to that side as there are different planes passing through three points of the
side, and the other diagram will be indefinite.

Relation between the Number of Edges, Summits, and Faces of Polyhedra.

It is manifest that after a closed surface has been divided into separate
faces by lines drawn upon it, every new line drawn from a point in the system.
either introduces one new point into the system, or divides a face into two
l>arts, according as it is drawn to an isolated point, or to a point already
connected with the system. Hence the sum of points and faces is increased
by one for every new line. If the closed surface is acyclic, or simply connected*,
like that of a solid body without any passage through it, then, if from any
point we draw a closed curve on the surface, we divide the surface into two
faces. We have here one line, one point, and two faces. Hence, if c be the
number of lines, s the number of points, and / the number of faces, then in
general

e — s —f— m

* See Riemann, Crelle's Journal, 1857, Lehrsdlte aus der analysis situs, for space of two dimen-
sions; .also Cayley on the Partitions of a Close, Phil. Mag. 1861; Helmholtz, Crelle's Journal, 1858,
Wirbclbfwguiig, for the application of the idea of multiple continuity to space of three dimensions ;
J. B. Listing, Gottingen Trans., 1861, Der Censut R&umlicher Complexe, a complete treatise on the
Kubject of Cyclosis and Periphraxy.

On the importance of this subject see Gauss, Werke, v. 605, "Von der Geometria Situs ilir
Leibnitz abnte und in die nur einem Paar Geometern (Euler und Vandermonde) einen schwachen
Blick ru thun vergbnnt war, wissen und haben wir nach anderthalbhundert Jahren noch nicht vicl
nifhr wie nichtfl."

Note added March 14, 1870. — Since this was written, I have seen Listing's Census. In his
notation, the surface of an w-ly connected body (a body with n - 1 holes through it) is (2/t-ii)
cyclic. If 2n — 2 = A', expresses the degree of cyclosis, then Listing's general equation is —

where « is the number of points, e the number of lines, A7", the number of endless curves, f the number
of faces, A't the number of degrees of cyclosis of the faces, wt the number of periphractic or closed
faces, v the number of regions of space, Kt the number of degrees of cyclosis, vr3 their number of
degrees of periphraxy or the number of regions which they completely surround, and to is to be put
• 1 or — 0, according as the system does or does not extend to infinity.

AND DIAGRAMS OF FORCES. 171

when m remains constant, however many lines be drawn. But in the case of a
simple closed surface

Mi-— 2,

If the closed surface is doubly connected, like that of a solid body with a
hole through it, then if we draw one closed curve round the hole, and another
closed curve through the hole, and round one side of the body, we shall have
e = 2, s=l, f—l, so that m = 0. If the surface is n-\y connected, like that of
a solid with n — 1 holes through it, then we may draw n closed curves round
the n — 1 holes and the outside of the body, and n — I other closed curves each
through a hole and round the outside of the body.

We shall then have 4(»— 1) segments of curves terminating in 2(n— 1)
points and dividing the surface into two faces, so that e = 4(n — 1), s = 2(n— 1),

and /= 2, and

e — s— f=2n — 4,

and this is the general relation between the edges, summits, and faces of a
polyhedron whose surface is n-ly connected.

The plane reciprocal diagrams, considered as plane projections of such poly-
hedra, have the same relation between the numbers of their lines, points, and
polygons. It is manifest that since

e^ = ea s, =/„ and /I = s2 ,
where the suffixes refer to the first and second diagrams respectively

nl = na,
or the two diagrams are connected to the same degree.

On the Degrees of Freedom and Constraint of Frames.

To determine the positions of s points in space, with reference to a given
origin and given axes, 3s data are required ; but since the position of the origin
and axes involve 6 data, the number of data required to determine the relative
position of s points is 3s — 6.

If, therefore, the lengths of 3s -6 lines joining selected pairs of a system
of s points be given, and if these lengths are all independent of each other, then
the distances between any other pair of points will be determinate, and the
system will be rigidly connected.

22—2

172 RECIPROCAL FIGURES, FRAMES,

If, however, the lines are so chosen that those which join pairs of points of
a system of «' of the points are more than 3*' -6 in number, the lengths of
these lines will not be independent of each other, and the lines of this partial
system will only give 3s' — 6 independent data to determine the complete system.

In a system of s points joined by e lines, there will in general be 3s — 6 — e
»» degrees of freedom, provided that in every partial system of s' points joined
by t lines, and having in itself p' degrees of freedom, p' is not negative. If in
any such system p is negative, we may put q = —p, and call q the number of
lingr Bi ii i of constraint, and there will be q equations connecting the lengths of
the lines ; and if the system is a material one, the stress along each piece will
be a function of q independent variables. Such a system may be said to have
q degrees of constraint. If p' is negative in any partial system, then the de-
grees of freedom of the complete system are p—p', where p and p' are got
from the number of points and lines in the complete and partial systems. If .«
points are connected by e lines, so as to form a polyhedron of f faces, enclosing
a space n times connected, and if each of the faces has m sides, then

mf=2e.

We have also e — s— f=2n — 4,

and 3s-e=p + Q,

(c \
2 -- }e.
m/

If all the faces of the polyhedron are triangles, m = 3, and we have

If n=l, or in the case of a simply connected polyhedron with triangular
faces, p = o, that is to say, such a figure is a rigid system, which would be no
longer rigid if any one of its lines were wanting. In such a figure, if made of
material rods forming a closed web of triangles, the tensions and pressures in
the rods would be completely determined by the external forces applied to the
figure, and if there were no external force, there would be no stress in the rods.

In a closed surface of any kind, if we cover the surface* with a system
of curves which do not intersect each other, and if we draw another system

* On the Bending of Surfaces, by J. Clerk Maxwell. Cambridge Transactions, 1856. [Vol. i. p. 80.]

AND DIAGRAMS OP FOKCES. 173

intersecting these, and a third system passing diagonally through the intersec-
tions of the other two, the whole surface will be covered with small curvilinear
triangles, and if we now substitute for the surface a system of rectilinear
triangles having the same angular points, we shall have a polyhedron with
triangular faces differing infinitely little from the surface, and such that the
length of any line on the surface differs infinitely little from that of the corre-
sponding line on the polyhedron. We may, therefore, in all questions about the
transformation of surfaces by bending, substitute for them such polyhedra with
triangular faces.

We thus find with respect to a simply connected closed inextensible surface
— 1st, That it is of invariable form*; 2nd, That the stresses in the surface
depend entirely on the external applied forcesf ; 3rd, That if there is no external
force, there is no stress in the surface.

In the limiting case of the curved surface, however, a kind of deformation
is possible, which is not possible in the case of the polyhedron. Let us suppose
that in some way a dimple has been formed on a convexo-convex part of the
surface, so that the edge of the dimple is a plane closed curve, and the
dimpled part is the reflexion in this plane of the original form of the surface.
Then the length of any line drawn on the surface will remain unchanged.

Now let the dimple be gradually enlarged, so that its edge continually
changes its position. Every line on the surface will still remain of the same
length during the whole process, so that the process is possible in the case of
an inextensible surface. In this way such a surface may be gradually turned
outside in, and since the dimple may be formed from a mere point, a pressure
applied at a single point on the outside of an inextensible surface will not be
resisted, but will form a dimple which will increase till one part of the surface
comes in contact with another.

In the case of closed surfaces doubly connected, p= -6, that is, such sur-
faces are not only rigid, but are capable of internal stress, independent of
external forces, and the expression of this stress depends on six independent
variables.

* This has been shewn by Professor Jellett, Trans. R.I. A., Vol. XXH. p. 377.
t On the Equilibrium of a Spherical Envelope, by J. C. Maxwell. Quarterly Journal of Mathe-
matics, 1867. [VoL ii. p. 86.]

174 RECIPROCAL FIGURES, FRAMES,

In a polyhedron with triangular faces, if a number of the edges be taken
away so as to form a hole with el sides, the number of degrees of freedom is

p = e, — Gn + 3.

Hence, in order to make an n-ly connected polyhedron simply rigid without
stress, we may cut out the edges till we have formed a hole having 6>/-:i
edges. The system will then be free from stress, but if any more edges be
removed, the system will no longer be rigid.

Since in the limiting case of the inextensible surface, the smallest hole
may be regarded as having an infinite number of sides, the smallest hole made
in a closed inextensible surface connected to any degree will destroy its rigidity.
Its flexibility, however, may be confined within very narrow limits.

In the case of a plane frame of * points, we have 2s data required to
determine the points with reference to a given origin and axes ; but since 3
arbitrary data are involved in the choice of origin and axis, the number of
data required to determine the relative position of * points in a plane is 2* — 3.

If we know the lengths of e lines joining certain pairs of these points,
then in general the number of degrees of freedom of the frame will be

p = 2s-e-3.

If, however, in any partial system of s' points connected by e' lines, the
quantity p' = 2s' — e' — 3 be negative, or in other words, if a part of the frame
be self-strained, this partial system will contribute only 2s' — 3 equations inde-
pendent of each other to the complete system, and the whole frame will have
p—p degrees of freedom.

In a plane frame, consisting of a single sheet, every element of which is
triangular, and in which the pieces form three systems of continuous lines, as
at p. 173, if the frame contains e pieces connecting s points, s' of which arc
on the circumference of the frame and s, in the interior, then

3s - s' = e + 3.
Hence p= —(« — *')= — s,,

a negative quantity, or such a frame is necessarily stiff; and if any of the
pouits are in the interior of the frame, the frame has as many degrees of
constraint as there are interior points — that is, the stresses in each piece will he

AND DIAGRAMS OF FORCES. 175

functions of s, variables, and *, pieces may be removed from the frame without
rendering it loose.

If there are n holes in the frame, so that s' points lie on the circum-
ference of the frame or on those of the holes, and s1 points lie in the interior,
the degree of stiffness will be

—p = sx + 3n.

If a plane frame be a projection of a polyhedron of / faces, each of m sides,
and enclosing a space n times connected, then

mf= 2e,
e — s— f=2n — 4,

whence p = 5 — 4n + ( 1 -- e.

If all the faces are quadrilaterals m = 4 and p = 5 — 4n, or a plane frame which
is the projection of a closed polyhedron with quadrilateral faces, has one degree
of freedom if the polyhedron is simply connected, as in the case of the pro-
jection of the solid bounded by six quadrilaterals, but if the polyhedron be
doubly connected, the frame formed by its plane projection will have three
degrees of stiffness. (See Diagram II.)

THEOREM. — If every one of a system of points in a plane is in equilibrium
under the action of tensions and pressures acting along the lines joining the
points, then if we substitute for each point a small smooth ring through which
smooth thin rods of indefinite length corresponding to the lines are compelled
to pass, then, if to each rod be applied a couple in the plane, whose moment
is equal to the product of the length of the rod between the points multiplied
by the tension or pressure in the former case, and tends to turn the rod in
the positive or the negative direction, according as the force was a tension or
a pressure, then every one of the system of rings will be in equilibrium. For
each ring is acted on by a system of forces equal to the tensions and pres-
sures in the former case, each to each, the whole system being turned round
a right angle, and therefore the equilibrium of each point is undisturbed.

THEOREM. — In any system of points in equilibrium in a plane under the
action of repulsions and attractions, the sum of the products of each attraction

176 RECIPROCAL FIGURES, FRAMES,

multiplied by the distance of the points between which it acts, is equal to
the sum of the products of the repulsions multiplied each by the distance of
the points between which it acts.

For since each point is in equilibrium under the action of a system of
attractions and repulsions in one plane, it will remain in equilibrium if the
system of forces is turned through a right angle in the positive direction. If
this operation is performed on the systems of forces acting on all the points,
then at the extremities of each line joining two points we have two equal
forces at right angles to that line and acting in opposite directions, fornnn<:
a couple whose magnitude is the product of the force between the points and
their distance, and whose direction is positive if the force be repulsive, and
negative if it be attractive. Now since every point is in equilibrium these two
systems of couples are in equilibrium, or the sum of the positive couples is
equal to that of the negative couples, which proves the theorem.

In a plane frame, loaded with weights in any manner, and supported by
vertical thrusts, each weight must be regarded as attracted towards a horizontal
base line, and each support of the frame as repelled from that line. Hence
the following rule:

Multiply each load by the height of the point at which it acts, and each
tension by the length of the piece on which it acts, and add all these
products together.

Then multiply the vertical pressures on the supports of the frame each by
the height at which it acts, and each pressure by the length of the piece on
which it acts, and add the products together. This sum will be equal to the
former sum.

If the thrusts which support the frame are not vertical, their horizontal
components must be treated as tensions or pressures borne by the foundations
of the structure, or by the earth itself.

The importance of this theorem to the engineer arises from the circum-
stance that the strength of a piece is in general proportional to its section, so
that if the strength of each piece is proportional to the stress which it has
to bear, its weight will be proportional to the product of the stress multiplied
by the length of the piece. Hence these sums of products give an estimate of
the total quantity of material which must be used in sustaining tension and
pressure respectively.

AND DIAGRAMS OF FORCES. 177

The following method of demonstrating this theorem does not require the
consideration of couples, and is applicable to frames in three dimensions.

Let the system of points be caused to contract, always remaining similar
to its original form, and with its pieces similarly situated, and let the same
forces continue to act upon it during this operation, so that every point is
always in equilibrium under the same system of forces, and therefore no work
is done by the system of forces as a whole.

Let the contraction proceed till the system is reduced to a point. Then
the work done by each tension is equal to the product of that tension by the
distance through which it has acted, namely, the original distance between the
points. Also the work spent in overcoming each pressure is the product of
that pressure by the original distance of the points between which it acts ; and
since no work is gained or lost on the whole, the sum of the first set of
products must be equal to the sum of the second set. In this demonstration
it is not necessary to suppose the points all in one plane. This demonstration
is mathematically equivalent to the following algebraical proof: —

Let the co-ordinates of the n different points of the system be x^zu
X3jfa xpypzp, &c., and let the force between any two points p, q, be Ppq,
and their distance rm, and let it be reckoned positive when it is a pressure,
and negative when it is a tension, then the equation of equilibrium of any
point p with respect to forces parallel to x is

(Xp-Xj -& + (xp-Xa) -^-f&C. + (xp-Xq)-^ + &C. = 0,
rpi rpt rpq

or generally, giving t all values from 1 to n,

Multiply this equation by xf. There are n such equations, so that if each is
multiplied by its proper co-ordinate and the sum taken, we get

P

and adding the corresponding equations in y and z, we get

1 1
which is the algebraic expression of the theorem.

VOL. ii. 23

178 RECIPROCAL FIGURES, FRAMES,

GENERAL THEORY OF DIAGRAMS OF STRESS IN THREE DIMENSIONS.

./ Method of Representing Stress in a Body.

DEFINITION*. A diagram of stress is a figure having such a relation to a
body under the action of internal forces, that if a surface A, limited by a
closed curve, is drawn in the body, and if the corresponding limited surface a
be drawn in the diagram of stress, then the resultant of the actual internal
forces on the positive side of the surface A in the body is equal and parallel
to the resultant of a uniform normal pressure p acting on the positive side of
the surface a in the diagram of stress.

Let x, y, z be the co-ordinates of any point in the body, £, 17, £ those of
the corresponding point in the diagram of stress, then £ 77, £ are functions of
x, y, z, the nature of which we have to ascertain, so that the internal forces
in the body may be in equilibrium. For the present we suppose no external
forces, such as gravity, to act on the particles of the body. We shall consider
such forces afterwards.

THEOREM 1. — If any closed surface is described in the body, and if the stress
on any element of that surface is equal and parallel to the pressure on the
corresponding element of surface in the diagram of stress, then the resultant stress
on the whole closed surface will vanish ; for the corresponding surface in the
diagram of stress is a closed surface, and the resultant of a uniform normal
pressure p on every element of a closed surface is zero by hydrostatics.

It does not, however, follow that the portion of the body within the
closed surface is in equilibrium, for the stress on its surface may have a re-
sultant moment.

THEOREM 2. — To ensure equilibrium of every part of the body, it is neces-
sary and sufficient that

f_dF _dF dF

*~dx' ^~dy' 4~dz'

where F is any function of x, y, and z.

Let us consider the elementary area in the body dydz. The stress acting
on this area will be a force equal and parallel to the resultant of a pressure p
acting on the corresponding element of area in the diagram of stress. Resolving

AND DIAGRAMS OF FORCES. 179

this pressure in the directions of the co-ordinate axes, we find the three com-
ponents of stress on dydz, which we may call p^dydz, p^dydz, and pmdydz,
each equal to p multiplied by the area of the projection of the corresponding
element of the diagram of stress on the three co-ordinate planes. Now, the
projection on the plane yz, is

dr)

Hence we find for the component of stress in the direction of x

Pxx~P \dydz dzdy)'
which we may write for brevity at present

Pzx=pJ(n, £; y, z).

Similarly,

P*y=PJ(t» £"> y> «), Px*=pJ(£, -n; y, z).

In the same way, we may find the components of stress on the areas
dzdx and dxdy —

; z> x)> PW

Now, consider the equilibrium of the parallelepiped dxdydz, with respect
to the moment of the tangential stresses about its axes.

The moments of the forces tending to turn this elementary parallelepiped
about the axis of x are

dzdxp^ . dy — dxdypzy . dz.

To ensure equilibrium as respects rotation about the axis of x, we must have

P*-fir

Similarly, for the moments about the axes of y and 2, we obtain the equa-

tions

Pzx=px* and
Now, let us assume for the present

23—2

180 RECIPROCAL FIGURES, FRAMES,

Then the equation ;>„-/>* becomes

d(dr,\ n(dldt dl
-3xd;)-p \dxdy Ty

or

Similarly, from the two other equations of equilibrium we should find

From these three equations it follows that

Ct = 0, C, = 0, C, = 0.

drl_dC <K_dj[ dj[_d7L
dz~dy' dx~dz' dy dx'

and (dx + rjdy + [dz is a complete differential of some function, F, of x, y and z,
whence it follows that

d_F _dF dF

*~dx' r}~ dy' 4~<fe '

F may be called the function of stress, because when it is known, the diagram
of stress may be formed, and the components of stress calculated. The form
of the function F is limited only by the conditions to be fulfilled at the
bounding surface of the body.

The six components of stress expressed in terms of F are

\ (#Fd*F

} ' Pm =P \ dz> dx* ~

_ _

~ ~P {dtf ~3£ ~ (dydz) } ' Pm =P \ dz> dx* ~ (dzdx) }'Pa~P\dy? d'/ (dxdy

(d'F d'F d'F d'F \ (d'F d'F d'F d'F \

*j — ** i __ _____ ___ 1 if\ — tin i rirT I

\dzdxdxdy dx* dydz/' "" "\dxdydydz dy* dzdx/'

(d'F d'F d'F d'F\
P*»~P (dydz dzdx dz' dxdy) '

If -f- = z, F becomes Airy's function of stress in two dimensions, and we have

d'F cTF d'F

The system of stress in three dimensions deduced in this way from any
function, F, satisfies the equations of equilibrium of internal stress. It is not,

AND DIAGRAMS OF FORCES. 181

however, a general solution of these equations, as may be easily seen by taking
the case in which p^ and pyz are both zero at all points. In this case, since
there is no tangential action in planes parallel to xy, the stresses pm> p,^ and
pm in each stratum must separately fulfil the conditions of equilibrium,

d d d d

The complete solution of these equations is, as we have seen,

_df df _d*f

~ Pxy~ dxdy' Pm~

where f is any function of x and y, the form of which may be different for
every different value of z, so that we may regard f as a perfectly general
function of x, y, and z.

Again, if we consider a cylindrical portion of the body with its generating
lines parallel to z, we shall see that there is no tangential action parallel to z
between this cylinder and the rest of the body. Hence the longitudinal stress
in this cylinder must be constant throughout its length, and is independent of
the stress in any other part of the body.

Hence pa = <$>(x,y),

where <f> is a function of x and y only, but may be any such function. But
expressing the stresses in terms of F under the conditions pxz = 0, pyi = Q, we
find that if F is a perfectly general function of x and y

,

-j— r = 0, and -T— y- = ,
dxdz dydz

d F dF

whence it follows that -j— and -7— are functions of x and y only, and that

dF

-7- is a function of z only. Hence

F=G + Z,

when G is a function of x and y only, and Z a function of z only, and the
components of stress are

<PG<PZ d*Gd?Z (d'GtfG (

~ - ~

RECIPROCAL FIGURES, FRAMES,

Here the function / which determines the stress in the strata parallel to xy is

Now, thia function is not sufficiently general, for instead of being any
function of x, y, and z, it is the product of a function of x and y multiplied
by a function of z.

Besides this, though the value of pa is, as it ought to be, a function of x
and y only, it is not of the most general form, for it depends on G, t he-
function which determines the stresses pa, p^, and pn, whereas the value of
pm may be entirely independent of the values of these stresses. In fact, the
equations give

This method, therefore, of representing stress in a body of three dimen-
sions is a restricted solution of the equations of equilibrium.

On Reciprocal Diagrams in Three Dimensions.

Let us consider figures in two portions of space, which we shall call respec-
tively the first and the second diagrams. Let the co-ordinates of any point in
the first diagram be denoted by x, y, z, and those of the corresponding point in
the second by £ i), £> measured in directions parallel to x, y, z respectively.
Let F be a quantity varying from point to point of the first figure in any
continuous manner ; that is to say, if A, B are two points, and Fu Ft the
values of F at those points ; then, if B approaches A without limit, the value
of F, approaches that of Ft without limit. Let the co-ordinates (f, rj, £) of a
point in the second diagram be determined from x, y, z, those of the corre-
sponding point in the first by the equations

, dF dF dF

This is equivalent to the statement, that the vector (p) of any point in the
second diagram represents in direction and magnitude the rate of variation of /'
at the corresponding point of the first diagram.

AND DIAGRAMS OF FORCES. 183

Next, let us determine another function, <£, from the equation

(2),

$, as thus determined, will be a function of x, y, and z, since £ 17, £ are
known in terms of these quantities. But, for the same reason, <f> is a function
of £ 17, £. Differentiate <£ with respect to £, considering x, y, and z functions
of £ r?, £,

dxdydz dF

_ .
d£~

Substituting the values of £ 77, £ from (l)

d<k dFdx dF dy dF dz dF

_ II — I* -L _ _ I _ *?, l

dg * dx d^ dy d£ + dz dg d£

_

*d£
= x.
Differentiating (f> with respect to 77 and £, we get the three equations

~' ~' ~

or the vector (r) of any point in the first diagram represents in direction and
magnitude the rate of increase of <f> at the corresponding point of the second
diagram.

Hence the first diagram may be determined from the second by the same
process that the second was determined from the first, and the two diagrams,
each with its own function, are reciprocal to each other.

The relation (2) between the functions expresses that the sum of the func-
tions for two corresponding points is equal to the product of the distances of
these points from the origin multiplied by the cosine of the angle between the
directions of these distances.

Both these functions must be of two dimensions in space. Let F' be a
linear function of xyz, which has the same value and rate of variation as F
has at the point x^y^

r.^+d.-^g+Or-rt^+ti-^J ............... (4

The value of F' at the origin is found by putting x, y, and z = 0

F = Ft-x£-yn-z£=-4> ......................... (5),

184 RECIPROCAL FIGURES, FRAMES,

or the value of F at the origin is equal and opposite to the value of <f> at
the point £, 17, {.

If the rate of variation of F is nowhere infinite, the co-ordinates £, 77, £,
of the second diagram must be everywhere finite, and vice versa. Beyond the
limits of the second diagram the values of x, y, z, in terms of £, 77, £, must
he impossible, and therefore the value of ^ is also impossible. Within the
limits of the second diagram, the function <£ has an even number of values at
every point, corresponding to an even number of points in the first diagram,
which correspond to a single point in the second.

To find these points in the first diagram, let p be the vector of a given
point in the second diagram, and let surfaces be drawn in the first diagram
for which F is constant, and let points be found in each of these surfaces at
which the tangent plane is perpendicular to p, these points will form one or
more curves, which must be either closed or infinite, and the points on these
curves correspond to the points in the second diagram which lie in the
direction of the vector p. If p be the perpendicular from a point in the first
diagram on a plane through the origin perpendicular to p, then all those points

on these curves at which ~r- = p correspond to the given point in the second
diagram. Now, since this point is within the second diagram, there are values

of p both greater and less than the given one ; and therefore -y- is neither an

absolute maximum nor an absolute minimum value. Hence there are in general
an even number of points on the curve or curves which correspond to the
given point. Some of these points may coincide, but at least two of them
must be different, unless the given point is at the limit of the second diagram.

Let us now consider the two reciprocal diagrams with their functions, and
ascertain in what the geometrical nature of their reciprocity consists.

(1) Let the first diagram be simply the point Pu (xu yu z,), at which
F=FU then in the other diagram

(6),

or a point in one diagram is reciprocal to a space in the other, in which the
function ^ is a linear function of the co-ordinates.

AND DIAGRAMS OF FORCES. 185

(2) Let the first diagram contain a second point P2, (xa yv za), at which
F—F,_, then we must combine equation (6) with

-F1 ............................ (7),

whence eliminating <£,

If r,2 is the length of the line drawn from the first point Pt to the second
Pt; and if lumunn are its direction cosines, this equation becomes

IJi+'mtf + nJL = -^ — ' ,

'13

or the reciprocal of the two points Pl and P2 is a plane, perpendicular to the
line joining them, and such that the perpendicular from the origin on the plane
multiplied by the length of the line PjP., is equal to the excess of Fn over F}.

(3) Let there be a third point P, in the first diagram, whose co-ordinates
are x3, yt, z, and for which F=FS; then we must combine with equations (6)
and (7)

F3 ............................ (8).

The reciprocal of the three points P,, P2, P, is a straight line perpen-
dicular to the plane of the three points, and such that the perpendicular on
this line from the origin represents, in direction and magnitude, the rate of
most rapid increase of F in the plane PfJP^, F being a linear function of the
co-ordinates whose values at the three points are those given.

(4) Let there be a fourth point P4 for which F=Ft.

The reciprocal of the four points is a single point, and the line drawn
from the origin to this point represents, in direction and magnitude, the rate
of greatest increase of F, supposing F such a linear function of xyz that its
values at the four points are those given. The value of <f> at this point is
that of F at the origin.

Let us next suppose that the value of F is continuous, that is, that F
does not vary by a finite quantity when the co-ordinates vary by infinitesimal
quantities, but that the form of the function <£ is discontinuous, being a different
linear function of xyz in different parts of space, bounded by definite surfaces.

VOL. II. 24

186 RECIPROCAL FIGURES, FRAMES,

The bounding surfaces of these parts of space must be composed of planes.
For let the linear functions of xyz in contiguous portions of space be

then at the bounding surface, where F1 = Ft

(a,-a.)a: + (A-A)y + (y,-y,)z = *,-6 .................. (9),

and this is the equation of a plane.

Hence the portion of space in which any particular form of the value of F
holds good must be a polyhedron or cell bounded by plane faces, and therefore
having straight edges meeting in a number of points or summits.

Every face is the boundary of two cells, every edge belongs to three or
more cells, and to two faces of each cell.

Every summit belongs to at least four cells, to at least three faces of each
cell, and to two edges of each face.

The whole space occupied by the diagram is divided into cells in two
different ways, so that every point in it belongs to two different cells, and
has two values of F and its derivatives.

The reciprocal diagram is made up of cells in the same way, and tlie
reciprocity of the two diagrams may be thus stated : —

1. Every summit in one diagram corresponds to a cell in the other.

The radius vector of the summit represents the rate of increase of the
function within the cell, both in direction and magnitude.

The value of the function at the summit is equal and opposite to the
value which the function in the cell would have if it were continued under the
same algebraical form to the origin.

2. Every edge in the one diagram corresponds to a plane face in the
other, which is the face of contact of the two cells corresponding to the two
extremities of the edge.

The edge in the one diagram is perpendicular to the face in the other.

The distance of the plane from the origin represents the rate of increase
of the function along the edge.

AND DIAGRAMS OF FORCES. 187

3. Every face in the one diagram corresponds to an edge in which as
many cells meet as there are angles in the face, that is, at least three. Every
face must belong to two, and only two cells, because the edge to which it
corresponds has two, and only two extremities.

4. Every cell in the one diagram corresponds to a summit in the other.
Every face of the cell corresponds and is perpendicular to an edge having an
extremity in the summit. Since every cell must have four or more faces, every
summit must have four or more edges meeting there.

Every edge of the cell corresponds to a face having an angle in the summit.
Since every cell has at least six edges, every summit must be the point of
concourse of at least six faces, which are the boundaries of cells.

Every summit of the cell corresponds to a cell having a solid angle at
the summit. Since every cell has at least four summits, every summit must be
the meeting place of at least four cells.

Mechanical Reciprocity of the Diagrams.

If along each of the edges meeting in a summit forces are applied pro-
portional to the areas of the corresponding faces of the cell in the reciprocal
diagram, and in a direction which is always inward with respect to the cell,
then these forces will be in equilibrium at the summit.

This is the "Polyhedron of Forces," and may be proved by hydrostatics.

If the faces of the cell form a single closed surface which does not inter-
sect itself, it is easy to understand what is meant by the inside and outside
of the cell ; but if the surface intersects itself, it is better to speak of the
positive and negative sides of the surface. A cell, or portion of a cell, bounded
by a closed surface, of which the positive side is inward, may be called a
positive cell. If the surface intersects itself, and encloses another portion of
space with its negative side inward, that portion of space forms a negative cell.
If any portion of space is surrounded by n sheets of the surface of the same
cell with their positive side inward, and by m sheets with their negative side
inward, the space enclosed in this way must be reckoned n — m times.

In passing to a contiguous cell, we must suppose that its face in contact
with the first cell has its positive surface on the opposite side from that of

24—2

188 RECIPROCAL FIGURES, FRAMES,

the first cell. In this way, by making the positive side of the surface con-
tinuous throughout each cell, and by changing it when we pass to the next
cell, we may settle the positive and negative side of every face of every cell,
the sign of every face depending on which of the two cells it is considered
for the moment to belong to.

If we now suppose forces of tension or pressure applied along every edge of
the first diagram, so that the force on each extremity of the edge is in the
direction of the positive normal to the corresponding face of the cell corre-
sponding to that extremity, and proportional to the area of the face, then these
pressures and tensions along the edges will keep every point of the diagram
in equilibrium.

Another way of determining the nature of the force along any edge of
the first diagram, is as follows : —

Round any edge of the first diagram draw a closed curve, embracing it
and no other edge. However small the curve is, it will enter each of the
cells which meet in the edge. Hence the reciprocal of this closed curve will
be a plane polygon whose angles are the points reciprocal to these cells taken
in order. The area of this polygon represents, both in direction and magnitude,
the whole force acting through the closed curve, that is, in this case the stress
along the edge. If, therefore, in going round the angles of the polygon, w»-
travel in the same direction of rotation in space as in going round the closed
curve,' the stress along the edge will be a pressure ; but if the direction is
opposite, the stress will be a tension.

This method of expressing stresses in three dimensions comprehends all
cases in which Rankine's reciprocal figures are possible, and is applicable to
certain cases of continuous stress. That it is not applicable to all such cases
is easily seen by the example of p (189).

On Reciprocal Diagrams in two Dimensions.

If we make F a function of x and y only, all the properties already
deduced for figures in three dimensions will be true in two ; but we may form
a more distinct geometrical conception of the theory by substituting cz for /'

AND DIAGRAMS OF FORCES. 189

and c£ for <j). We have then for the equations of relation between the two
diagrams

dz dz

.(10).

These equations are equivalent to the following definitions : —

Let z in the first diagram be given as a function of x and y, z will lie
on a surface of some kind. Let x0, y0 be particular values of x and y, and
let z0 be the corresponding value of z. Draw a tangent plane to the surface at
the point xa, y0, z0, and from the point f=0, 7? = 0, £= — c; in the second
diagram draw a normal to this tangent plane. It will cut the plane £ — 0 at
the point %, 77 corresponding to xy, and the value of £ is equal and opposite
to the segment of the axis of z cut off by the tangent plane. The two
surfaces may be defined as reciprocally polar (in the ordinary sense) with respect
to the paraboloid of revolution

(11),

and the diagrams are the projections on the planes of xy and &) of points and
lines on these surfaces.

If one of the surfaces is a plane-faced polyhedron, the other will also be a
plane-faced polyhedron, every face in the one corresponding to a point in the
other, and every edge in the one corresponding to the line joining the points
corresponding to the faces bounded by the edge. In the projected diagrams
every line is perpendicular to the corresponding line, and lines which meet in
a point in one figure form a closed polygon in the other.

These are the conditions of reciprocity mentioned at p. 169, and it now
appears that if either of the diagrams is a projection of a plane-faced poly-
hedron, the other diagram can be drawn. If the first diagram cannot be a
projection of a plane-faced polyhedron, let it be a projection of a polyhedron
whose faces are polygons not in one plane. These faces must be conceived to
be filled up by surfaces, which are either curved or made up of different plane
portions. In the first case the polygon will correspond not to a point, but to

19O RECIPROCAL FIGURES, FRAMES,

a finite portion of a surfece; in the second, it will correspond to several points,
HO that the lines, which correspond to the edges of such a polygon, will termi-
in several points, and not in one, as is necessary for reciprocity.

Second Method of representing Stress in a Body.

Let a, 6 be any two consecutive points in the first diagram, distant «,
and o. /8 the corresponding points in the second, distant <r, then if the direc-
tion cosines of the line al are I, m, n and those of aft, X, p., v

o-X = si -r- + &m -==• + sn -p-
ax dy dz

, dn (frj drj

<ru. = sl -f^ + sm -j1 +sn -j1

dx dy dz

, d[ dl, dt,

ov = 8l -j- + »m T- + sn -s-
dx dy dz

(12).

Hence

If we put l\ + mp. + nv = co&e, where e is the angle between s and o-, and if
we take three sets of values of I, m, n, corresponding to three directions at
right angles to each other, we find

oj er, _ , <r, _ di- dy , d£ d*F , d*F , d*F

Hence this quantity depends only on the position of the point, and not
on the directions of su su st or of x, y, z, let us call it A'F.

Now, let us take an element of area perpendicular to s, and let us sup-
pose that the stress on this element is compounded of a normal pressure =

;md a tension parallel to <r and equal to p - .

AND DIAGRAMS OF FORCES,

191

By the rules for the composition of stress, we have for the componentH
of the force on this element, in terms of the six components of stress,

X = lpx
Y=lpx

+ npxl=p
+ np,,, =p

+ npa =p (
\

Hence,

(15).

d'F d'

ld*F d*

d*F d*

d"F
Pvz~ Pdxdy'

~ P

'dzdx' Px»~ pdxdy
By substituting these values in the equations of equilibrium

n.v\ n.rn ri.v\

- = 0, &c

....(16).

•(17),

dx dy dz
it is manifest that they are fulfilled for any value of F.

The most general solution of these equations of equilibrium is contained
in the values

(18).

dx* dz2

dy*

dydz' dzdx' dxdy

By making A = B=C=pF we get a case which, though restricted in its
generality, has remarkable properties with respect to diagrams of stress. We
have seen that a distribution of stress according to the definition above (16),
is consistent with itself, and will keep a body in equilibrium. Since the
stresses are linear functions of F, any two systems of stress can be compounded
by adding their respective functions, a process not applicable to the first method
of representation by areas.

Let us ascertain what kind of stress is represented in this way in the
case of the system of cells already considered.

192 RECIPROCAL FIGURES, FRAMES,

Since /' in each cell is a linear function of x, y, z, there can be no stress
at any point within it. Let us take a and 6 two contiguous points in different
oells, then a and ft will be the points at a finite distance to which these cells

are reciprocal, and A*.F=^j, which becomes infinite when ab vanishes.

If a and b are hi the surface bounding the cells, a and ft coincide. Hence
there is a stress in this surface, uniform in all directions in the plane of the
surface, and such that the stress across unit of length drawn on the surface is
proportional to the distance between the points which are reciprocal to the two
cells bounded by the surface, and this stress is a tension or a pressure according
as the two points are similarly or oppositely situated to the two cells.

The kind of equilibrium corresponding to this case is therefore that of a
system of liquid films, each having a tension like that of a soap bubble,
depending on the nature of the fluid of which it is composed. If all the films
are composed of the same fluid, their tensions must be equal, and all the edges
of the reciprocal diagram must be equal.

On Airy's Function of Stress.

Mr Airy, in a paper " On the Strains in the Interior of Beams*," was, I
believe, the first to point out that, in any body in equilibrium under the action
of internal stress in two dimensions, the three components of the stress in any
two rectangular directions are the three second derivatives, with respect to these
directions, of a certain function of the position of a point in the body.

This important simplification of the theory of the equilibrium of stress in
two dimensions does not depend on any theory of elasticity, or on the mode
in which stress arises in the body, but solely on the two conditions of equi-
librium of an element of a body acted on only by internal stress

*-+*-°' and

whence it follows that

d'F

> and A. - ............... (20)l

* Phil. Trans. 1863.

AND DIAGRAMS OF FORCES. 193

where F is a function of x and y, the form of which is (as far as these
equations are concerned) perfectly arbitrary, and the value of which at any
point is independent of the choice of axes of co-ordinates. Since the stresses
depend on the second derivatives of F, any linear function of x and y may be
added to F without affecting the value of the stresses deduced from F. Also,
since the stresses are linear functions of F, any two systems of stress may be
mechanically compounded by adding the corresponding values of F.

The importance of Airy's function in the theory of stress becomes even
more manifest when we deduce from it the diagram of stress, the co-ordinates
of whose points are

dF dF

For if s be the length of any curve in the original figure, and cr that of the
corcesponding curve in the diagram of stress, and if Xds, Yds are the com-
ponents of the whole stress acting on the element ds towards the right hand
of the curve s

dy , d'F dy 1 dg dy , dg ,
Xds = pxx~r-ds= -y-; -f-ds = -rL -f ds = -f- da-
em ds dy2 ds dy ds da-

dx , dsF dx , d-n dx , d-n
and ---^

Hence the stress on the right hand side of the element ds of the original
curve is represented, both in direction and magnitude, by the corresponding
element da- of the curve in the diagram of stress, and, by composition, the
resultant stress on any finite arc of the first curve s is represented in direction
and magnitude by the straight line drawn from the beginning to the end of
the corresponding curve <r.

If Pv Pa are the principal stresses at any point, and if P, is inclined a
to the axis of x, then the component stresses are

jp^ = P, cos3 a + P3 sin2 a i

^^(P.-P^sinacosa .......................... (23).

pm = Pl sin1 a + Ps cos2 a J

VOL. II. 25

194

RECIPROCAL FIGURES, FRAMES,

Hence

tan -a

_

da* dy'
cPF

(24).

Consider the area bounded by a closed curve s, and let us determine the sur-
face integral of the sum of the principal stresses over the area within the curve.

The integral is

By a well-known theorem, corresponding in two dimensions to that of Green in
three dimensions, the latter expression becomes, when once integrated,

( (dF dx _dFdy\ . .

}\dy ds dxds)a

or

These line integrals are to be taken round the closed curve s. If we take
a point in the curve s as origin in the original body, and the corresponding
point in <r as origin in the diagram of stress, then £ and 17 are the compo-
nents of the whole stress on the right hand of the curve from the origin to
a given point. If p denote the line joining the origin with the point £17, then
p will represent in direction and magnitude the whole stress on the arc o:

The line integral may now be interpreted as the work done on a point
which travels once round the closed curve *, and is everywhere acted on by a
force represented in direction and magnitude by p. We may express this
quantity in terms of the stress at every point of the curve, instead of the
resultant stress on the whole arc, as follows:

For integrating (27) by parts it becomes,

,(28),

AND DIAGRAMS OP FORCES. 195

or if Rds is the actual stress on ds, and r is the radius vector of ds, and
if R makes with r an angle e, we obtain the result

ds ..................... (29).

This line integral, therefore, which depends only on the stress acting on
the closed curve s, is equal to the surface integral of the sum of the principal
stresses taken over the whole area within the curve.

If there is no stress on the curve s acting from without, then the surface
integral vanishes. This is the extension to the case of continuous stress of the
theorem, given at p. 176, that the algebraic sum of all the tensions multiplied
each by the length of the piece in which it acts is zero for a system in
equilibrium. In the case of a frame, the stress in each piece is longitudinal,
and the whole pressure or tension of the piece is equal to the longitudinal
stress multiplied by the section, so that the integral //(Pj + PJcZxcfo/ for each
piece is its tension multiplied by its length.

If the closed curve s is a small circle, the corresponding curve cr will be
an ellipse, and the stress on any diameter of the circle will be represented in
direction and magnitude by the corresponding diameter of the ellipse. Hence,
the principal axes of the ellipse represent in direction and magnitude the prin-
cipal stresses at the centre of the circle.

Let us next consider the surface integral of the product of the principal
stresses at every point taken over the area within the closed curve s.

= ( rii ±2 _ ^ 11V ] ^jjj^y

J J \rfx c% cfo/ cfay ' *"
or by transformation of variables

Hence the surface integral of the product of the principal stresses within
the curve is equal to the area of the corresponding curve cr in the diagram of
stress, and therefore depends entirely on the external stress on the curve s.
This is seen from the construction of the curve cr in the diagram of stress,
since each element do- represents the stress on the corresponding element ds of

the original curve.

25—2

RECIPROCAL FIGURES, FRAMES,

If p represents in direction and magnitude the resultant of the stress on
the curve * from the origin to a point which moves round the curve, then the
area traced out by p is equal to the surface- integral required. If Xds and ) "</.<
are the components of the stress on the element ds, and I the whole length of
the closed curve *, then the surface integral is equal to either of the quantities

P Y ('Xds .ds, or-fx (' Yds . ds.

In a frame the stress in each piece is entirely longitudinal, so that the
product of the principal stresses is zero, and therefore nothing is contributed to
the surface-integral except at the points where the pieces meet or cross each
other. To find the value of the integral for any one of these points, draw a
closed curve surrounding it and no other point, and therefore cutting all the
pieces which meet in that point in order. The corresponding figure in the
diagram of stress will be a polygon, whose sides represent in magnitude and
direction the tensions in the several pieces taken in order. The area of this
j>olygon, therefore, represents the value of ^P1Ptdxdy for the point of concourse,
and is to be considered positive or negative, according as the tracing point
travels round it in the positive or the negative cyclical direction.

Hence the following theorem, which is applicable to all plane frames, whether
a diagram of forces can be drawn or not.

For each point of concourse or of intersection construct a polygon, liy
drawing in succession lines parallel and proportional to the forces acting on the
point in the several pieces which meet in that point, taking the pieces in
cyclical order round the point. The area of this polygon is to be taken positive
or negative, according as it lies on the left or the right of the tracing point.

If, then, a closed curve be drawn surrounding the entire frame, and a
polygon be drawn by drawing in succession lines parallel and proportional to
all the external forces which act on the frame in the order in which their
lines of direction meet the closed curve, then the area of this polygon is equal
to the algebraic sum of the areas of the polygons corresponding to the various
joints of the frame.

In this theorem a polygon is to be drawn for every point, whether the
lines of the frame meet or intersect, whether they are really jointed together,

AND DIAGRAMS OF FORCES.

197

or whether two pieces simply cross each other without mechanical connection.
In the latter case the polygon is a parallelogram, whose sides are parallel and
proportional to the stresses in the two pieces, and it is positive or negative
according as these stresses are of the same or of opposite signs.

If three or more pieces intersect, it is manifestly the same whether they
intersect at one point or not, so that we have the following theorem :—

The area of a polygon of an even number of sides, whose opposite sides
are equal and parallel, is equal to the sum of the areas of all the different
parallelograms which can be formed with their sides parallel and equal to those
of the polygon.

This is easily shewn by dividing the polygon into the different parallelo-
grams.

On the Equilibrium of Stress in a Solid Body.

Let PQR be the longitudinal, and STU the tangential components of
stress, as indicated in the following table of stresses and strains, taken from
Thomson and Tait's Natural Philosophy, p. 511, § 669: —

Components of the

_ -x

Planes, of which
Relative Motion, or
across which Force,
is reckoned

Direction of
Relative Motion
or of Force

Strain

Stress

«

P

yz

X

/

Q

ZX

y

ff

R

xy

z

a

s

lyx
\zx

y

z

b

T

ley

\xy

z

X

c

U

\XZ

\y«

X

y

RECIPROCAL FIGURES, FRAMES,

Then the equations of equilibrium of an element of the body are, by § 697
of that work,

dP dU . dT

dx dy dz

JJl+dS + dR
dx dy dz

.(1).

If

we assume three functions A, B, C, such that
<I i/<l:' dzdx'

d*JB TT_ d*C
</:'/./• ' dxdy

v <IV dV dV

and put A = 5^« =dy~> Z=~dz' J

then a sufficiently general solution of the equations of equilibrium is gi
putting

(2),

given by

p=^

dz* + dy*

XA_
dz*

.(3).

dy* dx*

I am not aware of any method of finding other relations between the com-
ponents of stress without making further assumptions. The most natural assump-
tion to make is that the stress arises from elasticity in the body. I shall
confine myself to the case of an isotropic body, such that it can be deprived
of all stress and strain by a removal of the applied forces. In this case, if
y are the components of displacement, and n the coefficient of rigidity,

a,

the equations of tangential elasticity are, by equation (6), §§ 670 and 694 of
Thomson and Tait,

(I —

c//8 dy I „ 1

— j— H -- j— — — O — ~~

dz dy n n

1 d*A

-- i - }~

n dydz

.(4).

AND DIAGRAMS OF FORCES.

199

with similar equations for b and c. A sufficiently general solution of these
equations is given by putting

(5).

_LA

O/yj fi ty

£4 Iv ' ' -

The equations of longitudinal elasticity are of the form given in § 693,

•(6),

where k is the co-efficient of cubical elasticity, with similar equations for Q
and R. Substituting for P, a, ft and y in equation (6) their values from (3)
and (5),

1 \ a1?/2 ay dy* dz* aV dz~ / '
If we put

<PA cPB #C_ d^ <V_ <£__.*

0*35* dy dz* ' da? dy" dz"

this equation becomes

/ /* [ 4* YJ \ / A • A I A 2 /-^ 1 A 2 fi^ ^^ I 7i I I /»-) \ O.ii ^_ O/M ^^ — ty Yl \ A l7l

I A/ ^^ •q'/t I I — 1 ^i ^^ uA _/_^ ~y~ LA \_/ f IW ^T Q 't1 / «/-' ^~ £ It/ V ~~ ^ I tiA ^1 •••(••••(•••I/ I.

We have also two other equations differing from this only in having .&
and (7 instead of .4 on the right hand side. Hence equating the three expres-
sions on the right hand side we find

AM=A2JB = A2(7=Z)2, say, (8),

and P+Q + R = ^ ^,~2 =6&|>.~ ~ (10).

These equations are useful when we wish to determine the stress rather
than the strain in a body. For instance, if the co-efficients of elasticity, k

200 RECIPROCAL FIGURES, FRAMES,

and it, are increased in the same ratio to any extent, the displacements »i
tlu- body are proportionally diminished, but the stresses remain the same, and,
though their distribution depends essentially on the elasticity of the various
parts of the body, the values of the internal forces do not contain the co-
••thVients of elasticity as factors.

There are two cases in which the functions may be treated as functions of
two variables.

The first is when there is no stress, or a constant pressure in the direction
of 2, as in the case of a stratum originally of uniform thickness, in the di na-
tion of z, the thickness being small compared with the other dimensions of
the body, and with the rate of variation of strain.

The second is when there is no strain, or a uniform longitudinal strain in

the direction of z, as in the case of a prismatic body whose length in the

direction of z is very great, the forces on the sides being functions of x
and y only.

In both of these cases S=0 and T=0, so that we may write

' dxdy'

This method of expressing the stresses in two dimensions was first given
by the Astronomer Royal, in the Philosophical Transactions for 1863. "We shall
write F instead of C, and call it Airy's Function of Stress in Two Dimensions.

Let us assume two functions, G and H, such that

„ d*G T,

F= *** V=

dxdy> dxdy

then by Thomson and Tait, § 694, if a is the displacement in the direction of x

(13).
Case L— If # = 0 this becomes

Integrating with respect to a; we find the following equation for a —

7 ............ (14),

. AND DIAGRAMS OF FORCES. 201

where Y is a function of y only. Similarly for the displacement /8 in the
direction of y,

where X is a function of x only. Now the shearing stress U depends on the
shearing strain and the rigidity, or

Multiplying both sides of this equation by 2(er+l) and substituting from (11),
(14), and (15),

d*G d'G d'G d'G . ^ (#H d*H\ dX dY

. dY

Hence

an equation which must be fulfilled by G when the body is originally without
strain.

CASE II. — In the second case, in which there is no strain in the direction
of z, we have

Q ............................ (19).

Substituting for R in (13), and dividing by <r+l,

da d* r .<?(? d*G

/OM
......... (20),

with a similar equation for £. Proceeding as in the former case, we find

dy1) dx dy I — a- \doc* dy*/

This equation is identical with that of the first case, with the exception of

the coefficient of the part due to H, which depends on the density of the

body, and the value of cr, the ratio of lateral expansion to longitudinal com-
pression.

Hence, if the external forces are given in the two cases of no stress and
no strain in the direction of z, and if the density of the body or the intensity
VOL. ii. 26

Ml

RECIPROCAL FIGURES, FRAMES,

of the force acting on its substance is in the ratio of o- to (1— <r)' in the
two canon, the internal forces will be the same in every part, and will be
independent of the actual values of the coefficients of elasticity, provided the
strains are small. The solutions of the cases treated by Mr Airy, as given in
his paper, do not exactly fulfil the conditions deduced from the theory of
elasticity. In fact, the consideration of elastic strain is not explicitly introduced
into the investigation. Nevertheless, his results are statically possible, and
exceedingly near to the truth in the cases of ordinary beams.

As an illustration of the theory of Airy's Function, let us take the case of

J?T=— 7* cos 2p6 (22).

In this case we have for the co-ordinates of the point in the diagram corre-
ponding to (sy)

£=£c=r9-1co8(2p-l)0, vj = -r*-1sin(2p-l)0 (23),

and for the components of stress

<fF__ . ^ (24

If' we make G--rfcoap0, and H=-rpsmp0 (25),

i 1 * fl« L 7V~f **

then

"XX

(26).

Hence the curves for which G and H respectively are constant will be lines of
principal stress, and the stress at any point will be inversely as the square of
the distance between the consecutive curves G or //.

If we make
then we must have

=/3 Cos 0 and i; = psin <f> }

and ^-(2p-l)0J ....................

= *~l

If we put q for -^ then i + 1 = 2 and (2p - 1) (2q - 1) = 1,

AND DIAGRAMS OF FORCES. 203

so that if ft g, h in the diagram of stress correspond to F, G, H in the
original figure, we have

h=-p'1sinq<l> (28).

Case of a Uniform Horizontal Beam.

As an example of the application of the condition that the stresses must
be such as are consistent with an initial condition of no strain, let us take the
case of a uniform rectangular beam of indefinite length placed horizontally with
a load = h per unit of length placed on its upper surface, the weight of the
beam being k per unit of length. Let us suppose the beam to be supported
by vertical forces and couples in a vertical plane applied at the ends ; but
let us consider only the middle portion of the beam, where the conditions
applicable to the ends have no sensible effect. Let the horizontal distance x
be reckoned from the vertical plane where there is no shearing force, and let
the planes where there is no moment of bending be at distances ±«0 from
the origin. Let y be reckoned from the lower edge of the beam, and let b

be the depth of the beam. Then, if 17= — , , is the shearing stress, the total
vertical shearing force through a vertical section at distance x is

f03v-W -l^}

\ fl iT I \ CM IT I

and this must be equal and opposite to the weight of the beam and load
from 0 to x, which is evidently (h + k)x.

Hence, ^- = - (h + k) x<f> (y), where <£(&)-<£(0) = 1 ............... (29).

From this we find the vertical stress

The vertical stress is therefore a function of y only. It must vanish at
the lower side of the beam, where y = 0, and it must be —A on the upper side
of the beam, where y = b. The shearing stress U must vanish at both sides
of the beam, or <j>'(y) = Q, when y = 0, and when y = b.

26—2

RECIPROCAL FIGURES, FRAMES,

The simplest form of <j>(y) which will satisfy these conditions is

Hence we find the following expression for the function of stress by integrating
(29) with respect to x,

Y ...................... (30),

where a is a constant introduced in integration, and depends on the manner
in which the beam is supported. From this we obtain the values of the
vertical, horizontal, and shearing stresses,

The values of Q and of U, the vertical and the shearing stresses, as
given by these equations, are perfectly definite in terms of h and k, the load
and the weight of the beam per unit of length. The value of P, the hori-
zontal stress, however, contains an arbitrary function Y, which we propose to
find from the condition that the beam was originally unstrained. We therefore
determine a and /3, the horizontal and vertical displacement of any point (x, y),
by the method indicated by equations (13), (14), (15)

' (35),

where X' is a function of x only, and Y' of y only. Deducing from these
displacements the shearing strain, and comparing it with the value of the shearing
stress, U, we find the equation

(36).

Hence = n (by - y.) .................................... (37),

0 ........... (38).

AND DIAGRAMS OF FORCES. 205

If the total longitudinal stress across any vertical section of the beam is zero,

(IF
the value of -j- must be the same when y = 0 and when y = b. From this

condition we find the value of P by equation (32)

(39).

f

J«

The moment of bending at any vertical section of the beam is

• (40).

This becomes zero when x= ±aa where

«oS = «2-F (41).

If we wish to compare this case with that of a beam of finite length sup-
ported at both ends and loaded uniformly, we must make the moment of
bending zero at the supports, and the length of the beam between the supports
must therefore be 2a0. Substituting a0 for a in the value of P, we find

(42).

If we suppose the beam to be cut oft just beyond the supports, and sup-
ported by an intense pressure over a small area, we introduce conditions into
the problem which are not fulfilled by this solution, and the investigation of
which requires the use of Fourier's series. In order that our result may be
true, we must suppose the beam to extend to a considerable distance beyond
the supports on either side, and the vertical forces to be applied by means
of frames clamped to the ends of the beam, as in Diagram V«, so that the
stresses arising from the discontinuity at the extremities are insensible in the
part of the beam between the supports.

This expression differs from that given by Mr Airy only in the terms in
the longitudinal stress P depending on the function Y, which was introduced
in order to fulfil the condition that, when no force is applied, the beam is
unstrained. The effect of these terms is a maximum when y= '12788 b, and is
then equal to (h + k) '314, or less than a third of the pressure of the beam
and its load on a flat horizontal surface when laid upon it so as to produce a
uniform vertical pressure

206 RECIPROCAL FIGURES, FRAMES,

i:\ri. A NATION OF THE DIAGRAMS (PLATES I. II. III.).

Diagrams I a. and I b. illustrate the necessity of the condition of the possibility of reciprocal
diagrams, that each line must be a side of two, and only two, polygons. Diagram I a. is a skeleton of
a frame »uch, that if the force along any one piece be given, the force along any other piece may be
determined. But the piece N forms a side of four triangles, NFII, NGI, NJL, and NKM, so that if
there could be a reciprocal diagram, the line corresponding to N would have four extremities, which is
impossible. In this case we can draw a diagram of forces in which the forces //, /, J, and A' are each
represented by two parallel lines.

Diagrams Ila. and 116. illustrate the case of a frame consisting of thirty-two pieces, meeting four
and four in sixteen points, and forming sixteen quadrilaterals. Diagram II a. may be considered as a plane
projection of a polyhedron of double continuity, which we may describe as a quadrilateral frame consisting
of four quadrilateral rods, of which the ends are bevelled so as to fit exactly. The projection of this
frame, considered as a plane frame, has three degrees of stiffness, so that three of the forces may be
arbitrarily assumed

In the reciprocal diagram II b. the lines are drawn by the method given at p. 168, so that each line
is perpendicular to the corresponding line in the other figure. To make the corresponding lines parallel
we have only to turn one of the figures round a right angle.

Diagrams III a. and III b. illustrate the principle as applied to a bridge designed by Professor F.
Jenkin. The loads Qt Qs, &.C., are placed on the upper series of joints, and Jil Rf Ac., on the lower
series. The diagram III 6. gives the stresses due to both sets of loads, the vertical lines of loads being
different for the two series.

Diagrams IV a. and IV b. illustrate the application of Airy's Function to the construction of
diagrams of continuous stress.

IV a. represents a cylinder exposed to pressure in a vertical and horizontal direction, and to
tension in directions inclined 45* to these. The lines marked a, b, e, &c., are lines of pressure, and
those marked o, p, q, are lines of tension. In this case the lines of pressure and tension are rectangular
hyperbolas, the pressure is always equal to the tension, and varies inversely as the square of the
distance between consecutive curves, or, what is the same thing, directly as the square of the distance
from the centre.

IV b. represents the reciprocal diagram corresponding to the upper quadrant of the former one. The
stress on any line in the first diagram is represented in magnitude and direction by the corresponding line
in the second diagram, the correspondence being ascertained by that of the corresponding systems of lines
o, b, e, <fcc., and o, p, q, ic.

We may also consider IV b. as a sector of a cylinder of 270°, exposed to pressure along the lines
a, b, c, and to tension along o, p, q, the magnitude of the stress being in this case r~ . The upper
quadrant of IV a. is in this case the reciprocal figure. This figure illustrates the tendency of any
strained body to be ruptured at a re-entering angle, for it is plain that at the angle the stress becomes
indefinitely great.

VOL. II. PLATE XII. (i)

<•• pagt 206

Cambridge University Prcs*

VOL. 11 PLATE XIII. (ii)

«« ?»,7« 206

Cambridge University

in)

i 3

o

Cambridge University Press

AND DIAGRAMS OF FORCES. 207

In diagram IV a. : —

F=±r4co*40, G= \r* cos 20, 11 = ^ sin 20.

In diagram IV b. : —

/=fp* cos | <f>, ir=|plcos|(#,) & = ip!sinf<£.

Diagrams V a. and V 6. illustrate Airy's theory of stress in beams.

V a. is the beam supported at G and D by means of bent pieces clamped to the ends of the beam
at A and B, at such a distance from G and D, that the part of the beam between G and D is free from
the local effects of the pressures of the clamps at A. and B. The beam is divided into six strata by
horizontal dotted lines, marked 1, 2, 3, 4, 5, 6, and into sixteen vertical slices by vertical lines marked
a, b, c, &c.

The corresponding lines in the diagram V b. are marked with corresponding figures and letters. The
stress across any line joining any two points in V a. is represented in magnitude by the line in V 6., joining
corresponding points, and is perpendicular to it in direction.

These illustrations of the application of the graphic method to cases of continuous stress, are
intended rather to show the mathematical meaning of the method, than as practical aids to the engineer.
In calculating the stresses in frames, the graphic method is really useful, and is less liable to accidental
errors than the method of trigonometrical calculation. In cases of continuous stress, however, the
symbolical method of calculation is still the best, although, as I have endeavoured to show in this
paper, analytical methods may be explained, illustrated, and extended by considerations derived from
the graphic method.

[From the London Matftematical Society's Proceedings. Vol. IIL]

XL. On the Displacement in a, Case of Fluid Motion.

IN most investigations of fluid motion, we consider the velocity at any
point of the fluid as defined by its magnitude and direction, as a function i.i1
the coordinates of the point and of the time. We are supposed to be able to
take a momentary glance at the system at any time, and to observe the velo-
cities ; but are not required to be able to keep our eye on a particular molecule
during its motion. This method, therefore, properly belongs to the theory of a
continuous fluid alike in all its parts, in which we measure the velocity by the
volume which passes through unit of area rather than by the distance travci
by a molecule in unit of time. It is also the only method applicable to the
case of a fluid, the motions of the individual molecules of which are not expres-
sible as functions of their position, as in the motions due to heat and diffusion.
When similar equations occur in the theory of the conduction of heat or elec-
tricity, we are constrained to use this method, for we cannot even define what is
meant by the continued identity of a portion of heat or electricity.

The molecular theory, as it supposes each molecule to pi'eserve its identity,
requires for its perfection a determination of the position of each molecule at
any assigned time. As it is only in certain cases that our present mathematical
resources can effect this, I propose to point out a very simple case, with the
results.

Let a cylinder of infinite length and of radius a move with its axis parallel
to z, and always passing through the axis of x, with a velocity V, uniform or
variable, in the direction of x, through an infinite, homogeneous, incompressilile.
perfect fluid. Let r be the distance of any point in the fluid from the axis

THE DISPLACEMENT IN A CASE OF FLUID MOTION. 209

of the cylinder ; then it is easy to shew that, if x0 is the value of x for the
axis of the cylinder, and x that of the point, and

<h = -(x-x) and d-ll--

V(f> will satisfy the conditions of the velocity-potential, and V-fy that of the
stream function* ; and, since the expression for \|> does not contain the time,
its value will remain constant for a molecule during the whole of its motion.

If we consider the position of a particle as determined by the values of
z, r, and t/f, then z and $ will remain constant during the motion, and we have
only to find r in terms of the time. For this purpose we observe that, if we
put i|» in polar coordinates, it becomes

= ( 1 — 2 ) r sin 6,

dr V d^ I a*\

and -y • = - T£ = V 1 --^ cos 0.

dt r d6 \ ry

Expressing cos 6 in terms of r and i/>, this becomes

If we make v/(4a2 + 1/»2) + i/r = 2/3, and |-2 = c,

then /8 will be the value of y when the axis of the cylinder is abreast of the
particle, and

and if we now use instead of r a new anular variable sucn

s _

A r

f The velocity-potential is a quantity such that its rate of variation along any line is equal to
the velocity of the fluid resolved in the same direction. Whenever the motion of the fluid is irro-
tational, there is a velocity-potential.

The stream function exists in every case of the motion of an incompressible fluid in two dimen-
sions, and is such that the total instantaneous flow across any curve, referred to unit of time, is
equal to the difference of the values of the stream function at the extremities of the curve.

VOL. II. 27

210 THE DISPLACEMENT IN A CASE OF FLUID MOTION.

then we can express \Vdt or xt in terms of elliptic functions of the first and
second kinds,

win-re the position of the axis of the cylinder is expressed in terms of the
position of a molecule with respect to it.

Now let us take a molecule originally on the axis of y, at a distance ij
from the origin, and let the cylinder begin to move from an infinite distance
on the negative side of the axis of x\ then

$ = i}, and 2/J = ,/(4af- , &

and when the cylinder has passed from negative infinity to positive infinity in
the direction of x, then the coordinates of the molecule will be

a(l— c)

-~

It appears from this expression, that after the passage of the cylinder e\vrv
particle is at the same distance as at first from the plane of xz, but that it
is carried forward in the direction of the motion of the cylinder by a quantity
which is infinite when y = 0, but finite for all other values of y.

The motion of a particle at any instant is always inclined to the axis of
x at double the inclination of the line drawn to the axis of the cylinder.
Hence it is in the forward direction till the inclination of this line is 45",
backward from 45° to 135°, and forward again afterwards. The forward motion
is slower than the backward motion, but lasts for a longer time, and it appears
that the final displacement of eveiy particle is in the forward direction. It
follows from this that the condition fulfilled by the fluid at an infinite distance
is not that of being contained in a fixed vessel ; for in that case there would
have been, on the whole, a displacement backwards equal to that of the cylinder
forwards. The problem actually solved differs from this only by the application
of an infinitely small forward velocity to the infinite mass of fluid such as
generate a finite momentum.

In drawing the accompanying figures, I began by tracing the stream-lines
in Fig. 1, p. 211, by means of the intersections of a system of straight lines equi-
distant and parallel to the axis, with a system of circles touching the axis at

THE DISPLACEMENT IN A CASE OF FLUID MOTION.

211

FIG. 1.
Fluid flowing past a fixed cylinder.

27—2

212 THE DISPLACEMENT IN A CASE OF FLUID MOTION.

the origin and having their radii as the reciprocals of the natural numbers.
(See Prof. Rankine's Papers on Stream-Lines in the Phil. Ti-ans.)

The cylinder is f inch radius, and the stream-lines are originally ^ inch
apart.

I then calculated the coordinates, x and y, of the final form of a transverse
straight line from the values of the complete elliptic functions for values of c
corresponding to every 5*. The result is given in the continuous curve on the
left of Fig. 2, p. 213.

I then traced the path of a particle in contact with the cylinder from the
equation

where x=x9 + acos& and y = ctBm0.

The form of the path is the curve nearest the axis in Fig. 3. The dots
indicate the positions at equal intervals of time.

The paths of particles not in contact with the cylinder might be calculated
from Legendre's tables for incomplete functions, which I have not got.

I have therefore drawn them by eye so as to fulfil the following con-
ditions : —

The radius of curvature is £ -5—. .,fi —• > which, when y is large compared

ct sni. (7 ~T* y

with a, becomes nearly — .

2y

The paths of particles at a great distance from the axis are therefore very
nearly circles.

To draw the paths of intermediate particles, I observed that their two
extremities must lie at the same distance from the axis of x as the asymptote
of a certain stream-line, and the middle point of the path at a distance equal
to that of the same stream-line when abreast of the cylinder ; and, finally, that
the distance between the extremities is the same as that given in Fig. 2.

In this way I drew the paths of different particles in Fig. 3. I then
transferred these to Fig. 2, to shew the paths of a series of particles, originally
in a straight line, and finally in the curve already described.

THE DISPLACEMENT IN A CASE OF FLUID MOTION.

213

I then laid Fig. 1 on Fig. 2, and drew, through the intersections of the
stream-lines and the paths of the corresponding particles in the fluid originally

FIG. 2.
Paths of particles of the fluid -when a cylinder moves through it.

Fio. 3.
Paths of particles at different distances from the cylinder: radius of cylinder, 5 inch. At great

distances (ft) the path is a circle of radius ; „ and in this circle tan s = -^.

*P 2 P

THE DISPLACEMENT IN A CASE OF FLUID MOTION.

at rest, the lines which shew the form taken by a line of particles originally
Kt might as it flows past the cylinder. This method, however, does not give
the jxnnt where the line crosses the axis of x. I therefore calculated this
from the equation

i i r — a

x = r + yt log

calculating r for values of x differing by J inch.

The curves thus drawn appear to be as near the truth as I could get
without a much greater amount of labour.

If a maker of "marbled" paper were to rule the surface of his bath with
straight lines of paint at right angles, and then to draw a cylindrical ruler
through the bath up to the middle, and apply the painted lines to his paper,
he would produce the design of Fig. 1, p. 211.

[From the British Association Report, Vol. XL.]

XLI. Address to the Mathematical and Physical Sections of the British

Association.

[Liverpool, September 15, 1870.]

AT several of the recent Meetings of the British Association the varied
and important business of the Mathematical and Physical Section has been
introduced by an Address, the subject of which has been left to the selection
of the President for the time being. The perplexing duty of choosing a subject
has not, however, fallen to me.

Professor Sylvester, the President of Section A at the Exeter Meeting,
gave us a noble vindication of pure mathematics by laying bare, as it were,
the very working of the mathematical mind, and setting before us, not the
array of symbols and brackets which form the armoury of the mathematician,
or the dry results which are only the monuments of his conquests, but the
mathematician himself, with all his human faculties directed by his professional
sagacity to the pursuit, apprehension, and exhibition of that ideal harmony
which he feels to be the root of all knowledge, the fountain of all pleasure,
and the condition of all action. The mathematician has, above all things, an
eye for symmetry ; and Professor Sylvester has not only recognized the sym-
metry formed by the combination of his own subject with those of the former
Presidents, but has pointed out the duties of his successor in the following
characteristic note : —

" Mr Spottiswoode favoured the Section, in his opening Address, with a com-
bined history of the progress of Mathematics and Physics ; Dr. Tyndall's address
was virtually on the limits of Physical Philosophy ; the one here in print," says
Prof. Sylvester, "is an attempted faint adumbration of the nature of Mathe-
matical Science in the abstract. What is wanting (like a fourth sphere resting

216 ADDRESS TO THE MATHEMATICAL AND PHYSICAL SECTIONS

cm three other* in contact) to build up the Ideal Pyramid is a discourse on
the Relation of the two branches (Mathematics and Physics) to, their act inn
and reaction upon, one another, a magnificent theme, with which it is to be
hoped that some future President of Section A will crown the edifice and make
the Tetralogy (symbolizable by A + A', A, A', AA') complete."

The theme thus distinctly kid down for his successor by our late Presi
IB indeed a magnificent one, far too magnificent for any efforts of mine to
realize. I have endeavoured to follow Mr Spottiswoode, as with far-reaching
vision he distinguishes the systems of science into which phenomena, our k:
ledge of which is still in the nebulous stage, are growing. I have been carried
by the penetrating insight and forcible expression of Dr Tyndall into that
sanctuary of minuteness and of power where molecules obey the laws of their
••\istence, clash together in fierce collision, or grapple in yet more fierce embrace,
building up in secret the forms of visible things. I have been guided by Prof.
Sylvester towards those serene heights

" Where never creeps a cloud, or moves a wind,
Nor ever falls the least white star of snow,
Nor ever lowest roll of thunder moans,
Nor sound of human sorrow mounts to mar
Their sacred everlasting calm."

But who will lead me into that still more hidden and dimmer region where
Thought weds Fact, where the mental operation of the mathematician and the
physical action of the molecules are seen in their true relation ? Does not the
way 'to it pass through the very den of the metaphysician, strewed with the
remains of former explorers, and abhorred by every man of science ? It would
indeed be a foolhardy adventure for me to take up the valuable time of the
Section by leading you into those speculations which require, as we know,
thousands of years even to shape themselves intelligibly.

But we are met as cultivators of mathematics and physics. In our daily
work we are led up to questions the same in kind with those of metaphysics ;
and we approach them, not trusting to the native penetrating power of our
own minds, but trained by a long- continued adjustment of our modes of thought
to the facts of external nature.

As mathematicians, we perform certain mental operations on the symbols of
number or of quantity, and, by proceeding step by step from more simple to
more complex operations, we are enabled to express the same thing in many

OF THE BRITISH ASSOCIATION. 217

different forms. The equivalence of these different forms, though a necessary
consequence of self-evident axioms, is not always, to our minds, self-evident;
but the mathematician, who by long practice has acquired a familiarity with
many of these forms, and has become expert in the processes which lead from
one to another, can often transform a perplexing expression into another which
explains its meaning in more intelligible language.

As students of Physics we observe phenomena under varied circumstances,
and endeavour to deduce the laws of their relations. Every natural phenomenon
is, to our minds, the result of an infinitely complex system of conditions.
What we set ourselves to do is to unravel these conditions, and by viewing
the phenomenon in a way which is in itself partial and imperfect, to piece
out its features one by one, beginning with that which strikes us first, and
thus gradually learning how to look at the whole phenomenon so as to obtain
a continually greater degree of clearness and distinctness. In this process, the
feature which presents itself most forcibly to the untrained inquirer may not be
that which is considered most fundamental by the experienced man of science ;
for the success of any physical investigation depends on the judicious selection
of what is to be observed as of primary importance, combined with a voluntary
abstraction of the mind from those features which, however attractive they
appear, we are not yet sufficiently advanced in science to investigate with profit.

Intellectual processes of this kind have been going on since the first for-
mation of language, and are going on still. No doubt the feature which strikes
us first and most forcibly in any phenomenon, is the pleasure or the pain
which accompanies it, and the agreeable or disagreeable results which follow
after it. A theory of nature from this point of view is embodied in many of
our words and phrases, and is by no means extinct even in our deliberate
opinions.

It was a great step in science when men became convinced that, in order
to understand the nature of things, they must begin by asking, not whether
a thing is good or bad, noxious or beneficial, but of what kind is it ? and
how much is there of it ? Quality and Quantity were then first recognized as
the primary features to be observed in scientific inquiry.

As science has been developed, the domain of quantity has everywhere
encroached on that of quality, till the process of scientific inquiry seems to have
become simply the measurement and registration of quantities, combined with
a mathematical discussion of the numbers thus obtained. It is this scientific

VOL. II. 28

ADDRESS TO THE MATHEMATICAL AND PHYSICAL SECTIONS

method of directing our attention to those features of phenomena which may
be regarded as quantities which brings physical research under the influence of
mathematical reasoning. In the work of the Section we shall have abundant
examples of the successful application of this method to the most recent con-
queets of science; but I wish at present to direct your attention to some of
tin- reciprocal effects of the progress of science on those elementary conceptions
which are sometimes thought to be beyond the reach of change.

If the skill of the mathematician has enabled the experimentalist to see
that the quantities which he has measured are connected by necessary n Li:
the discoveries of physics have revealed to the mathematician new I'mi:
quantities which he could never have imagined for himself.

Of the methods by which the mathematician may make his labours
useful to the student of nature, that which I think is at present most im-
portant is the systematic classification of quantities.

The quantities which we study in mathematics and physics may be clas>
in two different ways.

The student who wishes to master any particular science must make hii
familiar with the various kinds of quantities which belong to that science. When
lie understands all the relations between these quantities, he regards tin:
forming a connected system, and he classes the whole system of quantities I
as belonging to that particular science. This classification is the most natural
from a physical point of view, and it is generally the first in order of time.

But when the student has become acquainted with several different seic-
he finds that the mathematical processes and trains of reasoning in one sc'
resemble those in another so much that his knowledge of the one science may
be made a most useful help in the study of the other.

When he examines into the reason of this, he finds that in the t\v<.
sciences he has been dealing with systems of quantities, in which the mathe-
matical forms of the relations of the quantities are the same in both -
though the physical nature of the quantities may be utterly different.

He is thus led to recognize a classification of quantities on a new principle,
according to which the physical nature of the quantity is subordinated t
mathematical form. This is the point of view which is characteristic of the
mathematician ; but it stands second to the physical aspect in order of time.
because the human mind, in order to conceive of different kinds of quant
must have them presented to it by nature.

OF THE BRITISH ASSOCIATION. 219

I do not here refer to the fact that all quantities, as such, are subject
to the rules of arithmetic and algebra, and are therefore capable of being sub-
mitted to those dry calculations which represent, to so many minds, their only
idea of mathematics.

The human mind is seldom satisfied, and is certainly never exercising its
highest functions, when it is doing the work of a calculating machine. What
the man of science, whether he is a mathematician or a physical inquirer,
aims at is, to acquire and develope clear ideas of the things he deals with.
For this purpose he is willing to enter on long calculations, and, to be for a
season a calculating machine, if he can only at last make his ideas clearer.

But if he finds that clear ideas are not to be obtained by means of pro-
cesses the steps of which he is sure to forget before he has reached the
conclusion, it is much better that he should turn to another method, and try
to understand the subject by means of well-chosen illustrations derived from
subjects with which he is more familiar.

We all know how much more popular the illustrative method of exposition
is found, than that in which bare processes of reasoning and calculation form
the principal subject of discourse.

Now a truly scientific illustration is a method to enable the mind to grasp
some conception or law in one branch of science, by placing before it a con-
ception or a law in a different branch of science, and directing the mind to
lay hold of that mathematical form which is common to the corresponding ideas
in the two sciences, leaving out of account for the present the difference
between the physical nature of the real phenomena.

The correctness of such an illustration depends on whether the two systems
of ideas which are compared together are really analogous in form, or whether,
in other words, the corresponding physical quantities really belong to the same
mathematical class. When this condition is fulfilled, the illustration is not only
convenient for teaching science in a pleasant and easy manner, but the recog-
nition of the formal analogy between the two systems of ideas leads to a
knowledge of both, more profound than could be obtained by studying each
system separately.

There are men who, when any relation or law, however complex, is put
before them in a symbolical form, can grasp its full meaning as a relation
among abstract quantities. Such men sometimes treat with indifference the
further statement that quantities actually exist in nature which fulfil this

28—2

ADDRESS TO THE MATHEMATICAL AND PHYSICAL SECTIONS

relation. The mental image of the concrete reality seems rather to disturb
thnn to assist their contemplations.

But the great majority of mankind are utterly unable, without long
training, to retain in their minds the unembodied symbols of the pure mathe-
matician, so that, if science is ever to become popular, and yet remain scientific,
it must be by a profound study and a copious application of those principles
of the mathematical classification of quantities which, as we have seen, lie at
the root of every truly scientific illustration.

There are, as I have said, some minds which can go on contemplating with
satisfaction pure quantities presented to the eye by symbols, and to the mind
in a form which none but mathematicians can conceive.

There are others who feel more enjoyment in following geometrical forms,
which they draw on paper, or build up in the empty space before them.

Others, again, are not content unless they can project their whole physical
energies into the scene which they conjure up. They learn at what a rate the
planets rush through space, and they experience a delightful feeling of exhila-
ration. They calculate the forces with which the heavenly bodies pull at one
another, and they feel their own muscles straining with the effort.

To such men momentum, energy, mass are not mere abstract expressions
of the results of scientific inquiry. They are words of power, which stir their
souls like the memories of childhood.

For the sake of persons of these different types, scientific truth should be
presented in different forms, and should be regarded as equally scientific, whether
it appears in the robust form and the vivid colouring of a physical illustration,
or in the tenuity and paleness of a symbolical expression.

Time would fail me if I were to attempt to illustrate by examples the
scientific value of the classification of quantities. I shall only mention the name
of that important class of magnitudes having direction in space which Hamilton
has called vectors, and which form the subject-matter of the Calculus of Qua-
ternions, a branch of mathematics which, when it shall have been thoroughly
understood by men of the illustrative type, and clothed by them with physical
imagery, will become, perhaps under some new name, a most powerful method
of communicating truly scientific knowledge to persons apparently devoid of the
calculating spirit.

The mutual action and reaction between the different departments of human
thought is so interesting to the student of scientific progress, that, at the risk

OF THE BKITISH ASSOCIATION. £21

of still further encroaching on the valuable time of the Section, I shall say a
few words on a branch of physics which not very long ago would have been
considered rather a branch of metaphysics. I mean the atomic theory, or, as
it is now called, the molecular theory of the constitution of bodies.

Not many years ago if we had been asked in what regions of physical
science the advance of discovery was least apparent, we should have pointed
to the hopelessly distant fixed stars on the one hand, and to the inscrutable
delicacy of the texture of material bodies on the other.

Indeed, if we are to regard Comte as in any degree representing the
scientific opinion of his time, the research into what takes place beyond our
own solar system seemed then to be exceedingly unpromising, if not altogether
illusory.

The opinion that the bodies which we see and handle, which we can set
in motion or leave at rest, which we can break in pieces and destroy, are
composed of smaller bodies which we cannot see or handle, which are always
in motion, and which can neither be stopped nor broken in pieces, nor in any
way destroyed or deprived of the least of their properties, was known by the
name of the Atomic theory. It was associated with the names of Democritus,
Epicurus, and Lucretius, and was commonly supposed to admit the existence
only of atoms and void, to the exclusion of any other basis of things from the
universe.

In many physical reasonings and mathematical calculations we are accustomed
to argue as if such substances as air, water, or metal, which appear to our
senses uniform and continuous, were strictly and mathematically uniform and
continuous.

We know that we can divide a pint of water into many millions of
portions, each of which is as fully endowed with all the properties of water
as the whole pint was ; and it seems only natural to conclude that we might
go on subdividing the water for ever, just as we can never come to a limit
in subdividing the space in which it is contained. We have heard how Faraday
divided a grain of gold into an inconceivable number of separate particles, and
we may see Dr Tyndall produce from a mere suspicion of nitrite of butyle
an immense cloud, the minute visible portion of which is still cloud, and there-
fore must contain many molecules of nitrite of butyle.

But evidence from different and independent sources is now crowding in upon
us which compels us to admit that if we could push the process of subdivision

ADDRESS TO THE MATHEMATICAL AND PHYSICAL SECTIONS

.-till further we should come to a limit, because each portion would then contain
onlv one molecule, an individual body, one and indivisible, unalterable by any
power in nature.

Kven in our ordinary experiments on very finely divided matter we find
that the substance is beginning to lose the properties which it exhibits when
in a large mass, and that effects depending on the individual action of mole-
cules are beginning to become prominent.

The study of these phenomena is at present the path which leads to the
development of molecular science.

That superficial tension of liquids which is called capillary attraction is one
of these phenomena. Another important class of phenomena are those which
are due to that motion of agitation by which the molecules of a liquid or gas
are continually working their way from one place to another, and continually
changing their course, like people hustled in a crowd.

On this depends the rate of diffusion of gases and liquids through each
other, to the study of which, as one of the keys of molecular science, that
unwearied inquirer into nature's secrets, the late Prof. Graham, devoted such
arduous labour.

The rate of electrolytic conduction is, according to Wiedemann's theory,
influenced by the same cause ; and the conduction of heat in fluids depends
probably on the same kind of action. In the case of gases, a molecular theory
has been developed by Clausius and others, capable of mathematical treatment,
and subjected to experimental investigation ; and by this theory nearly every
known mechanical property of gases has been explained on dynamical principles ;
so that the properties of individual gaseous molecules are in a fair way to
Income objects of scientific research.

Now Mr Stoney has pointed out'" that the numerical results of experiments
on gases render it probable that the mean distance of their particles at the
ordinary temperature and pressure is a quantity of the same order of magnitude
as a millionth of a millimetre, and Sir William Thomson has sincet shewn, by
several independent lines of argument, drawn from phenomena so different in
themselves as the electrification of metals by contact, the tension of soap-
bubbles, and the friction of air, that in ordinary solids and liquids the average
distance between contiguous molecules is less than the hundred-millionth, and
greater than the two-thousand-millionth of a centimetre.

* Phil. Mag. Aug. 1868. t Nature, March 31, 1870.

OF THE BRITISH ASSOCIATION. 223

These, of course, are exceedingly rough estimates, for they are derived from
measurements some of which are still confessedly very rough ; but if, at the
present time, we can form even a rough plan for arriving at results of this
kind, we may hope that, as our means of experimental inquiry become more
accurate and more varied, our conception of a molecule will become more definite,
so that we may be able at no distant period to estimate its weight with a
greater degree of precision.

A theory, which Sir W. Thomson has founded on Helmholtz's splendid
hydrodynamical theorems, seeks for the properties of molecules in the ring-
vortices of a uniform, frictionless, incompressible fluid. Such whirling rings may
l)e seen when an experienced smoker sends out a dexterous puff of smoke into
the still air, but a more evanescent phenomenon it is difficult to conceive.
This evanescence is owing to the viscosity of the air ; but Helmholtz has shewn
that in a perfect fluid such a whirling ring, if once generated, would go on
whirling for ever, would always consist of the very same portion of the fluid
which was first set whirling, and could never be cut in two by any natural
cause. The generation of a ring-vortex is of course equally beyond the power
of natural causes, but once generated, it has the properties of individuality,
permanence in quantity, and indestructibility. It is also the recipient of impulse
and of energy, which is all we can affirm of matter ; and these . ring- vortices
are capable of such varied connexions and knotted self-involutions, that the
properties of differently knotted vortices must be as different as those of diffe-
rent kinds of molecules can be.

If a theory of this kind should be found, after conquering the enormous
mathematical difficulties of the subject, to represent in any degree the actual
properties of molecules, it will stand in a very different scientific position from
those theories of molecular action which are formed by investing the molecule
with an arbitrary system of central forces invented expressly to account for the
observed phenomena.

In the vortex theory we have nothing arbitrary, no central forces or occult
properties of any other kind. We have nothing but matter and motion, and when
the vortex is once started its properties are all determined from the original
impetus, and no further assumptions are possible.

Even in the present undeveloped state of the theory, the contemplation
of the individuality and indestructibility of a ring-vortex in a perfect fluid

224 ADDRESS TO THE MATHEMATICAL AND PHYSICAL SECTIONS

cannot fail to disturb the commonly received opinion that a molecule, in order
to be permanent, must be a very hard body.

In fact one of the first conditions which a molecule must fulfil is, ap-
parently, inconsistent with its being a single hard body. We know from those
spectroscopic researches which have thrown so much light on different branches
of science, that a molecule can be set into a state of internal vibration, in
which it gives off to the surrounding medium light of definite refrangibility —
light, that is, of definite wave-length and definite period of vibration. The
fact that all the molecules (say, of hydrogen) which we can procure for our
experiments, when agitated by heat or by the passage of an electric spark,
vibrate precisely in the same periodic time, or, to speak more accurately, that
their vibrations are composed of a system of simple vibrations having always
the same periods, is a very remarkable fact.

I must leave it to others to describe the progress of that splendid series
of spectroscopic discoveries by which the chemistry of the heavenly bodies has
been brought within the range of human inquiry. I wish rather to direct
your attention to the fact that, not only has every molecule of terrestrial
hydrogen the same system of periods of free vibration, but that the spectro-
scopic examination of the light of the sun and stars shews that, in regions
the distance of which we can only feebly imagine, there are molecules vibrating
in as exact unison with the molecules of terrestrial hydrogen as two tuning-
forks tuned to concert pitch, or two watches regulated to solar time.

Now this absolute equality in the magnitude of quantities, occurring in all
parts of the universe, is worth our consideration.

The dimensions of individual natural bodies are either quite indeterminate,
as in the case of planets, stones, trees, &c., or they vary within moderate
limits, as in the case of seeds, eggs, &c. ; but even in these cases small quanti-
tative differences are met with which do not interfere with the essential
properties of the body.

Even crystals, which are so definite in geometrical form, are variable with
respect to their absolute dimensions.

Among the works of man we sometimes find a certain degree of uniformity.

There is a uniformity among the different bullets which are cast in the
same mould, and the different copies of a book printed from the same type.

If we examine the coins, or the weights and measures, of a civilized
country, we find a uniformity, which is produced by careful adjustment to

OF THE BRITISH ASSOCIATION. 225

standards made and provided by the state. The degree of uniformity of these
national standards is a measure of that spirit of justice in the nation which
has enacted laws to regulate them and appointed officers to test them.

This subject is one in which we, as a scientific body, take a warm
interest ; and you are all aware of the vast amount of scientific work which
has been expended, and profitably expended, in providing weights and measures
for commercial and scientific purposes.

The earth has been measured as a basis for a permanent standard of
length, and every property of metals has been investigated to guard against
any alteration of the material standards when made. To weigh or measure any
thing with modern accuracy, requires a course of experiment and calculation
in which almost every branch of physics and mathematics is brought into
requisition.

Yet, after all, the dimensions of our earth and its time of rotation, though,
relatively to our present means of comparison, very permanent, are not so by
any physical necessity. The earth might contract by cooling, or it might be
enlarged by a layer of meteorites falling on it, or its rate of revolution might
slowly slacken, and yet it would continue to be as much a planet as before.

But a molecule, say of hydrogen, if either its mass or its time of vibration
were to be altered in the least, would no longer be a molecule of hydrogen.

If, then, we wish to obtain standards of length, time, and mass which
shall be absolutely permanent, we must seek them not in the dimensions, or
the motion, or the mass of our planet, but in the wave-length, the period of
vibration, and the absolute mass of these imperishable and unalterable and
perfectly similar molecules.

When we find that here, and in the starry heavens, there are innumerable
multitudes of little bodies of exactly the same mass, so many, and no more,
to the grain, and vibrating in exactly the same time, so many times, and no more,
in a second, and when we reflect that no power in nature can now alter in
the least either the mass or the period of any one of them, we seem to have
advanced along the path of natural knowledge to one of those points at which
we must accept the guidance of that faith by which we understand that
" that which is seen was not made of things which do appear."

One of the most remarkable results of the progress of molecular science
is the light it has thrown on the nature of irreversible processes — processes,
that is, whidh always tend towards and never away from a certain limiting

VOL. II. 29

ADDRESS TO THE MAimniATICAL AND PHYSICAL SECTIONS

state. Thus, if two gasea be put into the same vessel, they become mixed,
and the mixture tends continually to become more uniform. If two unequally
heated portions of the same gas are put into the vessel, something of the
kind takes place, and the whole tends to become of the same temperature.
If two unequally heated solid bodies be placed in contact, a continual approxi-
mation of both to an intermediate temperature takes place.

In the case of the two gases, a separation may be effected by chemical means ;
l>ut in the other two cases the former state of things cannot be restored by
:iny natural process.

In the case of the conduction or diffusion of heat the process is not only
irreversible, but it involves the irreversible diminution of that part of the whole
stock of thermal energy which is capable of being converted into mechanical work.

This is Thomson's theory of the irreversible dissipation of energy, and it is
equivalent to the doctrine of Clausius concerning the growth of what he calk
Entropy.

The irreversible character of this process is strikingly embodied in Fourier's
theory of the conduction of heat, where the formulae themselves indicate, for
all positive values of the time, a possible solution which continually tends to
the form of a uniform diffusion of heat.

But if we attempt to ascend the stream of time by giving to its symbol
continually diminishing values, we are led up to a state of things in which
the formula has what is called a critical value ; and if we inquire into the
state of things the instant before, we find that the formula becomes absurd.

We thus arrive at the conception of a state of things which, cannot be
conceived as the physical result of a previous state of things, and we find
that this critical condition actually existed at an epoch not in the utmost
depths of a past eternity, but separated from the present time by a finite interval.

This idea of a beginning is one which the physical researches of recent
times have brought home to us, more than any observer of the course of
scientific thought in former times would have had reason to expect.

But the mind of man is not, like Fourier's heated body, continually settling
down into an ultimate state of quiet uniformity, the character of which we
can already predict; it is rather like a tree, shooting out branches which adapt
themselves to the new aspects of the sky towards which they climb, and roots
which contort themselves among the strange strata of the earth into which they
delve. To us who breathe only the spirit of our own age, and know only the

OF THE BRITISH ASSOCIATION. 227

characteristics of contemporary thought, it is as impossible to predict the general
tone of the science of the future as it is to anticipate the particular discoveries
which it will make.

Physical research is continually revealing to us new features of natural
processes, and we are thus compelled to search for new forms of thought
appropriate to these features. Hence the importance of a careful study of those
relations between Mathematics and Physics which determine the conditions under
which the ideas derived from one department of physics may be safely used in
forming ideas to be employed in a new department.

The figure of speech or of thought by which we transfer the language and
ideas of a familiar science to one with which we are less acquainted may be
called Scientific Metaphor.

Thus the words Velocity, Momentum, Force, &c. have acquired certain precise
meanings in Elementary Dynamics. They are also employed in the Dynamics
of a Connected System in a sense which, though perfectly analogous to the
elementary sense, is wider and more general.

These generalized forms of elementary ideas may be called metaphorical
terms in the sense in which every abstract term is metaphorical. The charac-
teristic of a truly scientific system of metaphors is that each term in its
metaphorical use retains all the formal relations to the other terms of the
system which it had in its original use. The method is then truly scientific- —
that is, not only a legitimate product of science, but capable of generating
science in its turn.

There are certain electrical phenomena, again, which are connected together
by relations of the same form as those which connect dynamical phenomena.
To apply to these the phrases of dynamics with proper distinctions and pro-
visional reservations is an example of a metaphor of a bolder kind ; but it is
a legitimate metaphor if it conveys a true idea of the electrical relations to
those who have been already trained in dynamics.

Suppose, then, that we have successfully introduced certain ideas belonging
to an elementary science by applying them metaphorically to some new class
of phenomena. It becomes an important philosophical question to determine in
what degree the applicability of the old ideas to the new subject may be
taken as evidence that the new phenomena are physically similar to the old.

The best instances for the determination of this question are those in
which two different explanations have been given of the same thing.

29—2

Ill ADDRESS TO THE MATHEMATICAL AND PHYSICAL SECTIONS

The most celebrated case of this kind is that of the corpuscular and the
undulatory theories of light. Up to a certain point the phenomena of light
are equally well explained by both ; beyond this point, one of them fails.

To understand the true relation of these theories in that part of the field
where they seem equally applicable we must look at them in the light which
Hamilton has thrown upon them by his discovery that to every brachistochrone
problem there corresponds a problem of free motion, involving different velocities
and times, but resulting in the same geometrical path. Professor Tait has
written a very interesting paper on this subject.

According to a theory of electricity which is making great progress in
Germany, two electrical particles act on one another directly at a distance,
hut with a force which, according to Weber, depends on their relative velocity,
and according to a theory hinted at by Gauss, and developed by Riemann,
Lorenz, and Neumann, acts not instantaneously, but after a time depending
on the distance. The power with which this theory, in the hands of these
eminent men, explains every kind of electrical phenomena must be studied in
order to be appreciated.

Another theory of electricity, which I prefer, denies action at a distance
and attributes electric action to tensions and pressures in an all-pervading medium,
these stresses being the same in kind with those familiar to engineers, and the
medium being identical with that in which light is supposed to be propagated.

Both these theories are found to explain not only the phenomena by the
aid of which they were originally constructed, but other phenomena, which
were not thought of or perhaps not known at the time ; and both have inde-
pendently arrived at the same numerical result, which gives the absolute
velocity of light in terms of electrical quantities.

That theories apparently so fundamentally opposed should have so large
a field of truth common to both is a fact the philosophical importance of which
we cannot fully appreciate till we have reached a scientific altitude from which
the true relation between hypotheses so different can be seen.

I shall only make one more remark on the relation between Mathematics
and Physics. In themselves, one is an operation of the mind, the other is a
dance of molecules. The molecules have laws of their own, some of which we
select as most intelligible to us and most amenable to our calculation. We
form a theory from these partial data, and we ascribe any deviation of the
actual phenomena from this theory to disturbing causes. At the same time we

OF THE BRITISH ASSOCIATION. 229

confess that what we call disturbing causes are simply those parts of the true
circumstances which we do not know or have neglected, and we endeavour in
future to take account of them. We thus acknowledge that the so-called dis-
turbance is a mere figment of the mind, not a fact of nature, and that in
natural action there is no disturbance.

But this is not the only way in which the harmony of the material with
the mental operation may be disturbed. The mind of the mathematician is
subject to many disturbing causes, such as fatigue, loss of memory, and hasty
conclusions ; and it is found that, from these and other causes, mathematicians
make mistakes.

I am not prepared to deny that, to some mind of a higher order than
ours, each of these errors might be traced to the regular operation of the laws
of actual thinking ; in fact we ourselves often do detect, not only errors of
calculation, but the causes of these errors. This, however, by no means alters
our conviction that they are errors, and that one process of thought is right
and another process wrong.

One of the most profound mathematicians and thinkers of our time, the
late George Boole, when reflecting on the precise and almost mathematical
character of the laws of right thinking as compared with the exceedingly per-
plexing though perhaps equally determinate laws of actual and fallible thinking,
was led to another of those points of view from which Science seems to look
out into a region beyond her own domain. .

"We must admit," he says, "that there exist laws" (of thought) "which
even the rigour of their mathematical forms does not preserve from violation.
We must ascribe to them an authority, the essence of which does not consist
in power, a supremacy which the analogy of the inviolable order of the natural
world in no way assists us to comprehend."

[From the Report of the British Association, 1870.]

XLII. On Colour-vision at different points of the Retina.

IT haa long been known that near that point of the retina where it is
intersected by the axis of the eye there is a yellowish spot, the existence of
which can be shewn not only by the ophthalmoscope, but by its effect on
vision. At the Cheltenham Meeting in 1856 the author pointed out a method
of seeing this spot by looking at that part of a very narrow spectrum which
lies near the line F. Since that time the spot has been described by Helmholtz
and others ; and the author has made a number of experiments, not yet pub-
lished, in order to determine its effects on colour-vision.

One of the simplest methods of seeing the spot was suggested to the
author by Prof. Stokes. It consists in looking at a white surface, such as that
of a white cloud, through a solution of chloride of chromium made so weak that
it appears of a bluish-green colour. If the observer directs his attention to
what he sees before him before his eyes have got accustomed to the new tone of
colour, he sees a pinkish spot like a wafer on a bluish-green ground; and this
spot is always at the place he is looking at. The solution transmits the red
end of the spectrum, and also a portion of bluish-green light near the line F.
The latter portion is partially absorbed by the spot, so that the red light has
the preponderance.

Experiments of a more accurate kind were made with an instrument the
original conception of which is due to Sir Isaac Newton, and is described in
his Lectiones Opticce, though it does not appear to have been actually constructed
till the author set it up in 1862, with a solid frame and careful adjustments.
It consists of two parts, side by side. In the first part, white light is dis-
persed by a prism so as to form a spectrum. Certain portions of this spectrum
are selected by being allowed to pass through slits in a screen. These selected
portions are made to converge on a second prism, which unites them into a

COLOUR- VISION AT DIFFERENT POINTS OF THE RETINA. 231

single beam of light, in which state they enter the eye. The second part of
the instrument consists of an arrangement by which a beam of light from the
very same source is weakened by two reflections from glass surfaces, and enters
the eye alongside of the beam of compound colours.

The instrument is formed of three rectangular wooden tubes, the whole
length being about nine feet. It contains two prisms, two mirrors, and six
lenses, which are so adjusted that, in spite of the very different treatment to
which the two portions of a beam of light are subjected, they shall enter the
eye so as to form exactly equal and coincident images of the source of light.
In fact, by looking through the instrument a man's face may be distinctly
seen by means of the red, the green, or the blue light which it emits, or by
any combination of these at pleasure.

The arrangement of the three slits is made by means, of six brass slides,
which can be worked with screws outside the instrument ; and the breadth of
the slits can be read off with a gauge very accurately.

In each observation three colours of the spectrum are mixed and so adjusted
that their mixture is so exactly equivalent to the white light beside it, that
the line which divides the two can no longer be seen.

It is found that in certain cases, when this adjustment is made so as to
satisfy one person, a second will find the mixed colour of a green hue, while
to a third it will appear of a reddish colour, compared with the white beam.

But, besides this, it is found that the mixed colour may be so adjusted
that, if we look directly at it, it appears red, while if we direct the eye away
from it, and cast a sidelong glance at it, we see it green. The cause of this
is the yellow spot, which acts somewhat as a piece of yellow glass would do,
absorbing certain kinds of light more than others ; and the difference between
different persons arises from different intensities of the absorbing spot. It is
found in persons of every nation, but generally stronger in those of dark com-
plexion. The degree of intensity does not seem to depend so much on the

COLOUR-VISION AT DIFFERENT POINTS OF THE RETINA.

colour of the hair or the iris of the individual, as to run through families
independent of outward complexion.

The same difference is found between different colour-blind persons; so that
in the comparison of their vision with that of the normal eye, persons should
be selected for comparison who have the yellow spot of nearly the same intensity.

In my own eye the part of the spectrum from A to E is seen decidedly
better by the central part of the retina than by the surrounding parts. Near /'
this is reversed, and the central part gives a sensation of about half the
intensity of the rest. Beyond G the central part is again the most sensitive,
and it is decidedly so near //.

Before I conclude I wish to direct the attention of those who wish to
study colour to the exceedingly simple and beautiful series of experiments
described by Mr W. Benson in his works on colour. By looking through a prism
at the black and white diagrams in his book, any one can see more of the
true relations of colour than can be got from the most elaborately coloured
theoretical arrangements of tints.

[From the Philosophical Magazine for December, 1870.]

XLIII. On Hills and Dales.

To the Editors of the Philosophical Magazine and Journal.

GENTLEMEN,

I FIND that in the greater part of the substance of the following
paper I have been anticipated by Professor Cayley, in a memoir " On Contour
and Slope Lines," published in the Philosophical Magazine in 1859 (S. 4. Vol.
xvin. p. 264). An exact knowledge of the first elements of physical geography,
however, is so important, and loose notions on the subject are so prevalent,
that I have no hesitation in sending you what you, I hope, will have no
scruple in rejecting if you think it superfluous after what has been done by
Professor Cayley.

I am, Gentlemen,

Your obedient Servant,

J. CLERK MAXWELL.

GLENLAIR, DALBEATTIE,

October 12, 1870.

1. ON CONTOUR-LINES AND MEASUREMENT OF HEIGHTS.

The results of the survey of the surface of a country are most conveniently
exhibited by means of a map on which are traced contour-lines, each contour-line
representing the intersection of a level surface with the surface of the earth,
and being distinguished by a numeral which indicates the level surface to
which it belongs.

When the extent of country surveyed is small, the contour-lines are defined
VOL. II. 30

; • BILLS AND DALES.

with sufficient accuracy by the number of feet above the mean level of the
M*; but when the survey is so extensive that the variation of the force of
gravity must be taken into account, we must adopt a new definition of the
height of a place in order to be mathematically accurate. If we could deter-
mine the exact form of the surface of equilibrium of the sea, so as to know
its position in the interior of a continent, we might draw a normal to this
surface from the top of a mountain, and call this the height of the mountain.
Tins would be perfectly definite in the case when the surface of equilibrium
is everywhere convex ; but the lines of equal height would not be level
surfaces.

Level surfaces are surfaces of equilibrium, and they are not equidistant.
The only thing which is constant is the amount of work required to rise from
one to another. Hence the only consistent definition of a level surface is
obtained by assuming a standard station, say, at the mean level of the sea
at a particular place, and defining every other level surface by the work
required to raise unit of mass from the standard station to that level surface.
This work must, of course, be expressed in absolute measure, not in local foot-
pounds.

At every step, therefore, in ascertaining the difference of level of two
places, the surveyor should ascertain the force of gravity, and multiply the
linear difference of level observed by the numerical value of the force of gravity.

The height of a place, according to this system, will be defined by a
number which represents, not a lineal quantity, but the half square of the
velocity which an unresisted body would acquire in sliding along any path from
that place to the standard station. This is the only definition of the height
of a place consistent with the condition that places of equal height should be
on the same level. If by any means we can ascertain the mean value of
gravity along the line of force drawn from the place to the standard level
surface, then, if we divide the number already found by this mean value, we
shall obtain the length of this line of force, which may be called the linear
height of the place.

On the Forms of Contour-lines.

Let us begin with a level surface entirely within the solid part of the
earth, and let us suppose it to ascend till it reaches the bottom of the deepest

HILLS AND DALES. 235

sea. At that point it will touch the surface of the earth; and if it continues
to ascend, a contour-line will be formed surrounding this bottom (or Immit, as
it is called by Professor Cayley) and enclosing a region of depression. As the
level surface continues to ascend, it will reach the next deepest bottom of the
sea; and as it ascends it will form another contour-line, surrounding this point,
and enclosing another region of depression below the level surface. As the level
surface rises these regions of depression will continually expand, and new ones
will be formed corresponding to the different lowest points of the earth's
surface.

At first there is but one region of depression, the whole of the rest of
the earth's surface forming a region of elevation surrounding it. The number
of regions of elevation and depression can be altered in two ways.

1st. Two regions of depression may expand till they meet and so run
into one. If a contour-line be drawn through the point where they meet, it
forms a closed curve having a double point at this place. This contour-line
encloses two regions of depression. We shall call the point where these two
regions meet a Bar.

It may happen that more than two regions run into each other at once.
Such cases are singular, and we shall reserve them for separate consideration.

2ndly. A region of depression may thrust out arms, which may meet each
other and thus cut off a region of elevation in the midst of the region of
depression, which thus becomes a cyclic region, while a new region of elevation
is introduced. The contour-line through the point of meeting cuts off two
regions of elevation from one region of depression, and the point itself is called
a Pass. There may be in singular cases passes between more than two regions
of elevation.

3rdly. As the level surface rises, the regions of elevation contract and at
last are reduced to points. These points are called Summits or Tops.

Relation between the Number of Summits and Passes.

At first the whole earth is a region of elevation. For every new region
of elevation there is a Pass, and for every region of elevation reduced to a
point there is a Summit. And at last the whole surface of the earth is a

30—2

._-.,; BILLS AND DALES.

region of depression. Hence the number of Summits is one more than the
number of Passes. If S is the number of Summits and P the number of passes,

Relation between the Number of Bottoms and Bars.

For every new region of depression there is a Bottom, and for every
diminution of the number of these regions there is a Bar. Hence the number
of Bottoms is one more than the number of Bars. If / is the number of
Bottoms or Immits and B the number of Bars, then

From this it is plain that if, in the singular cases of passes and bars, we
reckon a pass as single, double, or n-ple, according as two, three, or n + 1
regions of elevation meet at that point, and a bar as single, double, or ?«-ple,
aa two, three, or n+1 regions of depression meet at that point, then the
census may be taken as before, giving each singular point its proper number.
If one region of depression meets another in several places at once, one of
these must be taken as a bar and the rest as passes.

The whole of this theory applies to the case of the maxima and minima
of a function of two variables which is everywhere finite, determinate, and
continuous. The summits correspond to maxima and the bottoms to minima.
If there are p maxima and q minima, there must be p + q — 2 cases of stationary
values which are neither maxima nor minima. If we regard those points in
themselves, we cannot make any distinction among them ; but if we consider
the regions cut off by the curves of constant value of the function, we may
call p — 1 of them false maxima and q — 1 of them false minima.

On Functions of Three Variables.

If we suppose the three variables to be the three co-ordinates of a point,
and the regions where the function is greater or less than a given value to be
called the positive and the negative regions, then, as the given value increases,
for every negative region formed there will be a minimum, and the positive
region will have an increase of its periphraxy. For every junction of two dif-
ferent negative regions there will be a false minimum, and the positive region

HILLS AND DALES. 237

will have a diminution of its periphraxy. Hence if there are q true minima
there will be q — 1 false minima.

There are different orders of these stationary points according to the number
of regions which meet in them. The first order is when two negative regions
meet surrounded by a positive region, the second order when three negative
regions meet, and so on. Points of the second order count for two, those of
the third for three, and so on, in this relation between the true minima and
the false ones.

In like manner, when a negative region expands round a hollow part and
at last surrounds it, thus cutting off a new positive region, the negative region
acquires periphraxy, a new positive region is formed, and at the point of contact
there is a false maximum.

When any positive region is reduced to a point and vanishes, the negative
region loses periphraxy and there is a true maximum. Hence if there are p
maxima there are p — 1 false maxima.

But these are not the only forms of stationary points ; for a negative
region may thrust out arms which may meet in a stationary point. The negative
and the positive region both become cyclic. Again, a cyclic region may close
in so as to become acyclic, forming another kind of stationary point where the
ring first fills up. If there are r points at which cyclosis is gained and r'
points at which it is lost, then we know that

r = r' ;

but we cannot determine any relation between the number of these points and
that of either the true or the false maxima and minima.

If the function of three variables is a potential function, the true maxima
are points of stable equilibrium, the true minima points of equilibrium unstable
in every direction, and at the other stationary points the equilibrium is stable
in some directions and unstable in others.

On Lines of Slope.

Lines drawn so as to be everywhere at right angles to the contour-lines
are called lines of slope. At every point of such a line there is an upward
and a downward direction. If we follow the upward direction we shall in
general reach a summit, and if we follow the downward direction we shall in
general reach a bottom. In particular cases, however, we may reach a pass or
a bar.

H0 HILLS AND DALES.

On Hills and Dales.

Hence each point of the earth's surface has a line of slope, which begins
at a certain summit and ends in a certain bottom. Districts whose lines of
slope run to the same bottom are called Basins or Dales. Those whose lines
of slope come from the same summit may be called, for want of a better name,

Hills.

Hence the whole earth may be naturally divided into Basins or Dales, and
also, by an independent division, into hills, each point of the surface belonging
to a certain dale and also to a certain hill.

On Watersheds and Watercourses.

Dales are divided from each other by Watersheds, and Hills by Watercourses.

To draw these lines, begin at a pass or a bar. Here the ground is level,
so that we cannot begin to draw a line of slope ; but if we draw a very small
closed curve round this point, it will have highest and lowest points, the
number of maxima being equal to the number of minima, and each one more
than the index number of the pass or bar. From each maximum point draw
a line of slope upwards till it reaches a summit. This will be a line of Water-
shed. From each minimum point draw a line of slope downwards till it reaches
a bottom. This will be a line of Watercourse. Lines of Watershed are the
only lines of slope which do not reach a bottom, and lines of Watercourse
are the only lines of slope which do not reach a summit. All other lines of
slope diverge from some summit and converge to some bottom, remaining
throughout their course in the district belonging to that summit and that
bottom, which is bounded by two watersheds and two watercourses.

In the pure theory of surfaces there is no method of determining a line
of watershed or of watercourse, except by first finding a pass or a bar and
drawing the line of slope from that point. In nature, water actually trickles
down the lines of slope, which generally converge towards the mathematical
watercourses, though they do not actually join them ; but when the streams
increase in quantity, they join and excavate courses for themselves ; and these
actually run into the main watercourse which bounds the district, and so cut
out a river-bed, which, whether full or empty, forms a visible mark on the

HILLS AND DALES. 239

earth's surface. No such action takes place at a watershed, which therefore
generally remains invisible.

There is another difficulty in the application of the mathematical theory,
on account of the principal regions of depression being covered with water, so
that very little is known about the positions of the singular points from which
the lines of watershed must be drawn to the summits of hills near the coast.
A complete division of the dry land into districts, therefore, requires some
knowledge of the form of the bottom of the sea and of lakes.

On the Number of Natural Districts.

Let p, be the number of single passes, p^ that of double passes, and so on.
Let &„ &2, &c. be the numbers of single, double, &c. bars. Then the number
of summits will be, by what we have proved,

and the number of bottoms will be

The number of watersheds will be

W= 2 (b, +Pl) + 3 (b, +Pi) + &c.
The number of watercourses will be the same.

Now, to find the number of faces, we have by Listing's rule

where P is the number of points, L that of lines, F that of faces, and R
that of regions, there being in this case no instance of cyclosis or periphraxy.
Here R = 2, viz. the earth and the surrounding space ; hence

F=L-P + 2.

If we put L equal to the number of watersheds, and P equal to that of

summits, passes, and bars, then F is the number of Dales, which is evidently
equal to the number of bottoms.

If we put L for the number of watercourses, and P for the number of

passes, bars, and bottoms, then F is the number of Hills, which is evidently
equal to the number of summits.

240

HILLS AND DALES.

If we put L equal to the whole number of lines, and P equal to the
whole number of points, we find that F, the number of natural districts
named from a hill and a dale together, is equal to W, the number of water-
sheds or watercourses, or to the whole number of summits, bottoms, passes,
and bars diminished by 2.

Chart of an Inland Basin.

/,, /,, /,, /,. Lowest points, Bottoms or Immits.
St. Highest points, Tops or Summits.
Bars between regions of depression.

Slt *!?,,

£lt £t, Bf

Plt Pj Pp Pf Pf Passes between regions of elevation.
/, Bt It &c. Lines of Watercourse.

Sl P, St <fec. Lines of Watershed.
Dotted line. Contour-lines.

XLIV. Introductory Lecture on Experimental Physics.

THE University of Cambridge, in accordance with that law of its evolution,
by which, while maintaining the strictest continuity between the successive
phases of its history, it adapts itself with more or less promptness to the
requirements of the times, has lately instituted a course of Experimental
Physics. This course of study, while it requires us to maintain in action all
those powers of attention and analysis which have been so long cultivated in
the University, calls on us to exercise our senses in observation, and our hands
in manipulation. The familiar apparatus of pen, ink, and paper will no longer be
sufficient for us, and we shall require more room than that afforded by a seat at
a desk, and a wider area than that of the black board. We owe it to the
munificence of our Chancellor, that, whatever be the character in other respects
of the experiments which we hope hereafter to conduct, the material facilities
for their full development will be upon a scale which has not hitherto been
surpassed.

The main feature, therefore, of Experimental Physics at Cambridge is the
Devonshire Physical Laboratory, and I think it desirable that on the present
occasion, before we enter on the details of any special study, we should con-
sider by what means we, the University of Cambridge, may, as a living body,
appropriate and vitalise this new organ, the outward shell of which we expect
soon to rise before us. The course of study at this University has always
included Natural Philosophy, as well as Pure Mathematics. To diffuse a sound
knowledge of Physics, and to imbue the minds of our students with correct
dynamical principles, have been long regarded as among our highest functions,
and very few of us can now place ourselves in the mental condition in which
even such philosophers as the great Descartes were involved in the days before
Newton had announced the true laws of the motion of bodies. Indeed the
cultivation and diffusion of sound dynamical ideas has already effected a great
change in the language and thoughts even of those who make no pretensions

VOL. II. 31

INTRODUCTORY LECTURE ON EXPERIMENTAL PHYSICS.

to science, and we are daily receiving fresh proofs that the popularisation of
scientific doctrines is producing as great an alteration in the mental state of
society as the material applications of science are effecting in its outward lit',-.
Such indeed is the respect paid to science, that the most absurd opinions may
become current, provided they are expressed in language, the sound of which
recals some well-known scientific phrase. If society is thus prepared to receive
all kinds of scientific doctrines, it is our part to provide for the diffusion and
cultivation, not only of true scientific principles, but of a spirit of sound
criticism, founded on an examination of the evidences on which statements
apparently scientific depend.

When we shall be able to employ in scientific education, not only the
trained attention of the student, and his familiarity with symbols, but the
keenness of his eye, the quickness of his ear, the delicacy of his touch, and
the adroitness of his fingers, we shall not only extend our influence over a
class of men who are not fond of cold abstractions, but, by opening at once
all the gateways of knowledge, we shall ensure the association of the doctrines
of science with those elementary sensations which form the obscure background
of all our conscious thoughts, and which lend a vividness and relief to ideas,
which, when presented as mere abstract terms, are apt to fade entirely from
the memory.

In a course of Experimental Physics we may consider either the Physics
or the Experiments as the leading feature. We may either employ the experi-
ments to illustrate the phenomena of a particular branch of Physics, or we
may make some physical research in order to exemplify a particular experimental
method. In the order of time, we should begin, in the Lecture Room, with
a course of lectures on some branch of Physics aided by experiments of illus-
tration, and conclude, in the Laboratory, with a course of experiments of research.

Let me say a few words on these two classes of experiments, — Experiments
of Illustration and Experiments of Research. The aim of an experiment of
illustration is to throw light upon some scientific idea so that the student may
be enabled to grasp it. The circumstances of the experiment are so arranged
that the phenomenon which we wish to observe or to exhibit is brought into
prominence, instead of being obscured and entangled among other phenomena, as
it is when it occurs in the ordinary course of nature. To exhibit illustrative
experiments, to encourage others to make them, and to cultivate in every
way the ideas on which they throw light, forms an important part of our duty.

INTRODUCTORY LECTURE ON EXPERIMENTAL PHYSICS. 243

The simpler the materials of an illustrative experiment, and the more familiar
they are to the student, the more thoroughly is he likely to acquire the idea
which it is meant to illustrate. The educational value of such experiments is
often inversely proportional to the complexity of the apparatus. The student
who uses home-made apparatus, which is always going wrong, often learns more
than one who has the use of carefully adjusted instruments, to which he is apt
to trust, and which he dares not take to pieces.

It is very necessary that those who are trying to learn from books the
facts of physical science should be enabled by the help of a few illustrative
experiments to recognise these facts when they meet with them out of doors.
Science appears to us with a very different aspect after we have found out
that it is not in lecture rooms only, and by means of the electric light pro-
jected on a screen, that we may witness physical phenomena, but that we may
find illustrations of the highest doctrines of science in games and gymnastics,
in travelling by land and by water, in storms of the air and of the sea, and
wherever there is matter in motion.

This habit of recognising principles amid the endless variety of their action
can never degrade our sense of the sublimity of nature, or mar our enjoyment
of its beauty. On the contrary, it tends to rescue our scientific ideas from
that vague condition in which we too often leave them, buried among the
other products of a lazy credulity, and to raise them into their proper position
among the doctrines in which our faith is so assured, that we are ready at all
times to act on them.

Experiments of illustration may be of very different kinds. Some may be
adaptations of the commonest operations of ordinary life, others may be carefully
arranged exhibitions of some phenomenon which occurs only under peculiar
conditions. They all, however, agree in this, that their aim is to present some
phenomenon to the senses of the student in such a way that he may associate
with it the appropriate scientific idea. When he has grasped this idea, the
experiment which illustrates it has served its purpose.

In an experiment of research, on the other hand, this is not the principal
aim. It is true that an experiment, in which the principal aim is to see what
happens under certain conditions, may be regarded as an experiment of research
by those who are not yet familiar with the result, but in experimental
researches, strictly so called, the ultimate object is to measure something which
we have already seen — to obtain a numerical estimate of some magnitude.

31—2

244 INTRODUCTORY LBCTUBE ON EXPERIMENTAL PHYSICS.

Experiments of this class — those in which measurement of some kind is
involved, are the proper work of a Physical Laboratory. In every experiment
we have first to make our senses familiar with the phenomenon, but we must
not stop here, we must find out which of its features are capable of measure-
ment, and what measurements are required in order to make a complete
specification of the phenomenon. We must then make these measurements,
and deduce from them the result which we require to find.

This characteristic of modern experiments — that they consist principally of
measurements, — is so prominent, that the opinion seems to have got abroad,
that in a few years all the great physical constants will have been approxi-
mately estimated, and that the only occupation which will then be left to men
of science will be to carry on these measurements to another place of decimals.

If this is really the state of things to which we are approaching, our
Laboratory may perhaps become celebrated as a place of conscientious labour
and consummate skill, but it will be out of place in the University, and ought
rather to be classed with the other great workshops of our country, where equal
ability is directed to more useful ends.

But we have no right to think thus of the unsearchable riches of creation,
or of the untried fertility of those fresh minds into which these riches will
continue to be poured. It may possibly be true that, in some of those fields
of discovery which lie open to such rough observations as can be made without
artificial methods, the great explorers of former times have appropriated most
of what is valuable, and that the gleanings which remain are sought after,
rather for their abstruseness, than for their intrinsic worth. But the history
of science shews that even during that phase of her progress in which she
devotes herself to improving the accuracy of the numerical measurement of
quantities with which she has long been familiar, she is preparing the materials
for the subjugation of new regions, which would have remained unknown if
she had been contented with the rough methods of her early pioneers. I might
bring forward instances gathered from every branch of science, shewing how
the labour of careful measurement has been rewarded by the discovery of new
fields of research, and by the development of new scientific ideas. But the
history of the science of terrestrial magnetism affords us a sufficient example
of what may be done by Experiments in Concert, such as we hope some day
to perform in our Laboratory.

That celebrated traveller, Humboldt, was profoundly impressed with the

INTRODUCTORY LECTURE ON EXPERIMENTAL PHYSICS. 245

scientific value of a combined effort to be made by the observers of all nations,
to obtain accurate measurements of the magnetism of the earth ; and we owe
it mainly to his enthusiasm for science, his great reputation and his wide-
spread influence, that not only private men of science, but the governments of
most of the civilised nations, our own among the number, were induced to
take part in the enterprise. But the actual working out of the scheme, and
the arrangements by which the labours of the observers were so directed as to
obtain the best results, we owe to the great mathematician Gauss, working along
with Weber, the future founder of the science of electro-magnetic measurement,
in the magnetic observatory of Gottingen, and aided by the skill of the
instrument-maker Leyser. These men, however, did not work alone. Numbers
of scientific men joined the Magnetic Union, learned the use of the new instru-
ments and the new methods of reducing the observations ; and in every city
of Europe you might see them, at certain stated times, sitting, each in his
cold wooden shed, with his eye fixed at the telescope, his ear attentive to the
clock, and his pencil recording in his note-book the instantaneous position of
the suspended magnet.

Bacon's conception of " Experiments in concert " was thus realised, the
scattered forces of science were converted into a regular army, and emulation
and jealousy became out of place, for the results obtained by any one observer
were of no value till they were combined with those of the others.

The increase in the accuracy and completeness of magnetic observations
which was obtained by the new method, opened up fields of research which
were hardly suspected to exist by those whose observations of the magnetic
needle had been conducted in a more primitive manner. We must reserve for
its proper place in our course any detailed description of the disturbances to
which the magnetism of our planet is found to be subject. Some of these
disturbances are periodic, following the regular courses of the sun and moon.
Others are sudden, and are called magnetic storms, but, like the storms of
the atmosphere, they have their known seasons of frequency. The last and
the most mysterious of these magnetic changes is that secular variation by
which the whole character of the earth, as a great magnet, is being slowly
modified, while the magnetic poles creep on, from century to century, along
their winding track in the polar regions.

We have thus learned that the interior of the earth is subject to the
influences of the heavenly bodies, but that besides this there is a constantly

IRODUCTORY LECTURE ON EXPERIMENTAL PHYSICS.

progressive change going on, the cause of which is entirely unknown. In each
of the magnetic observatories throughout the world an arrangement is at work,
by means of which a suspended magnet directs a ray of light on a prepared
sheet of paper moved by clockwork. On that paper the never-resting heart of
the earth is now tracing, in telegraphic symbols which will one day be inter-
preted, a record of its pulsations and its fiutterings, as well as of that slow
but mighty working which warns us that we must not suppose that the inner
history of our planet is ended.

But this great experimental research on Terrestrial Magnetism produced
lasting effects on the progress of science in general. I need only mention one
or two instances. The new methods of measuring forces were successfully applied
by Weber to the numerical determination of all the phenomena of electricity,
and very soon afterwards the electric telegraph, by conferring a commercial
value on exact numerical measurements, contributed largely to the advancement,
as well as to the diffusion of scientific knowledge.

But it is not in these more modern branches of science alone that this
influence is felt. It is to Gauss, to the Magnetic Union, and to magnetic
observers in general, that we owe our deliverance from that absurd method of
estimating forces by a variable standard which prevailed so long even among
men of science. It was Gauss who first based the practical measurement of
magnetic force (and therefore of every other force) on those long established
principles, which, though they are embodied in every dynamical equation, have
been so generally set aside, that these very equations, though correctly given
in our Cambridge textbooks, are usually explained there by assuming, in addition
to the variable standard of force, a variable, and therefore illegal, standard of
mass.

Such, then, were some of the scientific results which followed in this case
from bringing together mathematical power, experimental sagacity, and manipu-
lative skill, to direct and assist the labours of a body of zealous observers. If
therefore we desire, for our own advantage and for the honour of our University,
that the Devonshire Laboratory should be successful, we must endeavour to
maintain it in living union with the other organs and faculties of our learned
body. We shall therefore first consider the relation in which we stand to
those mathematical studies which have so long flourished among us, which deal
with our own subjects, and which differ from our experimental studies only in
the mode in which they are presented to the mind*

INTRODUCTORY LECTURE ON EXPERIMENTAL PHYSICS. 247

There is no more powerful method for introducing knowledge into the
mind than that of presenting it in as many different ways as we can. When
the ideas, after entering through different gateways, effect a junction in the
citadel of the mind, the position they occupy becomes impregnable. Opticians
tell us that the mental combination of the views of an object which we obtain
from stations no further apart than our two eyes is sufficient to produce in
our minds an impression of the solidity of the object seen ; and we find that
this impression is produced even when we are aware that we are really looking
at two flat pictures placed in a stereoscope. It is therefore natural to expect that
the knowledge of physical science obtained by the combined use of mathematical
analysis and experimental research will be of a more solid, available, and enduring
kind than that possessed by the mere mathematician or the mere experimenter.

But what will be the effect on the University, if men pursuing that course
of reading which has produced so many distinguished Wranglers, turn aside to
work experiments ? Will not their attendance at the Laboratory count not
merely as time withdrawn from their more legitimate studies, but as the intro-
duction of a disturbing element, tainting their mathematical conceptions with
material imagery, and sapping their faith in the formulae of the textbooks ?
Besides this, we have already heard complaints of the undue extension of our
studies, and of the strain put upon our questionists by the weight of learning
which they try to carry with them into the Senate-House. If we now ask
them to get up their subjects not only by books and writing, but at the same
time by observation and manipulation, will they not break down altogether?
The Physical Laboratory, we are told, may perhaps be useful to those who
are going out in Natural Science, and who do not take in Mathematics, but to
attempt to combine both kinds of study during the time of residence at the
University is more than one mind can bear.

No doubt there is some reason for this feeling. Many of us have already
overcome the initial difficulties of mathematical training. When we now go on
with our study, we feel that it requires exertion and involves fatigue, but we
are confident that if we only work hard our progress will be certain.

Some of us, on the other hand, may have had some experience of the
routine of experimental work. As soon as we can read scales, observe times,
focus telescopes, and so on, this kind of work ceases to require any great
mental effort. We may perhaps tire our eyes and weary our backs, but we do
not greatly fatigue our minds.

•J(, INTRODUCTORY LECTURE ON EXPERIMENTAL PHYSICS.

It is not till we attempt to bring the theoretical part of our training into
contact with the practical that we begin to experience the full effect of what
Faraday has called "mental inertia" — not only the difficulty of recognising,
among the concrete objects before us, the abstract relation which we have
learned from books, but the distracting pain of wrenching the mind away from
the symbols to the objects, and from the objects back to the symbols. This
however is the price we have to pay for new ideas.

But when we have overcome these difficulties, and successfully bridged over
the gulph between the abstract and the concrete, it is not a mere piece of
knowledge that we have obtained : we have acquired the rudiment of a perma-
nent mental endowment. When, by a repetition of efforts of this kind, we have
more fully developed the scientific faculty, the exercise of this faculty in detecting
scientific principles in nature, and in directing practice by theory, is no longer
irksome, but becomes an unfailing source of enjoyment, to which we return so
often, that at last even our careless thoughts begin to run in a scientific
channel.

I quite admit that our mental energy is limited in quantity, and I know
that many zealous students try to do more than is good for them. But the
question about the introduction of experimental study is not entirely one of
quantity. It is to a great extent a question of distribution of energy. Some
distributions of energy, we know, are more useful than others, because they are
more available for those purposes which we desire to accomplish.

Now in the case of study, a great part of our fatigue often arises, not
from those mental efforts by which we obtain the mastery of the subject, but
from those which are spent in recalling our wandering thoughts ; and these
efforts of attention would be much less fatiguing if the disturbing force of
mental distraction could be removed.

This is the reason why a man whose soul is in his work always makes
more progress than one whose aim is something not immediately connected with
his occupation. In the latter case the very motive of which he makes use to
stimulate his flagging powers becomes the means of distracting his mind from
the work before him.

There may be some mathematicians who pursue their studies entirely for
their own sake. Most men, however, think that the chief use of mathematics
is found in the interpretation of nature. Now a man who studies a piece of
mathematics in order to understand some natural phenomenon which he has

INTRODUCTORY LECTURE ON EXPERIMENTAL PHYSICS. 249

seen, or to calculate the best arrangement of some experiment which he means
to make, is likely to meet with far less distraction of mind than if his sole
aim had been to sharpen his mind for the successful practice of the Law, or to
obtain a high place in the Mathematical Tripos.

I have known men, who when they were at school, never could see the
good of mathematics, but who, when in after life they made this discovery,
not only became eminent as scientific engineers, but made considerable progress
in the study of abstract mathematics. If our experimental course should help
any of you to see the good of mathematics, it will relieve us of much anxiety,
for it will not only ensure the success of your future studies, but it will make
it much less likely that they will prove injurious to your health.

But why should we labour to prove the advantage of practical science to
the University ? Let us rather speak of the help which the University may
give to science, when men well trained in mathematics and enjoying the advan-
tages of a well-appointed Laboratory, shall unite their efforts to carry out some
experimental research which no solitary worker could attempt.

At first it is probable that our principal experimental work must be the
illustration of particular branches of science, but as we go on we must add to
this the study of scientific methods, the same method being sometimes illustrated
by its application to researches belonging to different branches of science.

We might even imagine a course of experimental study the arrangement of
which should be founded on a classification of methods, and not on that of the
objects of investigation. A combination of the two plans seems to me better
than either, and while we take every opportunity of studying methods, we shall
take care not to dissociate the method from the scientific research to which it
is applied, and to which it owes its value.

We shall therefore arrange our lectures according to the classification of the
principal natural phenomena, such as heat, electricity, magnetism and so on.

In the laboratory, on the other hand, the place of the different instruments
will be determined by a classification according to methods, such as weighing
and measuring, observations of time, optical and electrical methods of observa-
tion, and so on.

The determination of the experiments to be performed at a particular time
must often depend upon the means we have at command, and in the case of
the more elaborate experiments, this may imply a long time of preparation, during

VOL. II. 32

250 UJTBODUCTORY LBCTUBB ON EXPERIMENTAL PHYSICS.

which the instrument*, the methods, and the observers themselves, are being
gradually fitted for their work. When we have thus brought together the
requisite*, both material and intellectual, for a particular experiment, it may
sometimes be desirable that before the instruments are dismounted and the
observers dispersed, we should make some other experiment, requiring the same
method, but dealing perhaps with an entirely different class of physical phe-

nomOMk

Our principal work, however, in the Laboratory must be to acquaint our-
selves with all kinds of scientific methods, to compare them, and to estimate
their value. It will, I think, be a result worthy of our University, and more
likely to be accomplished here than in any private laboratory, if, by the free
and full discussion of the relative value of different scientific procedures, uc
succeed in forming a school of scientific criticism, and in assisting the develop-
ment of the doctrine of method.

But admitting that a practical acquaintance with the methods of Physical
Science is an essential part of a mathematical and scientific education, we may
be asked whether we are not attributing too much importance to science alto-
gether as part of a liberal education.

Fortunately, there is no question here whether the University should con-
tinue to be a place of liberal education, or should devote itself to preparing
young men for particular professions. Hence though some of us may, I hope,
see reason to make the pursuit of science the main business of our lives, it
must be one of our most constant aims to maintain a living connexion between
our work and the other liberal studies of Cambridge, whether literary, philological,
historical or philosophical.

There is a narrow professional spirit which may grow up among men of
science, just as it does among men who practise any other special business.
But surely a University is the very place where we should be able to over-
come this tendency of men to become, as it were, granulated into small worlds,
which are all the more worldly for their very smallness. We lose the ad-
vantage of having men of varied pursuits collected into one body, if we do not
endeavour to imbibe some of the spirit even of those whose special branch of
learning is different from our own.

It is not so long ago since any man who devoted himself to geometry,
or to any science requiring continued application, was looked upon as necessarily
a misanthrope, who must have abandoned all human interests, and betaken

INTRODUCTORY LECTURE ON EXPERIMENTAL PHYSICS. 251

himself to abstractions so far removed from the world of life and action that
he has become insensible alike to the attractions of pleasure and to the claims
of duty.

In the present day, men of science are not looked upon with the same
awe or with the same suspicion. They are supposed to be in league with the
material spirit of the age, and to form a kind of advanced Radical party among
men of learning.

We are not here to defend literary and historical studies. We admit that
the proper study of mankind is man. But is the student of science to be
withdrawn from the study of man, or cut off from every noble feeling, so long
as he lives in intellectual fellowship with men who have devoted their lives to
the discovery of truth, and the results of whose enquiries have impressed them-
selves on the ordinary speech and way of thinking of men who never heard
their names ? Or is the student of history and of man to omit from his con-
sideration the history of the origin and diffusion of those ideas which have
produced so great a difference between one age of the world and another ?

It is true that the history of science is very different from the science of
history. We are not studying or attempting to study the working of those
blind forces which, we are told, are operating on crowds of obscure people,
shaking principalities and powers, and compelling reasonable men to bring events
to pass in an order laid down by philosophers.

The men whose names are found in the history of science are not mere
hypothetical constituents of a crowd, to be reasoned upon only in masses. We
recognise them as men like ourselves, and their actions and thoughts, being
more free from the influence of passion, and recorded more accurately than those
of other men, are all the better materials for the study of the calmer parts
of human nature.

But the history of science is not restricted to the enumeration of success-
ful investigations. It has to tell of unsuccessful inquiries, and to explain why
some of the ablest men have failed to find the key of knowledge, and how
the reputation of others has only given a firmer footing to the errors into
which they fell.

The history of the development, whether normal or abnormal, of ideas is
of all subjects that in which we, as thinking men, take the deepest interest.
But when the action of the mind passes out of the intellectual stage, in which
truth and error are the alternatives, into the more violently emotional states of

32—2

INTRODUCTORY LECTURE ON EXPERIMENTAL PHYSICS.

anger and passion, malice and envy, fury and madness; the student of science,
though he is obliged to recognise the powerful influence which these wild forces
have exercised on mankind, is perhaps in some measure disqualified from pursuing
the study of this part of human nature.

But then how few of us are capable of deriving profit from such studies.
W« cannot enter into full sympathy with these lower phases of our nature
without losing some of that antipathy to them which is our surest safeguard
against a reversion to a meaner type, and we gladly return to the company of
those illustrious men who by aspiring to noble ends, whether intellectual or
practical, have risen above the region of storms into a clearer atmosphere, where
there is no misrepresentation of opinion, nor ambiguity of expression, but where
one mind comes into closest contact with another at the point where both
approach nearest to the truth.

I propose to lecture during this term on Heat, and, as our facilities for
experimental work are not yet fully developed, I shall endeavour to place before
you the relative position and scientific connexion of the different branches of the
science, rather than to discuss the details of experimental methods.

We shall begin with Thermometry, or the registration of temperatures, and
Calorimetry, or the measurement of quantities of heat. We shall then go on
to Thermodynamics, which investigates the relations between the thermal pro-
perties of bodies and their other dynamical properties, in so far as these relations
may be traced without any assumption as to the particular constitution of these
bodies:

The principles of Thermodynamics throw great light on all the phenomena
of nature, and it is probable that many valuable applications of these principles
have yet to be made ; but we shall have to point out the limits of this
science, and to shew that many problems in nature, especially those in which
the Dissipation of Energy comes into play, are not capable of solution by the
principles of Thermodynamics alone, but that in order to understand them, we
are obliged to form some more definite theory of the constitution of bodies.

Two theories of the constitution of bodies have struggled for victory with
various fortunes since the earliest ages of speculation : one is the theory of a
universal plenum, the other is that of atoms and void.

The theory of the plenum is associated with the doctrine of mathe-
matical continuity, and its mathematical methods are those of the Differential

INTRODUCTORY LECTURE ON EXPERIMENTAL PHYSICS. 253

Calculus, which is the appropriate expression of the relations of continuous
quantity.

The theory of atoms and void leads us to attach more importance to the
doctrines of integral numbers and definite proportions ; but, in applying dynamical
principles to the motion of immense numbers of atoms, the limitation of our
faculties forces us to abandon the attempt to express the exact history of each
atom, and to be content with estimating the average condition of a group of
atoms large enough to be visible. This method of dealing with groups of atoms,
which I may call the statistical method, and which in the present state of our
knowledge is the only available method of studying the properties of real bodies,
involves an abandonment of strict dynamical principles, and an adoption of the
mathematical methods belonging to the theory of probability. It is probable
that important results will be obtained by the application of this method, which
is as yet little known and is not familiar to our minds. If the actual history
of Science had been different, and if the scientific doctrines most familiar to
us had been those which must be expressed in this way, it is possible that
we might have considered the existence of a certain kind of contingency a self-
evident truth, and treated the doctrine of -philosophical necessity as a mere
sophism.

About the beginning of this century, the properties of bodies were investi-
gated by several distinguished French mathematicians on the hypothesis that
they are systems of molecules in equilibrium. The somewhat unsatisfactory nature
of the results of these investigations produced, especially in this country, a
reaction in favour of the opposite method of treating bodies as if they were,
so far at least as our experiments are concerned, truly continuous. This method,
in the hands of Green, Stokes, and others, has led to results, the value of which
does not at all depend on what theory we adopt as to the ultimate constitution
of bodies.

One very important result of the investigation of the properties of bodies
on the hypothesis that they are truly continuous is that it furnishes us with
a test by which we can ascertain, by experiments on a real body, to what
degree of tenuity it must be reduced before it begins to give evidence that its
properties are no longer the same as those of the body in mass. Investigations
of this kind, combined with a study of various phenomena of diffusion and of
dissipation of energy, have recently added greatly to the evidence in favour of
the hypothesis that bodies are systems of molecules in motion.

L\, I INTRODUCTORY LBCTUBK ON EXPERIMENTAL PHYSICS.

I hope to be able to lay before you in the course of the term some of
tho evidence for the existence of molecules, considered as individual bodies
having definite properties. The molecule, as it is presented to the scientific
imagination, is a very different body from any of those with which experience
lias hitherto made us acquainted.

In the first place its mass, and the other constants which define its
properties, are absolutely invariable ; the individual molecule can neither grow
nor decay, but remains unchanged amid all the changes of the bodies of which
it may form a constituent.

In the second place it is not the only molecule of its kind, for there are
innumerable other molecules, whose constants ore not approximately, but abso-
lutely identical with those of the first molecule, and this whether they ore
found on the earth, in the sun, or in the fixed stars.

By what process of evolution the philosophers of the future will attempt
to account for this identity in the properties of such a multitude of bodies,
each of them unchangeable in magnitude, and some of them separated from
others by distances which Astronomy attempts in vain to measure, I cannot
conjecture. My mind is limited in its power of speculation, and I am forced
to believe that these molecules must have been made as they are from the
beginning of their existence.

I also conclude that since none of the processes of nature, during their
varied action on different individual molecules, have produced, in the course of
ages, the slightest difference between the properties of one molecule and those
of another, the history of whose combinations has been different, we cannot
ascribe either their existence or the identity of their properties to the operation
of any of those causes which we coll natural.

Is it true then that our scientific speculations have really penetrated
beneath the visible appearance of things, which seem to be subject to gene-
ration and corruption, and reached the entrance of that world of order and per-
fection, which continues this day as it was created, perfect in number and
measure and weight ?

We may be mistaken. No one has as yet seen or handled an individual
molecule, and our molecular hypothesis may, in its turn, be supplanted by some
new theory of the constitution of matter; but the idea of the existence of
unnumbered individual things, all alike and all unchangeable, is one which cannot
enter the human mind and remain without fruit.

INTRODUCTORY LECTURE ON EXPERIMENTAL PHYSICS. 255

But what if these molecules, indestructible as they are, turn out to be not
substances themselves, but mere affections of some other substance ?

According to Sir W. Thomson's theory of Vortex Atoms, the substance of
which the molecule consists is a uniformly dense plenum, the properties of which
are those of a perfect fluid, the molecule itself being nothing but a certain
motion impressed on a portion of this fluid, and this motion is shewn, by a
theorem due to Helmholtz, to be as indestructible as we believe a portion of
matter to be.

If a theory of this kind is true, or even if it is conceivable, our idea of
matter may have been introduced into our minds through our experience of those
systems of vortices which we call bodies, but which are not substances, but
motions of a substance ; and yet the idea which we have thus acquired of
matter, as a substance possessing inertia, may be truly applicable to that fluid
of which the vortices are the motion, but of whose existence, apart from the
vortical motion of some of its parts, our experience gives us no evidence
whatever.

It has been asserted that metaphysical speculation is a thing of the past,
and that physical science has extirpated it. The discussion of the categories of
existence, however, does not appear to be in danger of coming to an end in
our time, and the exercise of speculation continues as fascinating to every fresh
mind as it was in the days of Thales.

[From the Proceedings of the Cambridge Philosophical Society, VoL n.]

XLV. On the Solution of Electrical Problems by the Transformation

of Conjugate Functions*.

THE general problem in electricity is to determine a function which shall
have given values at the various surfaces which bound a region of space, and
which shall satisfy Laplace's partial differential equation at every point within
this region. The solution of this problem, when the conditions are arbitrarily
given, is beyond the power of any known method, but it is easy to find any
number of functions which satisfy Laplace's equation, and from any one of these
we may find the form of a system of conductors for which the function is a
solution of the problem.

The only known method for transforming one electrical problem into another
is that of Electric Inversion, invented by Sir William Thomson ; but in problems
involving only two dimensions, any problem of which we know the solution
may be made to furnish an inexhaustible supply of problems which we can solva

The condition that two functions a and /8 of x and y may be conjugate is

a + S^lfi = F (x + v^T y).

This condition may be expressed in the form of the two equations

da d/3 da. d/3

~T~ ~~ ~T~ •* "» J~ T ~7~ = U.

dx dy ay dx

If a denotes the " potential function," $ is the " function of induction."
As examples of the method, the theory of Thomson's Guard Ring and that
of a wire grating, used as an electric screen, were illustrated by drawings of
the lines of force and equipotential surfaces.

* [The author's treatment of this subject and a full explanation of the examples mentioned in
the text will be found in the chapter on Conjugate Functions in his treatise on Electricity and
Magnetism.]

[From the Proceedings of ike London Mathematical Society, Vol. in. No. 34.]

XL VI. On the Mathematical Classification of Physical Quantities.

THE first part of the growth of a physical science consists in the discovery
of a system of quantities on which its phenomena may be conceived to depend.
The next stage is the discovery of the mathematical form of the relations
between these quantities. After this, the science may be treated as a mathe-
matical science, and the verification of the laws is effected by a theoretical
investigation of the conditions under which certain quantities can be most
accurately measured, followed by an experimental realisation of these conditions,
and actual measurement of the quantities.

It is only through the progress of science in recent times that we have
become acquainted with so large a number of physical quantities that a classifi-
cation of them is desirable.

One very obvious classification of quantities is founded on that of the
sciences in which they occur. Thus temperature, pressure, density, specific heat,
latent heat, &c., are quantities occurring in the theory of the action of heat on
bodies.

But the classification which I now refer to is founded on the mathematical
or formal analogy of the different quantities, and not on the matter to which
they belong. Thus a finite straight line, a force, a velocity of rotation, &c.,
are quantities, differing in their physical nature, but agreeing in their mathe-
matical form. We may distinguish the two methods of classification by calling
the first a physical, and the second a mathematical classification of quantities.

A knowledge of the mathematical classification of quantities is of great use
both to the original investigator and to the ordinary student of the science.
The most obvious case is that in which we learn that a certain system of
quantities in a new science stand to one another in the same mathematical

VOL. ii. 33

258 MATHEMATICAL CLASSIFICATION

relations as a certain other system in an old science, which has already been
reduced to a mathematical form, and its problems solved by mathematicians.

Thus, when Mossotti observed that certain quantities relating to electro-
static induction in dielectrics had been shewn by Faraday to be analogous to
certain quantities relating to magnetic induction in iron and other bodies, he
was enabled to make use of the mathematical investigation of Poisson relative
to magnetic induction, merely translating it from the magnetic language into
the electric, and from French into Italian.

Another example, by no means so obvious, is that which was originally
pointed out by Sir William Thomson, of the analogy between problems in
attractions and problems in the steady conduction of heat, by the use of vhic-h
we are able to make use of many of the results of Fourier for heat in explaining
electrical distributions, and of all the results of Poisson in electricity in explaining
problems in heat.

But it is evident that all analogies of this kind depend on principles of a
more fundamental nature ; and that, if we had a true mathematical classification
of quantities, we should be able at once to detect the analogy between any
system of quantities presented to us and other systems of quantities in known
sciences, so that we should lose no time in availing ourselves of the mathe-
matical labours of those who had already solved problems essentially the same.

All quantities may be classed together in one respect, that they may be
defined by means of two factors, the first of which is a numerical quantity, and
the second is a standard quantity of the same kind with that to be defined.

Thus number may be said to rule the whole world of quantity, and the
four rules of arithmetic may be regarded as the complete equipment of the
mathematician.

Position and form, which were formerly supposed to be in the exclusive
possession of geometers, were reduced by Descartes to submit to the rules of
arithmetic by means of that ingenious scaffolding of co-ordinate axes which he
made the basis of his operations.

Since this great step was taken in mathematics, all quantities have been
treated in the same way, and presented to the mind by means of numbers, or
symbols which denote numbers, so that as soon as any science has been
thoroughly reduced to the mathematical form, the solution of problems in that
science, as a mental process, is supposed (at least by the outer world) to be
carried on without the aid of any of the physical ideas of the science.

OF PHYSICAL QUANTITIES. 259

I need not say that this is not true, and that mathematicians, in solving
physical problems, are very much aided by a knowledge of the science in which
the problems occur.

At the same time, I think that the progress of science, both in the way of
discovery, and in the way of diffusion, would be greatly aided if more attention
were paid in a direct way to the classification of quantities.

A most important distinction was drawn by Hamilton when he divided the
quantities with which he had to do into Scalar quantities, which are completely
represented by one numerical quantity, and Vectors, which require three numerical
quantities to define them.

The invention of the calculus of Quaternions is a step towards the know-
ledge of quantities related to space which can only be compared for its impor-
tance, with the invention of triple co-ordinates by Descartes. The ideas of this
calculus, as distinguished from its operations and symbols, are fitted to be of
the greatest use in all parts of science.

We may imagine another step in the advancement of science to be the
invention of a method, equally appropriate, of conceiving dynamical quantities.
As our conceptions of physical science are rendered more vivid by substituting
for the mere numerical ideas of Cartesian mathematics the geometrical ideas of
Hamiltonian mathematics, so in the higher sciences the ideas might receive a
still higher development if they could be expressed in language as appropriate
to dynamics as Hamilton's is to geometry.

Another advantage of such a classification is, that it guides us in the use
of the four rules of arithmetic. We know that we must not apply the rules
of addition or subtraction unless the quantities are of the same kind. In certain
cases we may multiply or divide one quantity by another, but in other cases
the result of the process is of no intellectual value.

It has been pointed out by Professor Rankine that the physical quantity
called Energy or Work can be conceived as the product of two factors in many
different ways.

The dimensions of this quantity are — ™- , where L, M, and T represent

the concrete units of length, time, and mass. If we divide the energy into two
factors, one of which contains L", both factors will be scalars. If, on the other
hand, both factors contain L, they will be both vectors. The energy itself is
always a scalar quantity.

33—2

MATHEMATICAL CLASSIFICATION

Thus, if we take the mass and the square of the velocity as the factors,
M » done in the ordinary definitions of vis-viva or kinetic energy, both factors
are acalar, though one of them, the square of the velocity, has no distinct physical

meaning.

Another division into apparently scalar factors is that into volume and
hydrostatic pressure, though here we must consider the volume, not in itself,
but as a quantity subject to increase and diminution, and this change of volume
can only occur at the surface, and is due to a variation of the surface in the
direction of the normal, so that it is not a scalar but a vector quantity. The pressure
also, though the abstract conception of a hydrostatic pressure is scalar, must be
conceived as applied at a surface, and thus becomes a directed quantity or vector.

The division of the energy into vector factors affords results always capable
of satisfactory interpretation. Of the two factors, one is conceived as a tendency
towards a certain change, and the other as that change itself.

Thus the elementary definition of Work regards it as the product of a force
into the distance through which its point of application moves, resolved in the
direction of the force. In the language of Quaternions, it is the scalar part of
the product of the force and the displacement.

These two vectors, the force and the displacement, may be regarded as
types of many other pairs of vectors, the products of which have for their
scalar part some form of energy.

Thus, instead of dividing kinetic energy into the factors " mass " and " square
of velocity," the latter of which has no meaning, we may divide it into "mo-
mentum " and " velocity," two vectors which, in the dynamics of a particle, are
in the same direction, but, in generalized dynamics, may be in different direc-
tions, so that in taking their product we must remember the rule for finding
the scalar part of it.

But it is when we have to deal with continuous bodies, and quantities
distributed in space, that the general principle of the division of energy into
two factors is most clearly seen.

When we regard energy as residing intrinsically in a body, we may measure
its intensity by the amount contained in unit of volume. This is, of course, a
scalar quantity.

Of the factors which compose it, one is referred to unit of length, and the
other to unit of area. This gives what I regard as a very important distinction
among vector quantities.

OF PHYSICAL QUANTITIES. 261

Vectors which are referred to unit of length I shall call Forces, using the
word in a somewhat generalized sense, as we shall see. The operation of taking
the integral of the resolved part of a force in the direction of a line for
every element of that line, has always a physical meaning. In certain cases the
result of the integration is independent of the path of the line between its
extremities. The result is then called a Potential.

Vectors which are referred to unit of area I shall call Fluxes. The opera-
tion of taking the integral of the resolved part of a flux perpendicular to a
surface for every element of the surface has always a physical meaning. In
certain cases the result of the integration over a closed surface is independent,
within certain restrictions, of the position of the surface. The result then
expresses the Quantity of some kind of matter, either existing within the surface,
or flowing out of it, according to the physical nature of the flux.

In many physical cases, the force and the flux are always in the same
direction, and proportional to each other. The one is therefore used as the
measure of the other, their symbols degenerate into one, and their ideas become
confounded together. One of the most important mathematical results of the
discovery of substances having different physical properties in different directions,
has been to enable us to distinguish between the force and the flux, by letting
us see that their directions may be different.

Thus, in the ordinary theory of fluids, in which the only motion considered
is that which we can directly perceive, we may define the velocity equally well
in two different ways. We may define it with reference to unit of length, as
the number of such units described by a particle in unit of time ; or we may
define it with reference to unit of area, as the volume of the fluid which
passes through unit of area in unit of time. If defined in the first way, it
belongs to the category of forces ; if defined in the second way, to the category
of fluxes.

But if we endeavour to develope a more complete theory of fluids, which
shall take into account the facts of diffusion, where one fluid has a different
velocity from another in the same place ; or if we accept the doctrine, that the
molecules of a fluid, in virtue of the heat of the substance, are in a state of
agitation ; then though we may give a definition of the velocity of a single mole-
cule with reference to unit of length, we cannot do so for the fluid ; and the only
way we have of defining the motion of the fluid is by considering it as a flux,
and measuring it by the mass of the fluid which flows through unit of area.

,, • MATHEMATICAL CLASSIFICATION

Thw distinction in still more necessary when we come to heat and electricity.
The flux of beat or of electricity cannot be even thought of in any way except
M the quantity which flows through a given area in a given time. To form a
conception of the velocity, properly so called, of either agent, would require
us to conceive heat or electricity as a continuous substance having a known

density.

We must therefore consider these quantities as fluxes. The forces corre-
sponding to them are, in the case of heat, the rate of variation of temperature.
and, in the case of electricity, the rate of variation of potential.

I have said enough to point out the distinction between forces and fluxes.
In statical electricity the resultant force at a point is the rate of variation of
potential, and the flux is a quantity, hitherto confounded with the force, which
I have called the electric displacement.

In magnetism the resultant force is also the rate of variation of the
potential, and the flux is what Faraday calls the magnetic induction, and is
measured, as Thomson has shewn, by the force on a unit pole placed in a narrow
crevasse cut perpendicular to the direction of magnetization of the magnet. I
shall not detain the Society with the explanation of these quantities, but I
must briefly state the nature of a ratio of a force to a flux in its most general
form.

When one vector is a function of another vector, the ratio of the first to
the second is, in general, a quaternion which is a function of the second
vector.

When, if the second vector varies in magnitude only, the first is always
proportional to it, and remains constant in direction, we have the important
case of the function being a linear one. The first vector is then said to be a
linear and vector function of the second. If a, /3, y are the Cartesian compo-
nents of the first vector, and a, b, c those of the second, then

where the coefficients p, q, r are constants. When the p'a are equal to the
corresponding 5*8, the function is said to be self-conjugate. It may then be
represented geometrically as the relation between the radius vector from tin-
centre of an ellipsoid, and the perpendicular on the tangent plane.

OF PHYSICAL QUANTITIES. 263

We may remark that even here, where we may seem to have reached a
purer air, uncontarninated by physical applications, one vector is essentially a
line, while the other is defined as the normal to a plane, as in all the other
pairs of vectors already mentioned*.

Another distinction among physical vectors is founded on a different prin-
ciple, and divides them into those which are defined with reference to translation
and those which are defined with reference to rotation. The remarkable analogies
between these two classes of vectors is well pointed out by Poins6t in his
treatise on the motion of a rigid body. But the most remarkable illustration of
them is derived from the two different ways in which it is possible to contem-
plate the relation between electricity and magnetism.

Helmholtz, in his great paper on Vortex Motion, has shewn how to construct
an analogy between electro-magnetic and hydro-kinetic phenomena, in which
magnetic force is represented by the velocity of the fluid, a species of translation,
while electric current is represented by the rotation of the elements of the fluid.
He does not propose this as an explanation of electro-magnetism ; for though
the analogy is perfect in form, the dynamics of the two systems are different.

According to Ampere and all his followers, however, electric currents are
regarded as a species of translation, and magnetic force as depending on rotation.
I am constrained to agree with this view, because the electric current is
associated with electrolysis, and other undoubted instances of translation, while
magnetism is associated with the rotation of the plane of polarization of light,
which, as Thomson has shewn, involves actual motion of rotation.

The Hamiltonian operator FV, applied to any vector function, converts it
from translation to rotation, or from rotation to translation, according to the kind
of vector to which it is applied.

I shall conclude by proposing for the consideration of mathematicians certain
phrases to express the results of the Hamiltonian operator A. I should be
greatly obliged to anyone who can give me suggestions on this subject, as I
feel that the onomastic power is very faint in me, and that it can be success-
fully exercised only in societies.

V is the operation ; _ +y _ + fc - ,

where i, j, k are unit vectors parallel to x, y, z respectively. The result of

* The subject of linear equations in quaternions has been developed by Professor Tait, in several
communications to the Royal Society of Edinburgh.

MATHEMATICAL CLASSIFICATION

performing thb operation twice on any subject is the well known operation (of

The discovery of the square root of this operation is due to Hamilton ;
but most of the applications here given, and the whole development of tin-
theory of this operation, are due to Prof. Tait, and are given in several papers,
of which the first is in the Proceedings of the Royal Society of Edu,l»ir<j1i,
April 28, 1 862, and the most complete is that on " Green's and other allied
Theorems," Transactions of the Royal Society of Edinburgh, 1869 — 70.

And, first, I propose to call the result of V* (Laplace's operation with the
negative sign) the Concentration of the quantity to which it is applied.

For if Q be a quantity, either scalar or vector, which is a function of the
position of a point; and if we find the integral of Q taken throughout the
volume of a sphere whose radius is r; then, if we divide this by the volume
of the sphere, we shall obtain Q, the mean value of Q within the sphere. If
Qt is the value of Q at the centre of the sphere, then, when r is small,

or the value of Q at the centre of the sphere exceeds the mean value of Q
within the sphere by a quantity depending on the radius, and on V'Q. Since,
therefore, V'Q indicates the excess of the value of Q at the centre above its
mean value in the sphere, I shall call it the concentration of Q.

If Q is a scalar quantity, its concentration is, of course, also scalar. Thus,
if Q is an electric potential, VQ is the density of the matter which produces
the potential.

If Q is a vector quantity, then both Q, and Q are vectors, and V'Q is
also a vector, indicating the excess of the uniform force Q, applied to the whole
substance of the sphere above the resultant of the actual force Q acting on all
the parts of the sphere.

Let us next consider the Hamiltonian operator V. First apply it to a scalar
function P. The quantity VP is a vector, indicating the direction in which P
decreases most rapidly, and measuring the rate of that decrease. I venture, with
much diffidence, to call this the slope of P. Lame* calls the magnitude of W
the "first differential parameter" of P, but its direction does not enter into his

OF PHYSICAL QUANTITIES. 265

conception. We require a vector word, which shall indicate both direction and
magnitude, and one not already employed in another mathematical sense. I have
taken the liberty of extending the ordinary sense of the word slope from topo-
graphy, where only two independent variables are used, to space of three
dimensions.

If <r represents a vector function, Vcr may contain both a scalar and a
vector part, which may be written SVa- and Fv<r.

I propose to call the scalar part the Convergence of cr, because, if a closed
surface be described about any point, the surface integral of a; which expresses
the effect of the vector cr considered as an inward flux through the surface, is
equal to the volume integral of Sv<r throughout the enclosed space. I think,
therefore, that the convergence of a vector function is a very good name for the
effect of that vector function in carrying its subject inwards towards a point.

But Vcr has, in general, also a vector portion, and I propose, but with
great diffidence, to call this vector the Curl or Version of the original vector
function.

It represents the direction and magnitude of the rotation of the subject
matter carried by the vector cr. I have sought for a word which shall neither,
like Rotation, Whirl, or Twirl, connote motion, nor, like Twist, indicate a helical
or screw structure which is not of the nature of a vector at all.

/' \ v

( " J -> V

CONVERGENCE. CURL. CONVERGENCE AND CURL.

If we subtract from the general value of the vector function cr its value
cr0 at the point P, then the remaining vector cr — o-0 will, when there is pure
convergence, point towards P. When there is pure curl, it will point tan-
gentially round P ; and when there is both convergence and curl, it will point
in a spiral manner.

The following statements are true : —

The slope of a scalar function has no curl.

The curl of a vector function has no convergence.

VOL. u. 34

MATUKMATKAL CLASSIFICATION OF PHYSICAL QUANTITIES.

The convergence of the slope of a scalar function is its concentration.

The concentration of a vector function is the slope of its convergence,
together with the curl of its curl.

The quaternion expressions, of which the above statements are a transla
won given by Prof. Tait, in his paper in the Proceedings of the Royal &••
of fhfinburyh, April 28th, 1862; but for the more complete mathematical treat-
ment of the operator v, see a very able paper by Prof. Tait, "On Green's
and other Allied Theorems" (Transactions of the Royal Society of Edinlmnjh,
1870), and another paper in the Proceedings of the Royal Society of E<1
for 1870—71, p. 318.

[From the Proceedings of the Royal Institution of Great Britain, Vol. vi.]

XL VII. On Colour Vision.

ALL vision is colour vision, for it is only by observing differences of colour
that we distinguish the forms of objects. I include differences of brightness
or shade among differences of colour.

It was in the Royal Institution, about the beginning of this century, that
Thomas Young made the first distinct announcement of that doctrine of the
vision of colours which I propose to illustrate. We may state it thus: — We
are capable of feeling three different colour-sensations. Light of different kinds
excites these sensations in different proportions, and it is by the different
combinations of these three primary sensations that all the varieties of visible
colour are produced. In this statement there is one word on which we must
fix our attention. That word is, Sensation. It seems almost a truism to say
that colour is a sensation ; and yet Young, by honestly recognising this ele-
mentary truth, established the first consistent theory of colour. So far as I
know, Thomas Young was the first who, starting from the well-known fact that
there are three primary colours, sought for the explanation of this fact, not in
the nature of light, but in the constitution of man. Even of those who have
written on colour since the time of Young, some have supposed that they ought
to study the properties of pigments, and others that they ought to analyse
the rays of light. They have sought for a knowledge of colour by examining
something in external nature — something out of themselves.

Now, if the sensation which we call colour has any laws, it must be
something in our own nature which determines the form of these laws ; and I
need not tell you that the only evidence we can obtain respecting ourselves
is derived from consciousness.

The science of colour must therefore be regarded as essentially a mental

34—2

COLOUR VISION.

tBMPOfr It differ* from the greater part of what is called mental science in
the Urge uae which it makes of the physical sciences, and in particular of
optics and anatomy. But it gives evidence that it is a mental science by the
numerous illustrations which it furnishes of various operations of the mind.

In this place we always feel on firmer ground when we are dealing with
pin-weal science. I shall therefore begin by shewing how we apply the dis-
coveries of Newton to the manipulation of light, so as to give you an oppor-
tunity of feeling for yourselves the different sensations of colour.

Before the time of Newton, white light was supposed to be of all ki
things the purest. When light appears coloured, it was supposed to have
baoome contaminated by coming into contact with gross bodies. We may still
think white light the emblem of purity, though Newton has taught us that
its purity does not consist in simplicity.

We now form the prismatic spectrum on the screen [exhibited]. These are
the simple colours of which white light is always made up. We can distin.
a great many hues in passing from the one end to the other; but it is when
we employ powerful spectroscopes, or avail ourselves of the labours of those
who have mapped out the spectrum, that we become aware of the immense
multitude of different kinds of light, every one of which has been the object
of special study. Every increase of the power of our instruments increas-
the same proportion the number of lines visible in the spectrum.

All light, as Newton proved, is composed of these rays taken in difl'i
proportions. Objects which we call coloured when illuminated by white light,
make a selection of these rays, and our eyes receive from them only a
• •I' the light which falls on them. But if they receive only the pure rays of
u single colour of the spectrum they can appear only of that colour. It I
place this disk, containing alternate quadrants of red and green paper, in the
red rays, it appears all red, but the red quadrants brightest. If I place it in
the green rays both papers appear green, but the red paper is now the dan
This, then, is the optical explanation of the colours of bodies when illuminated
with white light. They separate the white light into its component parts,
absorbing some and scattering others.

Here are two transparent solutions [exhibited]. One appears yellow, it
contains bichromate of potash; the other appears blue, it contains sulphate of
copper. If I transmit the light of the electric lamp through the two solutions
at once, the spot on the screen appears green. By means of the spectrum we

COLOUR VISION. 269

shall be able to explain this. The yellow solution cuts off the blue end of
the spectrum, leaving only the red, orange, yellow, and green. The blue solution
cuts off the red end, leaving only the green, blue, and violet. The only light
which can get through both is the green light, as you see. In the same way
most blue and yellow paints, when mixed, appear green. The light which falls
on the mixture is so beaten about between the yellow particles and the blue
that the only light which survives is the green. But yellow and blue light
when mixed do not make green, as you will see if we allow them to fall on
the same part of the screen together.

It is a striking illustration of our mental processes that many persons have
not only gone on believing, on the evidence of the mixture of pigments, that
blue and yellow make green, but that they have even persuaded themselves
that they could detect the separate sensations of blueness and of yellowness
in the sensation of green.

We have availed ourselves hitherto of the analysis of light by coloured
substances. We must now return, still under the guidance of Newton, to the
prismatic spectrum. Newton not only

" Untwisted all the shining robe of day,"

but shewed how to put it together again. We have here a pure spectrum,
but instead of catching it on a screen we allow it to pass through a lens
large enough to receive all the coloured rays. These rays proceed, according
to well-known principles in optics, to form an image of the prism on a screen
placed at the proper distance. This image is formed by rays of all colours,
and you see the result is white. But if I stop any of the coloured rays the
image is no longer white, but coloured ; and if I only let through rays of
one colour, the image of the prism appears of that colour.

I have here an arrangement of slits by which I can select one, two, or
three portions of the light of the spectrum, and allow them to form an image
of the prism while all the rest are stopped. This gives me a perfect command
of the colours of the spectrum, and I can produce on the screen every possible
shade of colour by adjusting the breadth and the position of the slits through
which the light passes. I can also, by interposing a lens in the passage of
the light, shew you a magnified image of the slits, by which you will see the
different kinds of light which compose the mixture.

The colours are at present red, green, and blue, and the mixture of the

;;, COLOUR VISION.

three colour* b, as you see, nearly white. Let us try the effect of mixing
two of those colours. Red and blue form a fine purple or crimson, green and
blue form a sea-green or sky-blue, red and green form a yellow.

Here Again we have a fact not universally known. No painter, wishing to
produce a fine yellow, mixes his red with his green. The result would be a
T«T dirty drab colour. He is furnished by nature with brilliant yellow pigments,
and he takes advantage of these. When he mixes red and green paint, the
red light scattered by the red paint is robbed of nearly all its brightnee
getting among particles of green, and the green light fares no better, fur it
is sure to fall in with particles of red paint. But when the pencil with which
we paint is composed of the rays of light, the effect of two coats of colour
is very different. The red and the green form a yellow of great splendour,
which may be shewn to be as intense as the purest yellow of the spectrum.

I have now arranged the slits to transmit the yellow of the spectrum.
You see it is similar in colour to the yellow formed by mixing red and gi
It differs from the mixture, however, in being strictly homogeneous in a ]»hy
point of view. The prism, as you see, does not divide it into two por
as it did the mixture. Let us now combine this yellow with the blue of the
spectrum. The result is certainly not green ; we may make it pink if our
yellow is of a warm hue, but if we choose a greenish yellow we can produce
a good white.

You have now seen the most remarkable of the combinations of colours—
the others differ from them in degree, not in kind. I must now ask you to
think no more of the physical arrangements by which you were enabled to see
these colours, and to concentrate your attention upon the colours you saw, that
is to say on certain sensations of which you were conscious. We are here
surrounded by difficulties of a kind which we do not meet with in purely
physical inquiries. We can all feel these sensations, but none of us can des«
them. They are not only private property, but they are incommunicable. \\ >•
have names for the external objects which excite our sensations, but not for
the sensations themselves.

When we look at a broad field of uniform colour, whether it is really
simple or compound, we find that the sensation of colour appears to our
sciousness as one and indivisible. We cannot directly recognise the elemen
sensations of which it is composed, as we can distinguish the component notes

COLOUR VISION. 271

of a musical chord. A colour, therefore, must be regarded as a single thing,
the quality of which is capable of variation.

To bring a quality within the grasp of exact science, we must conceive it
as depending on the values of one or more variable quantities, and the first
step in our scientific progress is to determine the number of these variables
which are necessary and sufficient to determine the quality of a colour. We
do not require any elaborate experiments to prove that the quality of colour
can vary in three and only in three independent ways.

One way of expressing this is by saying, with the painters, that colour
may vary in hue, tint, and shade.

The finest example of a series of colours, varying in hue, is the spectrum
itself. A difference in hue may be illustrated by the difference between adjoining
colours in the spectrum. The series of hues in the spectrum is not complete ;
for, in order to get purple hues, we must blend the red and the blue.

Tint may be defined as the degree of purity of a colour. Thus, bright
yellow, buff, and cream-colour, form a series of colours of nearly the same hue,
but varying in tint. The tints, corresponding to any given hue, form a series,
beginning with the most pronounced colour, and ending with a perfectly neutral
tint.

Shade may be defined as the greater or less defect of illumination. If
we begin with any tint of any hue, we can form a gradation from that colour
to black, and this gradation is a series of shades of that colour. Thus we
may say that brown is a dark shade of orange.

The quality of a colour may vary in three different and independent ways.
We cannot conceive of any others. In fact, if we adjust one colour to another,
so as to agree in hue, in tint, and in shade, the two colours are absolutely
indistinguishable. There are therefore three, and only three, ways in which a
colour can vary.

I have purposely avoided introducing at this stage of our inquiry anything
which may be called a scientific experiment, in order to shew that we may
determine the number of quantities upon which the variation of colour depends
by means of our ordinary experience alone.

Here is a point in this room : if I wish to specify its position, I may do
so by giving the measurements of three distances — namely, the height above the
floor, the distance from the wall behind me, and the distance from the wall
at my left hand.

:j COLOUR M8IOX.

This is only one of many ways of stating the position of a point, but
it is one of the most convenient. Now, colour also depends on three things.
If we call these the intensities of the three primary colour sensations, and it'
we are able in any way to measure these three intensities, we may consider
the colour as specified by these three measurements. Hence the specification of
a colour agrees with the specification of a point in the room in depending on
three measurements.

Let us go a step farther and suppose the colour sensations measured on
some scale of intensity, and a point found for which the three distances, or
co-ordinates, contain the same number of feet as the sensations contain dcL
of intensity. Then we may say, by a useful geometrical convention, that the
colour is represented, to our mathematical imagination, by the point so found
in the room; and if there are several colours, represented by several points, the
chromatic relations of the colours will be represented by the geometrical rela-
tions of the points. This method of expressing the relations of colours is a
great help to the imagination. You will find these relations of colours s*
in an exceedingly clear manner in Mr Benson's Manual of Colour one of the
very few books on colour in which the statements are founded on legitimate
experiments.

There is a still more convenient method of representing the relations of
colours by means of Young's triangle of colours. It is impossible to repn
on a plane piece of paper every conceivable colour, to do this requires space of
three dimensions. If, however, we consider only colours of the same shade —
that is, colours in which the sum of the intensities of the three sensations is
the same, then the variations in tint and in hue of all such colours may be
represented by points on a plane. For this purpose we must draw a plane
cutting off equal lengths from the three lines representing the primary sensat
The part of this plane within the space in which we have been distributing our
colours will be an equilateral triangle. The three primary colours will be at the
three angles, white or gray will be in the middle, the tint or degree of purity
of any colour will be expressed by its distance from the middle point, and its
hue will depend on the angular position of the line which joins it with the
middle point.

Thus the ideas of tint and hue can be expressed geometrically on You
triangle. To understand what is meant by shade we have only to suppose the
illumination of the whole triangle increased or diminished, so that by means of

COLOUR VISION. 273

this adjustment of illumination Young's triangle may be made to exhibit every
variety of colour. If we now take any two colours in the triangle and mix
them in any proportions, we shall find the resultant colour in the line joining
the component colours at the point corresponding to their centre of gravity.

I have said nothing about the nature of the three primary sensations, or
what particular colours they most resemble. In order to lay down on paper
the relations between actual colours, it is not necessary to know what the
primary colours are. We may take any three colours, provisionally, as the
angles of a triangle, and determine the position of any other observed colour
with respect to these, so as to form a kind of chart of colours.

Of all colours which we see, those excited by the different rays of the
prismatic spectrum have the greatest scientific importance. All light consists
either of some one kind of these rays, or of some combination of them. The
colours of all natural bodies are compounded of the colours of the spectrum. If
therefore we can form a chromatic chart of the spectrum, expressing the relations
between the colours of its different portions, then the colours of all natural
bodies will be found within a certain boundary on the chart defined by the
positions of the colours of the spectrum.

But the chart of the spectrum will also help us to the knowledge of the
nature of the three primary sensations. Since every sensation is essentially a
positive thing, every compound colour-sensation must be within the triangle of
which the primary colours are the angles. In particular, the chart of the spec-
trum must be entirely within Young's triangle of colours, so that if any colour
in the spectrum is identical with one of the colour-sensations, the chart of the
spectrum must be in the form of a line having a sharp angle at the point
corresponding to this colour.

I have already shewn you how we can make a mixture of any three of
the colours of the spectrum, and vary the colour of the mixture by altering
the intensity of any of the three components. If we place this compound
colour side by side with any other colour, we can alter the compound colour
till it appears exactly similar to the other. This can be done with the greatest
exactness when the resultant colour is nearly white. I have therefore constructed
an instrument which I may call a colour-box, for the purpose of making
matches between two colours. It can only be used by one observer at a time,
and it requires daylight, so I have not brought it with me to-night. It is
nothing but the realisation of the construction of one of Newton's propositions

VOL. n. 35

COLOUR VISION.

in

hi* Lectume* Optica, where he shews how to take a beam of light, to
separate it into its components, to deal with these components as we please by
means of slita, and afterwards to unite them into a beam again. The observer
looks into the box through a small slit. He sees a round field of light con-
sisting of two semicircles divided by a vertical diameter. The semicircle on the
left consists of light which has been enfeebled by two reflexions at the surface
of glass. That on the right is a mixture of colours of the spectrum, the positions
and intensities of which are regulated by a system of slits.

The observer forms a judgment respecting the colours of the two semicircles.
Suppose he finds the one on the right hand redder than the other, he says so,
and the operator, by means of screws outside the box, alters the breadth of
one of the slits, so as to make the mixture less red; and so on, till the right
semicircle is made exactly of the same appearance as the left, and the line of
separation becomes almost invisible.

When the operator and the observer have worked together for some time,
they get to understand each other, and the colours are adjusted much more
rapidly than at first.

When the match is pronounced perfect, the positions of the slits, as indicated
by a scale, are registered, and the breadth of each slit is carefully measured by
means of a gauge. The registered result of an observation is called a " colour
equation." It asserts that a mixture of three colours is, in the opinion of the
observer (whose name is given), identical with a neutral tint, which we shall call
Standard White. Each colour is specified by the position of the slit on the
scale, which indicates its position in the spectrum, and by the breadth of the
slit, which is a measure of its intensity.

In order to make a survey of the spectrum we select three points for
purposes of comparison, and we call these the three Standard Colours. The
standard colours are selected on the same principles as those which guide the
engineer in selecting stations for a survey. They must be conspicuous and
invariable and not in the same straight line.

In the chart of the spectrum you may see the relations of the various
colours of the spectrum to the three standard colours, and to each other. It
is manifest that the standard green which I have chosen cannot be one of the
true primary colours, for the other colours do not all lie within the triangle
formed by joining them. But the chart of the spectrum may be described as con-
sisting of two straight lines meeting in a point. This point corresponds to a green

COLOUB VISION. 275

about a fifth of the distance from 6 towards F. This green has a wave length
of about 510 millionths of a millimetre by Ditscheiner's measurement. This green
is either the true primary green, or at least it is the nearest approach to it
which we can ever see. Proceeding from this green towards the red end of
the spectrum, we find the different colours lying almost exactly in a straight
line. This indicates that any colour is chromatically equivalent to a mixture of
any two colours on opposite sides of it, and in the same straight line. The
extreme red is considerably beyond the standard red, but it is in the same
straight line, and therefore we might, if we had no other evidence, assume the
extreme red as the true primary red. We shall see, however, that the true
primary red is not exactly represented in colour by any part of the spectrum.
It lies somewhat beyond the extreme red, but in the same straight line.

On the blue side of primary green the colour equations are seldom so
accurate. The colours, however, lie in a line which is nearly straight. I have
not been able to detect any measurable chromatic difference between the extreme
indigo and the violet. The colours of this end of the spectrum are represented
by a number of points very close to each other. We may suppose that the
primary blue is a sensation differing little from that excited by the parts of the
spectrum near G.

Now, the first thing which occurs to most people about this result is that
the division of the spectrum is by no means a fair one. Between the red and
the green we have a series of colours apparently very different from either, and
having such marked characteristics that two of them, orange and yellow, have
received separate names. The colours between the green and the blue, on the
other hand, have an obvious resemblance to one or both of the extreme colours,
and no distinct names for these colours have ever become popularly recognised.

I do not profess to reconcile this discrepancy between ordinary and scientific
experience. It only shews that it is impossible by a mere act of introspection
to make a true analysis of our sensations. Consciousness is our only authority ;
but consciousness must be methodically examined in order to obtain any trust-
worthy results.

I have here, through the kindness of Professor Huxley, a picture of the
structure upon which the light falls at the back of the eye. There is a minute
structure of bodies like rods and cones or pegs, and it is conceivable that the
mode in which we become aware of the shapes of things is by a consciousness
which differs according to the particular rods on the ends of which the light

35—2

.7''. COLOUR VISION.

falls, just as the pattern on the web formed by a Jacquard loom depends on the
mode in which the perforated cards act on the system of moveable rods in that
machine. In the eye we have on the one hand light falling on this wonderful
structure, and on the other hand we have the sensation of sight. We cannot
compare these two things ; they belong to opposite categories. The whole of
Metaphysics lies like a great gulf between them. It is possible that discoveries
in physiology may be made by tracing the course of the nervous disturbance

"Up the fine fibres to the sentient brain;"

but this would make us no wiser than we are about those colour-sensations
which we can only know by feeling them ourselves. Still, though it is impossible
to become acquainted with a sensation by the anatomical study of the organ
with which it is connected, we may make use of the sensation as a means of
investigating the anatomical structure.

A remarkable instance of this is the deduction of Helmholtz's theory of
the structure of the retina from that of Young with respect to the sensation
of colour. Young asserts that there are three elementary sensations of colour;
Helmholtz asserts that there are three systems of nerves in the retina, each
of which has for its function, when acted on by light or any other disturbing
agent, to excite in us one of these three sensations.

No anatomist has hitherto been able to distinguish these three systems of
nerves by microscopic observation. But it is admitted in physiology that the
only way in which the sensation excited by a particular nerve can vary is by
degrees. of intensity. The intensity of the sensation may vary from the faintest
impression up to an insupportable pain ; but whatever be the exciting cause,
the sensation will be the same when it reaches the same intensity. If this
doctrine of the function of a nerve be admitted, it is legitimate to reason from
the fact that colour may vary in three different ways, to the inference that
these three modes of variation arise from the independent action of three differ-
ent nerves or sets of nerves.

Some very remarkable observations on the sensation of colour have been
made by M. Sigmund Exner in Professor Helmholtz's physiological laboratory at
Heidelberg. While looking at an intense light of a brilliant colour, he exposed
his eye to rapid alternations of light and darkness by waving his fingers before
his eyes. Under these circumstances a peculiar minute structure made its
appearance in the field of view, which many of us may have casually observed.

COLOUR VISION. 277

M. Exner states that the character of this structure is different according to
the colour of the light employed. When red light is used a veined structure
is seen ; when the light is green, the field appears covered with minute black
dots, and when the light is blue, spots are seen, of a larger size than the
dots in the green, and of a lighter colour.

Whether these appearances present themselves to all eyes, and whether
they have for their physical cause any difference in the arrangement of the
nerves of the three systems in Helmholtz's theory I cannot say, but I am
sure that if these systems of nerves have a real existence, no method is more
likely to demonstrate their existence than that which M. Exner has followed.

Colour Blindness.

The most valuable evidence which we possess with respect to colour vision
is furnished to us by the colour-blind. A considerable number of persons in
every large community are unable to distinguish between certain pairs of colours
which to ordinary people appear in glaring contrast. Dr Dalton, the founder
of the atomic theory of chemistry, has given us an account of his own case.

The true nature of this peculiarity of vision was first pointed out by Sir
John Herschel in a letter written to Dalton in 1832, but not known to the
world till the publication of Dalton's Life by Dr Henry. The defect consists
in the absence of one of the three primary sensations of colour. Colour-blind
vision depends on the variable intensities of two sensations instead of three.
The best description of colour-blind vision is that given by Professor Pole in
his account of his own case in the Phil. Trans., 1859.

In all cases which have been examined with sufficient care, the absent
sensation appears to resemble that which we call red. The point P on the
chart of the spectrum represents the relation of the absent sensation to the
9olours of the spectrum, deduced from observations with the colour box furnished
by Professor Pole.

If it were possible to exhibit the colour corresponding to this point on the
chart, it would be invisible, absolutely black, to Professor Pole. As it does
not lie within the range of the colours of the spectrum we cannot exhibit it ;
and, in fact, colour-blind people can perceive the extreme end of the spectrum
which we call red, though it appears to them much darker than to us, and
does not excite in them the sensation which we call red. In the diagram of
the intensities of the three sensations excited by different parts of the spectrum,

r, COLOUR VISION.

the upper figure, marked P, is deduced from the observations of Professor Pole ;
while the lower one. marked K, is founded on observations by a very accurate
observer of the normal type.

The only difference between the two diagrams is that in the upper one
the red curve is absent. The forms of the other two curves are nearly the
same for both observers. We have great reason therefore to conclude that tin-
colour sensations which Professor Pole sees are what we call green and blue.
This is the result of my calculations; but Professor Pole agrees with every
other colour-blind person whom I know in denying that green is one of his
sensations. The colour-blind are always making mistakes about green things and
confounding them with red. The colours they have no doubts about are cer-
tainly blue and yellow, and they persist in saying that yellow, and not green,
is the colour which they are able to see.

To explain this discrepancy we must remember that colour-blind persons
learn the names of colours by the same method as ourselves. They are told
that the sky is blue, that grass is green, that gold is yellow, and that soldiers'
coats are red. They observe difference in the colours of these objects, and they
often suppose that they see the same colours as we do, only not so well. But
if we look at the diagram we shall see that the brightest example of their
second sensation in the spectrum is not in the green, but in the part which
we call yellow, and which we teach them to call yellow. The figure of the
spectrum below Professor Pole's curves is intended to represent to ordinary eyes
what a colour-blind person would see in the spectrum. I hardly dare to draw your
attention to it, for if you were to think that any painted picture would enable you
to see with other people's vision I should certainly have lectured in vain.

On the Yellow Spot.

Experiments on colour indicate very considerable differences between the
vision of different persons, all of whom are of the ordinary type. A colour,
for instance, which one person on comparing it with white will pronounce
pinkish, another person will pronounce greenish. This difference, however, does
not arise from any diversity in the nature of the colour sensations in different
persons. It is exactly of the same kind as would be observed if one of the
persons wore yellow spectacles. In fact, most of us have near the middle of
the retina a yellow spot through which the rays must pass before they reach
the sensitive organ : this spot appears yellow because it absorbs the rays near

COLOUR VISION. 279

the line F, which are of a greenish-blue colour. Some of us have this spot
strongly developed. My own observations of the spectrum near the line F are
of very little value on this account. I am indebted to Professor Stokes for
the knowledge of a method by which anyone may see whether he has this
yellow spot. It consists in looking at a white object through a solution of
chloride of chromium, or at a screen on which light which has passed through
this solution is thrown [exhibited]. This light is a mixture of red light with
the light which is so strongly absorbed by the yellow spot. When it falls on
the ordinary surface of the retina it is of a neutral tint, but when it falls on
the yellow spot only the red light reaches the optic nerve, and we see a red
spot floating like a rosy cloud over the illuminated field.

Very few persons are unable to detect the yellow spot in this way. The
observer K, whose colour equations have been used in preparing the chart of
the spectrum, is one of the very few who do not see everything as if through
yellow spectacles. As for myself, the position of white light in the chart of
the spectrum is on the yellow side of true white even when I use the outer
parts of the retina ; but as soon as I look direct at it, it becomes much
yellower, as is shewn by the point WC. It is a curious fact that we do not
see this yellow spot on every occasion, and that we do not think white objects
yellow. But if we wear spectacles of any colour for some time, or if we live
in a room lighted by windows all of one colour, we soon come to recognize
white paper as white. This shews that it is only when some alteration takes
place in our sensations, that we are conscious of their quality.

There are several interesting facts about the colour sensation which I can
only mention briefly. One is that the extreme parts of the retina are nearly
insensible to red. If you hold a red flower and a blue flower in your hand
as far back as you can see your hand, you will lose sight of the red flower,
while you still see the blue one. Another is, that when the light is diminished
red objects become darkened more in proportion than blue ones. The third is,
that a kind of colour blindness in which blue is the absent sensation can be
produced artificially by taking doses of Santonine. This kind of colour blindness
is described by Dr Edmund Rose, of Berlin. It is only temporary, and does
not appear to be followed by any more serious consequences than headaches.
I must ask your pardon for not having undergone a course of this medicine,
even for the sake of becoming able to give you information at first hand about
colour blindness.

[From the Transaction! of the Royal Society of Edinburgh, Vol. xxvi.]
XLVIII. On the Geometrical Mean Distance of Two Figures on a Pl»,>:

[Received January 5th; read January 15th, 1872.]

THERE are several problems of great practical importance in electro-magnetic
measurements, in which the value of a quantity has to be calculated by taking
the sum of the logarithms of the distances of a system of parallel wires from a
given point. The calculation is in some respects analogous to that in which
we find the potential at a point due to a given system of equal particles, by
adding the reciprocals of the distances of the particles from the given point.
There is this difference, however, that whereas the reciprocal of a line is com-
pletely defined when we know the unit of length, the logarithm of a line has no
meaning till we know not only the unit of length, but the modulus of the sy
of logarithms.

In both cases, however, an additional clearness may be given to the state-
ment of the result by dividing, by the number of wires in the first case, and
by the number of particles in the second. The result in the first case is the
logarithm of a distance, and in the second it is the reciprocal of a distance ;
and in both cases this distance is such that, if the whole system were con-
centrated at this distance from the given point, it would produce the same
potential as it actually does.

In the first case, since the logarithm of the resultant distance is the arith-
metical mean of the logarithms of the distances of the various components of
the system, we may call the resultant distance the geometrical mean distance of
the system from the given point.

In the second case, since the reciprocal of the resultant distance is the
arithmetical mean of the reciprocals of the distances of the particles, we may

THE GEOMETRICAL MEAN DISTANCE OF TWO FIGURES ON A PLANE. 281

call the resultant distance the harmonic mean distance of the system from the
given point.

The practical use of these mean distances may be compared with that of
several artificial lines and distances which are known in Dynamics as the radius
of gyration, the length of the equivalent simple pendulum, and so on. The
result of a process of integration is recorded, and presented to us in a form
which we cannot misunderstand, and which we may substitute in those ele-
mentary formulae which apply to the case of single particles. If we have any
doubts about the value of the numerical co-efficients, we may test the expression
for the mean distance by taking the point at a great distance from the system,
in which case the mean distance must approximate to the distance of the centre
of gravity.

Thus it is well known that the harmonic mean distance of two spheres, each
of which is external to the other, is the distance between their centres, and that
the harmonic mean distance of any figure from a thin shell which completely
encloses it is equal to the radius of the shell.

I shall not discuss the harmonic mean distance, because the calculations
which lead to it are well known, and because we can do very well without it.
I shall, however, give a few examples of the geometric mean distance, in order
to shew its use in electro-magnetic calculations, some of which seem to me to
be rendered both easier to follow and more secure against error by a free use
of this imaginary line.

If the co-ordinates of a point in the first of two plane figures be x and y,
and those of a point in the second £ and TJ, and if r denote the distance between
these points, then R, the geometrical mean distance of the two figures, is
defined by the equation

log R . \\l\dxdy dgdrj = //// log r dxdydgdrj.

The following are some examples of the results of this calculation : —
(1) Let AB be a uniform line, and 0 a point
not in the line, and let OP be the perpendicular
from 0 on the line AB, produced if necessary, then
if R is the geometric mean distance of 0 from the
line AB,

AB . (log 72+ l) = PB . log OB-PA log OA + OP. AOB.
VOL. ii. 36

IM

THE (JEOMETRICAL MEAN DISTANCE

(2) The geometrical mean distance of P, a point in the line itself, from AB
U found from the equation

AB(\ogR+ l) = PB\ogPB-PA \ogPA.

When P lies between A and B, PA must be taken negative, but in taking the
logarithm of PA we regard PA as a positive numerical quantity.

(3) If R is the geometric mean distance between two finite lines AB and
CD, lying in the same straight line,

AB . CD (2 log R + 8) = AD* log AD + BC1 log BC- A C' log A C- BD* log BD.

(4) If AB coincides with CD, we find for the geometric mean distance
of all the points of AB from each other

B Q_

(5) If R is the geometric mean distance of the
rectangle ABCD from the point 0 in its plane, and
POR and QOS are parallel to the sides of the
rectangle through O,

ABCD (2 log R + 3) = 20P . OQ log OA + 20Q . OR log OB
+ 20R. OS log OC + 20S. OP log OD
+ OP*.D6A + 0&.A6B
+ OR* . BdC + OS1 . COD

(6) If R is the geometric mean of the distances

of all the points of the rectangle ABCD from each Bt
other,

A&, AC .5(7, AC

When the rectangle is a square, whose side = a,

= log a -0-8050866,
R = 0-44705 a.

OF TWO FIGURES ON A PLANE.

283

(7) The geometric mean distance of a circular line of radius a, from a
point in its plane at a distance r from the centre, is r if the point be without
the circle, and a if the point be within the circle.

(8) The geometric mean distance of any figure from a circle which com-
pletely encloses it is equal to the radius of the circle. The geometric mean
distance of any figure from the annular space between two concentric circles,
both of which completely enclose it, is R, where

(a,2 - a,1) (log R + 1) = a," log a, - a/ log a2,

a, being the radius of the outer circle, and «2 that of the inner. The geometric
mean distance of any figure from a circle or an annular space between two
concentric circles, the figure being completely external to the outer circle, is
the geometric mean distance of the figure from the centre of the circle.

(9) The geometric mean distance of all the points of the annular space
between two concentric circles from each other is R, where

(a,' - a,')' (log R - log a,) = } (3a,8 - a,') (a,' - a/) - < log J .

a,

When a,, the radius of the inner circle, vanishes, we find

When an the radius of the inner circle, becomes nearly equal to a1} that of the

outer circle,

R = ar

As an example of the application of this method, let us take the case of a
coil of wire, in which the wires are arranged so that the transverse section of
the coil exhibits the sections of the wires arranged in square order, the distance
between two consecutive wires being D, and the diameter of each wire d.

Let the whole section of the coil be of dimensions which are small com-
pared with the radius of curvature of the wires, and let
the geometrical mean distance of the section from itself
be R.

Let it be required to find the coefficient of induction
of this coil on itself, the number of windings being n.

1st. If we begin by supposing that the wires fill up
the whole section of the coil, without any interval of
insulating matter, then if M is the coefficient of induc-

36—2

o

o

o

o

o

o

o

o

o

THE GEOMETRICAL MEAN DISTANCE

tion of a linear circuit of the same shape as the coil on a similar parallel cir-
• at a distance R, the coefficient of induction of the coil on itself will be

n'Jf.

2nd. The current, however, is not uniformly distributed over the section.
It is confined to the wires. Now the coefficient of self-induction of a unit of

length of a conductor is

C-2\ogR,

where C is a constant depending on the form of the axis of the conductor, and
R is the mean geometric distance of the section from itself.

Now for a square of side Z>,

and for a circle of diameter d

log Rt = log d — log 2 — ^.

and the coefficient of self-induction of the cylindric wire exceeds that of the
square wire by

2 {log -T + 0-1380606}

cL

per unit of length.

3rd. We must also compare the mutual induction between the cylindric
wire and the other cylindric wires next it with that between the square wire
and the neighbouring square wires. The geometric mean distance of two
squares side by side is to the distance of their centres of gravity as 0"99401 is
to unity.

The geometric mean distance of two squares placed corner to corner is to
the distance between their centres of gravity as I'OOll is to unity.

Hence the correction for the eight wires nearest to the wire considered is

-2 x (0-01971).

OF TWO FIGURES ON A PLANE. 285

The correction for the wires at a greater distance is less than one-thousandth
per unit of length.

The total self-induction of the coil is therefore

n'M + 21 {log -j + 0-11835},

where n is the number of windings, and I the length of wire.

For a circular coil of radius = a,

M = 4ira (log 8a - log R - 2),
where R is the geometrical mean distance of the section of the coil from itself.

[From the Proceeding of the Royal Society, No. 132, 1872.]

XLIX. On the Induction of Elect™ Currents in an Infinite Plane Slifct of

Uniform Conductivity.

1. WHEN, on account of the motion or the change of strength of any
magnet or electro-magnet, a change takes place in the magnetic field, electro-
motive forces are called into play, and, if the material in which they act is a
conductor, electric currents are produced. This is the phenomenon of the induc-
tion of electric currents, discovered by Faraday.

I propose to investigate the case in which the conducting substance is in
the form of a thin stratum or sheet, bounded by parallel planes, and of indefi-
nite extent. A system of magnets or electro-magnets is supposed to exist on
the positive side of this sheet, and to vary in any way by changing its position
or its intensity. We have to determine the nature of the currents induced in
the sheet, and their magnetic efiect at any point, and, in particular, their
reaction on the electro-magnetic system which gave rise to them. The induced
currents are due, partly to the direct action of the external system, and partly
to their mutual inductive action ; so that the problem appears, at first sight,
somewhat difficult.

2. The result of the investigation, however, may be presented in a re-
markably simple form, by the aid of the principle of images which was first
applied to problems in electricity and hydrokinetics by Sir W. Thomson. The
essential part of this principle is, that we conceive the state of things on the
positive side of a certain closed or infinite surface (which is really caused by
actions having their seat on that surface) to be due to an imaginary system on
the negative side of the surface, which, if it existed, and if the action of the

ELECTRIC INDUCTION. 287

surface were abolished, would give rise to the actual state of things in the
space on the positive side of the surface.

The state of things on the positive side of the surface is expressed by a
mathematical function, which is different in form from that which expresses the
state of things on the negative side, but which is identical with that which
would be due to the existence, on the negative side, of a certain system which
is called the Image.

The image, therefore, is what we should arrive at by producing, as it were,
the mathematical function as far as it will go ; just as, in optics, the virtual
image is found by producing the rays, in straight lines, backwards from the place
where their direction has been altered by reflexion or refraction.

3. The position of the image of a point in a plane surface is found by
drawing a perpendicular from the point to the surface and producing it to an
equal distance on the other side of the surface. If the image is of the same
sign as the point, as it is in hydrokinetics when the surface is a rigid plane, it
is called a positive image. If it is of the opposite sign, as in statical electricity,
when the surface is a conductor, it is called a negative image. The image of a
conducting circuit is reckoned positive when the electric current flows in the
corresponding directions through corresponding parts of the object and the image.
The image is reckoned negative when the direction of the current is reversed.

In the case of the plane conducting sheet, the imaginary system on the
negative side of the sheet is not the simple image, positive or negative, of the
real magnet or electro-magnet on the positive side, but consists of a moving
train of images, the nature of which we now proceed to define.

4. Let the electric resistance of a rectangular portion of the sheet whose
length is a, and whose breadth is 2ira, be R.

R is to be measured on the electro-magnetic system, and is therefore a
velocity, the value of which is independent of the magnitude of the line a.
[If p denotes the specific resistance of the material of the sheet for a unit

cube, and if c is the thickness of the sheet, then R = ~— ; and if cr denotes

the specific resistance of the sheet for a unit (or any other) square, R = — .]

•>88 ELECTRIC INDUCTION.

5. Let UB begin by dividing the time into a num-
ber of equal intervals, each equal to St. The smaller
we take these intervals the more accurate will be the
definition of the train of images which we shall now
describe.

6. At a given time t, let a positive image of the
magnet or electro-magnet be formed on the negative
side of the sheet.

e

As soon as it is formed, let this image begin to
move away from the sheet in the direction of the O ^

normal, with the velocity R, its form and intensity
remaining constantly the same as that which the
magnet had at the time t.

After an interval 8t, that is to say, at the time t + St, let a negative
image, equal in magnitude and opposite in sign to this positive image, be formed
in the original position of the positive image, and let it then begin to move along
the normal, after the positive image, with the velocity R. The interval of time
between the arrival of these images at any point will be St, and the distance
between corresponding points will be RSt.

7. Leaving this pair of images to pursue their endless journey, let us
attend to the real magnet, or electro-magnet, as it is at the tune t + St. At
this instant let a new positive image be formed of the magnet in its new
position, and let this image also travel in the direction of the normal with the
velocity R, and be followed after an interval of time St by a corresponding
negative image. Let these operations be repeated at equal intervals of time,
each of these intervals being equal to St.

8. Thus at any given instant there will be a train or trail of images,
beginning with a single positive image, and followed by an endless succession of
pairs of images. This trail, when once formed, continues unchangeable in form
and intensity, and moves as a whole away from the conducting sheet with
the constant velocity R.

9. If we now suppose the interval of time St to be diminished without
limit, and the train to be extended without limit in the negative direction, so
as to include all the images which have been formed in all past time, the
magnetic effect of this imaginary train at any point on the positive side of the

ELECTRIC INDUCTION. 289

conducting sheet will be identical with that of the electric currents which
actually exist in the sheet.

Before proceeding to prove this statement, let us take notice of the form
which it assumes in certain cases.

10. Let us suppose the real system to be an electro-magnet, and that its
intensity, originally zero, suddenly becomes /, and then remains constant. At
this instant a positive image is formed, which begins to travel along the
normal with velocity R. After an interval 8t another positive image is formed ;
but at the same instant a second negative image is formed at the same place,
which exactly neutralizes its effect. Hence the result is, that a single positive
image travels by itself along the normal with velocity R. The magnetic effect of
this image on the positive side of the sheet is equivalent to that of the currents
of induction actually existing in the sheet, and the diminution of this effect, as
the image moves away from the sheet, accurately represents the effect of the
currents of induction, which gradually decay on account of the resistance of the
sheet. After a sufficient time, the image is so distant that its effects are no
longer sensible on the positive side of the sheet. If the current of the electro-
magnet be now broken, there will be no more images ; but the last negative
image of the train will be left unneutralized, and will move away from the sheet
with velocity R. The currents in the sheet will therefore be of the same
magnitude as those which followed the excitement of the electro-magnet, but in
the opposite direction.

11. It appears from this that, when the electro-magnet is increasing in
intensity, it will be acted on by a repulsive force from the sheet, and when
its intensity is diminishing, it will be attracted towards the sheet.

It also appears that if any system of currents is produced in the sheet
and then left to itself, the effect of the decay of the currents, as observed at
a point on the positive side of the sheet, will be the same as if the sheet, with
its currents remaining constant, had been carried away in the negative direction
with velocity R.

12. If a magnetic pole of strength m is brought from an infinite distance
along a normal to the sheet with a uniform velocity v towards the sheet, it
will be repelled with a force

m* v
4z> R + v'
where z is the distance from the sheet at the given instant.

VOL. ii. 37

i ELECTRIC DfDUCTIOX.

This formula will not apply to the caae of the pole moving away from the
•beet, because in that caae we must take account of the currents which are
excited when the pole begins to move, wliich it does when near the sheet

13. If the magnetic pole moves in a straight line parallel to the sheet,
with uniform velocity r, it will be acted on by a force in the opposite direct i
to its motion, and equal to

Besides this retarding force, it is acted on by a force repelling it from the
sheet, equal to

14. If the pole moves uniformly in a circle, the trail is in the form of a
helix, and the calculation of its effect is more difficult ; it is easy, however, to
see that, besides the retarding force and the repelling force, there is also a force
towards the centre of the circle.

15. It is shewn, in my treatise On Electricity and Magnetism (Vol. n.
Art. 600), that the currents in any system are the same, whether the conducting
system or the inducing system be in motion, provided the relative motion is
the same. Hence the results already given are directly applicable to the case
of Arago's rotating disk, provided the induced currents are not sensibly affected
by the limitation arising from the edge of the disk. These will introduce other
seta of images, which we shall not now investigate.

16. The greater the resistance of the sheet, whether from its thinness or
from the low conducting-power of its material, the greater is the velocity //.
Hence in most actual cases R is very great compared with v the velocity <•)'
the external system, and the trail of images is nearly normal to the sheet, and
the induced currents differ little from those which arise from the direct action
of the external system (see § 1).

17. If the conductivity of the sheet were infinite, or its resistance zero,
R would be zero. The images, once formed, would remain stationary, and all
except the bat formed positive image would be neutralized. Hence the trail

ELECTRIC INDUCTION. 291

would be reduced to a single positive image, and the sheet would exert a repul-

ffiYi^

give force : -„ on the pole, whether the pole be in motion or at rest.

4z

I need not say that this case does not occur in nature as we know it.
Something of the kind is supposed to exist in the interior of molecules in
Weber's Theory of Diamagnetism.

Mathematical Investigation.

18. Let the conducting sheet coincide with the plane of xy, and let its
thickness be so small that we may neglect the variation of magnetic force at
different points of the same normal within its substance, and that, for the same
reason, the only currents which can produce sensible effects are those which are
parallel to the surface of the sheet.

Current-function.

19. We shall define the currents in the sheet by means of the current-
function <f>. This function expresses the quantity of electricity which, in unit
of time, crosses from right to left a curve drawn from a point at infinity to
the point P.

This quantity will be the same for any two curves drawn from this point
to P, provided no electricity enters or leaves the sheet at any point between
these curves. Hence ^ is a single-valued function of the position of the point P.

The quantity which crosses the element ds of any curve from right to left is

*+A.

1 ' ' o.

as

By drawing ds first perpendicular to the axis of x, and then perpendicular
to the axis of y, we obtain for the components of the electric current in the
directions of x and of y respectively

u = f, v=-^.. (1).

dy dx

The curves for which <f> is constant are called current lines.

20. The annular portion of the sheet included between the current lines
$ and (f> + S(j) is a conducting circuit round which an electric current of strength

37—2

ELECTRIC INDUCTION.

&£ in flowing in the positive direction, that is, from x towards y. Such a
lit is equivalent in ita magnetic effects to a magnetic shell of strength 8<£,

having the circuit for ita edge*.

The whole system of electric currents in the sheet will therefore be equi-

valent to a complex magnetic shell, consisting of all the simple shells, defined

as above, into which it can be divided. The strength of the equivalent complex

shell at any point will be £

We may suppose this shell to consist of two parallel plane sheets of
imaginary magnetic matter at a very small distance c, the surface-density being

on the positive sheet, and —-on the negative sheet.

21. To find the magnetic potential due to this complex plane shell at
any point not in its substance, let us begin by finding P, the potential at the
point (£, i), C) due to a plane sheet of imaginary magnetic matter whose surface-
density is <f>, and which coincides with the plane of xy. The potential due to

the positive sheet whose surface-density is -, and which is at a distance £c on

C

the positive side of the plane of xy, is

dp

That due to the negative sheet, at a distance £c on the negative side of the
plane .of xy is

-l

Hence the magnetic potential of the shell is

V- -^

This, therefore, is the value of the magnetic potential of the current-sheet at
any given point on the positive side of it. Within the sheet there is no
magnetic potential, and at any point (f, 17, — £) on the negative side of the
sheet the potential is equal and of opposite sign to that at the point (f, 77, £)
on the positive side.

* W. Thomson, " Mathematical Theory of Magnetism," Phil. Tram. 1850.

ELECTRIC INDUCTION. 293

22. At the positive surface the magnetic potential is

F=-g = 2^ ................................ (3).

At the negative surface

The normal component of magnetic force at the positive surface is

dV

In the case of the magnetic shell, the magnetic force is discontinuous at
the surface ; but in the case of the current-sheet this expression gives the value
of y within the sheet itself, as well as in the space outside.

23. Let F, G, H be the components of the electro-magnetic momentum at
any point in the sheet, due to external electro-magnetic action as well as to
that of the currents in the sheet, then the electromotive force in the direction
of x is

__
~ dt dx'

where »/» is the electric potential* ; and by Ohm's law this is equal to <ru,
where cr is the specific resistance of the sheet.

dF dili

Hence <ru = — -j- — -r-

dt dx

0. ., ,
Similarly,

(6)
dG " '

Let the external system be such that its magnetic potential is represented
by — v-" , then the actual magnetic potential will be

(7),
and F=^(P. + P), G=-£x(P0 + P), tf=0 ............... (8).

* "Dynamical Theory of the Electromagnetic Field," Phil. Trans. 1865, p. 483.

,; JBKTRIO INDUCTION'.

Hence equations (6) become, by introducing the stream-function <f> from (l),

V •> = ~ ~I'A T (P» "^" P ) ~J~

ay «'<'.'/ W«B

^™ ^ " i ~ *
t**c

A solution of these equations is

d_.p

Substituting the value of <f> in terms of P, as given in equation (4),

i

i

The quantity — is evidently a velocity ; let us therefore for conciseness
call it R, then

24. Let P.' be the value of P, at the time t — r, and at a point on the
negative side of the sheet, whose co-ordinates are x, y, (z — Rr), and let

<?= ( P.'dr (13).

J o

At the upper limit when T is infinite P/ vanishes. Hence at the lower limit,
when r = 0 and P0' = P., we must have

but by equation (12)

~dt~ "dt
Hence the equation will be satisfied if we make

25. This, then, is a solution of the problem. Any other solution must
differ from this by a system of closed currents, depending on the initial state
of the sheet, not due to any external cause, and which therefore must decay

ELECTRIC INDUCTION. 295

rapidly. Hence, since we assume an eternity of past time, this is the only
solution of the problem.

This solution expresses P, a function due to the action of the induced
current, in terms of P0", and through this of P0, a function of the same kind
due to the external magnetic system. By differentiating P and P0 with respect
to z, we obtain the magnetic potential, and by differentiating them with respect
to t, we obtain, by equation (10), the current-function. Hence the relation
between P and P,, as expressed by equation (16), is similar to the relation
between the external system and its trail of images as expressed in the descrip-
tion of these images in the first part of this paper (§§ 6, 7, 8, 9), which is
simply an explanation of the meaning of equation (16) combined with the
definition of P,' in § 24.

NOTE TO THE PRECEDING PAPER.

At the time when this paper was written, I was not able to refer to two
papers by Prof. Felici, hi Tortolini's Annali di Scienze for 1853 and 1854, in
which he discusses the induction of currents in solid homogeneous conductors and
in a plane conducting sheet, and to two papers by E. Jochmann in Crelle's
Journal for 1864, and one in Poggendorffs Annalen for 1864, on the currents
induced in a rotating conductor by a magnet.

Neither of these writers have attempted to take into account the inductive
action of the currents on each other, though both have recognised the existence
of such an action, and given equations expressing it. M. Felici considers the
case of a magnetic pole placed almost in contact with a rotating disk. E.
Jochmann solves the case in which the pole is at a finite distance from the
pkne of the disk. He has also drawn the forms of the current-lines and of
the equipotential lines, in the case of a single pole, and in the case of two
poles of opposite name at equal distances from the axis of the disk, but on
opposite sides of it, and has pointed out why the current-lines are not, as
Matteucci at first supposed, perpendicular to the equipotential lines, which he
traced experimentally.

I am not aware that the principle of images, as described in the paper
presented to the Royal Society, has been previously applied to the phenomena
of induced currents, or that the problem of the induction of currents in an

EJ.ECTRIC INDUCTION.

infinite plane sheet has been solved, taking into account the mutual induction
«.f these currents, so as to make the solution applicable to a sheet of any degree
of conductivity.

The statement in equation (10), that the motion of a magnetic system
does not produce differences of potential in the infinite sheet, may appear some-
what strange, since we know that currents may be collected by electrodes
touching the sheet at different points. These currents, however, depend entirely
on the inductive action on the part of the circuit not included in the sheet;
for if the whole circuit lies in the plane of the sheet, but is so arranged as not
to interfere with the uniform conductivity of the sheet, there will be no dif-
ference of potential in any part of the circuit. This is pointed out by Feliri,
who shews that when the currents are induced by the instantaneous magnetiza-
tion of a magnet, these currents are not accompanied with differences of potential
in different parts of the sheet.

When the sheet is itself in motion, it appears, from Art. GOO of my treatise
On Electricity and Magnetism, that the electric potential of any point, as
measured by means of the electrodes of a fixed circuit, is

ff f

where -rr, '^, \\ are the components of the velocity of the part of the sheet

i ' i ' Ot

to which the electrode is applied.

In the case of a sheet revolving with velocity t» about the axis of z, tlii.s
becomes

dP d

Note 2. — The velocity R for a copper plate of best quality 1 millimetre in
thickness is about 25 metres per second. Hence it is only for very small velo-
cities of the apparatus that we can obtain any approximation to the true result
by neglecting the mutual induction of the currents. — Feb. 13.

[From the Proceedings of the London Mathematical Society, Vol. IV.]

L. On the Condition that, in the Transformation of any Figure by Curvilinear
Co-ordinates in Three Dimensions, every Angle in the new Figure shall be
equal to the corresponding Angle in the original Figure.

[Read May 9th, 1872.]

IN the corresponding problem in two dimensions, the only condition is

x+J — ly =/(£+</— 177) (1),

where x, y are the co-ordinates in one system, and £ 77 in the other.
In three dimensions the solution is more restricted.

Let x, y, z be functions of £ 77, £; then the point in the system x, y, z,
for which £ is constant, will be in a certain surface, and by giving £ a series
of values we obtain a series of such surfaces. There will be two other series of
surfaces, corresponding to 77 and £ respectively. If in the second system £, 77, £
are rectangular co-ordinates, the surfaces corresponding to £ 77, £ in the first
system will intersect at right angles. The condition of this is

dx dx dy dy dz dz _

~j 7 b T~ ~T •TZ "y" ~~i TTv ~~ *J ••••••••«•••••••••«•••»

(2),

with two other equations in £ and £ and £ and 77.
If we now write

o« =

'dx

'dx

\' + t*y\' + (*\'

/ \dr)/ \drj)

HD'+f

.(3);

VOL. n.

38

TRAX8FORMATIOX OF AlHT FIGURE BY

and r/f is the intercept of the line (17 = const., £ = const.) cut off between the
•or&oes € and £+</£

The angle t, at which a line whose co-ordinates are functions of p cuts a
line whose co-ordinate* are functions of q, is found from the equation

dx dx dy dy dz dz

~T~ ~T~ + -J~ -J~ + ~J^ TL

= C08C

Expressing this in terms of £ ij, and £, it becomes

*,*, ««

dpdq ' dpdq

In order that the angle e, at which these lines intersect in the sy-
(j-, y, z), should be always equal to the angle e, at which the corresponding
lines in the system (f, ij, £) intersect, it is necessary and sufficient that

a = £ = y (6),

for these are the conditions that equations (4) and (5) should be of the sanu-
form.

Now consider the quadrilateral ABCD, cut off from
the surface (£= const.) by the surfaces f, £+d£, 77, and

Since the three sets of surfaces are orthogonal, their intersections are lines
of curvature, by Dupin's Theorem. Hence AB is a line of curvature of the

CURVILINEAR CO-ORDINATES IN THREE DIMENSIONS. 299

surface 17, drawn parallel to a. Hence the normals at A and B will intersect
at some point 0. Let us call OA, the radius of curvature of 17 in the plane
of a, R^, then

AB : DC ':: R : R

Hence R^ = '' similarlJ ****•

These values of the radii of curvature are true for any system of orthogonal
surfaces, but since in this case a=y, we find

or the principal radii of curvature of the surface 77 are equal to each other at
every point of the surface. Hence the surfaces rj must be spherical.

Since a = j3 = y, the other families of surfaces f and £ must also be spherical.

Now take one of the points where three spherical surfaces meet at right
angles for the origin, and invert the system. These spheres when inverted
become three planes intersecting at right angles, and the other spheres become
spheres, each intersecting at right angles two of these planes. Hence their
centres lie in the lines of intersection of the planes. Let a, b, c be the distances,
from the point of intersection of the three planes, of the centres of spheres
belonging each to one of the three systems, and let their radii be p, q, r
respectively. Then the conditions that these spheres intersect at right angles

are

whence p' = a\ qs = b\ r2 = c2;

or all the spheres pass through the origin, and each system has a common
tangent plane. Hence if we take these three planes as co-ordinate planes, and
write

we have rp = R?, a constant quantity,

x _y _ z

?"c;

38—2

HI TRANSFORMATION OF ANY FIGURE, &C.

A: : - xmfj?' y = 1'^' Z=S*p» ;

If m If £ = z&.

4

-»-7*=-
~r~V"/? p*

It appears from this that the only method of transforming a figure of three
dimensions, so that every angle in the system shall remain of the same mag-
nitude, is the method of inversion or of reciprocal radii vectores.

[Reference may be made to "Me'thodes de Transformation en Gdomt-trie
et en Physique mathe'matique," par J. N. Haton de Goupilliere, especially § \ii.:
Journal de FEcole Polytechnique, Tom. xxv., p. 188.]

.
[From Nature, Vol. vn.]

LI. Reprint of Papers on Electrostatics and Magnetism. By Sir W. Thomson,
D.C.L., LL.D., F.E.S., F.R.S.E., Fellow of St Peter's College, Cambridge,
and Professor of Natural Philosophy in the University of Glasgow. (London:
Macmillan and Co., 1872.)

To obtain any adequate idea of the present state of electro-magnetic science
we must study these papers of Sir W. Thomson's. It is true that a great deal
of admirable work has been done, chiefly by the Germans, both in analytical
calculation and in experimental researches, by methods which are independent
of, or at least different from, those developed in these papers, and it is the
glory of true science that all legitimate methods must lead to the same final
results. But if we are to count the gain to science by the number and value
of the ideas developed in the course of the inquiry, which preserve the results
of former thought in a form capable of being employed in future investigation,
we must place Sir W. Thomson's contributions to electro-magnetic science on
the very highest level.

One of the most valuable of these truly scientific, or science-forming ideas,
is that which forms the subject of the first paper in this collection. Two
scientific problems, each of the highest order of difficulty, had hitherto been
considered from quite different points of view. Cavendish and Poisson had
investigated the distribution of electricity on conductors on the hypothesis that
the particles of electricity exert on each other forces which vary inversely as
the square of the distance between them. On the other hand Fourier had
investigated the laws of the steady conduction of heat on the hypothesis that
the flow of heat from the hotter parts of a body to contiguous parts which
are colder is proportional to the rate at which the temperature varies from
point to point of the body. The physical ideas involved in these two problems

; - ELECTROSTATICS AND MAGXET18M.

are quite different In the one we have an attraction acting instantaneously
at a distance, in the other heat creeping along from hotter to colder parts.
The methods of investigation were also different. In the one the force on a
given particle of electricity has to be determined as the resultant of the
attraction of all the other particles. In the other we have to solve a certain
partial differential equation which expresses a relation between the rates of
variation of temperature in passing along lines drawn in three different direc-
tion* through a point. Thomson, in this paper, points out that these two
problems, so different, both in their elementary ideas and their analytical
methods, are mathematically identical, and that, by a proper substitution of
electrical for thermal terms in the original statement, any of Fourier's wonderful
methods of solution may be applied to electrical problems. The electrician has
only to substitute an electrified surface for the surface through which heat is
supplied, and to translate temperature into electric potential, and he may at
once take possession of all Fourier's solutions of the problem of the uniform
flow of heat.

To render the results obtained in the prosecution of one branch of inquiry
available to the students of another is an important service done to science,
but it is still more important to introduce into a science a new set of ideas,
belonging, as in this case, to what was, till then, considered an entirely uncon-
nected science. This paper of Thomson's, published in February 1842, when lie
was a very young freshman at Cambridge, first introduced into mathematical
science that idea of electrical action carried on by means of a continuous
medium which, though it had been announced by Faraday, and used by him
as the guiding idea of his researches, had never been appreciated by other
of science, and was supposed by mathematicians to be inconsistent with the
laws of electrical action, as established by Coulomb, and built on by Poi-
It was Thomson who pointed out that the ideas employed by Faraday under
the names of Induction, Lines of Force, &c., and implying an action transmitted
from one part of a medium to another, were not only consistent with the
results obtained by the mathematicians, but might be employed in a mathe-
matical form so as to lead to new results. One of these new results, which
was, we have reason to believe, obtained by this method, though demonstrated
by Thomson by a very elegant adaptation of Newton's method in the theory
of attraction, is the "Method of Electrical Images," leading to the "Method
"f Electrical Inversion."

ELECTBOSTATICS AND MAGNETISM. 303

Poisson had already, by means of Laplace's powerful method of spherical
harmonics, determined, in the form of an infinite series, the distribution of
electricity on a sphere acted on by an electrified system. No one, however,
seems to have observed that when the external electrified system is reduced
to a point, the resultant external action is equivalent to that of this point,
together with an imaginary electrified point within the sphere, which Thomson
calls the electric image of the external point.

Now if in an infinite conducting solid heat is flowing outwards uniformly
from a very small spherical source, and part of this heat is absorbed at another
small spherical surface, which we may call a sink, while the rest flows out in
all directions through the infinite solid, it is easy, by Fourier's methods, to
calculate the stationary temperature at any point in the solid, and to draw
the isothermal surfaces. One of these surfaces is a sphere, and if, in the
electrical problem, this sphere becomes a conducting surface in connection with
the earth, and the external source of heat is transformed into an electrified
point, the sink will become the image of that point, and the temperature and
flow of heat at any point outside the sphere will become the electric potential
and resultant force.

Thus Thomson obtained the rigorous solution of electrical problems relating
to spheres by the introduction of an imaginaiy electrified system within the
sphere. But this imaginary system itself next became the subject of exami-
nation, as the result of the transformation of the external electrified system by
reciprocal radii vectores. By this method, called that of electrical inversion, the
solution of many new problems was obtained by the transformation of problems
already solved. A beautiful example of this method is suggested by Thomson
in a letter to M. Liouville, dated October 8, 1845, and published in Liouville's
Journal, for 1845, but which does not seem to have been taken up by any
mathematician, till Thomson himself, in a hitherto unpublished paper (No. xv.
of the book before us), wrote out the investigation complete. This, the most
remarkable problem of electrostatics hitherto solved, relates to the distribution
of electricity on a segment of spherical surface, or a bowl, as Thomson calls
it, under the influence of any electrical forces. The solution includes a very
important case of a flat circular dish, and of an infinite flat screen with a
circular hole cut out of it.

If, however, the mathematicians were slow in making use of the physical
method of electric inversion, they were more ready to appropriate the geometrical

304 KLBCTR08TATIC8 AND MAGNETISM.

idea of inversion by reciprocal radii vectors, which is now well known to all
geometers, having been, we suppose, discovered and re-discovered repeatedly,
though, unless we are mistaken, most of these discoveries are later than 1845,
the date of Thomson's paper.

But to return to physical science, we have in No. vn. a paper of even
earlier date (1843), in which Thomson shews how the force acting on an elec-
trified body can be exactly accounted for by the diminution of the atmospheric
pressure on its electrified surface, this diminution being everywhere proportional
to the square of the electrification per unit of area. Now this diminution of
pressure is only another name for that tension along the lines of electric force,
by means of which, in Faraday's opinion, the mutual action between electrified
bodies takes place. This short paper, therefore, may be regarded as the germ
of that course of speculation by which Maxwell has gradually developed the
mathematical significance of Faraday's idea of the physical action of the lines
of force.

We have dwelt, perhaps at too great length, on these youthful contributions
to science, in order to shew how early in his career, Thomson laid a solid
foundation for his future labours, both in the development of mathematical
theories and in the prosecution of experimental research. Mathematicians how-
ever will do well to take note of the theorem in No. xm., the applications
of which to various branches of science will furnish them, if they be diligent,
both occupation and renown for some time to come.

We must now turn to the next part of this volume, in which the
mathematical electrician, now established as a Professor at Glasgow, turns his
attention to the practical and experimental work of his science. In such work
the mathematician, if he succeeds at all, proves himself no mere mathematician,
but a thoroughly furnished man of science. And first we have an account of
that research into atmospheric electricity which created a demand for electro-
meters ; then a series of electrometers of gradually improving species ; and lastly,
an admirable report on electrometers and electrostatic measurements, in which
the results of many years' experience are given in a most instructive and
scientific form. In this report the different instruments are not merely described,
but classified, so that the student is furnished with the means of devising a
new instrument to suit his own wants. He may also study, in the recorded
history of electrometers, the principles of natural selection, the conditions of the
permanence of species, the retention of rudimentary organs in manufactured

ELECTROSTATICS AND MAGNETISM. 305

articles, and the tendency to reversion to older types in the absence of scientific
control.

A good deal of Sir W. Thomson's practical electrical work is not referred
to in this volume. It is to be hoped that he will yet find time to give some
account of his many admirable telegraphic contrivances in galvanometers, sus-
pended coils, and recording instruments, and to complete this collection by his
papers on electrolysis, measurement of resistance, electric qualities of metals,
thermo-electricity, and electro-magnetism in general*.

The second division of the book contains the theory of magnetism.

The first paper, communicated to the Royal Society in 1849 and 1850, is
the best introduction to the theory of magnetism that we know of. The
discussion of particular distributions of magnetisation is altogether original, and
prepares the way for the theory of electro-magnets which follows. This paper
on electro-magnets is interesting as having been in manuscript for twenty-three
years, during which time a great deal has been done both at home and abroad
on the same subject, but without in any degree trenching upon the ground
occupied by Thomson in 1847. Though in these papers we find several formidable
equations bristling with old English capitals, the reader will do well to observe
that the most important results are often obtained without the use of this
mathematical apparatus, and are always expressed in plain scientific English.

As regards the most interesting of all subjects, the history of the develop-
ment of scientific ideas — we know of few statements so full of meaning as the
note at p. 419 relating to Ampere's theory of magnetism, as depending on
electric currents, flowing in circuits within the molecules of the magnet ; he
goes on to say : " From twenty to five-and-twenty years ago, when the materials
of the present compilation were worked out, I had no belief in the reality of
this theory ; but I did not then know that motion is the very essence of what
has been hitherto called matter. At the 1847 meeting of the British Association
in Oxford, I learned from Joule the dynamical theory of heat, and was forced
to abandon at once many, and gradually from year to year all other, statical
preconceptions regarding the ultimate causes of apparently statical phenomena."

After a short, but sufficient, proof that the magnetic rotation of the plane
of polarised light discovered by Faraday implies an actual rotatory motion of
something, and that this motion is part of the phenomenon of magnetism, he

* [See Mathematical and Physical Papers, by Sir W. Thomson, Vol. i., Cambridge University
Pi-ess, 1832.]

VOL. H. 39

ELECTROSTATICS AKD MAONLTISM.

•deb: "The explanation of all phenomena of electro-magnetic attraction or
Ision, and of electro-magnetic induction, is to be looked for simply in tin-
inertia and pressure of the matter of which the motions constitute heat. Whether
thin matter ia or ia not electricity, whether it is a continuous fluid interpt-r
meating the spaces between molecular nuclei, or is itself molecularly grouped;
or whether all matter is continuous, and molecular heterogeneousness consists in
finite vortical or other relative motions of contiguous parts of a body; it is
impossible to decide, and perhaps in vain to speculate, in the present state of

The date of these remarks is 1856. In 1861 and 1862 appeared Maxwells
"theory of molecular vortices applied to magnetism, electricity, &c." which may
be considered as a development of Thomson's idea in a form which, though
rough and clumsy compared with the realities of nature, may have served its
turn as a provisional hypothesis.

The concluding sections of the book before us are devoted to illustration!
of magnetic force derived from the motion of a perfect fluid. They are not
put forward as explanations of magnetic force, for in fact the forces are of
the opposite kind to those of magnets. They belong more properly to that
remarkable extension of the science of hydrokinetics which was begun by
Helmholtz and so ably followed up by Thomson himself.

The conception of a perfectly homogeneous, incompressible frictionless fluid
ia as essential a part of pure dynamics as that of a circle is of pure geometry.
It is true that the motions of ordinary fluids are very imperfect illustration!
of those of the perfect fluid. But it is equally true that most of the objects
which we are pleased to call circles are very imperfect representations of a true
circle.

Neither a perfect fluid nor a perfect circle can be formed from the materials
which we deal with, for they are assemblages of molecules, and therefore not
homogeneous except when regarded roughly in large masses. The perfect circle
is truly continuous and the perfect fluid is truly homogeneous.

It follows, however, from the investigations of Helmholtz and Thomson that
if a motion of the kind called rotational is once set up in the fluid, the portion
of the fluid to which this motion is communicated, retains for ever, during all
its wanderings through the fluid mass, the character of the motion thus impressed
on it.

This vortex then, as Helmholtz calls it, be it large or small, possesses that

ELECTROSTATICS AND MAGNETISM. 307

character of permanence and individuality which we attribute to a material
molecule, while at the same time it is capable, while retaining its essential
characteristics unchanged both in nature and value, of changing its form in an
infinite variety of ways, and of executing the vibrations which excite those rays
of the spectrum by which the species of the molecule may be discovered. It
would puzzle one of the old-fashioned little round hard molecules to execute
vibrations at all. There was no music in those spheres.

But besides this application of hydrokinetics to this new conception of the
old atom, there is a vast field of high mathematical inquiry opened up by the
papers of Helmholtz and Thomson. It is to be hoped that the latter will soon
complete his papers on Vortex Motion and give them to the world. But why
does no one else work in the same field ? Has the multiplication of symbols
put a stop to the development of ideas ?

39—2

[From the Proceedingt of the Cambridge Philosophical Society, Vol. n, 1876.]

LII. On the Proof of the Equations of Motion of a Connected System.

To deduce from the known motions of a system the forces which act on it
is the primary aim of the science of Dynamics. The calculation of the motion
when the forces are known, though a more difficult operation, is not so impor-
tant, nor so capable of application to the analytical method of physical science.

The expressions for the forces which act on the system in terms of the
motion of the system were first given by Lagrange in the fourth section <>f
the second part of his Mdcanique Analytique. Lagrange's investigation may be
regarded from a mathematical point of view as a method of reducing the dy-
namical equations, of which there are originally three for every particle of the
system, to a number equal to that of the degrees of freedom of the system.
In other words it is a method of eliminating certain quantities called reactions
from the equations.

The aim of Lagrange was, as he tells us himself, to bring dynamics under
the power of the calculus, and therefore he had to express dynamical relations in
terms of the corresponding relations of numerical quantities.

In the present day it is necessary for physical inquirers to obtain clear ideas
in dynamics that they may be able to study dynamical theories of the physical
sciences. We must therefore avail ourselves of the labours of the mathematician.
and selecting from his symbols those which correspond to conceivable physical
quantities, we must retranslate them into the language of dynamics.

In this way our words will call up the mental image, not of certain
operations of the calculus, but of certain characteristics of the motion of bod

The nomenclature of dynamics has been greatly developed by those who in
recent times have expounded the doctrine of the Conservation of Energy, and
it will be seen that most of the following statement is suggested by the inv

PROOF OF THE EQUATIONS OF MOTION OF A CONNECTED SYSTEM. 309

gations in Thomson and Tait's Natural Philosophy, especially the method of
beginning with the case of impulsive forces.

*I have applied this method in such a way as to get rid of the explicit
consideration of the motion of any part of the system except the co-ordinates or
variables on which the motion of the whole depends. It is important to the
student to be able to trace the way in which the motion of each part is
determined by that of the variables, but I think it desirable that the final equa-
tions should be obtained independently of this process. That this can be done is
evident from the fact that the symbols by which the dependence of the motion
of the parts on that of the variables was expressed, are not found in the final
equations.

The whole theory of the equations of motion is no doubt familiar to mathe-
maticians. It ought to be so, for it is the most important part of their science
in its application to matter. But the importance of these equations does not
depend on their being useful in solving problems in dynamics. A higher function
which they must discharge is that of presenting to the mind in the clearest
and most general form the fundamental principles of dynamical reasoning.

In forming dynamical theories of the physical sciences, it has been a too
frequent practice to invent a particular dynamical hypothesis and then by means
of the equations of motion to deduce certain results. The agreement of these
results with real phenomena has been supposed to furnish a certain amount of
evidence in favour of the hypothesis.

The true method of physical reasoning is to begin with the phenomena and
to deduce the forces from them by a direct application of the equations of
motion. The difficulty of doing so has hitherto been that we arrive, at least
during the first stages of the investigation, at results which are so indefinite
that we have no terms sufficiently general to express them without introducing
some notion not strictly deducible from our premisses.

It is therefore very desirable that men of science should invent some method
of statement by which ideas, precise so far as they go, may be conveyed to the
mind, and yet sufficiently general to avoid the introduction of unwarrantable
details.

For instance, such a method of statement is greatly needed in order to
express exactly what is known about the undulatory theory of light.

* [In the Author's treatise On Electricity and Magnetism, Vol. II. Part IV. Chap, v., the reader
will find the subject treated at length from the point of view advocated in the text.]

[From the Prootediityf of the Can&ridge Philosophical Society, Vol. IL, 1876.]

LI 1 1. On « Problem in the Calculus of Variations in which the solution

is discontinuous.

THE rider on the third question in the Senate-House paper of Wednesday,
January 15, 1J to 4, was set as an example of discontinuity introduced into
a problem in a way somewhat different, I think, from any of those discussed
in Mr Todhunter's essay*. In some of Mr Todhunter's cases the discontinuity
was involved or its possibility implied in the statement of the problem, as when
a curve is precluded from transgressing the boundary of a given region, or wheiv
its curvature must not be negative. In the case of figures of revolution con-
sidered as generated by a plane curve revolving about a line in its plane, this
forms a boundary of the region within which the curve must lie, and therefore
often forms part of the curve required for the solution.

In the problem now before us there is no discontinuity in the stateu)t-nt.
and it is introduced into the problem by the continuous change of the
efficients of a certain equation as we pass along the curve. At a certain point
the two roots of this equation which satisfy the minimum condition coalesce
with, each other and with a maximum root. Beyond this point the root which
formerly indicated a maximum indicates a minimum, and the other two roots
become impossible.

* Rtttarthft in the Calculus of Variations, ttc.

[The question referred to was set in 1873, and is as follows : — If the velocity of a carriage along
• roftd i» proportional to the cube of the cosine of the inclination of the road to the horizon, determine
the jmth of quickest ascent from the bottom to the top of a hemispherical hill, and shew that it consist*
of the spherical curve described by a point of a great circle which rolls on a small circle described about

the pole with a radios ~ , together with an arc of a great circle. How is the discontinuity introdnnil
into tliu problem 1]

[From the . Proceedings of the Royal Institution of Great Britain, Vol. vn.]

LIV. On Action at a Distance.

I HAVE no new discovery to bring before you this evening. I must ask
you to go over very old ground, and to turn your attention to a question
which has been raised again and again ever since men began to think.

The question is that of the transmission of force. We see that two bodies
at a distance from each other exert a mutual influence on each other's motion.
Does this mutual action depend on the existence of some third thing, some
medium of communication, occupying the space between the bodies, or do the
bodies act on each other immediately, without the intervention of anything else ?

The mode in which Faraday was accustomed to look at phenomena of this
kind diners from that adopted by many other modern inquirers, and my special
aim will be to enable you to place yourselves at Faraday's point of view, and to
point out the scientific value of that conception of lines of force which, in his
hands, became the key to the science of electricity.

When we observe one body acting on another at a distance, before we
assume that this action is direct and immediate, we generally inquire whether
there is any material connection between the two bodies ; and if we find strings,
or rods, or mechanism of any kind, capable of accounting for the observed
action between the bodies, we prefer to explain the action by means of these
intermediate connections, rather than to admit the notion of direct action at a
distance.

Thus, wheii we ring a bell by means of a wire, the successive parts of
the wire are first tightened and then moved, till at last the bell is rung at
a distance by a process in which all the intermediate particles of the wire have
taken part one after the other. We may ring a bell at a distance in other
ways, as by forcing air into a long tube, at the other end of which is a
cylinder with a piston which is made to fly out and strike the bell. We

ACTION AT A DISTANCE.

may also use a wire; but instead of pulling it, we may connect it at one end
with a voltaic batter}', and at tbe other with an electro-magnet, and thus ring
the bell by electricity.

Here are three different ways of ringing a bell. They all agree, however,
in the circumstance that between the ringer and the bell there is an unbroken
line of communication, and that at every point of this line some physical
process goes on by which the action is transmitted from one end to the other.
The process of transmission is not instantaneous, but gradual ; so that then- is
an interval of time after the impulse has been given to one extremity of the
line of communication, during which the impulse is on its way, but has not
reached the other end.

It is clear, therefore, that in many cases the action between bodies ,
distance may be accounted for by a series of actions between each sum
pair of a series of bodies which occupy the intermediate space ; and it is a-
by the advocates of mediate action, whether, in those cases in which we ca
perceive the intermediate agency, it is not more philosophical to admit the
existence of a medium which we cannot at present perceive, than to assert
a body can act at a place where it is not.

To a person ignorant of the properties of air, the transmission of force by
means of that invisible medium would appear as unaccountable as any •
example of action at a distance, and yet in this case we can explain the whole
process, and determine the rate at which the action is passed on from one
portion to another of the medium.

Why then should we not admit that the familiar mode of communicating
motion by pushing and pulling with our hands is the type and exemplification
of all action between bodies, even in cases in which we can observe nothing
between the bodies which appears to take part in the action ?

Here for instance is a kind of attraction with which Professor Guthrie
has made us familiar. A disk is set in vibration, and is then brought m
light suspended body, which immediately begins to move towards the disk, as
if drawn towards it by an invisible cord. What is this cord ? Sir W. Thomson
has pointed out that in a moving fluid the pressure is least where the velocity
is greatest. The velocity of the vibratory motion of the air is greatest nearest
the disk. Hence the pressure of the air on the suspended body is less on the
side nearest the disk than on the opposite side, the body yields to the gn
pressure, and moves toward the disk.

ACTION AT A DISTANCE. 313

The disk, therefore, does not act where it is not. It sets the air next it
in motion by pushing it, this motion is communicated to more and more distant
portions of the air in turn, and thus the pressures on opposite sides of the
suspended body are rendered unequal, and it moves towards the disk in conse-
quence of the excess of pressure. The force is therefore a force of the old
school — a case of vis a tergo — a shove from behind.

The advocates of the doctrine of action at a distance, however, have not
been put to silence by such arguments. What right, say they, have we to
assert that a body cannot act where it is not ? Do we not see an instance of
action at a distance in the case of a magnet, which acts on another magnet not
only at a distance, but with the most complete indifference to the nature of
the matter which occupies the intervening space ? If the action depends on
something occupying the space between the two magnets, it cannot surely be a
matter of indifference whether this space is filled with air or not, or whether
wood, glass, or copper, be placed between the magnets.

Besides this, Newton's law of gravitation, which every astronomical obser-
vation only tends to establish more firmly, asserts not only that the heavenly
bodies act on one another across immense intervals of space, but that two
portions of matter, the one buried a thousand miles deep in the interior of
the earth, and the other a hundred thousand miles deep in the body of the
sun, act on one another with precisely the same force as if the strata beneath
which each is buried had been non-existent. If any medium takes part in
transmitting this action, it must surely make some difference whether the space
between the bodies contains nothing but this medium, or whether it is occupied
by the dense matter of the earth or of the sun.

But the advocates of direct action at a distance are not content with
instances of this kind, in which the phenomena, even at first sight, appear to
favour their doctrine. They push their operations into the enemy's camp, and
maintain that even when the action is apparently the pressure of contiguous
portions of matter, the contiguity is only apparent — that a space always inter-
venes between the bodies which act on each other. They assert, in short, that
so far from action at a distance being impossible, it is the only kind of action
which ever occurs, and that the favourite old vis a tergo of the schools has no
existence in nature, and exists only in the imagination of schoolmen.

The best way to prove that when one body pushes another it does not
touch it, is to measure the distance between them. Here are two glass lenses,

VOL. ii. 40

ACTION AT A DISTANCE.

one of which U prewed against the other by means of a weight. By n,
of the electric light we may obtain on the screen an image of the place via re
the one lens presses against the other. A series of coloured rings is formed on
the screen. These rings were first observed and first explained by Newton.
The particular colour of any ring depends on the distance between the surfaces
of the pieces of glass. Newton formed a table of the colours corresponding to
different distances, so that by comparing the colour of any ring with Newton's
table, we may ascertain the distance between the surfaces at that ring. The
colours are arranged in rings because the surfaces are spherical, and therefore
the interval between the surfaces depends on the distance from the line joining
the centres of the spheres. The central spot of the rings indicates the place
where the lenses are nearest together, and each successive ring corresponds to
an increase of about the 4000th part of a millimetre in the distance of the
surfaces.

The lenses are now pressed together with a force equal to the weight of
an ounce; but there is still a measurable interval between them, even at the
place where they are nearest together. They are not in optical contact. To
prove this, I apply a greater weight. A new colour appears at the central
spot, and the diameters of all the rings increase. This shews that the sur
are now nearer than at first, but they are not yet in optical contact, for if
they were, the central spot would be black. I therefore increase the weights,
so as to press the lenses into optical contact.

But what we call optical contact is not real contact. Optical contact indi-
cates ' only that the distance between the surfaces is much less than a wave-
length of light. To shew that the surfaces are not in real contact, I remove
the weights. The rings contract, and several of them vanish at the centre.
Now it is possible to bring two pieces of glass so close together, that they
will not tend to separate at all, but adhere together so firmly, that when torn
asunder the glass will break, not at the surface of contact, but at some other
place. The glasses must then be many degrees nearer than when in mere optical
contact.

Thus we have shewn that bodies begin to press against each other whilst
still at a measurable distance, and that even when pressed together with u
force they are not in absolute contact, but may be brought nearer still, aii'l
that by many degrees.

Why, then, say the advocates of direct action, should we continue to

ACTION AT A DISTANCE. 315

maintain the doctrine, founded only on the rough experience of a pre-scientific
age, that matter cannot act where it is not, instead of admitting that all the
facts from which our ancestors concluded that contact is essential to action were
in reality cases of action at a distance, the distance being too small to be
measured by their imperfect means of observation ?

If we are ever to discover the laws of nature, we must do so by obtaining
the most accurate acquaintance with the facts of nature, and not by dressing
up in philosophical language the loose opinions of men who had no know-
ledge of the facts which throw most light on these laws. And as for those
who introduce setherial, or other media, to account for these actions, without
any direct evidence of the existence of such media, or any clear understanding
of how the media do their work, and who fill all space three and four times
over with aethers of different sorts, why the less these men talk about their
philosophical scruples about admitting action at a distance the better.

If the progress of science were regulated by Newton's first law of motion,
it would be easy to cultivate opinions in advance of the age. We should only
have to compare the science of to-day with that of fifty years ago ; and by
producing, in the geometrical sense, the line of progress, we should obtain the
science of fifty years hence.

The progress of science in Newton's time consisted in getting rid of the
celestial machinery with which generations of astronomers had encumbered the
heavens, and thus " sweeping cobwebs off the sky."

Though the planets had already got rid of their crystal spheres, they were
still swimming in the vortices of Descartes. Magnets were surrounded by
effluvia, and electrified bodies by atmospheres, the properties of which resembled
in no respect those of ordinary effluvia and atmospheres.

When Newton demonstrated that the force which acts on each of the
heavenly bodies depends on its relative position with respect to the other
bodies, the new theory met with violent opposition from the advanced philoso-
phers of the day, who described the doctrine of gravitation as a return to the
exploded method of explaining everything by occult causes, attractive virtues,
and the like.

Newton himself, with that wise moderation which is characteristic of all his
speculations, answered that he made no pretence of explaining the mechanism
by which the heavenly bodies act on each other. To determine the mode in
which their mutual action depends on their relative position was a great step

40—2

ACTION AT A DISTANCE.

in science, and this step Newton asserted that he had made. To explain the
prooew by which this action is effected was a quite distinct step, and this
step Newton, in his Principia, does not attempt to make.

Bat BO far was Newton from asserting that bodies really do act on oue
another at a distance, independently of anything between them, that in a
letter to Bentley, which has been quoted by Faraday in this place, he says :—

"It is inconceivable that inanimate brute matter should, without the media-
tion of something else, which is not material, operate upon and affect other
matter without mutual contact, as it must do if gravitation, in the sense of

Epicurus, be essential and inherent in it That gravity should be innate,

inherent, and essential to matter, so that one body can act upon anotht •:
a distance, through a vacuum, without the mediation of anything else, by and
through which their action and force may be conveyed from one to another, is
to me so great an absurdity, that I believe no man who has in philosophical
matters a competent faculty of thinking can ever fall into it."

Accordingly, we find in his Optical Queries, and in his letters to P>
that Newton had very early made the attempt to account for gravitation by
means of the pressure of a medium, and that the reason he did not publish
these investigations "proceeded from hence only, that he found he was
able, from experiment and observation, to give a satisfactory account of this
medium, and the manner of its operation in producing the chief phenomena of
nature*."

The doctrine of direct action at a distance cannot claim for its author tin-
discoverer of universal gravitation. It was first asserted by Roger Cotes, in
his preface to the Principia, which he edited during Newton's life. According
to Cotes, it is by experience that we learn that all bodies gravitate. We do
not learn in any other way that they are extended, movable, or solid. Gravi-
tation, therefore, lias as much right to be considered an essential property of
matter as extension, mobility, or impenetrability.

And when the Newtonian philosophy gained ground in Europe, it was the
opinion of Cotes rather than that of Newton that became most prevalent, till
at last Boscovich propounded his theory, that matter is a congeries of mathe-
matical points, each endowed with the power of attracting or repelling the
others according to fixed laws. In his world, matter is unextended, and contact

* Maclaurin's Account of XtwtatCs Discoveriei.

ACTION AT A DISTANCE. 317

la impossible. He did not forget, however, to endow his mathematical points
with inertia. In this some of the modern representatives of his school have
thought that he " had not quite got so far as the strict modern view of
'matter* as being but an expression for modes or manifestations of 'force'""."

But if we leave out of account for the present the development of the
ideas of science, and confine our attention to the extension of its boundaries, we
shall see that it was most essential that Newton's method should be extended
to every branch of science to which it was applicable — that we should investi-
gate the forces with which bodies act on each other in the first place, before
attempting to explain how that force is transmitted. No men could be better
fitted to apply themselves exclusively to the first part of the problem, than
those who considered the second part quite unnecessary.

Accordingly Cavendish, Coulomb, and Poisson, the founders of the exact
sciences of electricity and magnetism, paid no regard to those old notions of
"magnetic effluvia" and "electric atmospheres," which had been put forth in
the previous century, but turned their undivided attention to the determination
of the law of force, according to which electrified and magnetized bodies attract
or repel each other. In this way the true laws of these actions were dis-
covered, and this was done by men who never doubted that the action took
place at a distance, without the intervention of any medium, and who would
have regarded the discovery of such a medium as complicating rather than as
explaining the undoubted phenomena of attraction.

We have now arrived at the great discovery by Orsted of the connection
between electricity and magnetism. Orsted found that an electric current acts
on a magnetic pole, but that it neither attracts it nor repels it, but causes it
to move round the current. He expressed this by saying that "the electric
conflict acts hi a revolving manner."

The most obvious deduction from this new fact was that the action of the
current on the magnet is not a push-and-pull force, but a rotatory force, and
accordingly many minds were set a-speculating on vortices and streams of sether
whirling round the current.

But Ampere, by a combination of mathematical skill with experimental
ingenuity, first proved that two electric currents act on one another, and then
analysed this action into the resultant of a system of push-and-pull forces
between the elementary parts of these currents.

* Review of Mrs Somerville, Saturday Review, Feb. 13, 1869.

31 g ACTION AT A DISTANCE.

The formula of Ampere, however, is of extreme complexity, as compared
with Newton's law of gravitation, and many attempts have been made to res..l\v
it into something of greater apparent simplicity.

I have no wish to lead you into a discussion of any of these attempts to
improve a mathematical formula. Let us turn to the independent inethi»
investigation employed by Faraday in those researches in electricity and i

which have made this Institution one of the most venerable shrines of

No man ever more conscientiously and systematically laboured to inn
all his powers of mind than did Faraday from the very beginning of his
BCTfrptifif! career. But whereas the general course of scientific method then
aisted in the application of the ideas of mathematics and astronomy to each nt-\v
investigation in turn, Faraday seems to have had no opportunity of acquiring
a technical knowledge of mathematics, and his knowledge of astronomy
mainly derived from books.

Hence, though he had a profound respect for the great discovery of
he regarded the attraction of gravitation as a sort of sacred mystery, which,
as he was not an astronomer, he had no right to gainsay or to don! it. his
duty being to believe it in the exact form in which it was delivered to Li in.
Such a dead faith was not likely to lead him to explain new phenomena I>y
means of direct attractions.

Besides this, the treatises of Poisson and Ampere are of so technical a
form, that to derive any assistance fi-om them the student must have been
thoroughly trained in mathematics, and it is very doubtful if such a training
can be begun with advantage in mature years.

Thus Faraday, with his penetrating intellect, his devotion to science, suul
his opportunities for experiments, was debarred from following the course of
thought which had led to the achievements of the French philosophers, and
was obliged to explain the phenomena to himself by means of a symbolism
which he could understand, instead of adopting what had hitherto btvn tin-
only tongue of the learned.

This new symbolism consisted of those lines of force extending thems<
in every direction from electrified and magnetic bodies, which Faraday in his
mind's eye saw as distinctly as the solid bodies from which they emanated.

The idea of lines of force and their exhibition by means of iron li!
was nothing new. They had been observed repeatedly, and investigated matin

ACTION AT A DISTANCE. 319

matically as an interesting curiosity of science. But let us hear Faraday
himself, as he introduces to his reader the method which in his hands became
so powerful*.

"It would be a voluntary and unnecessary abandonment of most valuable
aid if an experimentalist, who chooses to consider magnetic power as represented
by lines of magnetic force, were to deny himself the use of iron filings. By
their employment he may make many conditions of the power, even in com-
plicated cases, visible to the eye at once, may trace the varying direction of
the lines of force and determine the relative polarity, may observe in which
direction the power is increasing or diminishing, and in complex systems may
determine the neutral points, or places where there is neither polarity nor
power, even when they occur in the midst of powerful magnets. By their use
probable results may be seen at once, and many a valuable suggestion gained
for future leading experiments."

Experiment on Lines of Force.

In this experiment each filing becomes a little magnet. The poles of oppo-
site names belonging to different filings attract each other and stick together,
and more filings attach themselves to the exposed poles, that is, to the ends
of the row of filings. In this way the filings, instead of forming a confused
system of dots over the paper, draw together, filing to filing, till long fibres
of filings are formed, which indicate by their direction the lines of force in
every part of the field.

The mathematicians saw in this experiment nothing but a method of exhibit-
ing at one view the direction in different places of the resultant of two forces,
one directed to each pole of the magnet ; a somewhat complicated result of
the simple law of force.

But Faraday, by a series of steps as remarkable for their geometrical
definiteness as for their speculative ingenuity, imparted to his conception of these
lines of force a clearness and precision far in advance of that with which the
mathematicians could then invest their own formulae.

In the first place, Faraday's lines of force are not to be considered merely
as individuals, but as forming a system, drawn in space in a definite manner

* Exp. Res. 3284.

, ACTION AT A DISTANCE.

so that the number of the lines wliich pass through an area, say of one square
inch, indicates the intensity of the force acting through the area. Tims tin-
lines of force become definite in number. The strength of a magnetic pole i-
mosimrH by the number of lines which proceed from it; the electro-tonic state
of a circuit is measured by the number of lines which pass through it.

In the second place, each individual line has a continuous existence in
space and time. When a piece of steel becomes a magnet, or when an electric
current begins to flow, the lines of force do not start into existence each in
its own place, but as the strength increases new lines are developed within
the magnet or current, and gradually grow outwards, so that the whole system
expands from within, like Newton's rings in our former experiment. Thus every
line of force preserves its identity during the whole course of its existence,
though its shape and size may be altered to any extent.

I have no time to describe the methods by which every question relating
to the forces acting on magnets or on currents, or to the induction of currents
in conducting circuits, may be solved by the consideration of Faraday's lines of
force. In this place they can never be forgotten. By means of this ii-
symbolism, Faraday defined with mathematical precision the whole theory of
electro-magnetism, in language free from mathematical technicalities, and appli-
cable to the most complicated as well as the simplest cases. But Faraday did
not stop here. He went on from the conception of geometrical lines of force
to that of physical lines of force. He observed that the motion which the
magnetic or electric force tends to produce is invariably such as to shorten
the lines of force and to allow them to spread out laterally from each other.
He thus perceived in the medium a state of stress, consisting of a tension,
like that of a rope, in the direction of the lines of force, combined with a
pressure in all directions at right angles to them.

This is quite a new conception of action at a distance, reducing it to a
phenomenon of the same kind as that action at a distance which is exerted
by means of the tension of ropes and the pressure of rods. When the muscles
of our bodies are excited by that stimulus which we are able in some unknown
way to apply to them, the fibres tend to shorten themselves and at the same
time to expand laterally. A state of stress is produced in the muscle, and the
limb moves. This explanation of muscular action is by no means complete.
It gives no account of the cause of the excitement of the state of stress, nor
does it even investigate those forces of cohesion which enable the muscles to

ACTION AT A DISTANCE. 321

support this stress. Nevertheless, the simple fact, that it substitutes a kind of
action which extends continuously along a material substance for one of which
we know only a cause and an effect at a distance from each other, induces
us to accept it as a real addition to our knowledge of animal mechanics.

For similar reasons we may regard Faraday's conception of a state of stress
in the electro-magnetic field as a method of explaining action at a distance by
means of the continuous transmission of force, even though we do not know
how the state of stress is produced.

But one of Faraday's most pregnant discoveries, that of the magnetic
rotation of polarised light, enables us to proceed a step farther. The phe-
nomenon, when analysed into its simplest elements, may be described thus : —
Of two circularly polarised rays of light, precisely similar in configuration, but
rotating in opposite directions, that ray is propagated with the greater velocity
which rotates in the same direction as the electricity of the magnetizing
current.

It follows from this, as Sir W. Thomson has shewn by strict dynamical
reasoning, that the medium when under the action of magnetic force must be
in a state of rotation — that is to say, that small portions of the medium,
which we may call molecular vortices, are rotating, each on its own axis, the
direction of this axis being that of the magnetic force.

Here, then, we have an explanation of the tendency of the lines of mag-
netic force to spread out laterally and to shorten themselves. It arises from
the centrifugal force of the molecular vortices.

The mode in which electromotive force acts in starting and stopping the
vortices is more abstruse, though it is of course consistent with dynamical
principles.

We have thus found that there are several different kinds of work to be
done by the electro-magnetic medium if it exists. We have also seen that
magnetism has an intimate relation to light, and we know that there is a theory
of light which supposes it to consist of the vibrations of a medium. How is
this luminiferous medium related to our electro-magnetic medium ?

It fortunately happens that electro-magnetic measurements have been made
from which we can calculate by dynamical principles the velocity of progagation
of small magnetic disturbances in the supposed electro-magnetic medium.

This velocity is very great, from 288 to 314 millions of metres per second,
according to different experiments. Now the velocity of light, according to

VOL. II. 41

ACTION* AT A DISTANCE.

Fooeault's experiments, is 298 millions of metres per second. In fact, the different
determinations of either velocity differ from each other more than the estimated
velocity of light doe* from the estimated velocity of propagation of small electro-
magnetic disturbance. But if the luminiferous and the electro-magnetic media
occupy the same place, and transmit disturbances with the same velocity, what
nuinn have we to distinguish the one from the other? By considering them
as the same, we avoid at least the reproach of filling space twice over with
different kinds of tether.

Besides this, the only kind of electro-magnetic disturbances which can be
propagated through a non-conducting medium is a disturbance transverse to the
direction of propagation, agreeing in this respect with what we know of that
disturbance which we call light. Hence, for all we know, light also may be an
electro-magnetic disturbance in a non-conducting medium. If we admit this, the
electro-magnetic theory of light will agree in every respect with the undulatory
theory, and the work of Thomas Young and Fresnel will be established on a
firmer basis than ever, when joined with that of Cavendish and Coulomb by
the key-stone of the combined sciences of light and electricity — Faraday's great
discovery of the electro-magnetic rotation of light.

The vast interplanetary and interstellar regions will no longer be regarded
as waste places in the universe, which the Creator has not seen fit to fill with
the symbols of the manifold order of His kingdom. We shall find them to be
already full of this wonderful medium ; so full, that no human power can remove
it from the smallest portion of space, or produce the slightest flaw in its
infinite' continuity. It extends unbroken from star to star ; and when a molecule
of hydrogen vibrates in the dog-star, the medium receives the impulses of these
vibrations ; and after carrying them in its immense bosom for three years,
delivers them in due course, regular order, and full tale into the spectroscope
• t Mr Muggins, at Tulse HilL

But the medium has other functions and operations besides bearing light
from man to man, and from world to world, and giving evidence of the absolute
unity of the metric system of the universe. Its minute parts may have rotatory
as well as vibratory motions, and the axes of rotation form those lines of
magnetic force which extend in unbroken continuity into regions which no eye
has seen, and which, by their action on our magnets, are telling us in language
not yet interpreted, what is going on in the hidden underworld from minute
to minute and from century to century.

ACTION AT A DISTANCE. 323

And these lines must not be regarded as mere mathematical absti-actions.
They are the directions in which the medium is exerting a tension like that
of a rope, or rather, like that of our own muscles. The tension of the medium
in the direction of the earth's magnetic force is in this country one grain
weight on eight square feet. In some of Dr Joule's experiments, the medium
has exerted a tension of 200 Ibs. weight per square inch.

But the medium, in virtue of the very same elasticity by which it is able
to transmit the undulations of light, is also able to act as a spring. When
properly wound up, it exerts a tension, different from the magnetic tension, by
which it draws oppositely electrified bodies together, produces effects through
the length of telegraph wires, and when of sufficient intensity, leads to the
rupture and explosion called lightning.

These are some of the already discovered properties of that which has
often been called vacuum, or nothing at all. They enable us to resolve several
kinds of action at a distance into actions between contiguous parts of a con-
tinuous substance. Whether this resolution is of the nature of explication or
complication, I must leave to the metaphysicians.

41—2

[From Mature, Vol. vn.]

LV. Elements of Natural Philosophy. By Professors Sir W. Thomson and
P. G. Tait. Clarendon Press Series. (Macmillan and Co., 1873.)

NATURAL Philosophy, which is the good old English name for what is
now called Physical Science, has been long taught in two very different ways.
One method is to begin by giving the student a thorough training in pure
mathematics, so that when dynamical relations are afterwards presented to him
in the form of mathematical equations, he at once appreciates the language, if
not the ideas, of the new subject. The progress of science, according to this
method, consists in bringing the different branches of science in succession under
the power of the calculus. When this has been done for any particular science,
it becomes in the estimation of the mathematician like an Alpine peak which
lias been scaled, retaining little to reward original explorers, though perhaps
still of some use, as furnishing occupation to professional guides.

The other method of diffusing physical science is to render the senses
familiar with physical phenomena, and the ear with the language of science, till
the student becomes at length able both to perform and to describe experiments
of his own. The investigator of this type is in no danger of having no more
worlds to conquer, for he can always go back to his former measurements, and
carry them forward to another place of decimals.

Each of these types of men of science is of service in the great work of
subduing the earth to our use, but neither of them can fully accomplish the
still greater work of strengthening their reason and developing new powers of
thought. The pure mathematician endeavours to transfer the actual effort of
thought from the natural phenomena to the symbols of his equations, and the
pure experimentalist is apt to spend so much of his mental energy on matters
of detail and calculation, that he has hardly any left for the higher forms
of thought. Both of them are allowing themselves to acquire an unfruitful

ELEMENTS OF NATURAL PHILOSOPHY. 325

familiarity with the facts of nature, without taking advantage of the opportunity
of awakening those powers of thought which each fresh revelation of nature is
fitted to call forth.

There is, however, a third method of cultivating physical science, in which
each department in turn is regarded, not merely as a collection of facts to be
co-ordinated by means of the formulae laid up in store by the pure mathe-
maticians, but as itself a new mathesis by which new ideas may be developed.

Every science must have its fundamental ideas — modes of thought by which
the process of our minds is brought into the most complete harmony with the
process of nature — and these ideas have not attained their most perfect form
as long as they are clothed with the imagery, not of the phenomena of the
science itself, but of the machinery with which mathematicians have been
accustomed to work problems about pure quantities.

Poinsot has pointed out several of his dynamical investigations as instances
of the advantage of keeping before the mind the things themselves rather than
arbitrary symbols of them ; and the mastery which Gauss displayed over every
subject which he handled is, as he said himself, due to the fact that he never
allowed himself to make a single step, without forming a distinct idea of the
result of that step.

The book before us shews that the Professors of Natural Philosophy at
Glasgow and Edinburgh have adopted this third method of diffusing physical
science. It appears from their preface that it has been since 1863 a text-book
in then: classes, and that it is designed for use in schools and in the junior
classes ha Universities. The book is therefore primarily intended for students
whose mathematical training has not been carried beyond the most elementary
stage.

The matter of the book however bears but small resemblance to that of
the treatises usually put into the hands of such students. We are very soon
introduced to the combination of harmonic motions, to irrotational strains, to
Hamilton's characteristic function, &c., and in every case the reasoning is con-
ducted by means of dynamical ideas, and not by making use of the analysis
of pure quantity.

The student, if he has the opportunity of continuing his mathematical
studies, may do so with greater relish when he is able to see in the mathe-
matical equations the symbols of ideas which have been already presented to
his mind in the more vivid colouring of dynamical phenomena. The differential

OF NATURAL PHILOSOPHY.

caloulu*. for example, is at once recognised ad the method of reasoning applicable
to quantities in a state of continuous change. This is Newton's conception of
Fluxions, and all attempts to banish the ideas of time and motion from the
mind must fail, since continuity cannot be conceived by us except by following
in imagination the course of a point which continues to exist while it m

in

"The arrangement of the book differs from that which has hitherto been
adopted in text-books. It has been usual to begin with those parts of the
subject in which the idea of change, though implicitly involved in the very
conception of force, is not explicitly developed so as to bring into view the
different configurations successively assumed by the system. For this reason,
the first place has generally been assigned to the doctrine of the equilibrium
of forces and the equivalence of systems of forces. The science of pure statics,
as thus set forth, is conversant with the relations of forces and of systems of
forces to each other, and takes no account of the nature of the material systems
to which they may be applied, or whether these systems are at rest or in
motion. The concrete illustrations usually given relate to systems of forces in
equilibrium, acting on bodies at rest, but the equilibrium of the forces is
established by reasoning which has nothing to do with the nature of the body,
or with its being at rest.

The practical reason for beginning with statics seems to be that the student
is not supposed capable of following the changes of configuration which take
place in moving systems. He is expected, however, to be able to follow trains
of reasoning about forces, the idea of which can never be acquired apart from
that of motion, and which can only be thought of apart from motion by a
process of abstraction.

Professors Thomson and Tait, on the contrary, begin with kinematics, the
science of mere motion considered apart from the nature of the moving body
and the causes which produce its motion. This science differs from geometry
only by the explicit introduction of the idea of time as a measurable quantity.
(The idea of time as a mere sequence of ideas is as necessary in geometry as
in every other department of thought.) Hence kinematics, as involving the
smallest number of fundamental ideas, has a metaphysical precedence over statics,
which involves the idea of force, which in its turn implies the idea of m:
as well as that of motion.

In kinematics, the conception of displacement comes before that of velocity,

ELEMENTS OF NATURAL PHILOSOPHY. 327

which is the rate of displacement. And here we cannot but regret that the
authors, one of whom at least is an ardent disciple of Hamilton, have not at
once pointed out that every displacement is a vector, and taken the opportunity \
of explaining the addition of vectors as a process, which, applied primarily to \
displacements, is equally applicable to velocities, or the rates of change of \
displacement, and to accelerations, or the rates of change of velocities. For it
is only in this way that the method of Newton, to which we are glad to see
that our authors have reverted, can be fully understood, and the "parallelogram
of forces" deduced from the "parallelogram of velocities." Another conception
of Hamilton's, however, that of the hodograph, is early introduced and employed
with great effect. The fundamental idea of the hodograph is the same as that
of vectors in general. The velocity of a body, being a vector, is defined by
its magnitude and direction, so that velocities may be represented by straight
lines, and these straight lines may be moved parallel to themselves into whatever
position is most suitable for exhibiting their geometrical relations, as for instance
in the hodograph they are all drawn from one point. The same idea is made
use of in the theorems of the " triangle " and the " polygon " of forces, and
in the more general method of "diagrams of stress," in which the Hues which
represent the stresses are drawn, not in the positions in which they actually
exist, but in those positions which most fully exhibit their geometrical relations.
We are sorry that a certain amount of slight is thrown on these methods in
§ 411, where a different proposition is called the true triangle of forces.

It is when a writer proceeds to set forth the first principle of dynamics
that his true character as a sound thinker or otherwise becomes conspicuous.
And here we are glad to see that the authors follow Newton, whose Leges
Motus, more perhaps than any other part of his great work, exhibit the
unimproveable completeness of that mind without a flaw.

We would particularly recommend to writers on philosophy, first to deduce
from the best philosophical data at their command a definition of equal intervals
of time, and then to turn to § 212, where such a definition is given as a
logical conversion of Newton's First Law.

But it is in the exposition of the Third Law, which affirms that the actions
between bodies are mutual, that our authors have brought to light a doctrine,
which, though clearly stated by Newton, remained unknown to generations of
students and commentators, and even when acknowledged by the whole scientific
world was not known to be contained in a paragraph of the Princima till it

OF NATURAL PHILOSOPHY.

pointed out by our authors in an article on "Energy" in Good 1 1".. /•</.*,
October 1862.

Our limits forbid us from following the authors as they carry the student
through the theories of varying action, kinetic force, electric images, and elastic
solids. We can only express our sympathy with the efforts of men, thoroughly
conversant with all that mathematicians have achieved, to divest scientific truths
of that symbolic language in which the mathematicians have left them, and to
clothe them in words, developed by legitimate methods from our mother tongue,
but rendered precise by clear definitions, and familiar by well-rounded statements.
Mathematicians may flatter themselves that they possess new ideas which
mere human language is as yet unable to express. Let them make the effort
to express these ideas in appropriate words without the aid of symbols, and
if they succeed they will not only lay us laymen under a lasting obligation,
hut, we venture to say, they will find themselves very much enlightened during
the process, and will even be doubtful whether the ideas as expressed in
symbols had ever quite found their way out of the equations into their minds.

[From the Proceedings of the London Mathematical Society, VoL IV.]

LVI. On the Theory of a System of Electrified Conductors, and other Physical
Theories involving Homogeneous Quadratic Functions.

[Read April 10th, 1873.]

THE theory of homogeneous functions of the second degree is useful in
several parts of natural philosophy.

The general form of such a function may be written

V=^(Aux13 + 2A^clx3 + &c.) ........................... (1),

r=n i=n

or more concisely Vx = \ 2 2 (Arjcjct) ............................ (l*),

in which each term consists of a product of two out of n variables xu ... arn,
which may or may not be different, and of a coefficient An belonging to that
pair of variables.

Differentiating V with respect to each of the n variables in succession, we
get n new quantities £„ ... £„ of the form

(2).

Multiplying each £ by its corresponding x, and taking half the sum of the
products, we obtain a second expression for V,

F=iT(av£) ............ ...................... (3).

r—l

Since each of the n quantities £ is a linear function of the n variables x,
we may, by solving the n equations of the form (2), obtain expressions for
each x in terms of the fs

r=n

x,= 2 (a^fc) .................................... (4);

T"l

VOL. it. 42

330 THEORY OF A SYSTEM OF ELECTRIFIED CONDUCTORS.

«,„! by •ubMituting these valuee of the X'B in (3), we obtain a third expression

vt-iT*M.(,) (5).

f-l r I

Tb« 6«t and last of these expressions for V are distinguished by the
MiftixoH JT and (, in order to shew that the first is expressed in terms of the

.hies * and the last in terms of the variables £ The coefficients A and a
nmy be functions of any number, m, of variables. Let yt be one of
ramble*.

Since F,, V, and Vt are three different expressions for the same quant i

r2(xrfr) (r,).

T 1

If we now suppose the three sets of variables ar, f and y to vary in any
consistent manner, and remember that V, is a function of the x's and y's, I
tin- fs and y's, and V of the x's and fs, we find

The three sets of variations Sx, 8£ and Sy are not independent of each
nt her, for if the variations Sy and either of the other sets be given, the other
set may be determined. Hence we cannot immediately deduce any definite
results from this equation. But we know, from equation (2), that the coeffici
of the variations 8x vanish of themselves, and the remaining variations Sf and S//
are all independent of each other, so that we may equate the coefficient of •
tif them to zero. We thus obtain two sets of equations,

_dVt
'~'

and + = 0 ................................... (9).

In this purely algebraical theory of quadratic functions, of the two sets of
variables x and (, either may be taken as the primary set. In the physical
.i| -plications of the theory, however, the variables form two classes which are not
interchangeable.

Thus, in kinetics, the variables are the components of velocity and tli<-
imimentum; in the theory of elasticity, they are stresses and strains; in eld

THEORY OF A SYSTEM OF ELECTRIFIED CONDUCTORS. 331

magnetism, they are electric currents and the " electrotonic state" of circuits;
and in electrostatics, they are the potentials and the charges of conductors.

Now, if a change takes place in the configuration of the system, by which
the energy (whether potential or kinetic) of the system is diminished, while the
momenta, the strains, the electrotonic states, or the charges remain constant, the
displacement will be aided by a force of such a magnitude that the work done
by the force during the displacement is equal to the diminution of the energy
of the system.

For the momenta, the strains, the electrotonic states, and the charges do not
require for their maintenance the application of energy from without the system.

The same change of configuration, if effected under the condition that the
velocities, stresses, potentials, or electric currents remain constant, would, by
equation (9), increase the energy of the system by exactly the same amount as
it was diminished in the former case. The work done by the system during
the displacement will be the same as before, so that energy must in this case be
supplied from without to an amount double of this work, in order to satisfy
these new conditions.

For it is only by external compulsion that the velocities of the system can
be maintained constant when the configuration changes. Work must be done on
an elastic system to keep the stress constant while the strain varies. The
currents in a conducting circuit tend to vary when the electrotonic state of the
circuit varies, and can only be kept constant by battery power. The same
agency must be employed to maintain the potential of a conductor constant
during the displacement of the system of conductors.

Hence, when momenta, strains, electrotonic states, or charges are maintained
constant, the internal forces of the system tend to produce displacements which
would diminish the energy of the system.

If, on the other hand, the velocities, stresses, electric currents, or potentials
are maintained constant, the internal forces tend to produce displacements
which would increase the energy of the system.

This distinction between the two sets of physical quantities is of great
importance in the theory of classification of such quantities. The characteristic
of the first set is inherent persistence. Any persistence which we may observe
in the second set is only apparent, and arises from the second set being
functions of the first set when the configuration of the system either does
not vary or is not such as to cause a variation of the coefficients A and a.

42—2

[From the Proceedings of the London Mathematical Society, Vol. iv.]

LVII. On the Focal Lines of a, Refracted Pencil.
[Read April 10th, 1873.]

HAMILTON has shewn, by means of his Characteristic Function, that in
wliatever manner a pencil may be refracted, the rays always pass through t\v<«
focal lines, the planes through which and the axis of the pencil are at right
angles to each other. The same method leads to the following geometrical
construction for finding the focal lines of any thin pencil after refraction through
a curved surface, the focal lines of the incident pencil and the nature of the
curvature of the surface being given.

1. The characteristic function of a thin pencil whose axis coincides with
the axis of z, and whose rays pass through focal lines in the planes of xz and
yz at distances a and b from the origin respectively, is for points near the
origin

If we turn this system of rays round the axis of z, through an angle
reckoned from x towards y, the expression for V becomes

(2),

where

1

1

1 ' 1 '

A

0

6 ™

1

= — sin* <& +

1

— cos w

I*

Ctr

b

1

-I1 l\ '

a2*

(3).

FOCAL LINES OF A REFRACTED PENCIL. 333

If we now turn the system round the axis of y, through an angle 6
from z towards x, we find

"* 9 y"~ xv cos 9

(2x cos 6 + z sin 6} z sin 9 yz sin 6\ , .

~~~ ~~ '

2. Now consider two portions of a pencil, close to the origin but in
different media, whose indices of refraction are ft, ft respectively, and let the
coefficients belonging to these portions be distinguished by corresponding suffixes
(, and ,).

Let the media be separated by the surface whose equation is

Then, since the characteristic function is continuous at this surface, we
have the condition Vl— Fa = 0 when z has the value given in (5).

Substituting this value of z in the expressions for Vl and F2, and neglecting
terms of the third degree in x and y, we obtain

B

cos 0i ~ ^ cos ^ + x (f*1 sin 0l ~ ^ sn

ar'cos'g. , y1 ,xycoa0\ /x3 cos* 0, , f , xycos0,\ n , .
~2ZT f2^,4 2^~/"l"'l>\"pr 22T. ~2CT/

That the term in x may vanish, we must have

& tan 0! = ft sun 0, ................................ (7),

the ordinary law of refraction.

Equating to zero the coefficients of x*, y\ and xy, we find
cos'0, cos*0,

—

/

, sm ^, -4, sui

/ ,/) ,/j\ /0\

"TlOOtft — OOtft) ..................... (8),

A v

p- -S-JT -fl=-i(cot^-cot(?,) ..................... (9),

6, sin P! 5., sin P, 5 v

COt 0, COt 0,

3*4

FOCAL LINES OF A REFRACTED PENCIL.

3. These relations of the quantities A, B, C may be found by the ful-
lowing construction : —

Let IO be the incident and OR the refracted ray, and let ON be the
normal to the surface; NOI=e» NOR = 6r

Find the points A, B, C, whose co-ordinates are

for A,
for B,

x=A
z = 0,
x=<7

cos* 0,- cos' 0.
cot01-cot0, '

',-0080.
C0t0,-C0t0,'

z = A

z = <

cos* 0. cot 0, - cos* 0. cot 0.
cot 0, — cot 0,

cos 0, cot 0, — cos 02 cot 0,

(12);

cot 0j — cot 0a

The positions of these points depend only on the form of the surface and
on the directions of the axes of the incident and refracted pencil. The point
B is absolutely fixed, being the centre of curvature of a normal section of the
surface perpendicular to the plane of refraction.

Let OAt, OBlt 0(7, be the values of Au Bv (7, for the incident ray. To
find the corresponding quantities for the refracted ray, draw the straight lines
A A u BBlt (7(7, intersecting the refracted ray in Av Bu Ct. OA» OBV and 0(7,
are the required values of Av Bv Cr

When any of these quantities becomes infinite, the line must be drawn
in a given direction. For At, Bu or (7, infinite, it must be parallel to the
incident ray. For Av Bv or (7, infinite, it must be parallel to the refracted
ray. For B infinite, it must be parallel to the normal ; for A infinite, it must
make with the normal an angle whose tangent is

9,-cos*01

cos

cos' 0t cot 6l — cos8 0, cot 0, '

FOCAL LINES OF A EEFRACTED PENCIL.

335

and for C infinite the tangent of the angle must be

cos ! — cos

,(15).

COS #jCOt 0j — COS #2 COt ft, ' '

If the plane of refraction cuts the surface along a line of curvature,
C= so , and if one of the focal lines of the incident pencil is in the plane of
refraction, Cl = oo . The points At, Bl then coincide with the focal lines of the
incident pencil, and A*, B3 with those of the refracted pencil.

That A! may coincide with Bv and A3 with Blf the line joining both pairs
of points must be on the line AB. There is therefore one, and only one, point
on the axis of the incident pencil from which a pencil may diverge so that,
after refraction, it still diverges from or converges to a single point.

4. When the quantity C has a finite value, the plane of refraction is not
a plane of symmetry, and we have to deduce the quantities a, 6, <£ from
A, B, C. The following construction enables us to pass from either of these
systems to the other : —

Let OA, OB, OC be the values of A, B, C.

Draw A A' perpendicular and equal to AO. Join BA', and produce to D,
a point on the perpendicular to OA through 0. Cut off OD' equal to OD,

;,, JOCAL LINES OF A REFRACTED PENCIL.

but in the opposite direction. Join Z>% and produce to P, where VA
meets BD.

Join CD, and draw OQ perpendicular to CD from 0.

Make Od, OtT each equal to OQ. Draw dP, d'P cutting OA in b and a.
Bisect the angle DOQ by 0£.

Tlu-n Oa*o, 06 = 6, and DOE=<f>, the angle which the first focal line
makes with the plane of xz.

5. If a, b, and <f> are given, the construction is easily reversed, thus :

Let Oa = fi, 06 = 6, and DOE=<f>.

Draw aa' perpendicular and equal to aO. Draw ba cutting OD, the per-
pendicular to nO from 0 in d. Cut off Od' equal and opposite to Od. I
-/'•« cutting bd in P.

Draw OQ=Od so that the angle DOE=OEQ.

Draw CO.Z) perpendicular to 00., cutting Oa in (7 and Od in D.

Make OZX equal and opposite to OD. Draw DP, Z>'P cutting Oa in />'
and - J.

Then OA = A, OB = B, and OC=C.

6. If therefore the given data be the radii of curvature of the refra.
surface and the angle, <j>, which the plane of incidence makes with the prin-
cipal section whose curvature is a, we may determine A, B, C for the refracting
surface.

Then from a, and bu the distances of the focal lines of the incident pencil,
and ^, the angle which a, makes with the plane of incidence, we must find
A., BI, Ct for the incident pencil.

From these data, by § 4, we must determine At, Blt Ct for the refracted
pencil, and from these a,, 6,, and <£,.

I have not been able to obtain any simpler construction for the general
case of a refracted pencil.

7. For a pencil after passing through any series of surfaces the construction
is necessarily more complex, as ten constants are involved in the general term
of the second degree of the characteristic function, which is of the form

V= ^a^ + Iby? + clx^l + iajX,' + £&#,' + c^, +pxtxt + qxyt + rt/.x,

FOCAL LINES OF A REFRACTED PENCIL. 337

and if Vl =

and V3

and we put D = (A, + a,) (B, + bi)-(C1 + c,)*,

(At - o») D + p* (B1 + bl) + ri(Al + a,) - 2pr ( Ci + cx) = 0,

(O, - c.) D +pq (B, + b,) + rs (A, + a,) - ( ps + qr) (C, + c.) = 0.

These equations enable us to determine Aly Bt, C2 when Alf Bv C, are

given.

Here we must observe that the quantities A, B, C, &c., do not represent
lines, as in the first part of this paper, but the reciprocals of lines.

VOL. II. 43

[From Nature, Vol. vin.]

LVIII. An Fumy on the M<itJ,<-ntntical Principles of Physics, &c. By the Rev.
James Challis, M.A., F.R.S., F.R.A.S., Plumian Professor of Astronomy and
Experimental Philosophy in the University of Cambridge, and Fellow of
Trinity College. (Cambridge: Deighton, Bell, and Co., 1873.)

THIS Essay is a sort of abstract or general account of the mathemn
and physical researches on which the author has been so long engaged, portions
<>f which have appeared from time to time in the Philosophical Magazine, ami
also in his larger work on the " Principles of Mathematics and Physics." I ;
is always desirable that mathematical results should be expressed in intelligible
language, as well as in the symbolic form in which they were at first obtained.
and we have to thank Professor Challis for this Essay, which though, or rather
liecause, it hardly contains a single equation, sets forth his system more clearly
than has been done in some of his previous mathematical papers.

The aim of this Essay, and of the author's long-continued labours, i
advance the theoretical study of Physics. He regards the material universe as
"a vast and wonderful mechanism, of which not the least wonderful qualit-.
its being so constructed that we can understand it." The Book of Nature, in
fact, contains elementary chapters, and, to those who know where to look for
them, the mastery of one chapter is a preparation for the study of the next.
The discovery of the calculation necessary to determine the acceleration <>f a
particle whose position is given in terms of the time led to the Newtonian
epoch of Natural Philosophy. The study from the cultivation of which our
author looks for the " inauguration of a new scientific epoch," is that of tin-
motion of fluids, commonly called Hydrodynamics. The scientific method \\hic-h
he recommends is that described by Newton as the "foundation of all philo-
sophy," namely, that the properties which we attribute to the least parts »t

AN ESSAY ON THE MATHEMATICAL PRINCIPLES OF PHYSICS. 339

matter must be consistent with those of which experiments on sensible bodies
have made us cognizant.

The world, according to Professor Challis, is made up of atoms and aether.
The atoms are spheres, unalterable in magnitude, and endowed with inertia, but
with no other property whatever. The aether is a perfect fluid, endowed with
inertia, and exerting a pressure proportional to its density. It is truly con-
tinuous (and therefore does not consist of atoms), and it fills up all the
interstices of the atoms.

Here, then, we have set before us with perfect clearness the two con-
stituents of the universe : the atoms, which we can picture in our minds as
so many marbles ; and the aether, which behaves exactly as air would do if
Boyle's law were strictly accurate, if its temperature were invariable, if it were
destitute of viscosity, and if gravity did not act on it.

We have no difficulty, therefore, in forming an adequate conception of the
properties of the elements from which we have to construct a world. The
hypothesis is at least an honest one. It attributes to the elements of things
no properties except those which we can clearly define. It stands, therefore,
on a different scientific level from those waxen hypotheses in which the atoms
are endowed with a new system of attractive or repulsive forces whenever a
new phenomenon has to be explained.

But the task still before us is a herculean one. It is no less than to
explain all actions between bodies or parts of bodies, whether in apparent
contact or at stellar distances, by the motions of this all-embracing aether, and
the pressure thence resulting.

One kind of motion of the aether is evidently a wave-motion, like that of
sound-waves in air. How will such waves affect an atom ? Will they propel
it forward like the driftwood which is flung upon the shore, or will they draw
it back like the shingle which is carried out by the returning wave ? Or will
they make it oscillate about a fixed position without any advance or recession
on the whole ?

We have no intention of going through the calculations necessary to solve
this problem. They are not contained in this Essay, and Professor Challis
admits that he has been unable to determine the absolute amount of the
constant term which indicates the permanent effect of the waves on an atom.
This is unfortunate, as it gives us no immediate prospect of making those
numerical comparisons with observed facts which are necessary for the verification

43—2

340 AS ESSAY OX THE MATHEMATICAL PRINCIPLES OF PHYSICS.

of the theory. Let us, however, suppose this purely mathematical difficulty
surmounted, and let us admit with Professor Challis that if the wave-length
of the undulations is very small compared with the diameter of the atom, the
atom will be urged in the direction of wave-propagation, or in other words
repeUfd from the origin of the waves. If on the other liand the wave-length
is very great compared with the diameter of the atom, the atom will be urged
in the direction opposite to that in which the waves travel, that is, it will
be atti-acted towards the source of the waves.

The amount of this attraction or repulsion will depend on the mean »f
the square of the velocity of the periodic motion of the particles of the aether,
and since the amplitude of a diverging wave is inversely as the distance from
the centre of divergence, the force will be inversely as the square of this
distance, according to Newton's law.

We must remember, however, that the problem is only imperfectly solved,
as we do not know the absolute value of this force, and we have not yet
arrived at an explanation of the fact that the attraction of gravitation is in
exact proportion to the mass of the attracted body, whatever be its chemical
nature. (See p. 36.)

Admitting these results, and supposing the great ocean of aether to l)e
traversed by waves, these waves impinge on the atoms, and are reflected in
the form of diverging waves. These, in their turn, beat other atoms, and cause
attraction or repulsion, according as their wave-length is great or small. Thus
the waves of shortest period perform the office of repelling atom from atom,
and rendering their collision for ever impossible. Other waves, somewhat longer,
bind the atoms together in molecular groups. Others contribute to the elasticity
of bodies of sensible size, while the long waves are the cause of universal
gravitation, holding the planets in their courses, and preserving the most ancient
heavens in all their freshness and strength. Then besides the waves of aether,
our author contemplates its streams, spiral and otherwise, by which he accounts
for electric, magnetic, and galvanic phenomena.

Without pretending to have verified all or any of the calculations on which
this theory is based, or to have compared the electric, magnetic, and galvanic phe-
nomena, as described in the Essay, with those actually observed, we may venture
to make a few remarks upon the theory of action at a distance here put forth.

The explanation of any action between distant bodies by means of a clearly
conceivable process going on in the intervening medium is an achievement of

AN ESSAY ON THE MATHEMATICAL PRINCIPLES OF PHYSICS. 341

the highest scientific value. Of all such actions, that of gravitation is the most
universal and the most mysterious. Whatever theory of the constitution of
bodies holds out a prospect of the ultimate explanation of the process by which
gravitation is effected, men of science will be found ready to devote the whole
remainder of their lives to the development of that theory.

The only theory hitherto put forth as a dynamical theory of gravitation
is that of Lesage, who adopts the Lucretian theory of atoms and void.

Gravitation on this theory is accounted for by the impact of atoms of
incalculable minuteness, which are flying through the heavens with inconceivable
velocity and in every possible direction. These " ultramundane corpuscules "
falling on a solitary heavenly body would strike it on every side with equal
impetus, and would have no effect upon it in the way of resultant force. If,
however, another heavenly body were in existence, each would screen the other
from a portion of the corpuscular bombardment, and the two bodies would be
attracted to each other. The merits and the defects of this theory have been
recently pointed out by Sir W. Thomson. If the corpuscules are perfectly elastic
one body cannot protect the other from the storm, for it will reflect exactly
as many corpuscules as it intercepts. If they are inelastic, as Lesage supposes,
what becomes of them after collision ? Why are not bodies always growing by
the perpetual accumulation of them ? How do they get swept away ? and what
becomes of their energy ? Why do they not volatilise the earth in a few
minutes ? I shall not enter on Sir W. Thomson's improvement of this theory,
as it involves a different kind of hydrodynamics from that cultivated in the
Essay, but in whatever way we regard Lesage's theory, the cause of gravitation
in the universe can be represented only as depending on an ever fresh supply
of something from without.

Though Professor Challis has not, as far as we can see, stated in what
manner his sethereal waves are originally produced, it would seem that on his
theory also the primary waves, by whose action the waves diverging from the
atoms are generated, must themselves be propagated from somewhere outside the
world of stars.

On either theory, therefore, the universe is not even temporarily automatic,
but must be fed from moment to moment by an agency external to itself.

If the corpuscules of the one theory, or the aethereal waves of the other,
were from any cause to be supplied at a different rate, the value of every
force in the universe would suffer change.

1 .' AN ESSAY OS THE MATHEMATICAL PRINCIPLES OF PHYSICS.

On both theories, too, the preservation of the universe is effected only by the

unceasing expenditure of enormous quantities of work, so that the conservation

energy in physical operations, which has been the subject of so many

. ... . ; •;.. ,• ; iy of itiaA li:i- \>-<\ to so many disfovi-r'n-s, is

apparent only, and is merely a kind of "moveable equilibrium" between supply
nnd destruction.

It may seem a sort of anticlimax to descend from these highest heavens
invention down to the "equations of condition" of fluid motion. But it
would not be right to pass by the fact that the fluids treated of in this Essay
ore not in all respects similar to those met with elsewhere. In all their motions
they obey a law, which our author was the first to lay down, in addition—
or perhaps in some cases in opposition — to those prescribed for them by Lagrange,
Poisson, &c.

It is true that a perfect fluid, originally at rest, and afterwards acted on
only by such forces as occur in nature, will freely obey this law, and that not
only in the form laid down by Professor Challis, in which its rigour is partially
relaxed by the introduction of an arbitrary factor, but in its original severe
Minplifity, as the condition of the existence of a velocity-potential.

But, on the one hand, problems in which the motion is assumed to violate
this condition have been solved by Helmholtz and Sir W. Thomson, who tell
us what the fluid will then do; and, on the other hand, Professor ChallisV
fluid is able, in virtue of the new equation, to transmit plane waves consisting
of transverse displacements. As this is what takes place in the luminiferous
sether, other physicists refuse to regard that aether as a fluid, because, according
to their definition, the action between any contiguous portions of a fluid is
entirely normal to the surface which separates them.

It is not necessary, however, for us to say any more on this subject, as
the Essay before us does not contain, in an explicit form, the equation referred
to, but is devoted rather to the exposition of those wider theories of the
constitution of matter and the phenomena of nature, some of which we have
endeavoured to describe.

[From Nature, Vol. vin.]

LIX. On Loschmidfs Experiments on Diffusion in relation to the Kinetic

Theory of Gases.

THE kinetic theory asserts that a gas consists of separate molecules, each
moving with a velocity amounting, in the case of hydrogen, to 1,800 metres
per second. This velocity, however, by no means determines the rate at which
a group of molecules set at liberty in one part of a vessel full of the gas
will make their way into other parts. In spite of the great velocity of the
molecules, the direction of their course is so often altered and reversed by
collision with other molecules, that the process of diffusion is comparatively a
slow one.

The first experiments from which a rough estimate of the rate of diffusion
of one gas through another can be deduced are those of Graham "". Professor
Loschmidt, of Vienna, has recently f made a series of most valuable and accurate
experiments on the interdiffusion of gases in a vertical tube, from which he
has deduced the coefficient of diffusion of ten pairs of gases. These results I
consider to be the most valuable hitherto obtained as data for the construction
of a molecular theory of gases.

There are two other kinds of diffusion capable of experimental investigation,
and from which the same data may be derived, but in both cases the experi-
mental methods are exposed to much greater risk of error than in the case
of diffusion. The first of these is the diffusion of momentum, or the lateral
communication of sensible motion from one stratum of a gas to another. This
is the explanation, on the kinetic theory, of the viscosity or internal friction
of gases. The investigation of the viscosity of gases requires experiments of
great delicacy, and involving very considerable corrections before the true

* Brande's Journal for 1829, pt. ii. p. 74, "On the Mobility of Gases," Phil. Train. 1863.
t Sitzb. d. L Akad. d. Wissench. 10 Marz. 1870.

344 EXPERIMENTS ON DIFFUSION IN RELATION TO

of viscosity is obtained. Thus the numbers obtained by myself in 18G5

are nearly double of those calculated by Prof. Stokes from the experiments
of Boily on pendulums, but not much more than half those deduced by 0. E.
Meyer from his own experiments. The other kind of diffusion is that of the
energy of agitation of the molecules. This is called the conduction of heat
The experimental investigation of this subject is confessedly so difficult, that
it is only recently that Prof. Stefan of Vienna*, by means of a very ingenious
method, has obtained the first experimental determination of the conductivity
of air. This result is, as he says, in striking agreement with the kinetic
theory of gases.

The experiments on the interdiffusion of gases, as conducted by Prof.
Loachmidt and his pupils, appear to be far more independent of disturbing
causes than any experiments on viscosity or conductivity. The mterdifiuamg
gases are left to themselves in a vertical cylindrical vessel, the heavier gas
being underneath. No disturbing effect due to currents seems to exist, and the
results of different experiments with the same pair of gases appear to be
consistent with each other.

They prove conclusively that the coefficient of diffusion varies inversely as
the pressure, a result in accordance with the kinetic theory, whatever hypo-
thesis we adopt as to the nature of the mutual action of the molecules during
their encounters.

They also shew that the coefficient of diffusion increases as the temperature
rises, but the range of temperature in the experiments appears to be too small
to enable us to decide whether it varies as 2", as it should be according to the
theory of a force inversely as the fifth power of the distance adopted in my
paper in the Phil. Trans. 1866, or as jT* as it should do according to the
theory of elastic spherical molecules, which was the hypothesis originally developed
by Clausius, by myself in the Phil. Mag. 1860, and by O. E. Meyer.

In comparing the coefficients of diffusion of different pairs of gases, Prof.
Loschmidt has made use of a formula according to which the coefficient of
diffusion should vary inversely as the geometric mean of the atomic weiglits
of the two gases. I am unable to see any ground for this hypothesis in the
kinetic theory, which in fact leads to a different result, involving the diameters
of the molecules, as well as their masses. The numerical results obtained l>y

* Sitzb. d. k. Akad. Feb. 22, 1872.

THE KINETIC THEORY OF GASES. 345

Prof. Loschmidt do hot agree with his formula in a manner corresponding to
the accuracy of his experiments. They agree in a very remarkable manner with
the formula derived from the kinetic theory.

I have recently been revising the theory of gases founded on that of the
collisions of elastic spheres, using, however, the methods of my paper on the
dynamical theory of gases (Phil. Trans. 1866) rather than those of my first
paper in the Phil. Mag., 1860, which are more difficult of application, and
which led me into great confusion, especially in treating of the diffusion of
gases.

The coefficient of interdiffusion of two gases, according to this theory, is

D - -i=-Z

where wl and wt are the molecular weights of the two gases, that of hydrogen
being unity.

sa is the distance between the centres of the molecules at collision in centi-
metres.

V is the " velocity of mean square " of a molecule of hydrogen at 0° C.

r= I— = 185,900 centimetres per second.

P

N is the number of molecules in a cubic centimetre at 0° C. and 76 cm. B.
(the same for all gases).

Du is the coefficient of interdiffusion of the two gases in * — — ^ — -
measure.

We may simplify this expression by writing

Here o is a quantity the same for all gases, but involving the unknown
number N.

o- is a quantity which may be deduced from the corresponding experiment
of M. Loschmidt. We have thus

*i» = <urtt (3),

VOL. n. 44

346

EXPERIMENTS ON DIFFUSION IN RELATION TO

or the H--**"^> between the centres of the molecules at collision is proportional
to the quantity <r, which may be deduced from experiment.
If <*, and </, are the diameters of the two molecules,

Henoe if d = a£ ...... <ru = *(8, + 8,) ........................... (4).

Now M. Loachmidt has determined D for the six pairs of gases which
can be formed from Hydrogen, Oxygen, Carbonic Oxide, and Carbonic Acid. The
MX values of <r deduced from these experiments ought not to be independent,
since they may be deduced from the four values of 8 belonging to the two
Accordingly we find, by assuming

TABLE I.

8(H) =1-739
S(O) =2-283
8 (CO) =2-461
8 (CO.) = 2-775

Calculated

Observed

'a

4 {«,+»,)

\/^\/i+5i

For H and O

2-011

1-992

For H and CO

2-100

2-116

For H and CO,

2-257

2-260

For O and CO

3-372

2-375

For O and CO

2-629

2-545

For CO and CO,

2-618

2-599

Note. — These numbers must be multiplied by 0%6 to reduce them to (centi-
metre-second) measure from the (metre-hour) measure employed by Loschmidt.

The agreement of these numbers furnishes, I think, evidence of considerable
strength in favour of this form of the kinetic theory, and if it should be
confirmed by the comparison of results obtained from a greater number of pairs
of gases it will be greatly strengthened.

Evidence, however, of a higher order may be furnished by a comparison
between the results of experiments of entirely different kinds, as for instance,
the coefficients of diffusion and those of viscosity. If p. denotes the coefficient

THE KINETIC THEORY OF GASES.

347

of viscosity, and p the density of a gas at 0°C. and 760 mm. B., the theory
gives

J^^'i <5>'

so that the following relation exists between the viscosities of two gases and
their coefficient of interdiffusion—

ft p.

(6).

Calculating from the data of Table L, the viscosities of the gases, and com-
paring them with those found by 0. E. Meyer and by myself, and reducing all
to centimetre, gramme, second measure, and 0°C. —

TABLE II.
Coefficient of Viscosity.

Gas.

Loschmidt.

O. E. Meyer.

Maxwell.

H

0-000116

0-000134

0-000097

O

0 000270

0-000306

CO

0-000217

0-000266

CO,

0-000214

0-000231

0-000161

The numbers given by Meyer are greater than those derived from Loschmidt.
Mine, on the other hand, are much smaller. I think, however, that of the
three, Loschmidt's are to be preferred as an estimate of the absolute value
of the quantities, while those of Meyer, derived from Graham's experiments,
may possibly give the ratios of the viscosities of different gases more correctly.
Loschmidt has also given the coefficients of interdiffusion of four other pairs
of gases, but as each of these contains a gas not contained in any other pair,
I have made no use of them.

In the form of the theory as developed by Clausius, an important part
is played by a quantity called the mean length of the uninterrupted path of a

44—2

348 KXPKRIMKNTB OS DIFFUSION IN RELATION TO

nuJfculf, or, more concisely, the mean path. Its value, according to my cal-
culations, is

1*

Its value in tenth-metres (1 metre x 10~w) is

TABLE III.

For Hydrogen . . . 965 Tenth-metres at 0° C. and 760 B.

For Oxygen . . . 560

For Carbonic Oxide . . 482

For Carbonic Acid . . 430

(The wave-length of the hydrogen ray F is 4,861 tenth-metres, or about ten
times the mean path of a molecule of carbonic oxide.)

We may now proceed for a few steps on more hazardous ground, and
inquire into the actual size of the molecules. Prof. Loschmidt himself in his
paper "Zur Grosse der Luftmoleciile " (Acad. Vienna, Oct. 12, 1865), was the
first to make this attempt. Independently of him and of each other, Mr G. J.
Stoney (Phil Mag., Aug. 1868), and Sir W. Thomson (Nature, March 31, 1870),
have made similar calculations. We shall follow the track of Prof. Loschmidt.

TT

The volume of a spherical molecule is - s8, where s is its diameter. Hence
if N is the number of molecules in unit of volume, the space actually filled
by the molecules is ^Nsf.

This, then, would be the volume to which a cubic centimetre of the gas
would be reduced if it could be so compressed as to leave no room whatever
between the molecules. This, of course, is impossible ; but we may, for the
sake of clearness, call the quantity

(8)

• The difference between this value and that given by M. Clausius in his paper of 1858, arises
from his assuming that all the molecules have equal velocities, while I suppose the velocities to be
distributed according to the " law of errors."

THE KINETIC THEORY OF GASES. 349

the ideal coefficient of condensation. The actual coefficient of condensation, when
the gas is reduced to the liquid or even the solid form, and exposed to the
greatest degree of cold and pressure, is of course greater than c.

Multiplying equations (7) and (8), we find

s = 6*Jzd (9),

where s is the diameter of a molecule, c the coefficient of condensation, and
I the mean path of a molecule.

Of these quantities, we know I approximately already, but with respect
to e we only know its superior limit. It is only by ascertaining whether calcu-
lations of this kind, made with respect to different substances, lead to consistent
results, that we can obtain any confidence in our estimates of e.

M. Lorenz Meyer* has compared the " molecular volumes " of different
substances, as estimated by Kopp from measurements of the density of these
substances and their compounds, with the values of s3 as deduced from experi-
ments on the viscosity of gases, and has shewn that there is a considerable
degree of correspondence between the two sets of numbers.

The "molecular volume" of a substance here spoken of is the volume in
cubic centimetres of as much of the substance in the liquid state as contains
as many molecules as one gramme of hydrogen. Hence if pa denote the density
of hydrogen, and b the molecular volume of a substance, the actual coefficient
of condensation is

*' = pob (10).

These "molecular volumes" of liquids are estimated at the boiling-points of
the liquids, a very arbitrary condition, for this depends on the pressure, and
there is no reason in the nature of things for fixing on 760 mm. B. as a
standard pressure merely because it roughly represents the ordinary pressure
of our atmosphere. What would be better, if it were not impossible to obtain
it, would be the volume at — 273° C. and o>B.

But the volume relations of potassium with its oxide and its hydrated
oxide as described by Faraday seem to indicate that we have a good deal yet
to learn about the volumes of atoms.

* Annalen d. Chemie u. Pharmacie v. Supp. bd. 2, Heft (1867).

350 EXPERIMENTS ON DIFFUSION IN RELATION TO THE KINETIC THEORY OF GASES.

If, however, for our immediate purpose, we assume the smallest molecular
volume of oxygen given by Kopp as derived from a comparison of the volume
of tin with that of its oxide and put

b(0=16) = 27,
we find for the diameters of the molecules —

4

TABLE IV.

Hydrogen . . . . 5'8 tenth-metres.

Oxygen 7-6

Carbonic Oxide . . 8-3

Carbonic Acid ... 9-3

The mass of a molecule of hydrogen on this assumption is

4'6 x 10"" gramme.

The number of molecules in a cubic centimetre of any gas at 0"C. and

760 mm. B. is

AT= 19x10".

Hence the side of a cube which, on an average, would contain one molecule
would be

^V"* = 37 tenth-metres.

[From Nature, Vol. vm.]

LX. On the Final State of a System of Molecules in motion subject to forces

of any kind.

LET perfectly elastic molecules of different kinds be in motion within a
vessel with perfectly elastic sides, and let each kind of molecules be acted on
by forces which have a potential, the form of which may be different for
different kinds of molecules.

Let x, y, z be the co-ordinates of a molecule, M, and f, 17, £ the com-
ponents of its velocity, and let it be required to determine the number of
molecules of a given kind which, on an average, have their co-ordinates between
x and x + dx, y and y + dy, z and z + dz, and also their component velocities
between £ and £+cZ£ 17 and -rj + d-rj, and £ and £ + d£. This number must
depend on the co-ordinates and the components of velocities and on the limits
of these quantities. We may therefore write it

dN=f(x, y, z, £ TTJ, £) dxdydzdgdr)dt, (1).

We shall begin by investigating the manner in which this quantity depends
on the components of velocity, before we proceed to determine in what way it
depends on the co-ordinates.

If we distinguish by suffixes the quantities corresponding to different kinds
of molecules, the whole number of molecules of the first and second kind within
a given space which have velocities within given limits may be written

/, (£, i?,, £.) dfi« dr,,. d^ = n, (2),

and /s(£, 77,, £3) d^. d-r)3. d^ = n, (3).

The number of pairs which can be formed by taking one molecule of each
kind is nv nv

352 FINAL STATE OF A SYSTEM OF MOLECULES IN MOTION

Let a pair of molecules encounter each other, and after the encounter let
their component velocities be £', if,'. {,' and &', 17,', £,'. The nature of the
encounter U completely defined when we know £-£„ 17, -17,, £,-£, the velocity
of the second molecule relative to the first before the encounter, and SG.-X,,
y,-ym «,— r, the position of the centre of the second molecule relative to the
first at the instant of the encounter. When these quantities are given, £'-£,',
i),'-i)' and {,' — £,', the components of the relative velocity after the encounter,
are determinable.

Hence putting a, ft. y for these relative velocities, and a, b, c for t Ir-
relative positions, we find for the number of molecules of the first kind having
velocities between the limits f, and £, + </£ &c., which encounter molecules of
the second kind having velocities between the limits £ and £, + d£, &c., in such
a way that the relative velocities lie between o and a + da, &c., and the relative
positions between a and a+da, &c.

/, (£„ ?„ 0 dfadl.f, (£, 77, Q dtd-ndl.t (abcafr) dadbdcdadftdy... (4),
and after the encounter the velocity of 3f, will be between the limits f,' and
(\ + d(, Ac. and that of Mt between the limits £' and £' + d£, &c.

The differences of the limits of velocity are equal for both kinds of mole-
cules, and both before and after the encounter.

When the state of motion of the system is in its permanent condition,
as many pairs of molecules change their velocities from Vu Vt to F,', F,' as
from F,', F,' to Fw Vv and the circumstances of the encounter in the one case
are precisely similar to those in the second. Hence, omitting for the sake of
brevity the quantities d£, &c., and <f>, which are of the same value in the two
cases, we find

/, (& *, C,)/. (& * t) -/, (£» vS, O/, (&', V. 4') (5),

Anting log/(f, 17, 0 = ^^^, I, m, n) (6),

where I, m, n are the direction-cosines of the velocity, F, of the molecule M.

Taking the logarithm of both sides of equation (5),

F, (M, F.V.m.n.) + Ft (Mt F,*/,™,*,) = F, (M, F^XV) + *\ (Mn FtX«) • • • ( 7).

The only necessary rektion between the variables before and after the
encounter is

(8).

SUBJECT TO FORCES OF ANY KIND. 353

If the right-hand side of the equations (7) and (8) are constant, the left-
hand sides will also be constant ; and since lu m,, «j are independent of la mu n,
we must have

F^AMW and F, = AM,V? (9),

where A is a quantity independent of the components of velocity, or

° (10),

(n).

This result as to the distribution of the velocities of the molecules at a
given place is independent of the action of finite forces on the molecules during
their encounter, for such forces do not affect the velocities during the infinitely
short time of the encounter.

We may therefore write equation (1)

dN=Ceu"*+++*>d£dndldxdydz (12),

where C is a function of x, y, z which may be different for different kinds
of molecules, while A is the same for every kind of molecule, though it may,
for aught we know as yet, vary from one place to another.

Let us now suppose that the kind of molecules under consideration are
acted on by a force whose potential is \j>. The variations of x, y, z arising from
the motion of the molecules during a tune St are

&c = £&, By = T)8t, Sz = t,St (13),

and those of £ 17, £ in the same time due to the action of the force, are

«—£**•—$*«:- -g* <">•

If we make c = logC (15),

The variation of this quantity due to the variations 8xlt Sylt 8zlt S$u Si/,, S£l( is

(.. dc dc dc\ s AHflt d$ d$ r d\fi\ -
dx V dy dz) \ dx ^ dy dzj

.(17).

VOL. ii. 45

354 FINAL STATE OF A SYSTEM OP MOLECULES IN MOTION, &C.

Since the number of the molecules does not vary during their motion, this
quantity ia zero, whatever the values of f, 17, £. Hence we have in virtue of the

last term

dA dA dA .

or A is constant throughout the whole region traversed by the molecules.

Next, comparing the first and second terms, we find

c = AM(2$ + B) (19).

We thus obtain as the complete form of dN

G&zVj — c < t.i' 1 1 */ 1 iz< I c; < IT] ( ( {, • •••••••••••••• •••(20K

when A is an absolute constant, the same for every kind of molecule in the
vessel, but Bl belongs to the first kind only. To determine these constants,
we must integrate this quantity with respect to the six variables, and equate
the result to the number of molecules of the first kind. We must then, by
integrating dNl^Ml (£' + •»/!* + £' + 2^,) determine the whole energy of the system,
and equate it to the original energy. We shall thus obtain a sufficient number
of equations to determine the constant A, common to all the molecules, and
Bt, Bv &c. those belonging to each kind.

The quantity A is essentially negative. Its value determines that of the
mean kinetic energy of all the molecules in a given place, which is — f- -T- , and

therefore, according to the kinetic theory, it also determines the temperature
of the medium at that place. Hence, since Au in the permanent state of the
system, is the same for every part of the system, it follows that the tempera-
ture is everywhere the same, whatever forces act upon the molecules.

The number of molecules of the first kind in the element dxdydz,

(21).

The effect of the force whose potential is ^ is therefore to cause the
molecules of the first kind to accumulate in greater numbers in those parts of
the vessel towards which the force acts, and the distribution of each different
kind of molecules in the vessel is determined by the forces which act on them
in the same way as if no other molecules were present. This agrees with
Dalton's doctrine of the distribution of mixed gases.

s

-

•

•
like tl.

[From Nature, Vol. vin.]
LXI. Faraday.

[Michael Faraday, born September 22, 1791, died August 25, 1867.]

WITH this number of Nature we present to our subscribers the first of
what we hope will be a long series of Portraits of Eminent Men of Science.

This first portrait is one of Faraday, engraved on steel, by Jeens, from a
photograph by Watkins. Those who had the happiness of knowing Faraday
best will best appreciate the artist's skill — he has indeed surpassed himself, for'
the engraving is more life-like than the photograph. We could ill spare such
a memorial of such a man, one in which all the beautiful simplicity of his
life beams upon us. There is no posturing here !

There is no need that we should accompany the portrait with a memoir
of Faraday. Bence Jones, Tyndall, and Gladstone have already lovingly told the
story of the grand and simple life which has shed and will long continue to
shed such lustre on English Science, and their books have carried the story
home to millions ; nor is there any need that we should state why we have
chosen to commence our series with Faraday; everybody will acknowledge the
justice of our choice.

But there is great need just now that some of the lessons to be learnt
from Faraday's life should be insisted upon, and we regard it as a fortunate
circumstance that we have thus the opportunity of insisting upon them while
our Scientific Congress is in session, and before the echoes of the Address of
the President of the British Association for the Advancement of Science have
died away.

In the first place, then, we regard Faraday at once as the most useful
and the most noble type of a scientific man. The nation is bigger and stronger
in that Faraday has lived, and the nation would be bigger and stronger still
were there more Faraday s among us now. Professor Williamson, in his admirable

45—2

356 FARADAY.

address, acknowledges that the present time is "momentous." In truth the
question of the present condition of Science and the ways of improving it, is
occupying men's minds more than it has ever done before ; and it is now
conceded on all sides that this is a national question, and not only so, hut
one of fundamental importance. Now what is the present condition of English
Science? It is simply this, that while the numbers of our professors and their
emoluments are increasing, while the number of students is increasing, while
practical instruction is being introduced and text-books multiplied, while the
number and calibre of popular lecturers and popular writers in Science is
increasing, original research, the fountain-head of a nation's wealth, is decreasing.

Now a scientific man is useful as such to a nation according to the amount
of new knowledge with which he endows that nation. This is the test which
the nation, as a whole, applies, and Faraday's national reputation rests on it.
Let the nation know then that the real difficulty at present is this ; we want
more Faradays ; in other words more men working at new knowledge.

It is refreshing to see this want so clearly stated in the Presidential
Address :

"The first thing wanted for the work of advancing science is a supply of
well-qualified workers. The second thing is to place and keep them under the
conditions most favourable to their efficient activity. The most suitable men
must be found while still young, and trained to the work. Now I know only
one really effectual way of finding the youths who are best endowed by nature
for the purpose ; and that is to systematise and develop the natural conditions
which accidentally concur in particular cases, and enable youths to rise from
the crowd.

" Investigators, once found, ought to be placed in the circumstances most
favourable to their efficient activity.

" The first and most fundamental condition for this is, that their desire for
the acquisition of knowledge be kept alive and fostered. They must not merely
retain the hold which they have acquired on the general body of their science ;
they ought to strengthen and extend that hold, by acquiring a more complete
and accurate knowledge of its doctrines and methods ; in a word, they ought
to be more thorough students than during their state of preliminary training.

"They must be able to live by their work, without diverting any of their
energies to other pursuits; and they must feel security against want, in the
event of illness or in their old age.

FARADAY. 357

"They must be supplied with intelligent and trained assistants to aid in
the conduct of their researches, and whatever buildings, apparatus, and materials
may be required for conducting those researches effectively.

" The desired system must therefore provide arrangements favourable to the
maintenance and development of the true student-spirit in investigators, while
providing them with permanent means of subsistence, sufficient to enable them
to feel secure and tranquil in working at science alone, yet not sufficient to
neutralise their motives for exertion ; and at the same time it must give them
all external aids, in proportion to their wants and powers of making good use
of them."

Whether the scheme proposed by Dr Williamson to bring such a state of
things about will have the full success he anticipates is a matter of second-
rate importance ; what is of importance is, that the need of some scheme is
now fully recognised.

So far the remarks we have made have been suggested by Faraday's
usefulness. It is to be hoped that the nobleness of his simple, undramatic life,
will live as long in men's memories as the discoveries which have immortalised
his name. Here was no hunger after popular applause, no jealousy of other
men's work, no swerving from the well-loved, self-imposed task of "working,
finishing, publishing."

" The simplicity of his heart, his candour, his ardent love of the truth, his
fellow-interest in all the successes, and ingenuous admiration of all the discoveries
of others, his natural modesty in regard to what he himself discovered, his
noble soul — independent and bold — all these combined, gave an incomparable
charm to the features of the illustrious physicist."

Such was his portrait as sketched by Dumas, a man cast in the same
mould. All will recognise its truth. Can men of science find a nobler exemplar
on which to fashion their own life ? Nay, if it were more widely followed than
it is, should we not hear less of men falling away from the " brilliant promise "
of their youth, tempted by " fees," or the " applications of Science," or the
advantages attendant upon a popular exposition of other men's work ? Should
we not hear a little less- frequently than we do that research is a sham, and
that all attempts to aid it savour of jobbery ?

Lastly we may consider Faraday's place in the general history of Science ;
this is far from easy. Our minds are still too much occupied with the memory
of the outward form and expression of his scientific work to be able to compare

FARADAY.

him aright with the other great men among whom we shall have to place

him.

Every great man of the first rank is unique. Each has his own office and
his own place in the historic procession of the sages. That office did not exist
even in the imagination, till he came to fill it, and none can succeed to his
place when he has passed away. Others may gain distinction by adapting the
exposition of science to the varying language of each generation of students, but
their true function is not so much didactic as pa3dagogic — not to teach the use
of phrases which enable us to persuade ourselves that we understand a science,
but to bring the student into living contact with the two main sources of
mental growth, the fathers of the sciences, for whose personal influence over
the opening mind there is no substitute, and the material things to which
their labours first gave a meaning.

Faraday is, and must always remain, the father of that enlarged science of
electro-magnetism which takes in at one view, all the phenomena which former
inquirers had studied separately, besides those which Faraday himself discovered
by following the guidance of those convictions, which he had already obtained,
of the unity of the whole science.

Before him came the discovery of most of the fundamental phenomena, the
electric and magnetic attractions and repulsions, the electric current and its
effects. Then came Cavendish, Coulomb, and Poisson, who by following the
path pointed out by Newton, and making the forces which act between bodies
the principal object of their study, founded the mathematical theories of electric
and magnetic forces. Then Orsted discovered the cardinal fact of electro-magnetic
force, and Ampere investigated the mathematical laws of the mechanical action
between electric currents.

Thus the field of electro-magnetic Science was already very large when
Faraday first entered upon his public career. It was so large that to take in
at one view all its departments required a stretch of thought for which a
special preparation was necessary. Accordingly, we find Faraday endeavouring
in the first place to obtain, from each of the known sources of electric action,
all the phenomena which any one of them was able to exhibit. Having thus
established the unity of nature of all electric manifestations, his next aim was
to form a conception of electrification, or electric action, which would embrace
them all. For this purpose it was necessary that he should begin by getting
rid of those parasitical ideas, which are so apt to cling to every scientific term,

FARADAY. 359

and to invest it with a luxuriant crop of connotative meanings flourishing at
the expense of the meaning which the word was intended to denote. He
therefore endeavoured to strip all such terms as "electric fluid," "current," and
"attraction" of every meaning except that which is warranted by the phenomena
themselves, and to invent new terms, such as "electrolysis," "electrode," "di-
electric," which suggest no other meaning than that assigned to them by their
definitions.

He thus undertook no less a task than the investigation of the facts, the
ideas, and the scientific terms of electro-magnetism, and the result was the
remodelling of the whole according to an entirely new method.

That old and popular phrase, " electric fluid," which is now, we trust,
banished for ever into the region of newspaper paragraphs, had done what it
could to keep men's minds fixed upon those particular parts of bodies where
the "fluid" was supposed to exist.

Faraday, on the other hand, by inventing the word " dielectric," has
encouraged us to examine all that is going on in the air or other medium
between the electrified bodies.

It is needless to multiply instances of this kind. The terms, field of force,
lines of force, induction, &c., are sufficient to recall them. They all illustrate
the general principles of the growth of science, in the particular form of which
Faraday is the exponent.

We have, first, the careful observation of selected phenomena, then the
examination of the received ideas, and the formation, when necessary, of new
ideas ; and, lastly, the invention of scientific terms adapted for the discussion
of the phenomena in the light of the new ideas.

The high place which we assign to Faraday in electro-magnetic science may
appear to some inconsistent with the fact that electro-magnetic science is an
exact science, and that in some of its branches it had already assumed a
mathematical form before the tune of Faraday, whereas Faraday was not a
professed mathematician, and in his writings we find none of those integrations
of differential equations which are supposed to be of the very essence of an
exact science. Open Poisson and Ampere, who went before him, or Weber and
Neumann, who came after him, and you will find their pages full of symbols,
not one of which Faraday would have understood. It is admitted that Faraday
made some great discoveries, but if we put these aside, how can we rank his
scientific method so high without disparaging the mathematics of these eminent men ?

360 FARADAY.

It is true that no one can essentially cultivate any exact science without
understanding the mathematics of that science. But we are not to suppose that
the calculations and equations which mathematicians find so useful constitute
the whole of mathematics. The calculus is but a part of mathematics.

The geometry of position is an example of a mathematical science established
without the aid of a single calculation. Now Faraday's lines of force occupy
the same position in electro-magnetic science that pencils of lines do in the
geometry of position. They furnish a method of building up an exact mental
image of the thing we are reasoning about. The way in which Faraday made
use of his idea of lines of force in co-ordinating the phenomena of magneto-
electric induction* shews him to have been in reality a mathematician of a
very high order— one from whom the mathematicians of the future may derive
valuable and fertile methods.

For the advance of the exact sciences depends upon the discovery and
development of appropriate and exact ideas, by means of which we may form
a mental representation of the facts, sufficiently general, on the one hand, to
stand for any particular case, and sufficiently exact, on the other, to warrant
the deductions we may draw from them by the application of mathematical
reasoning.

From the straight line of Euclid to the lines of force of Faraday this has
been the character of the ideas by which science has been advanced, and by
the free use of dynamical as well as geometrical ideas we may hope for a
further advance. The use of mathematical calculations is to compare the results
of the application of these ideas with our measurements of the quantities
concerned in our experiments. Electrical science is now in the stage in which
such measurements and calculations are of the greatest importance.

We are probably ignorant even of the name of the science which will be
developed out of the materials we are now collecting, when the great philosopher
next after Faraday makes his appearance.

* To estimate the intensity of Faraday's scientific power, we cannot do better than read the first
and second series of his Researches and compare them, first, with the statements in Bence Jones's Lift
of Faraday, which tells us the tales of the first discovery of the facts, and of the final publication
of the results, and second, with the whole course of electro-magnetic science since, which has added
no new idea to those set forth, but has only verified the truth and scientific value of every one of

[From Nature, Vol. Till.]

LXII. Molecules*.

AN atom is a body which cannot be cut in two. A molecule is the
smallest possible portion of a particular substance. No one has ever seen or
handled a single molecule. Molecular science, therefore, is one of those branches
of study which deal with things invisible and imperceptible by our senses, and
which cannot be subjected to direct experiment.

The mind of man has perplexed itself with many hard questions. Is
space infinite, and if so in what sense ? Is the material world infinite in
extent, and are all places within that extent equally full of matter ? Do atoms
exist, or is matter infinitely divisible ?

The discussion of questions of this kind has been going on ever since men
began to reason, and to each of us, as soon as we obtain the use of our
faculties, the same old questions arise as fresh as ever. They form as essential
a part of the science of the nineteenth century of our era, as of that of the
fifth century before it.

We do not know much about the science organisation of Thrace twenty-
two centuries ago, or of the machinery then employed for diffusing an interest
in physical research. There were men, however, in those days, who devoted
their lives to the pursuit of knowledge with an ardour worthy of the most
distinguished members of the British Association ; and the lectures in which
Democritus explained the atomic theory to his fellow-citizens of Abdera realised,
not in golden opinions only, but in golden talents, a sum hardly equalled even
in America.

To another very eminent philosopher, Anaxagoras, best known to the world
as the teacher of Socrates, we are indebted for the most important service to

* A Lecture delivered before the British Association at Bradford.
VOL. II. 46

MOLECULES.

the atomic theory, which, after its statement by Democritus, remained to be
done. Anaxagoras, in fact, stated a theory which so exactly contradicts the
atomic theory of Democritus that the truth or falsehood of the one theory
implies the falsehood or truth of the other. The question of the existence or
non-existence of atoms cannot be presented to us this evening with gr.
clearness than in the alternative theories of these two philosophers.

Take any portion of matter, say a drop of water, and observe its properties.
Like every other portion of matter we have ever seen, it is divisible. Divide
it in two, each portion appears to retain all the properties of the original drop,
and among others that of being divisible. The parts are similar to the whole
in every respect except in absolute size.

Now go on repeating the process of division till the separate portions of
water are so small that we can no longer perceive or handle them. Still \ve
have no doubt that the sub-division might be carried further, if our senses
were more acute and our instruments more delicate. Thus far all are a^:
but now the question arises, Can this sub-division be repeated for ever ?

According to Democritus and the atomic school, we must answer in the
negative. After a certain number of sub-divisions, the drop would be di\
into a number of parts each of which is incapable of further sub-division. We
should thus, in imagination, arrive at the atom, which, as its name literally
signifies, cannot be cut in two. This is the atomic doctrine of Democritus,
Epicurus, and Lucretius, and, I may add, of your lecturer.

. According to Anaxagoras, on the other hand, the parts into which the drop
is divided are in all respects similar to the whole drop, the mere size of a
body counting for nothing as regards the nature of its substance. Hence if
the whole drop is divisible, so are its parts down to the minutest sub-divisions,
and that without end.

The essence of the doctrine of Anaxagoras is that parts of a body are
in all respects similar to the whole. It was, therefore, called the doctrine
of Homoiomereia. Anaxagoras did not of course assert this of the parts of
organised bodies such as men and animals, but he maintained that those inor-
ganic substances which appear to u's homogeneous are really so, and that the
universal experience of mankind testifies that every material body, without
exception, is divisible.

The doctrine of atoms and that of homogeneity are thus in direct con-
tradiction.

MOLECULES. 363

But we must now go on to molecules. Molecule is a modern word. It does
not occur in Johnson's Dictionary. The ideas it embodies are those belonging
to modern chemistry.

A drop of water, to return to our former example, may be divided into
a certain number, and no more, of portions similar to each other. Each of
these the modern chemist calls a molecule of water. But it is by no means
an atom, for it contains two different substances, oxygen and hydrogen, and by
a certain process the molecule may be actually divided into two parts, one
consisting of oxygen and the other of hydrogen. According to the received
doctrine, in each molecule of water there are two molecules of hydrogen and
one of oxygen. Whether these are or are not ultimate atoms I shall not
attempt to decide.

We now see what a molecule is, as distinguished from an atom.

A molecule of a substance is a small body such that if, on the one hand, a
number of similar molecules were assembled together they would form a mass
of that substance, while on the other hand, if any portion of this molecule
were removed, it would no longer be able, along with an assemblage of other
molecules similarly treated, to make up a mass of the original substance.

Every substance, simple or compound, has its own molecule. If this molecule
be divided, its parts are molecules of a different substance or substances from
that of which the whole is a molecule. An atom, if there is such a thing,
must be a molecule of an elementary substance. Since, therefore, every molecule
is not an atom, but every atom is a molecule, I shall use the word molecule
as the more general term.

I have no intention of taking up your time by expounding the doctrines
of modern chemistry with respect to the molecules of different substances. It
is not the special but the universal interest of molecular science which encourages
me to address you. It is not because we happen to be chemists or physicists
or specialists of any kind that we are attracted towards this centre of all
material existence, but because we all belong to a race endowed with faculties
which urge us on to search deep and ever deeper into the nature of things.

We find that now, as in the days of the earliest physical speculations,
all physical researches appear to converge towards the same point, and every
inquirer, as he looks forward into the dim region towards which the path of
discovery is leading him, sees, each according to his sight, the vision of the
same Quest.

46—2

364 MOLECULES.

One may see the atom as a material point, invested and surrounded by
potential forces. Another sees no garment of force, but only the bare and
utter hardness of mere impenetrability.

But though many a speculator, as he has seen the vision recede before
him into the innermost sanctuary of the inconceivably little, has had to confess
tliat the quest was not for him, and though philosophers in every age have
been exhorting each other to direct their minds to some more useful and
attainable aim, each generation, from the earliest dawn of science to the present
time, has contributed a due proportion of its ablest intellects to the quest of
the ultimate atom.

Our business this evening is to describe some researches in molecular
science, and in particular to place before you any definite information which
has been obtained respecting the molecules themselves. The old atomic theory,
as described by Lucretius and revived in modern times, asserts that the molecules
of all bodies are in motion, even when the body itself appears to be at rest.
These motions of molecules are in the case of solid bodies confined within so
narrow a range that even with our best microscopes we cannot detect that they
alter their places at all. In liquids and gases, however, the molecules are not
confined within any definite limits, but work their way through the whole mass,
even when that mass is not disturbed by any visible motion.

This process of diffusion, as it is called, which goes on in gases and liquids
and even in some solids, can be subjected to experiment, and forms one of the
most convincing proofs of the motion of molecules.

Now the recent progress of molecular science began with the study of the
mechanical effect of the impact of these moving molecules when they strike
against any solid body. Of course these flying molecules must beat against what-
ever is placed among them, and the constant succession of these strokes is, according
to our theory, the sole cause of what is called the pressure of air and other gases.

This appears to have been first suspected by Daniel Bernoulli, but he had not
the means which we now have of verifying the theory. The same theory was
afterwards brought forward independently by Lesage, of Geneva, who, however,
devoted most of his labour to the explanation of gravitation by the impact of
atoms. Then Herapath, in his Mathematical Physics, published in 1847, made a
much more extensive application of the theory to gases, and Dr Joule, whose
absence from our meeting we must all regret, calculated the actual velocity of
the molecules of hydrogen.

MOLECULES. 365

The further development of the theory is generally supposed to have
begun with a paper by Kro'nig, which does not, however, so far as I can see,
contain any improvement on what had gone before. It seems, however, to have
drawn the attention of Professor Clausius to the subject, and to him we owe
a very large part of what has been since accomplished.

We all know that air or any other gas placed in a vessel presses against
the sides of the vessel, and against the surface of any body placed within it.
On the kinetic theory this pressure is entirely due to the molecules striking
against these surfaces, and thereby communicating to them a series of impulses
which follow each other in such rapid succession that they produce an effect
which cannot be distinguished from that of a continuous pressure.

If the velocity of the molecules is given, and the number varied, then since
each molecule, on an average, strikes the sides of the vessel the same number
of times, and with an impulse of the same magnitude, each will contribute an
equal share to the whole pressure. The pressure in a vessel of given size is
therefore proportional to the number of molecules in it, that is to the quantity
of gas in it.

This is the complete dynamical explanation of the fact discovered by Robert
Boyle, that the pressure of air is proportional to its density. It shews also
that of different portions of gas forced into a vessel, each produces its own
part of the pressure independently of the rest, and this whether these portions
be of the same gas 'or not.

Let us next suppose that the velocity of the molecules is increased. Each
molecule will now strike the sides of the vessel a greater number of times in
a second, but, besides this, the impulse of each blow will be increased in the
same proportion, so that the part of the pressure due to each molecule
will vary as the square of the velocity. Now the increase of velocity cor-
responds, on our theory, to a rise of temperature, and in this way we can
explain the effect of warming the gas, and also the law discovered by Charles
that the proportional expansion of all gases between given temperatures is the
same.

The dynamical theory also tells us what will happen if molecules of different
masses are allowed to knock about together. The greater masses will go slower
than the smaller ones, so that, on an average, every molecule, great or small,
will have the same energy of motion.

The proof of this dynamical theorem, in which I claim the priority, has

366 MOLECULES.

recently been greatly developed and improved by Dr Ludwig Boltzmann. The
moat important consequence which flows from it is that a cubic centimetre of
every gas at standard temperature and pressure contains the same number of
molecules. This is the dynamical explanation of Gay Lussac's law of the
equivalent volumes of gases. But we must now descend to particulars, and
calculate the actual velocity of a molecule of hydrogen.

A cubic centimetre of hydrogen, at the temperature of melting ice, and at
a pressure of one atmosphere, weighs 0'00008954 grammes. We have to find
at what rate this small mass must move (whether altogether or in separate
molecules makes no difference) so as to produce the observed pressure on the
sides of the cubic centimetre. This is the calculation which was first made by
Dr Joule, and the result is 1,859 metres per second. This is what we are
accustomed to call a great velocity. It is greater than any velocity obtained
in artillery practice. The velocity of other gases is less, as you will see by
the table, but in all cases it is very great as compared with that of bullets.

We have now to conceive the molecules of the air in this hall flying about
in all directions, at a rate of about seventeen miles in a minute.

If all these molecules were flying in the same direction, they would con-
stitute a wind blowing at the rate of seventeen miles a minute, and the only
wind which approaches this velocity is that which proceeds from the mouth of
a cannon. How, then, are you and I able to stand here ? Only because the
molecules happen to be flying in different directions, so that those which strike
against our backs enable us to support the storm which is beating against our
faces. Indeed, if this molecular bombardment were to cease, even for an instant,
our veins would swell, our breath would leave us, and we should, literally,
expire. But it is not only against us or against the walls of the hall that
the molecules are striking. Consider the immense number of them, and the
fact that they are flying in every possible direction, and you will see that
they cannot avoid striking each other. Every time that two molecules come
into collision, the paths of both are changed, and they go off in new directions.
Thus each molecule is continually getting its course altered, so that in spite
of its great velocity it may be a long time before it reaches any great distance
from the point at which it set out.

I have here a bottle containing ammonia. Ammonia is a gas which you
can recognise by its smell. Its molecules have a velocity of six hundred metres
per second, so that if their course had not been interrupted by striking against

MOLECULES. 367

the molecules of air in the hall, everyone in the most distant gallery would
have smelt ammonia before I was able to pronounce the name of the gas. But
instead of this, each molecule of ammonia is so jostled about by the mole-
cules of air, that it is sometimes going one way and sometimes another, and
like a hare which is always doubling, though it goes a great pace, it makes
very little progress. Nevertheless, the smell of ammonia is now beginning to
be perceptible at some distance from the bottle. The gas does diffuse itself
through the air, though the process is a slow one, and if we could close up
every opening of this hall so as to make it air-tight, and leave everything to
itself for some weeks, the ammonia would become uniformly mixed through every
part of the air in the hall.

This property of gases, that they diffuse through each other, was first
remarked by Priestley. Dalton shewed that it takes place quite independently
of any chemical action between the inter-diffusing gases. Graham, whose
researches were especially directed towards those phenomena which seem to
throw light on molecular motions, made a careful study of diffusion, and obtained
the first results from which the rate of diffusion can be calculated.

Still more recently the rates of diffusion of gases into each other have
been measured with great precision by Professor Loschmidt of Vienna.

He placed the two gases in two similar vertical tubes, the lighter gas being
placed above the heavier, so as to avoid the formation of currents. He then
opened a sliding valve, so as to make the two tubes into one, and after leaving
the gases to themselves for an hour or so, he shut the valve, and determined
how much of each gas had diffused into the other.

As most gases are invisible, I shall exhibit gaseous diffusion to you by
means of two gases, ammonia and hydrochloric acid, which, when they meet,
form a solid product. The ammonia, being the lighter gas, is placed above the
hydrochloric acid, with a stratum of air between, but you will soon see that
the gases can diffuse through this stratum of air, and produce a cloud of white
smoke when they meet. During the whole of this process no currents or any
other visible motion can be detected. Every part of the vessel appears as calm
as a jar of undisturbed air.

But, according to our theory, the same kind of motion is going on in calm
air as in the inter-diffusing gases, the only difference being that we can trace
the molecules from one place to another more easily when they are of a different
nature from those through which they are diffusing.

MOLECTTLES.

If we wish to form a mental representation of what is going on among
the molecules in calm air, we cannot do better than observe a swarm of bees,
when every individual bee is flying furiously, first in one direction and then
in another, while the swarm, as a whole, either remains at rest, or sails slowly
through the air.

In certain seasons, swarms of bees are apt to fly off to a great distance,
and the owners, in order to identify their property when they find them on
other people's ground, sometimes throw handfulls of flour at the swarm. Now
let us suppose that the flour thrown at the flying swarm has whitened those
bees only which happened to be in the lower half of the swarm, leaving those
in the upper half free from flour.

If the bees still go on flying hither and thither in an irregular manner,
the floury bees will be found in continually increasing proportions in the upper
part of the swarm, till they have become equally diffused through every part
of it. But the reason of this diffusion is not because the bees were marked
with flour, but because they are flying about. The only effect of the marking
is to enable us to identify certain bees.

We have no means of marking a select number of molecules of air, so
as to trace them after they have become diffused among others, but we may
communicate to them some property by which we may obtain evidence of their
diffusion.

For instance, if a horizontal stratum of air is moving horizontally, molecules
diffusing out of this stratum into those above and below will carry their
horizontal motion with them, and so tend to communicate motion to the
neighbouring strata, while molecules diffusing out of the neighbouring strata
into the moving one will tend to bring it to rest. The action between the
strata is somewhat like that of two rough surfaces, one of which slides over
the other, rubbing on it. Friction is the name given to this action between
solid bodies ; hi the case of fluids it is called internal friction, or viscosity.

It is, in fact, only another kind of diffusion — a lateral diffusion of momentum,
and its amount can be calculated from data derived from observations of the
first kind of diffusion, that of matter. The comparative values of the viscosity
of different gases were determined by Graham in his researches on the tran-
spiration of gases through long narrow tubes, and their absolute values have
been deduced from experiments on the oscillation of discs by Oscar Meyer and
myself.

MOLECULES.

Another way of tracing the diffusion of molecules through calm air is to
heat the upper stratum of the air in a vessel, and to observe the rate at
which this heat is communicated to the lower strata. This, in fact, is a third
kind of diffusion — that of energy, and the rate at which it must take place
was calculated from data derived from experiments on< viscosity before any direct
experiments on the conduction of heat had been made. Professor Stefan, of
Vienna, has recently, by a very delicate method, succeeded in determining the
conductivity of air, and he finds it, as he tells us, in striking agreement with
the value predicted by the theory.

All these three kinds of diffusion — the diffusion of matter, of momentum,
and of energy — are carried on by the motion of the molecules. The greater
the velocity of the molecules and the further they travel before their paths are
altered by collision with other molecules, the more rapid will be the diffusion.
Now we know already the velocity of the molecules, and therefore, by experiments
on diffusion, we can determine how far, on an average, a molecule travels without
striking another. Professor Clausius, of Bonn, who first gave us precise ideas
about the motion of agitation of molecules, calls this distance the mean path
of a molecule. I have calculated, from Professor Loschmidt's diffusion experi-
ments, the mean path of the molecules of four well-known gases. The average
distance travelled by a molecule between one collision and another is given in
the table. It is a very small distance, quite imperceptible to us even with
our best microscopes. Roughly speaking, it is about the tenth part of the
length of a wave of light, which you know is a very small quantity. Of
course the time spent on so short a path by such swift molecules must be
very small. I have calculated the number of collisions which each must undergo
in a second. They are given in the table and are reckoned by thousands of mil-
lions. No wonder that the travelling power of the swiftest molecule is but small,
when its course is completely changed thousands of millions of times in a second.

The three kinds of diffusion also take place in liquids, but the relation
between the rates at which they take place is not so simple as in the case
of gases. The dynamical theory of liquids is not so well understood as that
of gases, but the principal difference between a gas and a liquid seems to be
that in a gas each molecule spends the greater part of its time in describing
its free path, and is for a very small portion of its time engaged in encounters
with other molecules, whereas, in a liquid, the molecule has hardly any free
path, and is always in a state of close encounter with other molecules.

VOL. II. 47

370 MOLECULES.

Henoe in a liquid the diffusion of motion from one molecule to another
take* place much more rapidly than the diffusion of the molecules themselves,
for the same reason that it is more expeditious in a dense crowd to pass on
a letter from hand to hand than to give it to a special messenger to work
hia way through the crowd. I have here a jar, the lower part of which contains
a solution of copper sulphate, while the upper part contains pure water. It
has been standing here since Friday, and you see how little progress the blue
liquid has made in diffusing itself through the water above. The rate of
diffusion of a solution of sugar has been carefully observed by Voit. Comparing
hia results with those of Loschmidt on gases, we find that about as much
diffusion takes place in a second in gases as requires a day in liquids.

The rate of diffusion of momentum is also slower in liquids than in gases,
but by no means in the same proportion. The same amount of motion takes
about ten times as long to subside in water as in air, as you will see by what
takes place when I stir these two jars, one containing water and the other air.
There is still less difference between the rates at which a rise of temperature
is propagated through a liquid and through a gas.

In solids the molecules are still in motion, but their motions are confined
within very narrow limits. Hence the diffusion of matter does not take place
in solid bodies, though that of motion and heat takes place very freely.
Nevertheless, certain liquids can diffuse through colloid solids, such as jelly and
gum, and hydrogen can make its way through iron and palladium.

We have no time to do more than mention that most wonderful molecular
motion which is called electrolysis. Here is an electric current passing through
acidulated water, and causing oxygen to appear at one electrode and hydrogen
at the other. In the space between, the water is perfectly calm ; and yet t\vu
opposite currents of oxygen and of hydrogen must be passing through it. The
physical theory of this process has been studied by Clausius, who has given
reasons for asserting that in ordinary water the molecules are not only moving,
hut every now and then striking each other with such violence that the ox
and hydrogen of the molecules part company, and dance about through tlu>
crowd, seeking partners which have become dissociated in the same way. In
ordinary water these exchanges produce, on the whole, no observable effect; l>ut
no sooner does the electromotive force begin to act than it exerts its guiding
influence on the unattached molecules, and bends the course of each toward its
proper electrode, till the moment when, meeting with an unappropriated molecule

MOLECULES. 371

of the opposite kind, it enters again into a more or less permanent union with
it till it is again dissociated by another shock. Electrolysis, therefore, is a
kind of diffusion assisted by electromotive force.

Another branch of molecular science is that which relates to the exchange
of molecules between a liquid and a gas. It includes the theory of evaporation
and condensation, in which the gas in question is the vapour of the liquid, and
also the theory of the absorption of a gas by a liquid of a different substance.
The researches of Dr Andrews on the relations between the liquid and the
gaseous state have shewn us that though the statements in our elementary
text-books may be so neatly expressed as to appear almost self-evident, their
true interpretation may involve some principle so profound that, till the right
man has laid hold of it, no one ever suspects that any thing is left to be
discovered.

These, then, are some of the fields from which the data of molecular
science are gathered. We may divide the ultimate results into three ranks,
according to the completeness of our knowledge of them.

To the first rank belong the relative masses of the molecules of different
gases, and their velocities in metres per second. These data are obtained from
experiments on the pressure and density of gases, and are known to a high
degree of precision.

In the second rank we must place the relative size of the molecules of
different gases, the length of their mean paths, and the number of collisions
in a second. These quantities are deduced from experiments on the three
kinds of diffusion. Their received values must be regarded as rough approxi-
mations till the methods of experimenting are greatly improved.

There is another set of quantities which we must place in the third rank,
because our knowledge of them is neither precise, as in the first rank, nor
approximate, as in the second, but is only as yet of the nature of a probable
conjecture. These are : — The absolute mass of a molecule, its absolute diameter,
and the number of molecules in a cubic centimetre. We know the relative
masses of different molecules with great accuracy, and we know their relative
diameters approximately. From these we can deduce the relative densities of
the molecules themselves. So far we are on firm ground.

The great resistance of liquids to compression makes it probable that their
molecules must be at about the same distance from each other as that at which
two molecules of the same substance in the gaseous form act on each other

47—2

.57.' MOLECULES.

during an encounter. This conjecture has been put to the test by Lorenz Mover.
who baa compared the densities of different liquids with the calculated relative
densities of the molecules of their vapours, and has found a remarkable cor-
respondence between them.

Now Loachmidt has deduced from the dynamical theory the following
remarkable proportion: — As the volume of a gas is to the combined volume of
all the molecules contained in it, so is the mean path of a molecule to one-
eighth of the diameter of a molecule.

Assuming that the volume of the substance, when reduced to the liquid
form, is not much greater than the combined volume of the molecules, we obtain
from this proportion the diameter of a molecule. In this way Loschmidt, in
1865, made the first estimate of the diameter of a molecule. Independently
of him and of each other, Mr Stoney in 1868, and Sir W. Thomson in 1870,
published results of a similar kind, those of Thomson being deduced not only
in this way, but from considerations derived from the thickness of soap-bul.
and from the electric properties of metals.

According to the Table, which I have calculated from Loschmidt's data,
the size of the molecules of hydrogen is such that about two millions of them
in a row would occupy a millimetre, and a million million million millio:
them would weigh between four and five grammes.

In a cubic centimetre of any gas at standard pressure and temperature
there are about nineteen million million million molecules. All these numbers
of the third rank are, I need not tell you, to be regarded as at piv
conjectural In order to warrant us in putting any confidence in nun.
obtained in this way, we should have to compare together a greater number
of independent data than we have as yet obtained, and to shew that they
lead to consistent results.

Thus far we have been considering molecular science as an inquiry
natural phenomena. But though the professed aim of all scientific work is to
unravel the secrets of nature, it has another effect, not less valuable, on the
mind of the worker. It leaves him in possession of methods which nothing
but scientific work could have led him to invent; and it places him in a
position from which many regions of nature, besides that which he has been
studying, appear under a new aspect.

The study of molecules has developed a method of its own, and it
also opened up new views of nature.

MOLECULES. 373

When Lucretius wishes us to form a mental representation of the motion
of atoms, he tells us to look at a sunbeam shining through a darkened room
(the same instrument of research by which Dr Tyndall makes visible to us the
dust we breathe), and to observe the motes which chase each other in all
directions through it. This motion of the visible motes, he tells us, is but a
result of the far more complicated motion of the invisible atoms which knock
the motes about. In his dream of nature, as Tennyson tells us, he

"Saw the flaring atom-streams
And torrents of her myriad universe,
Ruining along the illimitable inane,
Fly on. to clash together again, and make
Another and another frame of things
For ever."

And it is no wonder that he should ,have attempted to burst the bonds of
Fate by making his atoms deviate from their courses at quite uncertain times
and places, thus attributing to them a kind of irrational free will, which on
his materialistic theory is the only explanation of that power of voluntary
action of which we ourselves are conscious.

As long as we have to deal . with only two molecules, and have all the
data given us, we can calculate the result of their encounter ; but when we
have to deal with millions of molecules, each of which has millions of encounters
in a second, the complexity of the problem seems to shut out all hope of a
legitimate solution.

The modern atomists have therefore adopted a method which is, I believe,
new in the department of mathematical physics, though it has long been in
use in the section of Statistics. When the working members of Section F get
hold of a report of the Census, or any other document containing the numerical
data of Economic and Social Science, they begin by distributing the whole
population into groups, according to age, income-tax, education, religious belief,
or criminal convictions. The number of individuals is far too great to allow of
their tracing the history of each separately, so that, in order to reduce their
labour within human limits, they concentrate their attention on a small number
of artificial groups. The varying number of individuals in each group, and not
the varying state of each individual, is the primary datum from which they work.

This, of course, is not the only method of studying human nature. We
may observe the conduct of individual men and compare it with that conduct

374 MOLECULES.

which their previous character and their present circumstances, according to the
beat existing theory, would lead us to expect. Those who practise this method
endeavour to improve their knowledge of the elements of human nature in much
the same way as on astronomer corrects the elements of a planet by comparing
its actual position with that deduced from the received elements. The study
of human nature by parents and schoolmasters, by historians and statesmen, is
therefore to be distinguished from that carried on by registrars and tabulators,
and by those statesmen who put their faith in figures. The one may be called
the historical, and the other the statistical method.

The equations of dynamics completely express the laws of the historical
method as applied to matter, but the application of these equations implies a
perfect knowledge of all the data. But the smallest portion of matter which
we can subject to experiment consists of millions of molecules, not one of which
ever becomes individually sensible to us. We cannot, therefore, ascertain the
actual motion of any one of these molecules ; so that we are obliged to abandon
the strict historical method, and to adopt the statistical method of dealing with
large groups of molecules.

The data of the statistical method as applied to molecular science are the
sums of large numbers of molecular quantities. In studying the relations between
quantities of this kind, we meet with a new kind of regularity, the regularity
of averages, which we can depend upon quite sufficiently for all practical
purposes, but which can make no claim to that character of absolute precision
which belongs to the laws of abstract dynamics.

Thus molecular science teaches us that our experiments can never give us
anything more than statistical information, and that no law deduced from them
can pretend to absolute precision. But when we pass from the contemplation
of our experiments to that of the molecules themselves, we leave the world of
chance and change, and enter a region where everything is certain and im-
mutable.

The molecules are conformed to a constant type with a precision which is
not to be found in the sensible properties of the bodies which they constitute.
In the first place, the mass of each individual molecule, and all its other
properties, are absolutely unalterable. In the second place, the properties of
all molecules of the same kind are absolutely identical.

Let us consider the properties of two kinds of molecules, those of oxygen
and those of hydrogen.

MOLECULES. 375

We can procure specimens of oxygen from very different sources — from the
air, from water, from rocks of every geological epoch. The history of these
specimens has been very different, and if, during thousands of years, difference
of circumstances could produce difference of properties, these specimens of oxygen
would shew it.

In like manner we may procure hydrogen from water, from coal, or, as
Graham did, from meteoric iron. Take two litres of any specimen of hydrogen,
it will combine with exactly one litre of any specimen of oxygen, and will
form exactly two litres of the vapour of water.

Now if, during the whole previous history of either specimen, whether
imprisoned in the rocks, flowing in the sea, or careering through unknown
regions with the meteorites, any modification of the molecules had taken place,
these relations would no longer be preserved.

But we have another and an entirely different method of comparing the
properties of molecules. The molecule, though indestructible, is not a hard rigid
body, but is capable of internal movements, and when these are excited, it
emits rays, the wave-length of which is a measure of the time of vibration
of the molecule.

By means of the spectroscope the wave-lengths of different kinds of light
may be compared to within one ten-thousandth part. In this way it has been
ascertained, not only that molecules taken from every specimen of hydrogen in
our laboratories have the same set of periods of vibration, but that light, having
the same set of periods of vibration, is emitted from the sun and from the
fixed stars.

We are thus assured that molecules of the same nature as those of our
hydrogen exist in those distant regions, or at least did exist when the light
by which we see them was emitted.

From a comparison of the dimensions of the buildings of the Egyptians
with those of the Greeks, it appears that they have a common measure. Hence,
even if no ancient author had recorded the fact that the two nations employed
the same cubit as a standard of length, we might prove it from the buildings
themselves. We should also be justified in asserting that at some time or other
a material standard of length must have been carried from one country to the
other, or that both countries had obtained their standards from a common source.

But in the heavens we discover by their light, and by their light alone,
stars so distant from each other that no material thing can ever have passed

376 MOLECULES.

from one to another; and yet this light, which is to us the sole evidence of
the existence of these distant worlds, tells us also that each of them is built
up of molecules of the same kinds as those which we find on earth. A
molecule of hydrogen, for example, whether in Sirius or in Arcturus, executes
its vibrations in precisely the same time.

Each molecule, therefore, throughout the universe, bears impressed on it the
stamp of a metric system as distinctly as does the metre of the Archives at
Paris, or the double royal cubit of the Temple of Karnac.

No theory of evolution can be formed to account for the similarity of
molecules, for evolution necessarily implies continuous change, and the molecule
is incapable of growth or decay, of generation or destruction.

None of the processes of Nature, since the time when Nature began, hn
produced the slightest difference in the properties of any molecule. We
therefore unable to ascribe either the existence of the molecules or the identity
of their properties to the operation of any of the causes which we call natural.

On the other hand, the exact equality of each molecule to all others of
the same kind gives it, as Sir John Herschel has well said, the essential
character of a manufactured article, and precludes the idea of its being eternal
and self-existent.

Thus we have been led, along a strictly scientific path, very near to the
point at which Science must stop. Not that Science is debarred from study i
the internal mechanism of a molecule which she cannot take to pieces, any
more than from investigating an organism which she cannot put together. But
in tracing back the history of matter Science is arrested when she assures
herself, on the one hand, that the molecule has been made, and on the other,
that it has not been made by any of the processes we call natural

Science is incompetent to reason upon the creation of matter itself out of
nothing. We have reached the utmost limit of our thinking faculties when we
have admitted that because matter cannot be eternal and self-existent it must
have been created.

It is only when we contemplate, not matter in itself, but the form in
which it actually exists, that our mind finds something on which it can lay
hold.

That matter, as such, should have certain fundamental properties — that it
should exist in space and be capable of motion, that its motion should be
persistent, and so on, are truths which may, for anything we know, be of

MOLECULES. 377

the kind which metaphysicians call necessary. We may use our knowledge of
such truths for purposes of deduction, but we have no data for speculating as
to their origin.

But that there should be exactly so much matter and no more in every
molecule of hydrogen is a fact of a very different order. We have here a
particular distribution of matter — a collocation — to use the expression of Dr
Chalmers, of things which we have no difficulty in imagining to have been
arranged otherwise.

The form and dimensions of the orbits of the planets, for instance, are not
determined by any law of nature, but depend upon a particular collocation of
matter. The same is the case with respect to the size of the earth, from
which the standard of what is called the metrical system has been derived.
But these astronomical and terrestrial magnitudes are far inferior in scientific
importance to that most fundamental of all standards which forms the base
of the molecular system. Natural causes, as we know, are at work, which
tend to modify, if they do not at length destroy, all the arrangements and
dimensions of the earth and the whole solar system. But though in the course
of ages catastrophes have occurred and may yet occur in the heavens, though
ancient systems may be dissolved and new systems evolved out of their ruins,
the molecules out of which these systems are built — the foundation stones of
the material universe — remain unbroken and unworn.

They continue this day as they were created — perfect in number and
measure and weight, and from the ineffaceable characters impressed on them
we may learn that those aspirations after accuracy in measurement, truth in
statement, and justice in action, which we reckon among our noblest attributes
as men, are ours because they are essential constituents of the image of Him
who in the beginning created, not only the heaven and the earth, but the
materials of which heaven and earth consist.

VOL. n. 48

178

MOLECULES.

Table of Molecular Data.

Hydrogen.

Oxygen.

Carbonic
oxide.

Carbonic
acid.

Mass of molecule (hydrogen = 1)

1

16

14

22

Rank I.

Velocity (of mean square) metres per )
second at 0*C. j

1859

465

497

396

Mean path, tenth-metres

965

560

482

379

Rank II.

Collisions in a second (millions)

17750

7646

9489

9720

Diameter, tenth-metres

5-8

7-6

8-3

9-3

Rank III.

Maw, twenty-fifth grammes

46

736

644

1012

* MTWWW VI -txt/f UOlVfl. — UlCitOUlC.

Second

Calculated.

Observed.

II A- o

0-7086

0-7214\

H 4 CO

0-6519

0-6422

HA CO,
04 CO

0-5575
0-1807

0-1802 Diffusion of matter observed by

Loschmidt

O&CO,

0-1427

0-1409

CO 4 CO,

0-1386

0-1406'

H

1-2990

1-49 \

0
CO

0-1884
0-1748

0-213
0'912 Diffusion of momentum (Graham

and Meyer).

CO.

0-1087

0-117

Air

0-256J

Copper
Iron

1-077> Diffusion of temperature observed
0-183)

by Stefan.

Cane Sugar in wi
(Or in a day
Salt in water

iter 0-00000365) „ ..
0-3144) / Volt
0-00000116 Pick.

[From the Proceedings of the Royal Society, No. 148, 1873.]

LXIII. On Double Refraction in a Viscous Fluid in Motion.

ACCORDING to Poisson's* theory of the internal friction of fluids, a viscous
fluid behaves as an elastic solid would do if it were periodically liquefied for an
instant and solidified again, so that at each fresh start it becomes for the
moment like an elastic solid free from strain. The state of strain of certain
transparent bodies may be investigated by means of their action on polarized
light. This action was observed by Brewster, and was shewn by Fresnel to be
an instance of double refraction.

In 1866 I made some attempts to ascertain whether the state of strain in
a viscous fluid in motion could be detected by its action on polarized light. I
had a cylindrical box with a glass bottom. Within this box a solid cylinder
could be made to rotate. The fluid to be examined was placed in the annular
space between this cylinder and the sides of the box. Polarized light was
thrown up through the fluid parallel to the axis, and the inner cylinder was
then made to rotate. I was unable to obtain any result with solution of gum or
sirup of sugar, though I observed an effect on polarized light when I compressed
some Canada balsam which had become very thick and almost solid in a bottle.

It is easy, however, to observe the effect in Canada balsam, which is so
fluid that it very rapidly assumes a level surface after being disturbed. Put
some Canada balsam in a wide-mouthed square bottle ; let light, polarized in a
vertical plane, be transmitted through the fluid ; observe the light through a
Nicol's prism, and turn the prism so as to cut off the light; insert a spatula
in the Canada balsam, in a vertical plane passing through the eye. Whenever
the spatula is moved up or down in the fluid, the light reappears on both
sides of the spatula ; this continues only so long as the spatula is in motion.
As soon as the motion stops, the light disappears, and that so quickly that I
have hitherto been unable to determine the rate of relaxation of that state of
strain which the light indicates.

If the motion of the spatula in its own plane, instead of being in the plane

* Journal de VEcole Polytechnique, tome xiii. cah. xx. (1829).

48—2

380 DOUBLE REFRACTION IN A VISCOUS FLUID IN MOTION.

of polarization, is inclined 45* to it, no effect is observed, shewing that the axes
of strain are inclined 45* to the plane of shearing, as indicated by the theory.

I am not aware that this method of rendering visible the state of strain
of a viscous fluid has been hitherto employed ; but it appears capable of furnishing
important information as to the nature of viscosity in different substances.

Among transparent solids there is considerable diversity in their action on
polarized light. If a small portion is cut from a piece of unannealed glass at a
place where the strain is uniform, the effect on polarized light vanishes as soon
as the glass is relieved from the stress caused by the unequal contraction of
the parts surrounding it.

But if a plate of gelatine is allowed to dry under longitudinal tension, a
small piece cut out of it exhibits the same effect on light as it did before,
shewing that a state of strain can exist without the action of stress. A film of
gutta percha which has been stretched in one direction has a similar action on
light. If a circular piece is cut out of such a stretched film and warmed, it
contracts in the direction in which the stretching took place.

The body of a sea-nettle has all the appearance of a transparent jelly ; and
at one time I thought that the spontaneous contractions of the living animal
might be rendered visible by means of polarized light transmitted through its body.
But I found that even a very considerable pressure applied to the sides of the
sea-nettle produced no effect on polarized light, and I thus found, what I might
have learned by dissection, that the sea-nettle is not a true jelly, but consists
of cells filled with fluid.

•On the other hand, the crystalline lens of the eye, as Brewster observed,
has a strong action on polarized light when strained either by external pressure
or by the unequal contraction of its parts as it becomes dry.

I have enumerated these instances of the application of polarized light to
the study of the structure of solid bodies as suggestions with respect to the
application of the same method to liquids so as to determine whether a given
liquid differs from a solid in having a very small "rigidity," or in having a
small "time of relaxation"*, or in both ways. Those which, like Canada balsam, act
strongly on polarized light, have probably a small " rigidity," but a sensible " time
of relaxation." Those which do not shew this action are probably much more
" rigid," and owe their fluidity to the smallness of their " time of relaxation."

* The " time of relaxation " of a substance strained in a given manner is the time required
for the complete relaxation of the strain, supposing the rate of relaxation to remain the same as
at the beginning of tliii time.

[From the Proceedings of the London Mathematical Society, Vol. vi.]
LXIV. On Hamilton's Characteristic Function for a Narrow Beam of Light.

[Read January 8th, 1874.]

HAMILTON'S characteristic function V is an expression for the time of pro-
pagation of light from the point whose co-ordinates are xlt y1} z^ to the point
whose co-ordinates are xt, y3, z,. It is a function of these six co-ordinates of
the two points. The axes to which the co-ordinates are referred may be
different for the two points.

In isotropic media the differential equation of V may be written

where /i is the slowness of propagation at a point in the medium whose co"
ordinates are x, y, z, and is a function of these co-ordinates. If the time of
propagation through the unit of length in vacuum be taken as the unit of
time, then \L is the index of refraction of the medium.

The form of the equation in doubly refracting media, as given by Hamilton,
is not required for our present purpose.

Let OPQR be the path of a ray of light. Let the part OP be in a homo-
geneous medium whose index of refraction is /*,, and

let QR be in a homogeneous medium whose index r ,

of refraction is /n2. Between P and Q the ray may o P\ " la &

pass through any combination of media, singly or
doubly refracting.

Let us consider the characteristic function from a point near to OP in the
first medium to a point near to QR in the second.

382

HAMILTON'S CHARACTERISTIC FUNCTION

Let the position of the first point be referred to rectangular axes, the
origin of which is at P, and the axis of z, drawn in the direction PO. The
axes of x, and y, may be turned at pleasure round that of z, into the position
most suitable for our calculations.

Let the position of the second point be referred to Q as origin, to QR
as axis of z,, and to axes of ic, and y, the position of which is of course
independent of that chosen for xlt y,.

Let the ray from the first point & fa, ylt z,) to the second point R fa, y,, z,)
pass through Pf(fl, 17,, 0), and <?(&, 17,, 0).

We have then VffK= VCfP.+ Vp,y + VyK

Here

and
Also

(2).
•(3),
.(4).

Lif

^

(5).

+ terms involving higher powers and products of £,, 77,, £,, 1

Writing, for the sake of brevity, single symbols for the differential coefficients,
we obtain for the value of Vffg up to the terms of the second degree inclusive

+ 517,77,

This is the value of VffP+ Vry+ VyKt supposing the course of the ray to be
broken at P and <7 in an arbitrary manner.

FOR A NARROW BEAM OF LIGHT.

383

For the actual course of the ray the value of V0,R, must be stationary as
regards variations of f,, ijlt £, 7jt. Hence, differentiating with respect to these
variables, we obtain the four equations

if.

fli + 7

*!

«!
= /*!-•

X,

Since OPQR is a ray of the system, the co-ordinates xlt y,; f,, ^; |,, ^4;
xt, y., vanish together. Hence f1 = g1=ft = gt = Q, and if we write A for the deter-
minant

(8).

Oti "^ C

D <7

Zl

f7 a

c 6 + —

r s

zi

j3 r <

*+5 Cj

2 s

Cj ^3 + ^

Then

(9).

Substitutuig in (6), we obtain

^o'fi- = VOE + W* + W> + iaA1 + Si

I

38-4

HAMILTON'S CHARACTERISTIC FUNCTION

where

« =£? — —

a, % ..

.,

'

rfA

2z,z,A

©=-

n n

.c? _ i-i-

z.

6,= -

_Mi_ Mi
'-z, z,'A

.(11).

This ia the most general form of Hamilton's characteristic function for a pair
of points, each of which is near the principal ray. It is a homogeneous
function of the second degree in x,, ylt xt> yt, the coefficients of which are
functions of z, and z,.

By turning the axes of re, and y, about z,, and those of <c, and yt about z,,
we may get rid of two of the ten terms, and so reduce the expression to
eight.

We may, for example, get rid of c, and c,, and so of Sj and g,; but since,
in the theory of pencils having two focal lines, terms may enter which must
be added to 6, and (£„ this transformation is not of much use.

It is better to begin by getting rid of q and r, by turning xl and y,
round z,, through an angle 0V such that

and also turning a^ and yt round z,, through an angle #,, such that

(13).

For these new axes the values of q and r are reduced to zero.

As an instance of the use of the characteristic function, let us find the
form of the emergent pencil when that of the incident pencil is given.

The general form of the characteristic function of a pencil, whose axis is the
axis of z, is

FOR A NARROW BEAM OF LIGHT.

385

as in the Proceedings of the London Mathematical Society, Vol. iv., p. 337 (1873);
where A, B, C are lines from which, by the construction there given, a and b,
the co-ordinates of the focal lines, and <j>, the angle which the line a makes
with the plane xz, may be deduced.

The quantities a1} &c., which occur as the coefficients of the characteristic
function, are the reciprocals of lines.

Let

also let

Jl] O^j

1 1

" T ' & = "2 + -R- > y, =
At ±S,

(15).

The condition that the incident pencil defined by Alt Blt Cl should be
conjugate to the emergent pencil defined by A,,, B.,, Ct, is that the value of
the characteristic function must be the same for all rays of the pencil. Now
a particular ray of the pencil may be defined either by the co-ordinates aj,, yl} zt
of a point through which it passes in the first medium, or by ox,, y2, z2 those of
a point through which it passes in the second medium. In the first case, the
coefficients of x*, a^y,, and y* will vanish, and in the second those of a;/, x^.
and y,\ If the one set of conditions is fulfilled, the other set will be fulfilled
also. Hence we may write the conditions either in the form

or in the form
where

da,

-i-
da,

A=

dy

,(16),

W,

' dy

•" l*«/l

1

a

D O

(18)

^^

i

Jr TL

y.

A

V Q

p

r

Oj y,

q

s

ya A

A - cw5 + 20,7,7-5 -

— 2y,y2(ps + 2r)
(ps-2r)2

VOL. n.

...(19).
49

HAMILTON'S CHARACTERISTIC FUNCTION

The conditions of conjugate pencils may therefore be written, either

0,8,=

(20),

or in a form derived from this by exchanging the suffixes t and ,.
If the axes of co-ordinates are turned so that q and r vanish,

A-8&-o1ag8»-2y1yjw-&&/*+j>V (21),

and oA-fcp' 1

yA--r\f»[ • (22).

£A = a/ J

If we write al = Xli>, &=F,s, y^

a, = Xtp, A=l>. r, =
then Xlt Ylt Z^ will be inverse to Xt, Ylf Zt, and will satisfy the equations

xtx3+ztzt=i, zA+r.z.-oi
z& + r.r,- 1, j^z.+^r,- oJ

Fig. 2.

Fig. 3.

B i

The relations between the quantities Xlt Tlt Z, and Xit Y3, Z3, are she\\n
in the annexed figure (Fig. 2).

Let AR = X! and RB=Yl in the same straight line, and RO = Zl perpen-
dicular to AB. With 0 as centre and unity as radius describe a circle.

Draw CB1 the polar of A, and C A' the polar of B, with respect to this
circle. These lines meet BO and AO in B' and A' respectively. Join J'//'
cutting J20 in R. Then RK = X,, RA'=Y,, and OR' = Z1.

FOR A NARROW BEAM OF LIGHT. 387

Since OR is measured downwards, Z3 is negative in the ease represented
by the figure. It is manifest that Xlt Ylt and Z^ may be found from X,, F2>
Z2, by the same process.

The relation between the quantities A, B, C and X, Y, Z is shewn in
Fig. 3.

Let a be one of the first three or the last three of the ten coefficients of
the characteristic function Vpy.

Since a is the reciprocal of a line, let BP represent the line - .

GV

Draw PT perpendicular to BP so that PT is to the unit line as « to p.

Then, if PA=A, the line ATC will cut off from the line EC perpen-
dicular to BA a part BC equal to X. For

-- .

p A.p p

In this way X may be found when A is known, or A when X is known.
The same method gives the relations between B and Y and between C and Z.

The geometrical process for finding the focal lines of the emergent pencil,
when those of the incident pencil are given, is therefore as follows : —

From the distances at and &, of the focal lines, and the angle <^l between
the first of them and the plane of x^, deduce, by the method given in a
former communication (Vol. iv., p. 337), Alt Blt and Cj*.

From Alt Blt Ct, find, by the construction of Fig. 3, Xlt Ylf and Z^,

From Xlt Yj, Zlt find, by the construction of Fig. 2, Xt, Yt, and Z2.

From X,, Y,, Z,, find At, JS19 C,.

From AS, Bz, (75, find a2, 62, <f>2.

Thus far we have been considering the most general case of a pencil
passing through any number of media between P and Q, through surfaces of
any form, the media before incidence at P and after emergence at Q being
isotropic. When some of these media are doubly refracting, there may be two
or more emergent pencils corresponding to one incident pencil. Our investigation
is applicable only to one of these emergent pencils at a time. Each emergent
pencil must be treated separately.

* [Page 337, Vol. 11. of this Edition.]

49—2

388

HAMILTON'S CHARACTERISTIC FUNCTION

In certain cases of practical importance the characteristic function may be
greatly simplified. For instance, when the axis ray is refracted in one plane
through the prisms of a spectroscope, the same positions of the axes of x and y
which make «/ and r vanish, also make c, and c, vanish. The determinant A may
now be written as the product of two factors

,

«•+*) («•+*)-

+•

....(26).

If we write

MI= -r-1 -» «» =

PA

v,=

i ~ — - — ft ^— ^_ ' *

7l s'-feA ' g*~ t-\

(27),

the characteristic function becomes

(28).

Since in this case the terms of the characteristic function which involv,-
xt and xt are separated from those which involve yl and y,, we may consider

FOR A NARROW BEAM OF LIGHT. 389

either apart from the other. In the case of an optical instrument symmetrical
about its axis,

«,=«„ u2 = v,, fi = g» and /,=& (29).

Let llt In be the tangents of the angles which the incident and emergent rays,
projected on the plane of xz, make with the axis of z,

dV

7 (Zl ~ Ul) Xt ~JlXl /q-|\

- -"

If the incident ray is parallel to the axis, Zj = 0, and the equation of the

emergent ray is

(z1-ui)x1-fax1 = Q .............................. (32).

The emergent ray cuts the plane of yz where

z3 = tt3 ...................................... (33).

This therefore is the position of the second principal focus.

When z» = ut+f,> x^ = xt ................................. (34),

or the ray is at the same distance from the plane of yz as before incidence.
This gives the position of the second principal plane.

Its distance from the second principal focus is f3, which is called the
second principal focal length.

When 2j = «i+/1 and xt = 0, ^ — l., .......................... (35),

or every ray which passes through this point is equally inclined to the plane
of yz before and after passing through the instrument. This point is called the
second focal centre.

The distance of the emergent ray from the axis of z, when z = z2, is given
by the equation

/A = (Z» - M.) *1 - *1 {(21 - «0 (Z* - «») -/I/*} .................. (36)'

When k-tOk-tO-/./. = 0 ............................ (3?)>

the term multiplied by ^ vanishes. Hence all the rays which pass through the
point (Xj, z,) pass through (x,, z,), whatever their inclination to the axis. The
points xlt z1 and x2, z, are therefore conjugate foci, and x, is the image of xv

390 HAMILTON'S CHARACTERISTIC FUNCTION FOB A NABROW BEAM OF LIGHT.

<38>;

or. in woixls, — The distance of the, object from the axis is to the distance of
the uiHiyc from the axis as the distance of the object from the first principal
focus is to the first principal focal length, or as the second principal focal
length is to the distance of the image from the second principal focus.

Let A, be the distance at which an object of diameter x, must be placed
from the eye that it may subtend an angle equal to that which it subtends
when placed at z,, and seen through the instrument by an eye at z,,

/r, = TT when xt = Q, or A, = ' fU ................... (39),

or

The quantity /&, is that which occurs in Cotes' Theorem, and to which
Smith gives the name of the "apparent distance."

Differentiating ht with respect to zl and z,, we find

dh_l_z,-u? dhl_z1- u, d'h, 1 ,

~ - ' --

When the focal length is infinite, the instrument becomes a telescope, and the
characteristic function is

IT- IT- . , .1 iu,LLJlt . .,

7= F. + ^z. + f^j + l - r-^r1 + a sim Jar term in y.

--*-

Here m is the angular magnification, and ult w, are the co-ordinates of any

two conjugate foci. The linear magnification is -^- and the elongation is -~

-

[From the Cambridge Philosophical Proceedings, Vol. II.]

LXV. On the Relation of Geometrical Optics to other parts of Mathematics

and Physics.

THE study of geometrical optics may be made more interesting to the
mathematician by treating the relation between the object and the image by
the methods used in the geometry of homographic figures. The whole theory
of images formed by simple or compound instruments when aberration is not
considered is thus reduced to simple proportion, and- this is found very con-
venient in the practical work of arranging lenses for an experiment, in order
to produce a given effect.

As a preparation for physical optics the same elementary problems may be
treated by Hamilton's method of the Characteristic Function. This function
expresses, in terms of the co-ordinates of two points, the time taken by light
in travelling from the one to the other, or more accurately the distance through
which light would travel in a vacuum during this time, which we may call
the reduced path of the light between the two points. The relation between
this reduced path and the quantity which occurs in Cotes' celebrated but little
known theorem, is called by Dr Smith the "apparent distance." The relations
between the " apparent distance " and the positions of the foci conjugate to the
two points, the principal foci and the principal focal lengths, were explained * ;
and the general form of the characteristic function for a narrow pencil in the
plane of xr was shewn to be

V= 7i + MlrI + /v. + * -• • -' ~ + &c.,

* [i.e. to the Cambridge Philosophical Society at a meeting when the results of the foregoing
paper LXIV. were communicated.]

392 TUB RELATION OF GEOMETRICAL OPTICS TO MATHEMATICS AND PHYSICS.

where r,, r, are measured from the instrument in opposite directions along the
axis of the pencil in the media /*,, j*,, respectively, and x,, o^ are perpendicular
to the axis.

a,, a, are the values of r,, rt, for the principal foci, and /„ /„ the prin-
cipal focal lengths, and /,/t, =

If V5-^-!?'

/I »l — «i ^t

the last term of V assumes the form - , and an infinite number of possible

paths exist between the points («,, r,), and (a^, r,), which are therefore con-
jugate foci.

Differentiating V with respect to a^ and xt we obtain

1

Z> is the quantity in Cotes' Theorems which Dr Smith calls the Apparent
Distance, or the distance at which the object must be placed that it may
subtend the same angle as when viewed through the instrument.

We have also

dD . dD

[From Nature, Vol. x.]

LXVI. Plateau on Soap- Bubbles?- '.

Ox an Etruscan vase in the Louvre figures of children are seen blowing
bubbles. Those children probably enjoyed their occupation just as modern
children do. Our admiration of the beautiful and delicate forms, growing and
developing themselves, the feeling that it is our breath which is turning dirty
soap-suds into spheres of splendour, the fear lest by an irreverent touch we
may cause the gorgeous vision to vanish with a sputter of soapy water in our
eyes, our wistful gaze as we watch the perfected bubble when it sails away
from the pipe's mouth to join, somewhere in the sky, all the other beautiful
things that have vanished before it, assure us that, whatever our nominal age
may be we are of the same family as those Etruscan children.

Here, for instance, we have a book, in two volumes, octavo, written by
a distinguished man of science, and occupied for the most part with the theory
and practice of bubble-blowing. Can the poetry of bubbles survive this ? Will
not the lovely visions which have floated before the eyes of untold generations
collapse at the rude touch of Science, and "yield their place to cold material
laws " ? No, we need go no further than this book and its author to learn
that the beauty and mystery of natural phenomena may make such an impres-
sion on a fresh and open mind that no physical obstacle can ever check the
course of thought and study which it has once called forth.

M. Plateau in all his researches seems to have selected for his study those
phenomena which exhibit some remarkable beauty of form or colour. In the
zeal with which he devoted himself to the investigation of the laws of the
subjective impressions of colour, he exposed his eyes to an excess of light, and

* Slatique experimental* et tfteoriqrue des Liquides soumis aux seules Forces moleculaires. Par J.
Plateau, Professeur a 1' University de Gand, <fcc. (Paris, Gauthier-Villars ; London, Trubner & Co.;
Gand et Leipzig, F. Clemm. 1873.)

VOL. IL 50

394 PLATEAU ON SOAP-BUBBLES.

has ever since been blind. But in spite of this great loss he lias continued
for many yean to carry on experiments such as those described in this book,
..n the forma of liquid masses and films, which he himself can never either see
«>r handle, but from which he gathers the materials of science as they are
furnished to him by the hands, eyes, and minds of devoted friends.

So perfect has been the co-operation with which these experiments have
been carried out, that there is hardly a single expression in the book to
indicate that the measures which he took and the colours with which he was
charmed were observed by him, not in the ordinary way, but through the
mediation of other persons.

Which, now, is the more poetical idea — the Etruscan boy blowing bubl.Ks
for himself, or the blind man of science teaching his friends how to blow tlirm,
and making out by a tedious process of question and answer the conditions
>»f the forms and tints which he can never see ?

But we must now attempt to follow our author as he passes from phe-
nomena to ideas, from experiment to theory.

The surface which forms the boundary between a liquid and its vapour is
the seat of phenomena on the careful study of which depends much of our
future progress in the knowledge of the constitution of bodies. To take the
simplest case, that of a liquid, say water, placed in a vessel which it does not
fill, but which contains nothing else. The water lies at the bottom of the
vessel, and the upper part, originally empty, becomes rapidly filled with the
vapour of water. The temperature and the pressure — the quantities on which
the thermal and statical relations of any body to external bodies depend — are
the same for the water and its vapour, but the energy of a milligramme of
the vapour greatly exceeds that of a milligramme of the water. Hence the energy
of a milligramme of water-substance is much greater when it happens to be in
the upper part of the vessel in the state of vapour, than when it hapj>en8
to be in the lower part of the vessel in the state of water.

Now we find by experiment that there is no difference between the phe-
nomena in one part of the liquid and those in another part except in a region
close to the surface and not more than a thousandth or perhaps a millionth
of a millimetre thick. In the vapour also, everything is the same, except
perhaps in a very thin stratum close to the surface. The change in the value
of the energy takes place in the very narrow region between water and vapour.
Hence the energy of a milligramme of water is the same all through the mass

PLATEAU ON SOAP-BUBBLES. 395

of the water except in a thin stratum close to the surface, where it is some-
what greater ; and the energy of a milligramme of vapour is the same all
through the mass of vapour except close to the surface, where it is probably
less.

The whole energy of the water is therefore, in the first place, that due
to so many milligrammes of water ; but besides this, since the water close to
the surface has an excess of energy, a correction, depending on this excess,
must be added. Thus we have, besides the energy of the water reckoned per
milligramme, an additional energy to be reckoned per square millimetre of
surface.

The energy of the vapour may be calculated in the same way at so much
per milligramme, with a deduction of so much per square millimetre of surface.
The quantity of vapour, however, which lies within the region in which the
energy is beginning to change its value is so small that this deduction per
square millimetre is always much smaller than the addition which has to be
made on account of the liquid. Hence the whole energy of the system may
be divided into three parts, one proportional to the mass of liquid, one to the
mass of vapour, and the third proportional to the area of the surface which
separates the liquid from the vapour.

If the system is displaced by an external agent in such a way that the
area of the surface of the liquid is increased, the energy of the system is
increased, and the only source of this increase of energy is the work done by
the external agent. There is therefore a resistance to any motion which causes
the extension of the surface of a liquid.

On the other hand, if the liquid moves in such a way that its surface
diminishes, the energy of the system diminishes, and the diminution of energy
appears in the form of work done on the external agent which allows the
surface to diminish. Now a surface which tends to diminish in area, and which
thus tends to draw together any solid framework which forms its boundary,
is said to have surface-tension. Surface-tension is measured by the force acting
on one millimetre of the boundary edge. In the case of water at 20°C., the
tension is, according to M. Quincke, a force of 8*253 milligrammes weight per
millimetre.

M. Plateau hardly enters into the theoretical deduction of the surface-
tension from hypotheses respecting the constitution of bodies. We have there-
fore thought it desirable to point out how the fact of surface-tension may be

50—2

396 PLATEAU ON SOAP-BUBBLES.

deduced from the known fact that there is a difference in energy between a
liquid and its vapour, combined with the hypothesis, that as a milligramme of
the substance passes from the state of a liquid within the liquid mass, to that
of a vapour outside it, the change of its energy takes place, not instantaneously,
but in a continuous manner.

M. van der Waals, whose academic thesis, Over de Continuiteit van </«/<

en Vlocutofloestand*, is a most valuable contribution to molecular physics,

has attempted to calculate approximately the thickness of the stratum within

which this continuous change of energy is accomplished, and finds it for water

about 0*0000003 millimetre.

Whatever we may think of these calculations, it is at least manifest that
the only path in which we may hope to arrive at a knowledge of the size
of the molecules of ordinary matter is to be traced among those phenomena
which come into prominence when the dimensions of bodies are greatly reduced,
as in the superficial layer of a liquid.

But it is in the experimental investigation of the effects of surface-tension
on the form of the surface of a liquid that the value of M. Plateau's book is to
be found. He uses two distinct methods. In the first he prepares a mixture
of alcohol and water which has the same density as olive oil, then introducing
some oil into the mixture and waiting till it has, by absorption of a small
portion of alcohol into itself, become accommodated to its position, he obtains a
mass of oil no longer under the action of gravity, but subject only to the
surface-tension of its boundary. Its form is therefore, when undisturbed,
spherical, but by means of rings, disks, &c., of iron, he draws out or com-
presses his mass of oil into a number of different figures, the equilibrium and
stability of which are here investigated, both experimentally and theoretically.

The other method is the old one of blowing soap-bubbles. M. Plateau,
however, has improved the art, first by finding out the best kind of soap and
the best proportion of water, and then by mixing his soapy water with
glycerine. Bubbles formed of this liquid will last for hours, and even days.

By forming a frame of iron wire and dipping it into this liquid he forms
a film, the figure of which is that of the surface of minimum area which has
the frame for its boundary. This is the case when the air is free on both
sides of the film. If, however, the portions of air on the two sides of the film

* Leiden: A. W. Sijthoff, 1873.

PLATEAU ON SOAP-BUBBLES. 397

are not in continuous communication, the film is no longer the surface of absolute
minimum area, but the surface which, with the given boundary, and inclosing a
given volume, has a minimum area.

M. Plateau has gone at great length into the interesting but difficult ques-
tion of the conditions of the persistence of liquid films. He shews that the
surface of certain liquids has a species of viscosity distinct from the interior
viscosity of the mass. This surface-viscosity is very remarkable in a solution of
saponine. There can be no doubt that a property of this kind plays an impor-
tant part in determining the persistence or collapse of liquid films. M. Plateau,
however, considers that one of the agents of destruction is the surface-tension,
and that the persistence mainly depends on the degree in which the surface-
viscosity counteracts the surface-tension. It is plain, however, that it is rather
the inequality of the surface-tension than the surface-tension itself which acts as
a destroying force.

It has not yet been experimentally ascertained whether the tension varies
according to the thickness of the film. The variation of tension is certainly
insensible in those cases which have been observed.

If, as the theory seems to indicate, the tension diminishes when the thick-
ness of the film diminishes, the film must be unstable, and its actual persistence
would be unaccountable. On the other hand, the theory has not as yet been
able to account for the tension increasing as the thickness diminishes.

One of the most remarkable phenomena of liquid films is undoubtedly the
formation of the black spots, which were described in 1672 by Hooke, under
the name of holes.

Fusinieri has given a very exact account of this phenomenon as he observed
it in a vertical film protected from currents of air. As the film becomes thinner,
owing to the gradual descent of the liquid of which it is formed, certain portions
become thinner than the rest, and begin to shew the colours of thin plates.
These little spots of colour immediately begin to ascend, dragging after them
a sort of train like the tail of a tadpole. These tadpoles, as Fusinieri calls
them, soon begin to accumulate near the top of the film, and to range them-
selves in horizontal bands according to their colours, those which have the
colour corresponding to the smallest thickness ascending highest.

In this way the colours become arranged in horizontal bands in beautiful
gradation, exhibiting all the colours of Newton's scale. When the frame of the
film is made to oscillate, these bands oscillate like the strata formed by a

398 PLATEAU OX SOAP-BUBBLES.

of liquids of different densities. This shews that the film is subject to
dynamical conditions similar to those of such a liquid system. The liquid is
subject to the condition that the volume of each portion of it is invariable,
and the motion arises from the fact that by the descent of the denser portions
(which is necessarily accompanied by the rise of the rarer portions) the gravi-
tational potential energy of the system is diminished. In the case of the film,
the condition which determines that the descent of the thicker portions shall
entail the rise of the thinner portions must be that each portion of the film
offers a special resistance to an increase or diminution of area. This resistance
probably forms a large part of the superficial viscosity investigated by M. PlaU
which retards the motion of his magnetic needle, and evidently is far gren
than the viscosity of figure, in virtue of which the film resists a shearing motion.

The coloured bands gradually descend from the top of the film, presei
at first a continuous gradation of colour, but soon a remarkable black, or nearly
black, band begins to form at the top of the film, and gradually to extend
itself downwards. The lower boundary of this black band is sharply detii:
There is not a continuous gradation of colour according to the arrangement in
Newton's table, but the blaok appears in immediate contact with the wL
or even the yellow of the first order, and M. Fusinieri has even observed it
in contact with bands of the third order.

Nothing can shew more distinctly that there is some remarkable change
in the physical properties of the film, when it is of a thickness somewhat greater
than that of the black portion. And in fact the black part of the film is in
many other respects different from the rest. It is easy, as Leidenfrost tells us,
to pass a solid point through the thicker part of the film, and to withdi;
it, without bursting the film, but if anything touches the black part, the film
is shattered at once. The black portion does not appear to possess the mobility
which is so apparent in the coloured parts. It behaves more like a brittle solid,
such as a Prince Rupert's drop, than a fluid. Its edges are often very irre^ii
and when the curvature of the film is made to vary, the black portions some-
times seem to resist the change, so that their surface has no longer the same
continuous curvature as the rest of the bubble. We have thus numerous indi-
cations of the great assistance which molecular science is likely to derive from
the study of liquid films of extreme tenuity.

We have no time or inclination to discuss M. Plateau's work in a critical
spirit. The directions for making the experiments are very precise, and if some-

PLATEAU ON SOAP-BUBBLES. 399

times they appear tedious on account of repetitions, we must remember that it
is by words, and words alone, that the author can learn the details of the
experiment which he is performing by means of the hands of his friends, and
that the repetition of phrases must in his case take the place of the ordinary
routine of a careful experimenter. The description of the results of mathematical
investigation, which is a most difficult but at the same time most useful species
of literary composition, is a notable feature of this book, and could hardly be
better done. The mathematical researches of Lindelof, Lamarle, Scherk, Blemann,
&c., on surfaces of minimum area, deserve to be known to others besides pro-
fessed mathematicians, and M. Plateau deserves our thanks for giving us an
intelligible account of them, and still more for shewing us how to make them
visible with his improved soap-suds.

In the speculative part of the book, where the author treats of the causes
of the phenomena, there is of course more room for improvement, as there
always must be when a physicist is pushing his way into the unknown regions
of molecular science. In such matters everything human, at least in our century,
must be very imperfect, but for the same reason any real progress, however
small, is of the greater value.

[From yature, Vol. X.]

LXVII. Grove's " Correlation of Physical Forces*"

THKKE are few instances in which anyone whose life has not been exclu-
sively scientific has made such valuable contributions to science as those of Sir
W. R. Grove. His nitric acid battery, to the invention of which he was led,
not by accident, but by a course of reasoning, which in the year 1839 was as
new as it was original, is a contribution to science the value of which is proved
l>y its still surviving and continuing in daily use in every laboratory as the
most powerful generator of electric currents, while hundreds of batteries invented
since that of Grove have fallen into disuse, and become extinct in the struggle
for scientific existence.

The gas battery, though not of such practical importance, is still of great
scientific interest, and the collection which we have before us of those contri-
butions to science which took the form of papers, tempts us to indulge in
speculations as to the magnitude of the results which would have accrued
to science if so powerful a mind could have been continuously directed with
undivided energy towards some of the great questions of physics.

But the main feature of the volume is that from which it takes its name,
the essay on the Correlation of Physical Forces, the views contained in which
were first advanced in a lecture at the London Institution in January 1842,
printed by the proprietors, and subsequently more fully developed in a course
of lectures in 1843, published in abstract in the Literary Gazette. This essay
has a value peculiar to itself. Though it has long ago accomplished the main
point of its scientific mission to the world, it will always retain its place in

* Tlte Correlation of Physical Forces. Sixth edition. With other Contributions to Science. 15y
the Hon. Sir W. R. Grove, M.A., F.R.S., one of the judges of the Court of Common Pleas. (London:
Longmans, 1874.)

GROVES CORRELATION OP PHYSICAL FORCES. 401

the memory of the student of human thought, as one of the documents which
serve for the construction of the history of science.

It is not by discoveries only, and the registration of them by learned
societies, that science is advanced. The true seat of science is not in the
volume of Transactions, but in the living mind, and the advancement of science
consists in the direction of men's minds into a scientific channel ; whether this
is done by the announcement of a discovery, the assertion of a paradox, the
invention of a scientific phrase, or the exposition of a system of doctrine. It
is for the historian of science to determine the magnitude and direction of the
impulse communicated by either of these means to human thought.

But what we require at any given epoch for the advancement of science
is not merely to set men thinking, but to produce a concentration of thought
in that part of the field of science which at that particular season ought to be
cultivated. In the history of science we find that effects of this kind have often
been produced by suggestive books, which put into a definite, intelligible, and
communicable form, the guiding ideas that are already working in the minds
of men of science, so as to lead them to discoveries, but which they cannot yet
shape into a definite statement.

In the first half of the present century, when what is now called the
principle of the conservation of energy was as yet unknown by name, it "flung
its vague shadow back from the depths of futurity," and those who had greater
or less understanding of the times sketched out with greater or less clearness
their view of the form into which science was shaping itself.

Some of these addressed themselves to the advanced cultivators of science,
speaking, of course, in learned phraseology ; but others appealed to a larger
audience, and spoke in language which they could understand. Mrs Somerville's
book on the "Connection of the Physical Sciences" was published in 1834, and
had reached its eighth edition in 1849. This fact is enough to shew that there
already existed a widespread desire to be able to form some notion of physical
science as a whole.

But when we examine her book in order to find out the nature of the
connexion of the physical sciences, we are at first tempted to suppose that it is
due to the art of the bookbinder, who has bound into one volume such a quantity
of information about each of them. What we find, in fact, is a series of expo-
sitions of different sciences, but hardly a word about their connexion. The little
that is said about this connexion has reference to the mutual dependence of

VOL. II. 51

402 GROVE'S CORRELATION OF PHYSICAL FORCES.

the different sciences on each other, a knowledge of the elements of one being
essential to the successful prosecution of another. Thus physical astronomy
requires a knowledge of dynamics, and the practical astronomer must learn a
certain amount of optics in order to understand atmospheric refraction and the
adjustment of telescopes. The sciences are also shewn to have a common method,
namely, mathematical analysis ; so that analytical methods invented for the in-
vestigation of one science are often useful in another.

The unity shadowed forth in Mrs Somerville's book is therefore a unity of
the method of science, not a unity of the processes of nature.

Sir W. Grove's essay may be fairly called a popular book, as it has reached
its sixth edition. It is, therefore, not merely a record of the speculations nt
the author, but an index of the state of scientific thought among a large number
of readers. It has not the universal facility and occasional felicity of exposition
which distinguish Mrs Somerville's writings. No one could use it as a text-
book of any science, or even as an aid to the cultivation of the art of scientific
conversation. The design of the book is to shew that of the various forms of
energy existing in nature, any one may be transformed into any other, the one
form appearing as the other disappears. This is what is meant in the essay
by the "correlation of the physical forces," and the whole essay is an exposition
of this fact, each of the physical forces in turn being taken as the starting-
point, and employed as the source of all the others.

We are sorry that we are not at present able to refer to the early reviews
of the essay as indicating the reception given to the doctrine by the literary
and scientific public at the time of its original publication. It has certainly
exercised a very considerable effect in moulding the mass of what is called
scientific opinion, that is to say, the influence which determines what a scientific
man shall say when he has to make a statement about a science which he does
not understand. Many things in the essay which were then considered contrary
to scientific opinion, and were therefore objected to, have since then become
themselves part of scientific opinion, so that the objections now appear unin-
telligible to the rising generation of the scientific public.

Helmholtz's essay "On the Conservation of Force," published in 1847, un-
doubtedly masters a far greater step in science, but the immediate influence
was confined to a small number of trained men of science, and it had little
direct effect on the public mind.

The various papers of Mayer contain matter calculated to awaken an interest

GROVE'S CORBEL ATION OF PHYSICAL FORCES. 403

in the transformation of energy even in persons not exclusively devoted to science,
but they were long unknown in this country, and produced little direct effect,
even in Germany, at the time of their publication.

The rapid development of thermodynamics, and of other applications of the
principle of the conservation of energy, at the beginning of the second half of
this century, belongs to a later stage of the history of science than that with
which we have to do.

To form a just estimate of the value of Sir W. Grove's work we must
regard it as the instrument by which certain scientific ideas were diffused over
a large area, in language sufficiently appropriate to prevent misapprehension, and
yet sufficiently familiar to be listened to by persons who would recoil with horror
from any statement in which literary convention is sacrificed to precision.

It is worth while, however, to take note of the progress of evolution by
which the words of ordinary language are gradually becoming differentiated and
rendered scientifically precise. The fathers of dynamical science found a number
of words in common use expressive of action and the results of action, such as
force, power, action, impulse, impetus, stress, strain, work, energy, &c. They also
had in their minds a number of ideas to be expressed, and they appropriated
these words as they best could to express these ideas. But the equivalent
words Force Vis, Kraft, came most easily to hand, so that we find them com-
pelled to carry almost all the ideas above mentioned, while the other words
which might have borne a portion of the load were long left out of scientific
language, and retained only their more or less vague meanings as ordinary words.

Thus we have the expressions Vis acceleratrix, Vis motrix, Vis viva, Vis'
mortua, and even Vis inertia, in every one of which, except the second and
fourth, the word Vis is used in a sense radically different from that in which
it is used in the other expressions.

Confusion may perhaps be avoided in scientific works when read by scientific
students, by means of a careful appropriation of epithets such as those which
distinguish the meanings of the word Vis, but as soon as science becomes popu-
larised, unless its nomenclature is reformed and arranged upon a better principle,
the ideas of popular science will be more confused than those of so-called popular
ignorance.

Thus the " Physical Forces," whose correlation is discussed in the essay
before us, are Motion, Heat, Electricity, Light, Magnetism, Chemical Affinity, and
" other modes of force." According to the definition of force, as it has been

51—2

404 GROVE'S CORRELATION OF PHYSICAL FORCES.

laid down during the last two centuries in treatises on dynamics, not one of
these, except perhaps chemical affinity, can be admitted as a force. According
to that definition, " force is that which produces change of motion, and is
measured by the change of motion produced."

Newton himself reminds us that force exists only so long as it acts. Its
effects may remain, but the force itself is essentially transitive. Hence, when
we meet with such phrases as Conservation of Force, Persistence of Force,
and the like, we must suppose the word Force to be used in a sense radically
different from that adopted by scientific men from Newton downwards. In all
these cases, and in the phrase "The Physical Forces" as applied to heat, we
are now, thanks to Dr Thomas Young, able to use the word Energy instead
of Force, for this word, according to its scientific definition as " the capacity
for performing work," is applicable to all these cases. The confusion has extended
even to the metaphorical use of the word Force. Thus, it may be a legitimate
metaphor to speak of the force of public opinion as being brought to bear on
a statesman so as to exert an overpowering pressure upon him, because here
we have an action tending to produce motion in a particular direction ; but
when we speak of "the Queen's Forces," we use the term in a sense as
unscientific as when we speak of the Physical Forces. The author, in his con-
cluding remarks, points out the confusion of terms which embarrassed him in
his endeavours to enunciate scientific propositions, on account of the imper-
fection of scientific language. This, he tells us, " cannot be avoided without a
neology which I have not the presumption to introduce or the authority to
enforce."

Such a confession, proceeding from so great a master of the art of " putting
things," is a most valuable testimony to the importance of the study and
special cultivation of scientific language ; and a comparison of many passages in
the essay with the corresponding statements in more recent books of far inferior
power, will shew how much may be gained by the successful introduction of
appropriate neologies. What appeared mysterious and even paradoxical to the
giant, labouring among rough-hewn words, dwindles into a truism in the eyes
of the child, born heir to the palace of truth, for the erection of which the
giant has furnished the materials.

Thus the appropriation of the word " Mass " to denote the quantity of
matter as defined by the amount of force required to produce a given accelera-
tion, has placed the students of the present day on a very different level from

CORRELATION OF PHYSICAL FORCES. 405

those who had to puzzle out the meaning of the phrase Vis inertia by com-
bining the explanation of Vis as force, with that of inertia as laziness. In the
same way the word " stress " as an equivalent for " action and reaction," and
as a generic name for pressure, tension, &c., will save future generations a great
deal of trouble ; and the distinction between the possession of energy and the
act of doing work, which is now so familiar to us, would have obviated several
objections to the doctrine of the essay, which are founded on statements in
which the production of one form of energy and the maintenance of another are
treated as if they were operations of the same kind. We read at p. 163 : —
Thus, "a voltaic battery, decomposing water in a voltameter, while the same
current is employed at the same time to make (maintain) an electromagnet,
gives nevertheless in the voltameter an equivalent of gas, or decomposes an
equivalent of an electrolyte for each equivalent of decomposition in the battery
cells, and will give the same ratios if the electro-magnet be removed."

Here the maintenance of a magnet is a thing of a different order from
the decomposition of an electrolyte ; the first is maintenance of energy, the other
is doing work. This is well explained in the essay ; but if appropriate language
had been used from the first, the objection could never have been put into form.

[From Mature, Vol. x.]

I. XVIII. OH the application of Kirchhojfs Rules for Electric Circuit* t<> the

Solution of a Geometrical Problem.

THE geometrical problem is as follows : — Let it be required to arrange a
system of points so that the straight lines joining them into rows and columns
shall form a network such that the sum of the squares of all these joining
lines shall be a minimum, the first and last points of the first and last row
l»eing any four points given in space. The network may be regarded as a
kind of extensible surface, each thread of which has a tension in each segment
proportioned to the length of the segment. The problem is thus expressed as
a statical problem, but the direct solution would involve the consideration of a
large number of unknown quantities.

This number may be greatly reduced by means of the analogy between
this problem and the electrical problem of determining the currents and po-
tentials in the case of a network of wire having square meshes, one corner of
which is kept at a unit potential, while that of the other three corners is zero.
This problem having been solved by KirchhofFs method, the position of any
point P in the geometrical problem with reference to the given points A, B,
C, D, is by finding, the values of the potentials pa, pt, pe, pd of the corre-
s]x>nding point in the electric problem when the corners o, b, c, d respectively
are those of unit potential. The position of P is then found by supposing
Pa> Pb> Pe> Pd placed at A, B, C, D respectively, and determining P as the
centre of gravity of the four masses.

[From Nature, Vol. x.]

LXIX. Van der Wacds on the Continuity of the Gaseous and Liquid States*.

THAT the same substance at the same temperature and pressure can exist
in two very different states, as a liquid and as a gas, is a fact of the highest
scientific importance, for it is only by the careful study of the difference be-
tween these two states, the conditions of the substance passing from one to the
other, and the phenomena which occur at the surface which separates a liquid
from its vapour, that we can expect to obtain a dynamical theory of liquids.
A dynamical theory of " perfect " gases is already in existence ; that is to say,
we can explain many of the physical properties of bodies when in an extremely
rarefied state by supposing their molecules to be in rapid motion, and that
they act on one another only when they come very near one another. A
molecule of a gas, according to this theory, exists in two very different states
during alternate intervals of time. During its encounter with another molecule,
an intense force is acting between the two molecules, and producing changes
in the motion of both. During the time of describing its free path, the mole-
cule is at such a distance from other molecules that no sensible force acts
between them, and the centre of mass of the molecule is therefore moving with
constant velocity and in a straight line.

If we define as a perfect gas a system of molecules so sparsely scattered
that the aggregate of the time which a molecule spends in its encounters with
other molecules is exceedingly small compared with the aggregate of the time
which it spends in describing its free paths, it is not difficult to work out the
dynamical theory of such a system. For in this case the vast majority of the
molecules at any given instant are describing their free paths, and only a small

* Over de Continniteit van den Gas en Vloeistofloestand. Academisch pro?fsclirift. Door Johannes
Diderik van der Waals. (Leiden: A. W. Sijthoff, 1873.)

408 VAN DEU WAALS OX THE CONTINUITY

fraction of them are in the act of encountering each other. We know that
during an encounter action and reaction are equal and opposite, and -we assume,
with Clausius, that on an average of a large number of encounters the pro-
portion in which the kinetic energy of a molecule is divided between motion
«.f translation of its centre of mass and motions of its parts relative to this
point approaches some definite value. This amount of knowledge is by no moans
sufficient as a foundation for a complete dynamical theory of what takes ]>la.o
during each encounter, but it enables us to establish certain relations between
the changes of velocity of two molecules before and after their encounter.

While a molecule is describing its free path, its centre of mass is moving
witli constant velocity in a straight line. The motions of parts of the molecule
relative to the centre of mass depend, when it is describing its free path, only
on the forces acting between these parts, and not on the forces acting between
them and other molecules which come into play during an encounter. Hence
the theory of the motion of a system of molecules is very much simplified if
we suppose the space within which the molecules are free to move to be so
large that the number of molecules which at any instant are in the act of
encountering other molecules is exceedingly small compared with the number of
molecules which are describing their free paths. The dynamical theory of such
i\ system is in complete agreement with the observed properties of gases when
in an extremely rare condition.

But if the space occupied by a given quantity of gas is diminished more
and, more, the lengths of the free paths of its molecules will also be diminished,
and the number of molecules which are in the act of encounter will bear a
larger proportion to the number of those Avhich are describing free paths, till at
length the properties of the substance will be determined far more by the
nature of the mutual action between the encountering molecules than by the
nature of the motion of a molecule when describing its free path. And \\e
actually find that the properties of the substance become very different alter
it has reached a certain degree of condensation. In the rarefied state its pro-
])erties may be defined with considerable accuracy in terms of the laws of Boyle,
Charles, Gay-Lussac, Dulong and Petit, &c., commonly called the " gaseous
laws." In the condensed state the properties of the substance are entirely
different, and no mode of stating these properties has yet been discovered
having a simplicity and a generality at all approaching to that of the " gaseous
laws." According to the dynamical theory this is to be expected, because in

OF THE GASEOUS AND LIQUID STATES. 409

the condensed state the properties of the substance depend on the mutual
action of molecules when engaged in close encounter, and this is determined
by the particular constitution of the encountering molecules. We cannot there-
fore extend the dynamical theory from the rarer to the denser state of sub-
stances without at the same time obtaining some definite conception of the
nature of the action between molecules when they are so closely packed that
each molecule is at every instant so near to several others that forces of great
intensity are acting between them.

The experimental data for the study of the mutual action of molecules
are principally of two kinds. In the first place we have the experiments of
Regnault and others on the relation between the density, temperature, and
pressure of various gases. The field of research has been recently greatly
enlarged by Dr Andrews in his exploration of the properties of carbonic acid
at very high pressures. Experiments of this kind, combined with experiments
on specific heat, on the latent heat of expansion, or on the thermometric effect
on gases passing through porous plugs, furnish us with the complete theory
of the substance, so far as pure thermodynamics can carry us.

For the further study of molecular action we require experiments on the
rate of diffusion. There are three kinds of diffusion — that of matter, that of
visible motion, and that of heat. The inter-diffusion of gases of different kinds,
and the viscosity and thermal conductivity of a gaseous medium, pure or mixed,
enable us to estimate the amount of deviation which each molecule experiences
on account of its encounter with other molecules.

M. Van der Waals, in entering on this very difficult inquiry, has shewn
his appreciation of its importance in the present state of science ; many of his
investigations are conducted in an extremely original and clear manner ; and
he is continually throwing out new and suggestive ideas ; so that there can be
no doubt that his name will soon be among the foremost in molecular science.

He does not, however, seem to be equally familiar, as yet, with all parts
of the subject, so that in some places, where he has borrowed results from
Clausius and others, he has applied them in a manner which appears to me
erroneous.

He begins with the very remarkable theorem of Clausius, that in stationary
motion the mean kinetic energy of the system is equal to the mean virial.
As in this country the importance of this theorem seems hardly to be appre-
ciated, it may be as well to explain it a little more fully.

VOL. II. 52

410 VAX DKB WAALS ON THE CONTINUITY

When the motion of a material system is such that the sum of the
momenta of inertia of the system about three axes at right angles to each
•ither through ita centre of mass does not vary by more than small quantities
from a constant value, the system is said to be in a state of stationary motion.
The motion of the solar system satisfies this condition, and so does the motion
of the molecules of a gas contained in a vessel.

The kinetic energy of a particle is half the product of its mass into the
aquare of ita velocity, and the kinetic energy of a system is the sum of the
kinetic energy of its parts.

When an attraction or repulsion exists between two points, half the pro-
duct of this stress into the distance between the two points is called the
Virial of the stress, and is reckoned positive when the stress is an attraction,
and negative when it is a repulsion. The virial of a system is the sum of
the virial of the stresses which exist in it.

If the system is subjected to the external stress of the pressure of the
•ides of a vessel in which it is contained, the amount of virial due to this
external stress is three halves of the product of the pressure into the volume
of the vessel.

The virial due to internal stresses must be added to this.

The theorem of Clausius may now be written —

12 (in?) = $pV+&$(Rr).
The left-hand member denotes the kinetic energy.

On the right hand, in the first term, p is the external pressure on unit
of area, and V is the volume of the vessel.

The second term represents the virial arising from the action between every
pair of particles, whether belonging to different molecules or to the same mole-
cule. K is the attraction between the particles, and r is the distance between
them. The double symbol of summation is used because every pair of points
must be taken into account, those between which there is no stress contributing,
of course, nothing to the virial.

As an example of the generality of this theorem, we may mention that
in any framed structure consisting of struts and ties, the sum of the products
of the pressure in each strut into its length, exceeds the sum of the products
of the tension of each tie into its length, by the product of the weight of
the whole structure into the height of its centre of gravity above the founda-

OF THE GASEOUS AND LIQUID STATES. 411

tions. (See a paper on "Reciprocal Figures, &c." Trans. R. S. Edin., Vol. xxvi.
p. 14. 1870.)*

In gases the virial is very small compared with the kinetic energy. Hence,
if the kinetic energy is constant, the product of the pressure and the volume
remains constant. This is the case for a gas at constant temperature. Hence
we might be justified in conjecturing that the temperature of any one gas is
determined by the kinetic energy of unit of mass.

The theory of the exchange of the energy of agitation from one body to
another is one of the most difficult parts of molecular science. If it were
fully understood, the physical theory of temperature would be perfect. At
present we know the conditions of thermal equilibrium only in the case of
gases in which encounters take place between only a pair of molecules at once.
In this case the condition of thermal equilibrium is that the mean kinetic
energy due to the agitation of the centre of mass of a molecule is the same,
whatever be the mass of the molecule, the mean velocity being consequently
less for the more massive molecules.

With respect to substances of more complicated constitution, we know, as
yet, nothing of the physical condition on which their temperature depends,
though the researches of Boltzmann on this subject are likely to result in some
valuable discoveries.

M. Van der Waals seems, therefore, to be somewhat too hasty in assuming
that the temperature of a substance is in every case measured by the energy
of agitation of its individual molecules, though this is undoubtedly the case
with substances in the gaseous state.

Assuming, however, for the present that the temperature is measured by
the mean kinetic energy of a molecule, we obtain the means of determining the
virial by observing the deviation of the product of the pressure and volume
from the constant value given by Boyle's law.

It appears by Dr Andrews' experiments that when the volume of carbonic
acid is diminished, the temperature remaining constant, the product of the
volume and pressure at first diminishes, the rate of diminution becoming more
and more rapid as the density increases. Now, the virial depends on the
number of pairs of molecules which are at a given instant acting on one
another, and this number in unit of volume is proportional to the square of
the density. Hence the part of the pressure depending on the virial increases

* [Vol. n. p. 176.]

52—2

412 VAN DEB WAALS ON THB CONTINUITY

aft the square of the density, and since, in the case of carbonic acid, it
dimmudu* U»e pressure, it must be of the positive sign, that is, it roust arise
from at I ruction between the molecules.

Hut if the volume is still further diminished, at a certain point lique-
faction begins, and from this point till the gas is all liquefied no increase of
measure takes place. As soon, however, as the whole substance is in the liquid
condition, any further diminution of volume produces a great rise of pressure, so
that the product of pressure and volume increases rapidly. This indicates negative
vi rial, and shews that the molecules are now acting on each other by repulsion.

This is what takes place in carbonic acid below the temperature of 30'92°C.
Above that temperature there is first a positive and then a negative virial,
but no sudden liquefaction.

Similar phenomena occur in all the liquefiable gases. In other gases we are
able to trace the existence of attractive force at ordinary pressures, though the
compression has not yet been carried so far as to shew any repulsive force.
In hydrogen the repulsive force seems to prevail even at ordinary pressures.
This gas has never been liquefied, and it is probable that it never will be
liquefied, as the attractive force is so weak.

We have thus evidence that the molecules of gases attract each other at
a certain small distance, but when they are brought still nearer they repel
each other. This is quite in accordance with Boscovich's theory of atoms as
massive centres of force, the force being a function of the distance, and
changing from attractive to repulsive, and back again several times, as the
distance diminishes. If we suppose that when the force begins to be repulsive
it increases very rapidly as the distance diminishes, so as to become enormous
if the distance is less by a very small quantity than that at which the force
first begins to be repulsive, the phenomena will be precisely the same as those
of smooth elastic spheres.

M. Van der Waals makes his molecules elastic spheres, which, when not in
contact, attract each other. His treatment of the "molecular pressure" arising
from their attraction seems ingenious, and on the whole satisfactory, though he
has not attempted a complete calculation of the attractive virial in terms of
the law of force.

His treatment of the repulsive virial, however, shews a departure from the
principles on which his investigation is founded. He considers the effect of the
size of the molecules in diminishing the length of their "free paths," and he

OF THE GASEOUS AND LIQUID STATES. 413

shews that this effect, in the case of very rare gases, is the same as if the
volume of the space in which the molecules are free to move had been
diminished by four times the sum of the volumes of the molecules themselves.
He then substitutes for V, the volume of the vessel in Clausius' formula, this
volume diminished by four times the molecular volume, and thus obtains the
equation —

where p is the externally applied pressure, -=j is the molecular pressure

arising from attraction between the molecules, which varies as the square of
the density, or inversely as the square of the volume. The first factor is thus
what he considers the total effective pressure. V is the volume of the vessel,
and b is four times the volume of the molecules. The second factor is there-
fore the "effective volume" within which the molecules are free to move.

The right-hand member expresses the kinetic energy, represented by the
absolute temperature, multiplied by a quantity, R, constant for each gas.

The results obtained by M. Van der Waals by a comparison of this equa-
tion with the determinations of Regnault and Andrews are very striking, and
would almost persuade us that the equation represents the true state of the
case. But though this agreement would be strong evidence in favour of the
accuracy of an empirical formula devised to represent the experimental results,
the equation of M. Van der Waals, professing as it does to be derived from
the dynamical theory, must be subjected to a much more severe criticism.

It appears to me that the equation does not agree with the theorem of
Clausius on which it is founded.

In that theorem p is the pressure of the sides of the vessel, and V is
the volume of the vessel. Neither of these quantities is subject to correction.

The assumption that the kinetic energy is determined by the temperature
is true for perfect gases, and we have no evidence that any other law holds
for gases, even near their liquefying point.

The only source of deviation from Boyle's law is therefore to be looked
for in the term %Z2<(Rr), which expresses the virial. The effect of the repul-
sion of the molecules, causing them to act like elastic spheres, is therefore to
be found by calculating the virial of this repulsion.

4|4 VAN DER WAALS ON THE CONTINUITY

Neglecting the effect of attraction, I find that the effect of the impulsive
repulsion reduces the equation of Clausius to the form—

where tr is the density of the molecules and p the mean density of the medium.

The form of this equation is quite different from that of M. Van der
Waala, though it indicates the effect of the impulsive force in increasing the
pressure. It takes no account of the attractive force, a full discussion of which
\\.iiild carry us into considerable difficulties.

At a constant temperature the effect of the attractive virial is to diminish
the pressure by a quantity varying as the square of the density, as long as
the encounters of the molecules are, on the whole, between two at a time,
and not between three or more. The effect of the attraction in deflecting the
paths of the molecules is to make the number of molecules which at any given
instant are at distances between r and r + dr of each other greater than the
number in an equal volume at a greater distance in the proportion of the
velocities corresponding to these distances. As the temperature rises, the volume
being constant, the ratio of these velocities approaches to unity, so that the
distribution of molecules according to distance becomes more uniform, and the
virial is thus diminished.

If there is a virial arising from repulsive forces acting through a finite
distance, a rise of temperature will increase the amount of this kind of virial.

Hence a rise of temperature at constant volume will produce a greater
increase of pressure than that given by the law of Charles.

The isothermal lines at higher temperatures will exhibit less of the dimi-
nution of pressure due to attraction, and as the density increases will shew
more of the increase of pressure due to repulsion.

I must not, however, while taking exception to part of the work of
M. Van der Waals, forget to add that to him alone are due the suggestions
which led me to examine the theory of virial more carefully in order to explore
the continuity of the liquid and the gaseous states.

I cannot now enter into the comparison of his theoretical results with the
experiments of Andrews, but I would call attention to the able manner in
which he expounds the theory of capillarity, and to the remarkable phenomena
of the surface-tension of gases which he tells (p. 38) has been observed by

OF THE GASEOUS AND LIQUID STATES. 415

Bosscha in tobacco smoke. As tobacco smoke is simply warm air with a slight
excess of carbonic acid carrying solid particles along with it, the change of
properties at the surface of the cloud must be very slight compared with that
at the surface where two really different gases first come together. If, there-
fore, the phenomenon observed by Bosscha is a true instance of surface-tension,
we may expect to discover much more striking phenomena at the meeting-place
of different gases, if we can make our observations before the surface of dis-
continuity has been obliterated by the inter-diffusion of the gases.

[From Cambridge PhUotophical Society Proceedings, VoL H. 365.]

LXX. On the Centre of Motion of the Eye.

THE aeries of positions which the eye assumes as it is rolled horizontally
have been investigated by Dondere (Donders and Doijer, Derde JaarUjlc.^/,
I'rrslag betr. het Nederlandsch Gasthuis voor Ooglijders. Utrecht, 1862), and
recently by Mr J. L. Tupper (Proc. R. S., June 18, 1874). The chief difficulty
in the investigation consists in fixing the head while the eyeball moves.
The only satisfactory method of obtaining a system of co-ordinates fixed with
reference to the skull is that adopted by Hehnholtz (Handbuch der Physiolo-
gixhen Optit, p. 517), and described in his Croonian Lecture.

A piece of wood, part of the upper surface of which is covered with
warm sealingwax, is placed between the teeth and bitten hard till the sealing-
wax sets and forms a cast of the upper teeth. By inserting the teeth into
their proper holes in the sealingwax the piece of wood may at any time be
placed in a determinate position relatively to the skull.

By this device of Helmholtz the patient is relieved from the pressure of
Hcrews and clamps applied to the skin of his head, and he becomes free
to move his head as he likes, provided he keeps the piece of wood between
his teeth.

If we can now adjust another piece of wood so that it shall always have
a determinate position with respect to the eyeball, we may study the motion
«>f the one piece of wood with respect to the other as the eye moves about.

For this purpose a small mirror is fixed to a board, and a dot is marked
on the mirror. If the eye, looking straight at the image of its own pupil
in the mirror, sees the dot in the centre of the pupil, the normal to the
mirror through the dot is the visual axis of the eye — a determinate line.

A right-angled prism is fixed to the board near the eye in such a position
that the eye sees the image of its own cornea in profile by reflexion, first

THE CENTRE OF MOTION OF THE EYE. 417

at the prism, and then at the mirror. A vertical line is drawn with black
sealingwax on the surface of the prism next the eye, and the board is moved
towards or from the eye till this line appears as a tangent to the front of
the cornea, while the dot still is seen to cover the centre of the image of
the pupil. The only way in which the position of the board can now vary
with respect to the eye is by turning round the line of vision as an axis,
and this is prevented by the board being laid on a horizontal platform carried
by the teeth.

If now the eye is brought into two different positions and the board
moved on the platform, so as to be always in the same position relative to
the eye, we have to find the centre about which the board might have
turned so as to get from one position to the other.

For this purpose two holes are made in the platform, and a needle thrust
through the holes is made to prick a card fastened to the upper board. We
thus obtain two pairs of points, AB for the first position, and ab for the
second.

The ordinary rule for determining the centre of motion is to draw lines
bisecting Aa and Bb at right angles. The intersection of these is the centre
of motion. This construction fails when the centre of motion is in or near
the line AB, for then the two lines coincide. In this case we may produce
AB and ab till they meet, and draw a line bisecting the angle externally.
This line will pass through the centre of motion as well as the other two,
and when they coincide it intersects them at right angles.

VOL. II.

53

[From Katun, Vol. xi.]

LXXI. On the Dynamical Evidence of the Molecular Constitution of Bodies*.

WHEN any phenomenon can be described as an example of some general
principle which is applicable to other phenomena, that phenomenon is said to be
explained. Explanations, however, are of very various orders, according to the
degree of generality of the principle which is made use of. Thus the person
who first observed the effect of throwing water into a fire would feel a certain
amount of mental satisfaction when he found that the results were always
similar, and that they did not depend on any temporary and capricious anti-
pathy between the water and the fire. This is an explanation of the lowest
order, in which the class to which the phenomenon is referred consists of other
phenomena which can only be distinguished from it by the place and time of
their occurrence, and the principle involved is the very general one that place
and time are not among the conditions which determine natural processes. On
the other hand, when a physical phenomenon can be completely described as a
change in the configuration and motion of a material system, the dynamical
explanation of that phenomenon is said to be complete. We cannot conceive
any further explanation to be either necessary, desirable, or possible, for as soon
as we know what is meant by the words configuration, motion, mass, and force,
we see that the ideas which they represent are so elementary that they cannot
be explained by means of anything else.

The phenomena studied by chemists are, for the most part, such as have
not received a complete dynamical explanation.

Many diagrams and models of compound molecules have been constructed.
These are the records of the efforts of chemists to imagine configurations of
material systems by the geometrical relations of which chemical phenomena may

• A lecture delivered at the Chemical Society, Feb. 18, by Prof. Clerk-Maxwell, F.R.S.

THE DYNAMICAL EVIDENCE OF THE MOLECULAR CONSTITUTION OF BODIES. 419

be illustrated or explained. No chemist, however, professes to see in these
diagrams anything more than symbolic representations of the various degrees of
closeness with which the different components of the molecule are bound
together.

In astronomy, on the other hand, the configurations and motions of the
heavenly bodies are on such a scale that we can ascertain them by direct
observation. Newton proved that the observed motions indicate a continual
tendency of all bodies to approach each other, and the doctrine of universal
gravitation which he established not only explains the observed motions of our
system, but enables us to calculate the motions of a system in which the
astronomical elements may have any values whatever.

When we pass from astronomical to electrical science, we can still observe
the configuration and motion of electrified bodies, and thence, following the
strict Newtonian path, deduce the forces with which they act on each other ;
but these forces are found to depend on the distribution of what we call
electricity. To form what Gauss called a " construirbar Vorstellung " of the in-
visible process of electric action is the great desideratum in this part of science.

In attempting the extension of dynamical methods to the explanation of
chemical phenomena, we have to form an idea of the configuration and motion
of a number of material systems, each of which is so small that it cannot be
directly observed. We have, in fact, to determine, from the observed external
actions of an unseen piece of machinery, its internal construction.

The method which has been for the most part employed in conducting
such inquiries is that of forming an hypothesis, and calculating what would
happen if the hypothesis were true. If these results agree with the actual
phenomena, the hypothesis is said to be verified, so long, at least, as some one
else does not invent another hypothesis which agrees still better with the phe-
nomena.

The reason why so many of our physical theories have been built up by
the method of hypothesis is that the speculators have not been provided with
methods and terms sufficiently general to express the results of their induction
in its early stages. They were thus compelled either to leave their ideas vague
and therefore useless, or to present them in a form the details of which could
be supplied only by the illegitimate use of the imagination.

In the meantime the mathematicians, guided by that instinct which teaches
them to store up for others the irrepressible secretions of their own minds,

53—2

TOE DYNAMICAL EVIDENCE OF THE

had developed with the utmost generality the dynamical theory of a material

•yatem.

Of all hypotheses as to the constitution of bodies, that is surely the most
warrantable which assumes no more than that they are material systems, and
im|i|ujju to Deduce from the observed phenomena just as much information about
the conditions and connections of the material system as these phenomena can

legitimately furnish.

When examples of this method of physical speculation have been properly
act forth and explained, we shall hear fewer complaints of the looseness of the
reasoning of men of science, and the method of inductive philosophy will no
longer be derided as mere guess-work.

It is only a small part of the theory of the constitution of bodies which
has as yet been reduced to the form of accurate deductions from known facts.
To conduct the operations of science in a perfectly legitimate manner, by means
of methodised experiment and strict demonstration, requires a strategic skill
which we must not look for, even among those to whom science is most in-
debted for original observations and fertile suggestions. It does not detract from
the merit of the pioneers of science that their advances, being made on unknown
ground, are often cut off, for a time, from that system of communications
with an established base of operations, which is the only security for any per-
manent extension of science.

In studying the constitution of bodies we are forced from the very be-
ginning to deal with particles which we cannot observe. For whatever may be
our ultimate conclusions as to molecules and atoms, we have experimental proof
that bodies may be divided into parts so small that we cannot perceive them.

Hence, if we are careful to remember that the word particle means a small
part of a body, and that it does not involve any hypothesis as to the ultimate
divisibility of matter, we may consider a body as made up of particles, and
we may also assert that in bodies or parts of bodies of measurable dimensions,
the number of particles is very great indeed.

The next thing required is a dynamical method of studying a material
system consisting of an immense number of particles, by forming an idea of their
configuration and motion, and of the forces acting on the particles, and deducing
from the dynamical theory those phenomena which, though depending on the
configuration and motion of the invisible particles, are capable of being observed
in visible portions of the system.

MOLECULAR CONSTITUTION OF BODIES. 421

The dynamical principles necessary for this study were developed by the
fathers of dynamics, from Galileo and Newton to Lagrange and Laplace ; but
the special adaptation of these principles to molecular studies has been to a
great extent the work of Prof. Clausius of Bonn, who has recently laid us
under still deeper obligations by giving us, in addition to the results of his
elaborate calculations, a new dynamical idea, by the aid of which I hope we
shall be able to establish several important conclusions without much symbolical
calculation.

The equation of Clausius, to which I must now call your attention, is of
the following form :

Here p denotes the pressure of a fluid, and V the volume of the vessel
which contains it. The product pV, in the case of gases at constant tempera-
ture, remains, as Boyle's Law tells us, nearly constant for different volumes
and pressures. This member of the equation, therefore, is the product of two
quantities, each of which can be directly measured.

The other member of the equation consists of two terms, the first depending
on the motion of the particles, and the second on the forces with which they
act on each other.

The quantity T is the kinetic energy of the system, or, in other words,
that part of the energy which is due to the motion of the parts of the
system.

The kinetic energy of a particle is half the product of its mass into the
square of its velocity, and the kinetic energy of the system is the sum of the
kinetic energy of its parts.

In the second term, r is the distance between any two particles, and R
is the attraction between them. (If the force is a repulsion or a pressure, R
is to be reckoned negative.)

The quantity $Rr, or half the product of the attraction into the distance
across which the attraction is exerted, is defined by Clausius as the virial of
the attraction. (In the case of pressure or repulsion, the virial is negative.)

The importance of this quantity was first pointed out by Clausius, who, by
giving it a name, has greatly facilitated the application of his method to phy-
sical exposition.

The virial of the system is the sum of the virials belonging to every pair
of particles which exist in the system. This is expressed by the double sum

THE DYNAMICAL EVIDENCE OF THE

which indicate that the value of Jflr is to be found for every pair
of particles, and the result* added together.

CUuaius has established this equation by a very simple mathematical pro-
CDM ^th wljich I need not trouble you, as we are not studying mathematics
to-night. We may see, however, that it indicates two causes which may affect
the pressure of the fluid on the vessel which contains it: the motion of its
particles, which tends to increase the pressure, and the attraction of its particles,
which tends to increase the pressure.

We may therefore attribute the pressure of a fluid either to the motion
of its particles or to a repulsion between them.

Let us test by means of this result of Clausius the theory that the pres-
sure of a gas arises entirely from the repulsion which one particle exerts on
another, these particles, in the case of gas in a fixed vessel, being really at rest.

In this case the virial must be negative, and since by Boyle's Law the
product of pressure and volume is constant, the virial also must be constant,
whatever the volume, in the same quantity of gas at constant temperature.
It follows from this that Rr, the product of the repulsion of two particles
into the distance between them, must be constant, or in other words that the
repulsion must be inversely as the distance, a law which Newton has shewn
to be inadmissible in the case of molecular forces, as it would make the action
of the distant parts of bodies greater than that of contiguous parts. In fact,
we have only to observe that if Rr is constant, the virial of every pair of
particles must be the same, so that the virial of the system must be propor-
tional to the number of pairs of particles in the system — that is, to the
square of .the number of particles, or in other words to the square of the
quantity of gas in the vessel The pressure, according to this law, would not
be the same in different vessels of gas at the same density, but would be
greater in a large vessel than in a small one, and greater in the open air than
in any ordinary vessel.

The pressure of a gas cannot therefore be explained by assuming repulsive
forces between the particles.

It must therefore depend, in whole or in part, on the motion of the particles.

If we suppose the particles not to act on each other at all, there will be
no virial, and the equation will be reduced to the form

MOLECULAR CONSTITUTION OF BODIES. 423

If M is the mass of the whole quantity of gas, and c is the mean square of
the velocity of a particle, we may write the equation —

or in words, the product of the volume and the pressure is one-third of the
mass multiplied by the mean square of the velocity. If we now assume, what
we shall afterwards prove by an independent process, that the mean square of
the velocity depends only on the temperature, this equation exactly represents
Boyle's Law.

But we know that most ordinary gases deviate from Boyle's Law, especially
at low temperatures and great densities. Let us see whether the hypothesis
of forces between the particles, which we rejected when brought forward as the
sole cause of gaseous pressure, may not be consistent with experiment when
considered as the cause of this deviation from Boyle's Law.

When a gas is in an extremely rarefied condition, the number of particles
within a given distance of any one particle will be proportional to the density
of the gas. Hence the virial arising from the action of one particle on the
rest will vary as the density, and the whole virial in unit of volume will vary
as the square of the density.

Calling the density p, and dividing the equation by V, we get —

where A is a quantity which is nearly constant for small densities.

Now, the experiments of Regnault shew that in most gases, as the density
increases the pressure falls below the value calculated by Boyle's Law. Hence
the virial must be positive ; that is to say, the mutual action of the particles
must be in the main attractive, and the effect of this action in diminishing
the pressure must be at first very nearly as the square of the density.

On the other hand, when the pressure is made still greater the substance
at length reaches a state in which an enormous increase of pressure produces
but a very small increase of density. This indicates that the virial is now
negative, or, in other words, the action between the particles is now, in the
main, repulsive. We may therefore conclude that the action between two par-
ticles at any sensible distance is quite insensible. As the particles approach
each other the action first shews itself as an attraction, which reaches a maxi-
mum, then diminishes, and at length becomes a repulsion so great that no
attainable force can reduce the distance of the particles to zero.

4*4 THB DYNAMICAL EVIDENCE OF TOE

The relation between pressure and density arising from such an action
between the (articles is of this kind.

As the density increases from zero, the pressure at first depends almost
entirely on the motion of the particles, and therefore varies almost exactly as
the prvwure, according to Boyle's Law. As the density continues to increase,
the effect of the mutual attraction of the particles becomes sensible, and this
causes the rise of pressure to be less than that given by Boyle's Law. If
the temperature is low, the effect of attraction may become so large in pro-
portion to the effect of motion that the pressure, instead of always rising as
the density increases, may reach a maximum, and then begin to diminish.

At length, however, as the average distance of the particles is still further
liiminiahed, the effect of repulsion will prevail over that of attraction, and the
pressure will increase so as not only to be greater than that given by Boyle's
Law, but so that an exceedingly small increase of density will produce an
enormous increase of pressure.

Hence the relation between pressure and volume may be represented by
the curve ABCDEFO, where the horizontal ordinate represents the volume,
and the vertical ordinate represents the pressure.

the volume diminishes, the pressure increases up to the point C, then
diminishes to the point E, and finally increases without limit as the volume
diminishes.

We have hitherto supposed the experiment to be conducted in such a
way that the density is the same in every part of the medium. This, how-
ever, is impossible in practice, as the only condition we can impose on the
medium from without is that the whole of the medium shall be contained

MOLECULAR CONSTITUTION OF BODIES. 425

within a certain vessel. Hence, if it is possible for the medium to arrange
itself so that part has one density and part another, we cannot prevent it from
doing so.

Now the points B and F represent two states of the medium in which
the pressure is the same but the density very different. The whole of the
medium may pass from the state B to the state F, not through the inter-
mediate states CDE, but by small successive portions passing directly from the
state B to the state F. In this way the successive states of the medium as
a whole will be represented by points on the straight line BF, the point B
representing it when entirely in the rarefied state, and F representing it when
entirely condensed. This is what takes place when a gas or vapour is liquefied.

Under ordinary circumstances, therefore, the relation between pressure and
volume at constant temperature is represented by the broken line ABFG. If,
however, the medium when liquefied is carefully kept from contact with vapour,
it may be preserved in the liquid condition and brought into states represented
by the portion of the curve between F and E. It is also possible that methods
may be devised whereby the vapour may be prevented from condensing, and
brought into states represented by points in BC.

The portion of the hypothetical curve from C to E represents states which
are essentially unstable, and which cannot therefore be realised.

Now let us suppose the medium to pass from B to F along the hypo-
thetical curve BCDEF in a state always homogeneous, and to return along
the straight line FB in the form of a mixture of liquid and vapour. Since
the temperature has been constant throughout, no heat can have been trans-
formed into work. Now the heat transformed into work is represented by the
excess of the area FDE over BCD. Hence the condition which determines
the maximum pressure of the vapour at given temperature is that the line
BF cuts off equal areas from the curve above and below.

The higher the temperature, the greater the part of the pressure which
depends on motion, as compared with that which depends on forces between
the particles. Hence, as the temperature rises, the dip in the curve becomes
less marked, and at a certain temperature the curve, instead of dipping, merely
becomes horizontal at a certain point, and then slopes upward as before. This
point is called the critical point. It has been determined for carbonic acid
by the masterly researches of Andrews. It corresponds to a definite temperature,
pressure and density.

VOL. II. 54

THE DYNAMICAL EVIDENCE OF THE

At higher temperatures the curve slopes upwards throughout, and there is
nothing corresponding to liquefaction in passing from the rarest to the densest

•fete.

The molecular theory of the continuity of the liquid and gaseous states
forma the subject of an exceedingly ingenious thesis by Mr Johannes Diderick
ran der Waals*, a graduate of Leyden. There are certain points in which I
think he has fallen into mathematical errors, and his final result is certainly
not a complete expression for the interaction of real molecules, but his attack
on this difficult question is so able and so brave, that it cannot fail to give
a notable impulse to molecular science. It has certainly directed the attention
of more than one inquirer to the study of the Low-Dutch language in which
it is written.

The purely thermodynamical relations of the different states of matter do
not belong to our subject, as they are independent of particular theories about
molecules. I must not, however, omit to mention a most important American
contribution to this part of thermodynamics by Prof. Willard Gibbsf, of Yale
College, U.S., who has given us a remarkably simple and thoroughly satisfactory
method of representing the relations of the different states of matter by means
of a model. By means of this model, problems which had long resisted the
efforts of myself and others may be solved at once.

Let us now return to the case of a highly rarefied gas in which the
pressure is due entirely to the motion of its particles. It is easy to calculate
the mean square of the velocity of the particles from the equation of Clausius,
since the volume, the pressure, and the mass are all measurable quantities.
Supposing the velocity of every particle the same, the velocity of a molecule
of oxygen would be 461 metres per second, of nitrogen 492, and of hydrogen
1844, at the temperature of 0°C.

The explanation of the pressure of a gas on the vessel which contains it
by the impact of its particles on the surface of the vessel has been suggested
at various times by various writers. The fact, however, that gases are not ob-
served to disseminate themselves through the atmosphere with velocities at all
approaching those just mentioned, remained unexplained, till Clausius, by a

• Ovtr de continuitfit van den gat en vloeistoftoestand. (Leiden : A. W. Sijthoff, 1873.)
t " A method of geometrical representation of the thermodynamic properties of substances by means
of Burfacea." Trantactions of the Connecticut Academy of Arts and Sciences, Vol. n. Part 2.

MOLECULAB, CONSTITUTION OF BODIES. 427

thorough study of the motions of an immense number of particles, developed
the methods and ideas of modern molecular science.

To him we are indebted for the conception of the mean length of the
path of a molecule of a gas between its successive encounters with other
molecules. As soon as it was seen how each molecule, after describing an
exceedingly short path, encounters another, and then describes a new path in
a quite different direction, it became evident that the rate of diffusion of gases
depends not merely on the velocity of the molecules, but on the distance they
travel between each encounter.

I shall have more to say about the special contributions of Clausius to
molecular science. The main fact, however, is, that he opened up a new field
of mathematical physics by shewing how to deal mathematically with moving
systems of innumerable molecules.

Clausius, in his earlier investigations at least, did not attempt to determine
whether the velocities of all the molecules of the same gas are equal, or whether,
if unequal, there is any law according to which they are distributed. He
therefore, as a first hypothesis, seems to have assumed that the velocities are
equal. But it is easy to see that if encounters take place among a great
number of molecules, their velocities, even if originally equal, will become un-
equal, for, except under conditions which can be only rarely satisfied, two
molecules having equal velocities before their encounter will acquire unequal
velocities after the encounter. By distributing the molecules into groups ac-
cording to their velocities, we may substitute for the impossible task of following
every individual molecule through all its encounters, that of registering the
increase or decrease of the number of molecules in the different groups.

By following this method, which is the only one available either experi-
mentally or mathematically, we pass from the methods of strict dynamics to
those of statistics and probability.

When an encounter takes place between two molecules, they are transferred
from one pair of groups to another, but by the time that a great many en-
counters have taken place, the number which enter each group is, on an average,
neither more nor less than the number which leave it during the same time.
When the system has reached this state, the numbers in each group must be
distributed according to some definite law.

As soon as I became acquainted with the investigations of Clausius, I
endeavoured to ascertain this law.

54—2

THE DYXAJUCAL EVIDENCE OF THE

The result which I published in 1860 has since been subjected to a more
•trict investigation by Dr Ludwig Boltzmann, who has also applied his method

the study of the motion of compound molecules. The mathematical investi-
gation, though, like all part* of the science of probabilities and statistics, it is
MBewhat difficult, does not appear faulty. On the physical side, however, it
leads to consequences, some of which, being manifestly true, seem to indicate
that the hypotheses are well chosen, while others seem to be so irreconcilable
with known experimental results, that we are compelled to admit that some-
thing essential to the complete statement of the physical theory of molecular
encounters must have hitherto escaped us.

I must now attempt to give you some account of the present state of these
investigations, without, however, entering into their mathematical demonstration.

I must begin by stating the general law of the distribution of velocity
among molecules of the same kind.

If we take a fixed point in this diagram and draw from this point a line
representing in direction and magnitude the velocity of a molecule, and make
a dot at the end of the line, the position of the dot will indicate the state of
motion of the molecule.

If we do the same for all the other molecules, the diagram will be dotted
all over, the dots being more numerous in certain places than in others.

The law of distribution of the dots may be shewn to be the same as that
which prevails among errors of observation or of adjustment.

The dots in the diagram before you may be taken to represent the velocities
of molecules, the different observations of the position of the same star, or the
bullet-holes round the bull's-eye of a target, all of which are distributed in the
same mariner.

The velocities of the molecules have values ranging from zero to infinity,
so that in speaking of the average velocity of the molecules we must define
what we mean.

The most useful quantity for purposes of comparison and calculation is called
the "velocity of mean square." It is that velocity whose square is the average
of the squares of the velocities of all the molecules.

This is the velocity given above as calculated from the properties of different
gases. A molecule moving with the velocity of mean square has a kinetic
energy equal to the average kinetic energy of all the molecules in the medium,
and if a single mass equal to that of the whole quantity of gas were moving

MOLECULAR CONSTITUTION OF BODIES. 429

with this velocity, it would have the same kinetic energy as the gas actually
has, only it would be in a visible form and directly available for doing work.

If in the same vessel there are different kinds of molecules, some of greater
mass than others, it appears from this investigation that their velocities will be
so distributed that the average kinetic energy of a molecule will be the same,
whether its mass be great or small.

Diagram of Velocities.

Here we have perhaps the most important application which has yet been
made of dynamical methods to chemical science. For, suppose that we have
two gases in the same vessel. The ultimate distribution of agitation among the
molecules is such that the average kinetic energy of an individual molecule is
the same in either gas. This ultimate state is also, as we know, a state of
equal temperature. Hence the condition that two gases shall have the same
temperature is that the average kinetic energy of a single molecule shall be the
same in the two gases.

THK DYNAMICAL KVIDBNCK OF THE

Now. we have already shewn that the pressure of a gas is two-thirds of the
kinetic energy in unit of volume. Henoe, if the pressure as well as the tem-
penUurv be the some in the two gases, the kinetic energy per unit of volume
m the •"•fj, as well as the kinetic energy per molecule. There must, therefore,
be the mm** number of molecules in unit of volume in the two gases.

Thi« result coincides with the law of equivalent volumes established by Gay
LUBMC. This law, however, lias hitherto rested on purely chemical evidence,
the relative masses of the molecules of different substances having been deduced
from the proportions in which the substances enter into chemical combination.
It i* now demonstrated on dynamical principles. The molecule is defined as
that *rn*l\ portion of the substance which moves as one lump during the motion
of agitation. This is a purely dynamical definition, independent of any experi-
menta on combination.

The density of a gaseous medium, at standard temperature and pressure,
is proportional to the mass of one of its molecules as thus defined.

We have thus a safe method of estimating the relative masses of molecules
of different substances when in the gaseous state. This method is more to be
depended on than those founded on electrolysis or on specific heat, because our
knowledge of the conditions of the motion of agitation is more complete than
our knowledge of electrolysis, or of the internal motions of the constituents of
a molecule.

I must now say something about these internal motions, because the greatest
difficulty which the kinetic theory of gases has yet encountered belongs to this
part of the subject.

We have hitherto considered only the motion of the centre of mass of the
molecule. We have now to consider the motion of the constituents of the
molecule relative to the centre of mass.

If we suppose that the constituents of a molecule are atoms, and that each
atom is what is called a material point, then each atom may move in three
different and independent ways, corresponding to the three dimensions of space,
so that the number of variables required to determine the position and con-
figuration of all the atoms of the molecule is three times the number of atoms.

It is not essential, however, to the mathematical investigation to assume
that the molecule is made up of atoms. All that is assumed is that the position
and configuration of the molecule can be completely expressed by a certain
number of variables.

MOLECULAR CONSTITUTION OF BODIES. 431

Let us call this number n,

Of these variables, three are required to determine the position of the
centre of mass of the molecule, and the remaining n — 3 to determine its con-
figuration relative to its centre of mass.

To each of the n variables corresponds a different kind of motion.

The motion of translation of the centre of mass has three components.

The motions of the parts relative to the centre of mass have n — 3 com-
ponents.

The kinetic energy of the molecule may be regarded as made up of two
parts — that of the mass of the molecule supposed to be concentrated at its
centre of mass, and that of the motions of the parts relative to the centre
of mass. The first part is called the energy of translation, the second that of
rotation and vibration. The sum of these is the whole energy of motion of the
molecule.

The pressure of the gas depends, as we have seen, on the energy of translation
alone. The specific heat depends on the rate at which the whole energy, kinetic
and potential, increases as the temperature rises.

Clausius had long ago pointed out that the ratio of the increment of the
whole energy to that of the energy of translation may be determined if we
know by experiment the ratio of the specific heat at constant pressure to that
at constant volume.

He did not. however, attempt to determine & priori the ratio of the two
parts of the energy, though he suggested, as an extremely probable hypothesis,
that the average values of the two parts of the energy in a given substance
always adjust themselves to the same ratio. He left the numerical value of this
ratio to be determined by experiment.

In 1860 I investigated the ratio of the two parts of the energy on the
hypothesis that the molecules are elastic bodies of invariable form. I found, to
my great surprise, that whatever be the shape of the molecules, provided they
are not perfectly smooth and spherical, the ratio of the two parts of the energy
must be always the same, the two parts being in fact equal.

This result is confirmed by the researches of Boltzmann, who has worked
out the general case of a molecule having n variables.

He finds that while the average energy of translation is the same for
molecules of all kinds at the same temperature, the whole energy of motion
is to the energy of translation as n to 3.

THB DYNAMICAL EVIDENCE OF THE

For a rigid body n-6, which makes the whole energy of motion twice the

w of translation.

*But if the molecule is capable of changing its form under the action of
impressed forces, it must be capable of storing up potential energy, and if the
forces are such as to ensure the stability of the molecule, the average potential

will increase when the average energy of internal motion increases.
Hence, as the temperature rises, the increments of the energy of translation,
the energy of internal motion, and the potential energy are as 3, (n-3), and e
respectively, where e is a positive quantity of unknown value depending on the
law of the force which binds together the constituents of the molecule.

When the volume of the substance is maintained constant, the effect of
the application of heat is to increase the whole energy. We thus find for the
specific heat of a gas at constant volume—

where pt and Vt are the pressure and volume of unit of mass at zero centi-
grade, or 273* absolute temperature, and J is the dynamic equivalent of heat.
The specific heat at constant pressure is

In gases whose molecules have the same degree of complexity the value of
n is the same, and that of e may be the same.

If this is the case, the specific heat is inversely as the specific gravity,
according to the law of Dulong and Petit, which is, to a certain degree of
approximation, verified by experiment.

But if we take the actual values of the specific heat as found by Regnault
and compare them with this formula, we find that n + e for air and several
other gases cannot be more than 4-9. For carbonic acid and steam it is greater.
We obtain the same result if we compare the ratio of the calculated specific
heats

2 + n + c
n + e

with the ratio as determined by experiment for various gases, namely, T408.

MOLECULAR CONSTITUTION OF BODIES. 433

And here we are brought face to face with the greatest difficulty which
the molecular theory has yet encountered, namely, the interpretation of the
equation n + e = 4'9.

If we suppose that the molecules are atoms — mere material points, incapable
of rotatory energy or internal motion — then n is 3 and e is zero, and the ratio
of the specific heats is 1*66, which is too great for any real gas.

But we learn from the spectroscope that a molecule can execute vibrations
of constant period. It cannot therefore be a mere material point, but a system
capable of changing its form. Such a system cannot have less than six variables.
This would make the greatest value of the ratio of the specific heats 1'33, which
is too small for hydrogen, oxygen, nitrogen, carbonic oxide, nitrous oxide, and
hydrochloric acid.

But the spectroscope tells us that some molecules can execute a great many
different kinds of vibrations. They must therefore be systems of a very con-
siderable degree of complexity, having far more than six variables. Now, every
additional variable introduces an additional amount of capacity for internal
motion without affecting the external pressure. Every additional variable, there-
fore, increases the specific heat, whether reckoned at constant pressure or at
constant volume.

So does any capacity which the molecule may have for storing up energy
in the potential form. But the calculated specific heat is already too great
when we suppose the molecule to consist of two atoms only. Hence every
additional degree of complexity which we attribute to the molecule can only
increase the difficulty of reconciling the observed with the calculated value of
the specific heat.

I have now put before you what I consider to be the greatest difficulty
yet encountered by the molecular theory. Boltzmann has suggested that we
are to look for the explanation in the mutual action between the molecules
and the Eetherial medium which surrounds them. I am afraid, however, that
if we call in the help of this medium, we shall only increase the calculated
specific heat, which is already too great.

The theorem of Boltzmann may be applied not only to determine the dis-
tribution of velocity among the molecules, but to determine the distribution of
the molecules themselves in a region in which they are acted on by external

forces. It tells us that the density of distribution of the molecules at a point

_!
where the potential energy of a molecule is t/f, is proportional to e «» where 6 is

VOL. II. 55

; . TOT DYNAMICAL EVIDENCE OF THE

the absolute temperature, and it is a constant for all gases. It follows from
this, that if several gases in the same vessel are subject to an external force
like that of gravity, the distribution of each gas is the same as if no other
gu were present This result agrees with the law assumed by Dalton, according
to which the atmosphere may be regarded as consisting of two independent
atmospheres, one of oxygen, and the other of nitrogen; the density of the
oxygen diminishing faster than that of the nitrogen, as we ascend.

This would be the case if the atmosphere were never disturbed, but the
^jfftnfc of winds is to mix up the atmosphere and to render its composition more
uniform than it would be if left at rest.

Another consequence of Boltzmann's theorem is, that the temperature tends
to become equal throughout a vertical column of gas at rest.

In the case of the atmosphere, the effect of wind is to cause the tem-
perature to vary as that of a mass of air would do if it were carried vertically
upwards, expanding and cooling as it ascends.

But besides these results, which I had already obtained by a less elegant
method and published in 1866, Boltzmann's theorem seems to open up a path
into a region more purely chemical. For if the gas consists of a number of
similar systems, each of which may assume different states having different
amounts of energy, the theorem tells us that the number in each state is

proportional to e"s where ^ is the energy, 0 the absolute temperature, and K a
constant.

It is easy to see that this result ought to be applied to the theory of
the states of combination which occur in a mixture of different substances. But
as it is .only during the present week that I have made any attempt to do
so, I shall not trouble you with my crude calculations.

I have confined my remarks to a very small part of the field of molecular
investigation. I have said nothing about the molecular theory of the diffusion
of matter, motion, and energy, for though the results, especially in the diffusion
"f matter and the transpiration of fluids are of great interest to many chemists,
and though from them we deduce important molecular data, they belong to a
part of our study the data of which, depending on the conditions of the en-
counter of two molecules, are necessarily very hypothetical. I have thought it
better to exhibit the evidence that the parts of fluids are in motion, and to
describe the manner in which that motion is distributed among molecules of
different masses.

MOLECULAR CONSTITUTION OP BODIES. 435

To shew that all the molecules of the same substance are equal in mass,
we may refer to the methods of dialysis introduced by Graham, by which two
gases of different densities may be separated by percolation through a porous
plug.

If in a single gas there were molecules of different masses, the same process
of dialysis, repeated a sufficient number of times, would furnish us with two
portions of the gas, in one of which the average mass of the molecules would
be greater than in the other. The density and the combining weight of these
two portions would be different. Now, it may be said that no one has carried
out this experiment in a sufficiently elaborate manner for every chemical sub-
stance. But the processes of nature are continually carrying out experiments
of the same kind ; and if there were molecules of the same substance nearly
alike, but differing slightly in mass, the greater molecules would be selected in
preference to form one compound, and the smaller to form another. But hy-
drogen is of the same density, whether we obtain it from water or from a
hydrocarbon, so that neither oxygen nor carbon can find in hydrogen molecules
greater or smaller than the average.

The estimates which have been made of the actual size of molecules are
founded on a comparison of the volumes of bodies in the liquid or solid state,
with their volumes in the gaseous state. In the study of molecular volumes
we meet with "many difficulties, but at the same time there are a sufficient
number of consistent results to make the study a hopeful one.

The theory of the possible vibrations of a molecule has not yet been studied
as it ought, with the help of a continual comparison between the dynamical
theory and the evidence of the spectroscope. An intelligent student, armed
with the calculus and the spectroscope, can hardly fail to discover some important
fact about the internal constitution of a molecule.

The observed transparency of gases may seem hardly consistent with the
results of molecular investigations.

A model of the molecules of a gas consisting of marbles scattered at dis-
tances bearing the proper proportion to their diameters, would allow very little
light to penetrate through a hundred feet.

But if we remember the small size of the molecules compared with the
length of a wave of light, we may apply certain theoretical investigations of
Lord Kayleigh's about the mutual action between waves and small spheres,
which shew that the transparency of the atmosphere, if affected only by the

55—2

. I TBK DYNAMICAL EVIDENCE OF THE

tUMonoo of molecules, would be far greater than we have any reason to believe

it to be.

A much more difficult investigation, which has hardly yet been attempted,
to the electric properties of gases. No one has yet explained why dense
are such good insulators, and why, when rarefied or heated, they permit
the discharge of electricity, whereas a perfect vacuum is the best of all insulators.

It is true that the diffusion of molecules goes on faster in a rarefied gas,
&e mean path of a molecule is inversely as the density. But the
difference between dense and rare gas appears to be too great to be
accounted for in this way.

But while I think it right to point out the hitherto uncouquered difficulties
of this molecular theory, I must not forget to remind you of the numerous
farta which it satisfactorily explains. We have already mentioned the gaseous
laws, as they are called, which express the relations between volume, pressure,
and temperature, and Gay Lussac's very important law of equivalent volumes.
The explanation of these may be regarded as complete. The law of molecular
specific heats is less accurately verified by experiment, and its full explanation
depends on a more perfect knowledge of the internal structure of a molecule
than we as yet possess.

But the most important result of these inquiries is a more distinct con-
ception of thermal phenomena. In the first place, the temperature of the
medium is measured by the average kinetic energy of translation of a single
molecule of the medium. In two media placed in thermal communication, the
temperature as thus measured tends to become equal.

In the next place, we learn how to distinguish that kind of motion which
we call heat from other kinds of motion. The peculiarity of the motion called
heat is that it is perfectly irregular ; that is to say, that the direction and
magnitude of the velocity of a molecule at a given time cannot be expressed as
depending on the present position of the molecule and the time.

In the visible motion of a body, on the other hand, the velocity of the
re of mass of all the molecules in any visible portion of the body is the
observed velocity of that portion, though the molecules may have also an irregular
depending agitation on account of the body being hot.

In the transmission of sound, too, the different portions of the body have
a motion which is generally too minute and too rapidly alternating to be directly
observed. But in the motion which constitutes the physical phenomenon of sound,

MOLECULAR CONSTITUTION OP BODIES. 437

the velocity of each portion of the medium at any time can be expressed as
depending on the position and the time elapsed ; so that the motion of a medium
during the passage of a sound-wave is regular, and must be distinguished from
that which we call heat.

If, however, the sound-wave, instead of travelling onwards in an orderly
manner and leaving the medium behind it at rest, meets with resistances which
fritter away its motion into irregular agitations, this irregular molecular motion
becomes no longer capable of being propagated swiftly in one direction as sound,
but lingers in the medium in the form of heat till it is communicated to
colder parts of the medium by the slow process of conduction.

The motion which we call light, though still more minute and rapidly
alternating than that of sound, is, like that of sound, perfectly regular, and
therefore is not heat. What was formerly called Radiant Heat is a phenomenon
physically identical with light.

When the radiation arrives at a certain portion of the medium, it enters
it and passes through it, emerging at the other side. As long as the medium
is engaged in transmitting the radiation it is in a certain state of motion,
but as soon as the radiation has passed through it, the medium returns to its
former state, the motion being entirely transferred to a new portion of the
medium.

Now, the motion which we call heat can never of itself pass from one
body to another unless the first body is, during the whole process, hotter than
the second. The motion of radiation, therefore, which passes entirely out of one
portion of the medium and enters another, cannot be properly called heat.

We may apply the molecular theory of gases to test those hypotheses about
the luminiferous aether which assume it to consist of atoms or molecules.

Those who have ventured to describe the constitution of the luminiferous
aether have sometimes assumed it to consist of atoms or molecules.

The application of the molecular theory to such hypotheses leads to rather
startling results.

In the first place, a molecular aether would be neither more nor less than
a gas. We may, if we please, assume that its molecules are each of them
equal to the thousandth or the millionth part of a molecule of hydrogen, and
that they can traverse freely the interspaces of all ordinary molecules. But, as
we have seen, an equilibrium will establish itself between the agitation of the
ordinary molecules and those of the aether. In other words, the aether and the

438 TU« DYNAMICAL 111 !»•*• OF THE MOLECULAR CONSTITUTION OF BODIES.

in it will tend to equality of temperature, and the aether will be subject
to the ordinary gaseous laws as to pressure and temperature.

Among other properties of a gas, it will have that established by Dulong
and Petit, BO that the capacity for heat of unit of volume of the aether
muxt be equal to that of unit of volume of any ordinary gas at the same
It* presence, therefore, could not fail to be detected in our experi-
oo specific heat, and we may therefore assert that the constitution of
the a-ther is not molecular.

[From the Proceedings of the London Mathematical Society, Vol. vi. No. 83.]

LXXII. On the Application of Hamilton's Characteristic Function to the Theory
of an Optical Instrument symmetrical about its axis.

[Read April 8th, 1875.]

WHEN a ray of light passes from the point (xlt ylt zj to the point
(xv yt, z2) through any series of media, the line-integral V—\p,ds may be defined
as the distance which light would travel in vacuum in the same tune as it
travels from (#„ ylt z^ to (x3, y3, za).

Calling this the reduced distance between (x1} ylt Zj) and (x2, y2, z2), Hamil-
ton's Characteristic Function may be defined as the value of the reduced distance
between two points expressed in terms of the co-ordinates of these points.

It is not necessary that the co-ordinates of the two points should be
referred to the same system of axes. In treating of optical instruments we
shall reckon zt and za in opposite directions along the axis, and from different
planes of reference. We shall, however, make the axes of xl and a?2 parallel to
each other in what follows.

Let a ray from the point Pi(xl, 0, zj pass through the first plane of
reference at the point J2, (a,, &l} 0), through the second plane of reference at

&„ 0), and reach the point P2 (a;2, 0, z2);

then, putting
and

= (x2 — atf+bf+zf^',

THE THEORY OP OPTICAL INSTRUMENTS.

and the characteristic function from 72, to 72, = 7 (a function of a,, 6,, a,, &,),
the reduced path from P, to P, is

U=nlrl+ F+/i,r, (2).

HIM quantity is stationary with respect to variations of a,, 6,, a,, 6,; therefore,
differentiating with respect to these variables, we get

-.SCS+SE.O u^+^-oi

r' ^ r' A (3).

If P, and P, are conjugate foci for rays in the plane of xz, the reduced
jwth id stationary in passing from one ray to the next by simultaneous variation
of a, and <v

„ T /I (a.-z.tt tfFl . d'V i

Hence a, P^ + j— : <&*i + -i — ?— da* = 0

L VI ri / «iJ ^i^

(4).

[/A I tc« ^^ *' j / \ ^* ' I i Ct r ••

i£_ I ^~ ^^ • -• - I J" - — I fjuti- ^^ (JLCL "^ 0

Eliminating efct, and da, from these equations, and putting 6 for the angle
between the ray and the axis, we find

an equation connecting the values of r, and r, for conjugate primary focal lines
formed by rays in the plane of xz.

For rays in a plane perpendicular to that of xz, we obtain the relation
of r,' and r,' for the secondary focal lines, by passing from one ray to the next
Itv simultaneous variation of 6, and 6,,

(6),

whence

THE THEORY OF OPTICAL INSTRUMENTS.

441

The form of this equation differs from that for the primary focal lines
only by the omission of cos2 6, and by substituting b for a.

From these equations we obtain the following values of the cardinal points
of the ray in the primary and secondary plane :

1

Let

A' =

db?

, B =

i

B' =

=' D=

' TV

= 5&7' = B'^-A'C'

(8).

Primary.

Secondary.

(9)-

Value of r\ at 1st principal focus gl = ^l(

Value of r., at 2nd principal focus g.2 = ^AD cos202

1st principal focal length £ — ^BD cos Ql cos #2

2nd principal focal length f3 — ^BD cos 01 cos 0.2 f.{ = p.2l

In tracing the axis of the pencil, we may, as a first approximation, neglect
the square of 6.

Let it cross the axis before incidence at H1} and after emergence at £T2,

where H, is -ft beyond the first principal focus, and H, is a/, beyond the
a

second principal focus.

Let it cut the first and second principal planes at a distance f from the
axis. Then, since these planes are at distances / and /2 within the principal foci,

fa „ £ 1

l+a

.(10),
.(11).

Let the object and image be respectively -Rf, and $£ beyond the principal

foci,

•V0Tfl/'

<12>'

X1 =

We shall next determine the relation between the curvature of the object
and that of the image.

VOL. IL

56

TIIK THEORY OF OPTICAL INSTRUMENTS.

Let their curvatures be concave towards the instrument, their radii being
, and /£, respectively, then,

We must now insert these values in equation (5), but, for the sake of
Minplicity, we shall suppose the planes of reference to pass through the principal

•

We now Hnd ft = * /, = £

Jj jj

'* 2tf,

°'"i+i' a'=ITi <17)-

Hamilton's function for a symmetrical instrument is of the form

tf+V)

K + 63')'
(tt.rt, + 6,6,) («,' + 6,') + ±q («,' + 6,') (a./ + 6,J)

+ r (ai* + 6,s) (a,a, + 6,6,) (18).

hiffcrentiiiting. and making 6, and 6, zero, we find

- 1 = 81 + 3«a,s + 6ra,a, + (6 + f/) «,*

B-84-3fa1i+(26+2j)a1a»-J-3po,f (19),

C = 6 + (6 + q) a? + 6pa,a, + Sea,1

x4' = 21 + art,1 + 2raja, + qa? \

1? = Q + ra* + b<ilti, + j>af I (20).

C = G + ^a,1 + 2paa,J + ca,' )
It' tlu- pliiiies of reference pass through the principal foci,

21 = 0 and 6 = 0 (21).

Substituting in ^nations (5) and (7), and putting

THE THEORY OF OPTICAL INSTRUMENTS.

443

— H

W for the primary image ............ (24),

p7H -- p-,= U+ W for the secondary image ........... (25).

we find

The condition of distinctness is U=Q.

That the image of a flat object may be flat as well as distinct,

U=0 and TF=0 (26).

Form of the CJiaracteristic Function for a Spherical Refracting Surface.

Let the planes of reference pass through the p/

principal foci.

Let BC=s be the radius, then

AJ}_ /M T)A . W

.ii J.7 — j j^yj-2 ~~ •

Li, , ™~ r*i /* ~~ /^l

II n

If the co-ordinates of P, are alt bl} — -,

P7 P'z^

1 » ^2> ^2> ~ >

P „ a, 6, z,
and if PlP = rl and PPs = r2, then

F- /*!«•,

under the condition that F is stationary with respect to variations of a and
b, when

This gives the following values of a arid b :

6 = 6, + b, + XA + XA + &c.,

when

X, =

X, =

«* /*!

a,*

56—2

444 THE THEORY OF OPTICAL INSTRUMENTS.

whence we get, as the value of V to terms of the fourth order,

' + M.') («.«. + &&) (a,1 + 6,')
')(«.' + M(«.f + ^)

fl(«tf+V)(«A+W

f n
• M

'e the coefficients in the general equations (18), (19), (2<>), (±2), (-2:]) are
as follows :

a=o, s=*^£, 6=0,

[From the Encyclopaedia Britannica.]

LXXIII. Atom.

ATOM (aro/Ltos) is a body which cannot be cut in two. The atomic theory
is a theory of the constitution of bodies, which asserts that they are made up
of atoms. The opposite theory is that of the homogeneity and continuity of
bodies, and asserts, at least in the case of bodies having no apparent organisa-
tion, such, for instance, as water, that as we can divide a drop of water into
two parts which are each of them drops of water, so we have reason to believe
that these smaller drops can be divided again, and the theory goes on to assert
that there is nothing in the nature of things to hinder this process of division
from being repeated over and over again, times without end. This is the
doctrine of the infinite divisibility of bodies, and it is in direct contradiction
with the theory of atoms.

The atomists assert that after a certain number of such divisions the parts
would be no longer divisible, because each of them would be an atom. The
advocates of the continuity of matter assert that the smallest conceivable body
has parts, and that whatever has parts may be divided.

In ancient times Democritus was the founder of the atomic theory, while
Anaxagoras propounded that of continuity, under the name of the doctrine of
homo3omeria ('Oftoio//,e)oia), or of the similarity of the parts of a body to the
whole. The arguments of the atomists, and their replies to the objections of
Anaxagoras, are to be found in Lucretius.

In modern times the study of nature has brought to light many properties
of bodies which appear to depend on the magnitude and motions of their
ultimate constituents, and the question of the existence of atoms has once more
become conspicuous among scientific inquiries.

We shall begin by stating the opposing doctrines of atoms and of con-
tinuity before giving an outline of the state of molecular science as it now
exists. In the earliest times the most ancient philosophers whose speculations

446 ATOM.

•i* known to us seem to have discussed the ideas of number and of continuous

magnitude, of space and time, of matter and motion, with a native power of

thought which haa probably never been surpassed. Their actual knowledge,

however, and their scientific experience were necessarily limited, because in tin ii

dar* the records of human thought were only beginning to accumulate. It is

probable that the first exact notions of quantity were founded on the considera-

n of number. It is by the help of numbers that concrete quantities are

practically measured and calculated. Now, number is discontinuous. We pass

from one numU'r to the next per WM//I. The magnitudes, on the other hand,

which we meet with in geometry, are essentially continuous. The attempt to

apply numerical methods to the comparison of geometrical quantities led to the

doctrine of incommensurables, and to that of the infinite divisibility of space.

Mtitnwhilr, the same considerations had not been applied to time, so that in

the days of Zeno of Elea time was still regarded as made up of a finite number

Miim-nts." while space was confessed to be divisible without limit. This

was the state of opinion when the celebrated arguments against the possibility

..f in<>ti»n. of which that of Achilles and the tortoise is a specimen, were

propounded by Zeno, and such, apparently, continued to be the state of opinion

till Aristotle pointed out that time is divisible without limit, in precisely the

aame sense that space is. And the slowness of the development of scientific

ideas may be estimated from the fact that Bayle does not see any force in

this statement of Aristotle, but continues to admire the paradox of Zeno.

(Bayle's Dictionary, art. "Zeno".) Thus the direction of true scientific progress

was for many ages towards the recognition of the infinite divisibility of space

and time.

It was easy to attempt to apply similar arguments to matter. If matter
is extended and fills space, the same mental operation by which we recognise
tin- divisibility of space may be applied, in imagination at least, to the matter
which occupies space. From this point of view the atomic doctrine might be
regarded as a relic of the old numerical way of conceiving magnitude, and the
opposite doctrine of the infinite divisibility of matter might appear for a time
the most scientific. The atomists, on the other hand, asserted very strongly
th«- distinction between matter and space. The atoms, they said, do not fill
up the universe; there are void spaces between them. If it were not so,
Lucretius u-lls us, there could be no motion, for the atom which gives way
first must have some empty place to move into.

ATOM. 447

"Quapropter locus est intactus, inane, vacansque.
Quod si non esset, nulla rations moveri
Res possent; nainque, officium quod corporis exstat,
OfEcere atque obstare, id in omni tempore adesset
Omnibus : baud igitur quicquam prooedere posset,
Principium quoniam cedendi nulla daret res." De Rerun, Natura, I. 335.

The opposite school maintained then, as they have always done, that there
is no vacuum — that every part of space is full of matter, that there is a
universal plenum, and that all motion is like that of a fish in the water,
which yields in front of the fish because the fish leaves room for it behind.

"Cedere squamigeris latices nitentibus aiunt
Et liquidas aperire vias, quia post loca pisces
Linquant, quo possint cedentes confluere undse." I. 373.

In modern times Descartes held that, as it is of the essence of matter to
be extended in length, breadth, and thickness, so it is of the essence of
extension to be occupied by matter, for extension cannot be an extension of
nothing.

"Ac proinde si quaeratur quid fiet, si Deus auferat omne corpus quod in aliquo vase continetur,
et nullum aliud in ablati locum venire permittat? respondendum est, vasis latera sibi invicem hoc ipso
fore contigua. Cum enim inter duo corpora nihil interjacet, necesse est ut se mutuo tangant, at-
manifesto repugnat ut distent, sive ut inter ipsa sit distantia, et tamen ut ista distantia sit nihil ;
quia ouinis distautia est modus extensionis, et ideo sine substantia extensa esse non potest."

Principia, n. 18.

This identification of extension with substance runs through the whole of
Descartes's works, and it forms one of the ultimate foundations of the system
of Spinoza. Descartes, consistently with this doctrine, denied the existence of
atoms as parts of matter, which by their own nature are indivisible. He seems
to admit, however, that the Deity might make certain particles of matter
indivisible in this sense, that no creature should be able to divide them. These
particles, however, would be still divisible by their own nature, because the
Deity cannot diminish his own power, and therefore must retain his power of
dividing them. Leibnitz, on the other hand, regarded his monad as the ultimate
element of everything.

There are thus two modes of thinking about the constitution of bodies,
which have had their adherents both in ancient and in modern times. They
correspond to the two methods of regarding quantity — the arithmetical and the
geometrical. To the atomist the true method of estimating the quantity of

•.

•>:• M.

matter in a body is to count the atoms in it The void spaces between the
mom* count for nothing. To those who identify matter with extension, tl,«-
volume of space occupied by a body is the only measure of the quantity of

mutter in it

Of the different forms of the atomic theory, that of Boscovich may be tak.-n
M an example of the purest raonadism. According to Boscovich matter is made
up of atoms. Each atom is an indivisible point, having position in space, capal.U-
of motion in a continuous path, and possessing a certain mass, whereby a certain
Amount of force is required to produce a given change of motion. Besides this
the atom fa endowed with potential force, that is to say, that any two atoms
attract or repel each other with a force depending on their distance apart. The
law of this force, for all distances greater than say the thousandth of an inch,
• an attraction varying as the inverse square of the distance. For smaller
distances the force is an attraction for one distance and a repulsion for another,
according to some law not yet discovered. Boscovich himself, in order to obviate
the possibility of two atoms ever being in the same place, asserts that the
ultimate force is a repulsion which increases without limit as the distance
diminishes without limit, so that two atoms can never coincide. But this
tflflffv an unwarrantable concession to the vulgar opinion that two bodies cannot
co-exist in the same place. This opinion is deduced from our experience of the
behaviour of bodies of sensible size, but we have no experimental evidence that
two atoms may not sometimes coincide. For instance, if oxygen and hydrogen
combine to form water, we have no experimental evidence that the molecule of
oxygen is not in the very same place with the two molecules of hydrogen. Many
persons1 cannot get rid of the opinion that all matter is extended in length,
breadth, and depth. This is a prejudice of the same kind with the last, arising
from our experience of bodies consisting of immense multitudes of atoms. The
system of atoms, according to Boscovich, occupies a certain region of space in
virtue of the forces acting between the component atoms of the system and
any other atoms when brought near them. No other system of atoms can
occupy the same region of space at the same time, because, before it could do
so, the mutual action of the atoms would have caused a repulsion between the
two systems insuperable by any force which we can command. Thus, a number
of soldiers with firearms may occupy an extensive region to the exclusion of
the enemy's armies, though the space filled by their bodies is but small. In
this way Boscovich explained the apparent extension of bodies consisting of

ATOM. 449

atoms, each of which is devoid of extension. According to Boscovich's theory,
all action between bodies is action at a distance. There is no such thing in
nature as actual contact between two bodies. When two bodies are said in
ordinary language to be in contact, all that is meant is that they are so near
together that the repulsion between the nearest pairs of atoms belonging to
the two bodies is very great.

Thus, in Boscovich's theory, the atom has continuity of existence in time
and space. At any instant of time it is at some point of space, and it is never
in more than one place at a time. It passes from one place to another along
a continuous path. It has a definite mass which cannot be increased or di-
minished. Atoms are endowed with the power of acting on one another by
attraction or repulsion, the amount of the force depending on the distance
between them. On the other hand, the atom itself has no parts or dimensions.
In its geometrical aspect it is a mere geometrical point. It has no extension
in space. It has not the so-called property of Impenetrability, for two atoms
may exist in the same place. This we may regard as one extreme of the
various opinions about the constitution of bodies.

The opposite extreme, that of Anaxagoras — the theory that bodies apparently
homogeneous and continuous are so in reality — is, in its extreme form, a theory
incapable of development. To explain the properties of any substance by this
theory is impossible. We can only admit the observed properties of such sub-
stance as ultimate facts. There is a certain stage, however, of scientific progress
in which a method corresponding to this theory is of service. In hydrostatics,
for instance, we define a fluid by means of one of its known properties, and
from this definition we make the system of deductions which constitutes the
science of hydrostatics. In this way the science of hydrostatics may be built
upon an experimental basis, without any consideration of the constitution of a
fluid as to whether it is molecular or continuous. In like manner, after the
French mathematicians had attempted, with more or less ingenuity, to construct
a theory of elastic solids from the hypothesis that they consist of atoms in
equilibrium under the action of their mutual forces, Stokes and others shewed
that all the results of this hypothesis, so far at least as they agreed with
facts, might be deduced from the postulate that elastic bodies exist, and from
the hypothesis that the smallest portions into which we can divide them are
sensibly homogeneous. In this way the principle of continuity, which is the
basis of the method of Fluxions and the whole of modern mathematics, may

VOL. II. 5?

ATOJL

be applied to the analyii* of problema connected with material bodies by
•Morning tJm*t for the purpose of this analysis, to be homogeneous. All that
m required to make the results applicable to the real case is that the smallest
portion* of the substance of which we take any notice shall be sensibly of the
^yp^ ijjj,^ Thua, if a railway contractor has to make a tunnel through a hill
of gravel, and if one cubic yard of the gravel is so like another cubic yard
that for the purposes of the contract they may be taken as equivalent, then,
in estimating the work required to remove the gravel from the tunnel, he may,
without fear of error, make his calculations as if the gravel were a continuous
But if a worm has to make his way through the gravel, it makes

the greatest possible difference to him whether he tries to push right against
a piece of gravel, or directs his course through one of the intervals between
the pieces; to him, therefore, the gravel is by no means a homogeneous and
continuous substance.

In the same way, a theory that some particular substance, say water, is
homogeneous and continuous may be a good working theory up to a certain
point, but may fail when we come to deal with quantities so minute or so
attenuated that their heterogeneity of structure comes into prominence. Whether
this heterogeneity of structure is or is not consistent with homogeneity and
continuity of substance is another question.

The extreme form of the doctrine of continuity is that stated by Descartes,
who maintains that the whole universe is equally full of matter, and that this
matter is all of one kind, having no essential property besides that of extension.
All the properties which we perceive in matter he reduces to its parts being
movable among one another, and so capable of all the varieties which we can
perceive to follow from the motion of its parts (Principia, IL 23). Descartes's
own attempts to deduce the different qualities and actions of bodies in this way
are not of much value. More than a century was required to invent methods
of investigating the conditions of the motion of systems of bodies such as
Descartes imagined. But the hydrodynamical discovery of Helmholtz that a
vortex in a perfect liquid possesses certain permanent characteristics, has been
applied by Sir W. Thomson to form a theory of vortex atoms in a homogeneous,
incompressible, and frictionless liquid, to which we shall return at the proper
time.

ATOM. 451

Outline of Modern Molecular Science, and in particular of the Molecular Theory

of Gases.

We begin by assuming that bodies are made up of parts, each of which
is capable of motion, and that these parts act on each other in a manner
consistent with the principle of the conservation of energy. In making these
assumptions, we are justified by the facts that bodies may be divided into
smaller parts, and that all bodies with which we are acquainted are conservative
systems, which would not be the case unless their parts were also conservative
systems.

We may also assume that these small parts are in motion. This is the
most general assumption we can make, for it includes, as a particular case, the
theory that the small parts are at rest. The phenomena of the diffusion of
gases and liquids through each other shew that there may be a motion of the
small parts of a body which is not perceptible to us.

We make no assumption with respect to the nature of the small parts —
whether they are all of one magnitude. We do not even assume them to have
extension and figure. Each of them must be measured by its mass, and any
two of them must, like visible bodies, have the power of acting on one another
when they come near enough to do so. The properties of the body, or medium,
are determined by the configuration and motion of its small parts.

The first step in the investigation is to determine the amount of motion
which exists among the small parts, independent of the visible motion of the
medium as a whole. For this purpose it is convenient to make use of a
general theorem in dynamics due to Clausius.

When the motion of a material system is such that the time average of
the quantity 2 (waf) remains constant, the state of the system is said to be
that of stationary motion. When the motion of a material system is such that
the sum of the moments of inertia of the system, about three axes at right
angles through its centre of mass, never varies by more than small quantities
from a constant value, the system is said to be in a state of stationary motion.

The kinetic energy of a particle is half the product of its mass into the
square of its velocity, and the kinetic energy of a system is the sum of the
kinetic energy of all its parts.

57—2

. ATOM.

When an attraction or repulsion exists between two points, half the
product of this atwei into the distance between the two points is called the
onu/ of the ihMB, and is reckoned positive when the stress is an attraction,
and negative when it is a repulsion. The virial of a system is the sum of
the ririala of the stresses which exist in it. If the system is subjected to
the external stress of the pressure of the sides of a vessel in which it is
contained, this stress will introduce an amount of virial $pV, where p is the
imniiirn on unit of area and V is the volume of the vessel.

The theorem of Clausius may now be stated as follows : — In a material
ayvtem in ft state of stationary motion the time-average of the kinetic energy
it equal to the time-average of the virial. In the case of a fluid enclosed in a

where the first term denotes the kinetic energy, and is half the sum of
the product of each mass into the mean square of its velocity. In the
second term, p is the pressure on unit of surface of the vessel, whose volume
is F, and the third term expresses the virial due to the internal actions
between the parts of the system. A double symbol of summation is used,
because every pair of parts between which any action exists must be taken
into account. We have next to shew that in gases the principal part of the
pressure arises from the motion of the small parts of the medium, and not
from a repulsion between them.

In the first place, if the pressure of a gas arises from the repulsion of
its parts, the law of repulsion must be inversely as the distance. For, con-
sider a cube filled with the gas at pressure p, and let the cube expand till
each side is n times its former length. The pressure on unit of surface ac-
cording to Boyle's law is now ^ , and since the area of a face of the cube

is «' times what it was, the whole pressure on the face of the cube is - of

n

it* original value. But since everything has been expanded symmetrically, the
distance between corresponding parts of the air is now n times what it was,
and the force is n times less than it was. Hence the force must vary inversely
as the distance.

But Newton has shewn (Prindpia, Book i. Prop. 93) that this law is
inadmissible, as it makes the effect of the distant parts of the medium on a

ATOM. 453

particle greater than that of the neighbouring parts. Indeed, we should arrive
at the conclusion that the pressure depends not only on the density of the air
but on the form and dimensions of the vessel which contains it, which we
know not to be the case.

If, on the other hand, we suppose the pressure to arise entirely from the
motion of the molecules of the gas, the interpretation of Boyle's law becomes
very simple. For, in this case

The first term is the product of the pressure and the volume, which ac-
cording to Boyle's law is constant for the same quantity of gas at the same
temperature. The second term is two-thirds of the kinetic energy of the system,
and we have every reason to believe that in gases when the temperature is
constant the kinetic energy of unit of mass is also constant. If we admit
that the kinetic energy of unit of mass is in a given gas proportional to the
absolute temperature, this equation is the expression of the law of Charles as
well as of that of Boyle, and may be written —

pV=Rd,

where 0 is the temperature reckoned from absolute zero, and R is a constant.
The fact that this equation expresses with considerable accuracy the relation
between the volume, pressure, and temperature of a gas when in an extremely
rarified state, and that as the gas is more and more compressed the deviation
from this equation becomes more apparent, shews that the pressure of a gas is
due almost entirely to the motion of its molecules when the gas is rare, and
that it is only when the density of the gas is considerably increased that the
effect of direct action between the molecules becomes apparent.

The effect of the direct action of the molecules on each other depends on
the number of pairs of molecules which at a given instant are near enough to
act on one another. The number of such pairs is proportional to the square
of the number of molecules in unit of volume, that is, to the square of the
density of the gas. Hence, as long as the medium is so rare that the encounter
between two molecules is not affected by the presence of others, the deviation
from Boyle's law will be proportional to the square of the density. If the
action between the molecules is on the whole repulsive, the pressure will be
greater than that given by Boyle's law. If it is, on the whole, attractive, the
pressure will be less than that given by Boyle's law. It appears, by the ex-

454 ATOM.

perimente of Rtgnault and others, that the pressure does deviate from Boyle's
kw when the density of the gas is increased. In the case of carbonic acid
and other gases which are easily liquefied, this deviation is very great. In ail
case*, howerer, except that of hydrogen, the pressure is less than that given
by Boyle's law, shewing that the virial is on the whole due to attractive forces
between the molecules.

Another kind of evidence as to the nature of the action between the
molecules is furnished by an experiment made by Dr Joule. Of two vessels,
one was exhausted and the other filled with a gas at a pressure of 20 atmo-
spheres; and both were placed side by side in a vessel of water, which was
constantly stirred. The temperature of the whole was observed. Then a com-
munication was opened between the vessels, the compressed gas expanded to
twice its volume, and the work of expansion, which at first produced a strong
current in the gas, was soon converted into heat by the internal friction of
the gas. When all was again at rest, and the temperature uniform, the
temperature was again observed. In Dr Joule's original experiments the ob-
served temperature was the same as before. In a series of experiments, con-
ducted by Dr Joule and Sir W. Thomson on a different plan, by which the
thermal effect of free expansion can be more accurately measured, a slight
cooling effect was observed in all the gases examined except hydrogen. Since
the temperature depends on the velocity of agitation of the molecules, it
appears that when a gas expands without doing external work the velocity of
agitation is not much affected, but that in most cases it is slightly diminished.
Now, if the molecules during their mutual separation act on each other, their
velocity will increase or diminish according as the force is repulsive or attractive.
It appears, therefore, from the experiments on the free expansion of gases,
that the force between the molecules is small but, on the whole, attractive.

Having thus justified the hypothesis that a gas consists of molecules in
motion, which act on each other only when they come very close together
during an encounter, but which, during the intervals between their encounters
which constitute the greater part of their existence, are describing free paths,
and are not acted on by any molecular force, we proceed to investigate the
motion of such a system.

The mathematical investigation of the properties of such a system of
molecules in motion is the foundation of molecular science. Clausius was the
first to express the relation between the density of the gas, the length of

ATOM. 455

the free paths of its molecules, and the distance at which they encounter
each other. He assumed, however, at least in his earlier investigations, that
the velocities of all the molecules are equal. The mode in which the veloci-
ties are distributed was first investigated by the present writer, who shewed
that in the moving system the velocities of the molecules range from zero to
infinity, but that the number of molecules whose velocities lie within given
limits can be expressed by a formula identical with that which expresses in the
theory of errors the number of errors of observation lying within corresponding
limits. The proof of this theorem has been carefully investigated by Boltz-
mann1, who has strengthened it where it appeared weak, and to whom the method
of taking into account the action of external forces is entirely due.

The mean kinetic energy of a molecule, however, has a definite value, which
is easily expressed in terms of the quantities which enter into the expression
for the distribution of velocities. The most important result of this investi-
gation is that when several kinds of molecules are in motion and acting on
one another, the mean kinetic energy of a molecule is the same whatever be
its mass, the molecules of greater mass having smaller mean velocities. Now,
when gases are mixed their temperatures become equal. Hence we conclude
that the physical condition which determines that the temperature of two
gases shall be the same is that the mean kinetic energies of agitation of the
individual molecules of the two gases are equal. This result is of great im-
portance in the theory of heat, though we are not yet able to establish any
similar result for bodies in the liquid or solid state.

In the next place, we know that in the case in which the whole pressure
of the medium is due to the motion of its molecules, the pressure on unit
of area is numerically equal to two-thirds of the kinetic energy in unit of
volume. Hence, if equal volumes of two gases are at equal pressures the
kinetic energy is the same in each. If they are also at equal temperatures
the mean kinetic energy of each molecule is the same in each. If, therefore,
equal volumes of two gases are at equal temperatures and pressures, the
number of molecules in each is the same, and therefore, the masses of the two
kinds of molecules are in the same ratio as the densities of the gases to
which they belong.

This statement has been believed by chemists since the time of Gay-
Lussac, who first established that the weights of the chemical equivalents of
* Sitzungsberichte der K. K. Akad., Wien, 8th Oct. 1868.

; ATOM.

different substance* are proportional to the densities of these substances when in
the form of gas. The definition of the word molecule, however, as employed
in the statement of Gay-Luasac's law is by no means identical with the defi-
nition of the same word as in the kinetic theory of gases. The chemists
••certain by experiment the ratios of the masses of the different substances in
a compound. From these they deduce the chemical equivalents of the different
substances, that of a particular substance, say hydrogen, being taken as unity.
The only evidence made use of is that furnished by chemical combinations. It
is also assumed, in order to account for the facts of combination, that the reason
why substances combine in definite ratios is that the molecules of the substances
are in the ratio of their chemical equivalents, and that what we call combination
is an action which takes place by a union of a molecule of one substance to
a molecule of the other.

Tim kind of reasoning, when presented in a proper form and sustained by
proper evidence, has a high degree of ^cogency. But it is purely chemical
reasoning; it is not dynamical reasoning. It is founded on chemical experience,
not on the laws of motion.

Our definition of a molecule is purely dynamical. A molecule is that
minute portion of a substance which moves about as a whole, so that its
parts, if it has any, do not part company during the motion of agitation of
the gas. The result of the kinetic theory, therefore, is to give us information
about the relative masses of molecules considered as moving bodies. The con-
sistency of this information with the deductions of chemists from the phenomena
of combination, greatly strengthens the evidence in favour of the actual existence
and motion of gaseous molecules.

Another confirmation of the theory of molecules is derived from the experi-
ments of Dulong and Petit on the specific heat of gases, from which they
deduced the law which bears their name, and which asserts that the specific
heats of equal weights of gases are inversely as their combining weights, or, in
other words, that the capacities for heat of the chemical equivalents of different
gases are equal. We have seen that the temperature is determined by the
kinetic energy of agitation of each molecule. The molecule has also a certain
amount of energy of internal motion, whether of rotation or of vibration, but
the hypothesis of Clausius, that the mean value of the internal energy always
bears a proportion fixed for each gas to the energy of agitation, seems highly
probable and consistent with experiment The whole kinetic energy is there-

ATOM. 457

fore equal to the energy of agitation multiplied by a certain factor. Thus the
energy communicated to a gas by heating it is divided in a certain proportion
between the energy of agitation and that of the internal motion of each
molecule. For a given rise of temperature the energy of agitation, say of a
million molecules, is increased by the same amount whatever be the gas. The
heat spent in raising the temperature is measured by the increase of the
whole kinetic energy. The thermal capacities, therefore, of equal numbers of
molecules of different gases are in the ratio of the factors by which the energy
of agitation must be multiplied to obtain the whole energy. As this factor
appears to be nearly the same for all gases of the same degree of atomicity,
Dulong and Petit's law is true for such gases.

Another result of this investigation is of considerable importance in rela-
tion to certain theories *, which assume the existence of aethers or rare media
consisting of molecules very much smaller than those of ordinary gases. Ac-
cording to our result, such a medium would be neither more nor less than a gas.
Supposing its molecules so small that they can penetrate between the molecules
of solid substances such as glass, a so-called vacuum would be full of this
rare gas at the observed temperature, and at the pressure, whatever it may
be, of the setherial medium in space. The specific heat, therefore, of the
medium in the so-called vacuum will be equal to that of the same volume of
any other gas at the same temperature and pressure. Now, the purpose for
which this molecular sether is assumed in these theories is to act on bodies by
its pressure, and for this purpose the pressure is generally assumed to be very
great. Hence, according to these theories, we should find the specific heat of
a so-called vacuum very considerable compared with that of a quantity of air
filling the same space.

We have now made a certain definite amount of progress towards a com-
plete molecular theory of gases. We know the mean velocity of the molecules
of each gas in metres per second, and we know the relative masses of the
molecules of different gases. We also know that the molecules of one and the
same gas are all equal in mass. For if they are not, the method of dialysis,
as employed by Graham, would enable us to separate the molecules of smaller
mass from those of greater, as they would stream through porous substances
with greater velocity. We should thus be able to separate a gas, say hydrogen,
into two portions, having different densities and other physical properties,
* See Gustav Hausemann, Die Atome und ihre Bewegungen. 1871. (H. G. Mayer.)

VOL. II. 58

. •

ATOM.

different combining weights, and probably different chemical properties of other
k Aa no chemist has yet obtained specimens of hydrogen differing in this
way faxn other •|HHff'M"-, we conclude that all the molecules of hydrogen
are of wnsibly the same mass, and not merely that their mean mass is a statis-
tical constant of great stability.

But as yet we have not considered the phenomena which enable us to
form an estimate of the actual mass and dimensions of a molecule. It is to
Clau*iu* that we owe the first definite conception of the free path of a molecule
and of the mean distance travelled by a molecule between successive encounters.
He shewed that the number of encounters of a molecule in a given time is
proportional to the velocity, to the number of molecules in unit of volume,
and to the square of the distance between the centres of two molecules when
they act on one another so as to have an encounter. From this it appears
that if we call this distance of the centres the diameter of a molecule, and
the volume of a sphere having this diameter the volume of a molecule, and
the sum of the volumes of all the molecules the molecular volume of the gas,
then the diameter of a molecule is a certain multiple of the quantity obtained
by diminishing the free path in the ratio of the molecular volume of the gas
t-i the whole volume of the gas. The numerical value of this multiple differs
slightly, according to the hypothesis we assume about the law of distribution
of velocities. It also depends on the definition of an encounter. When the
molecules are regarded as elastic spheres we know what is meant by an
encounter, but if they act on each other at a distance by attractive or
repulsive forces of finite magnitude, the distance of their centres varies during
an encounter, and is not a definite quantity. Nevertheless, the above state-
ment of Clausius enables us, if we know the length of the mean path and
the molecular volume of gas, to form a tolerably near estimate of the diameter
of the sphere of the intense action of a molecule, and thence of the number
• •f molecules in unit of volume and the actual mass of each molecule. To
complete the investigation we have, therefore, to determine the mean path and
the molecular volume. The first numerical estimate of the mean path of a
gaseous molecule was made by the present writer from data derived from the
internal friction of air. There are three phenomena which depend on the
length of the free path of the molecules of a gas. It is evident that the
greater the free path the more rapidly will the molecules travel from one
part of the medium to another, because their direction will not be so often

ATOM. 459

altered by encounters with other molecules. If the molecules in different parts
of the medium are of different kinds, their progress from one part of the
medium to another can be easily traced by analysing portions of the medium taken
from different places. The rate of diffusion thus found furnishes one method
of estimating the length of the free path of a molecule. This kind of diffusion
goes on not only between the molecules of different gases, but among the
molecules of the same gas, only in the latter case the results of the diffusion
cannot be traced by analysis. But the diffusing molecules carry with them
in their free paths the momentum and the energy which they happen at a
given instant to have. The diffusion of momentum tends to equalise the
apparent motion of different parts of the medium, and constitutes the pheno-
menon called the internal friction or viscosity of gases. The diffusion of energy
tends to equalise the temperature of different parts of the medium, and
constitutes the phenomenon of the conduction of heat in gases.

These three phenomena — the diffusion of matter, of motion, and of heat
in gases — have been experimentally investigated, — the diffusion of matter by
Graham and Loschmidt, the diffusion of motion by Oscar Meyer and Clerk
Maxwell, and that of heat by Stefan.

These three kinds of experiments give results which in the present im-
perfect state of the theory and the extreme difficulty of the experiments,
especially those on the conduction of heat, may be regarded as tolerably con-
sistent with each other. At the pressure of our atmosphere, and at the
temperature of melting ice, the mean path of a molecule of hydrogen is about
the 10,000th of a millimetre, or about the fifth part of a wave-length of
green light. The mean path of the molecules of other gases is shorter than
that of hydrogen.

The determination of the molecular volume of a gas is subject as yet to
considerable uncertainty. The most obvious method is that of compressing the
gas till it assumes the liquid form. It seems probable, from the great resist-
ance of liquids to compression, that their molecules are about the same
distance from each other as that at which two molecules of the same substance
in the gaseous form act on each other during an encounter. If this is the
case, the molecular volume of a gas is somewhat less than the volume of the
liquid into which it would be condensed by pressure, or, in other words, the
density of the molecules is somewhat greater than that of the liquid.

Now, we know the relative weights of different molecules with great

KQ O

«J U^^Ld

.

aeeoracY, and. from a knowledge of the mean path, we can calculate their
relative diameter* approximately. From these we can deduce the relative
deneittea of different kinds of molecules. The relative densities so calculated
bar* been flump***1 by Lorenz Meyer with the observed densities of the liquids
into which the gases may be condensed, and he finds a remarkable correspond-
ence between them. There is considerable doubt, however, as to the relation
between the molecules of a liquid and those of its vapour, so that till a
larger number of comparisons have been made, we must not place too much
reliance on the calculated densities of molecules. Another, and perhaps a more
refined, method is that adopted by M. Van der Waals, who deduces the
molecular volume from the deviations of the pressure from Boyle's law as the
gas is compressed.

The first numerical estimate of the diameter of a molecule was that made
by Losclunidt in 1865 from the mean path and the molecular volume. Inde-
pendently of him and of each other, Mr Stoney, in 1868, and Sir W. Thomson,
in 1870, published results of a similar kind — those of Thomson being deduced
not onlv in this way, but from considerations derived from the thickness of
soap bubbles, and from the electric action between zinc and copper.

The diameter and the mass of a molecule, as estimated by these methods,
are, of course, very small, but by no means infinitely so. About two millions
of molecules of hydrogen in a row would occupy a millimetre, and about two
hundred million million million of them would weigh a milligramme. These
numbers must be considered as exceedingly rough guesses ; they must be
corrected by more extensive and accurate experiments as science advances ;
but the, main result, which appears to be well established, is that the deter-
mination of the mass of a molecule is a legitimate object of scientific research,
and that this mass is by no means immeasurably small.

Loschmidt illustrates these molecular measurements by a comparison with
the smallest magnitudes visible by means of a microscope. Nobert, he tells us,
can draw 4000 lines in the breadth of a millimetre. The intervals between
these lines can be observed with a good microscope. A cube, whose side is
the 4000th of a millimetre, may be taken as the minimum visibile for observers
of the present day. Such a cube would contain from GO to 100 million
molecules of oxygen or of nitrogen; but since the molecules of organised
substances contain on an average about 50 of the more elementary atoms, we
may assume that the smallest organised particle visible under the microscope

ATOM. 461

contains about two million molecules of organic matter. At least half of every
living organism consists of water, so that the smallest living being visible
under the microscope does not contain more than about a million organic
molecules. Some exceedingly simple organism may be supposed built up of
not more than a million similar molecules. It is impossible, however, to conceive
so small a number sufficient to form a being furnished with a whole system
of specialised organs.

Thus molecular science sets us face to face with physiological theories.
It forbids the physiologist from imagining that structural details of infinitely
small dimensions can furnish an explanation of the infinite variety which exists
in the properties and functions of the most minute organisms.

A microscopic germ is, we know, capable of development into a highly
organised animal. Another germ, equally microscopic, becomes, when developed,
an animal of a totally different kind. Do all the differences, infinite in number,
which distinguish the one animal from the other, arise each from some dif-
ference in the structure of the respective germs ? Even if we admit this as
possible, we shall be called upon by the advocates of Pangenesis to admit
still greater marvels. For the microscopic germ, according to this theory, is
no mere individual, but a representative body, containing members collected
from every rank of the long-drawn ramification of the ancestral tree, the number
of these members being amply sufficient not only to furnish the hereditary-
characteristics of every organ of the body and every habit of the animal from
birth to death, but also to afford a stock of latent gemmules to be passed
on in an inactive state from germ to germ, till at last the ancestral peculiarity
which it represents is revived in some remote descendant.

Some of the exponents of this theory of heredity have attempted to elude
the difficulty of placing a whole world of wonders within a body so small
and so devoid of visible structure as a germ, by using the phrase structureless
germs*. Now, one material system can differ from another only in the con-
figuration and motion which it has at a given instant. To explain differences
of function and development of a germ without assuming differences of structure
is, therefore, to admit that the properties of a germ are not those of a purely

material system.

The evidence as to the nature and motion of molecules, with which we
have hitherto been occupied, has been derived from experiments upon gaseous
* See F. Galton, "On Blood Relationship," Proc. Roy. Soc., June 13, 1872.

ATOM.

media, the amalleflt aensible portion of which contains millions of millions of
The constancy and uniformity of the properties of the gaseous
medium * the direct result of the inconceivable irregularity of the motion of
agitation of it« molecule*. Any cause which could introduce regularity into the
motion of agitation, and marshal the molecules into order and method in tin ir
vrolutiuna, might olut-k or even reverse that tendency to diffusion of matter,
motion, and energy, which is one of the most invariable phenomena of nature,
nnd to v hich Thomson has given the name of the dissipation of energy.

Thus, when a sound-wave is passing through a mass of air, this motion
is of a certain definite type, and if left to itself the whole motion is passed on
to other masses of air, and the sound-wave passes on, leaving the air behind
it at rest Heat, on the other hand, never passes out of a hot body except
to enter a colder body, so that the energy of sound-waves, or any other form
of energy which is propagated so as to pass wholly out of one portion of
the medium and into another, cannot be called heat.

\V-- have now to turn our attention to a class of molecular motions, which
are as remarkable for their regularity as the motion of agitation is for its
irregularity.

It has been found, by means of the spectroscope, that the light emitted
l)\- incandescent substances is different according to their state of condensation.
When they are in an extremely rarefied condition the spectrum of their light
consists of a set of sharply-defined bright lines. As the substance approaches
a denser condition the spectrum tends to become continuous, either by the
lines becoming broader and less defined, or by new lines and bands appearing
between them, till the spectrum at length loses all its characteristics and
becomes identical with that of solid bodies when raised to the same tem-
perature.

I Luce the vibrating systems, which are the source of the emitted light,
must be vibrating in a different manner in these two cases. When the spectrum
consists of a number of bright lines, the motion of the system must be com-
pounded of a corresponding number of types of harmonic vibration.

In order that a bright line may be sharply defined, the vibratory motion
which produces it must be kept up in a perfectly regular manner for some
hundreds or thousands of vibrations. If the motion of each of the vibrating
bodies is kept up only during a small number of vibrations, then, however
regular may be the vibrations of each body while it lasts, the resultant dis-

ATOM. 463

turbance of the luminiferous medium, when analysed by the prism, will be
found to contain, besides the part due to the regular vibrations, other motions,
depending on the starting and stopping of each particular vibrating body,
which will become manifest as a diffused luminosity scattered over the whole
length of the spectrum. A spectrum of bright lines, therefore, indicates that
the vibrating bodies when set in motion are allowed to vibrate in accordance
with the conditions of their internal structure for some time before they are
again interfered with by external forces.

It appears, therefore, from spectroscopic evidence that each molecule of a
rarefied gas is, during the greater part of its existence, at such a distance
from all other molecules that it executes its vibrations in an undisturbed and
regular manner. This is the same conclusion to which we were led by con-
siderations of another kind at p. 452.

We may therefore regard the bright lines in the spectrum of a gas as
the result of the vibrations executed by the molecules while describing their
free paths. When two molecules separate from one another after an encounter,
each of them is in a state of vibration, arising from the unequal action on
different parts of the same molecule during the encounter. Hence, though the
centre of mass of the molecule describing its free path moves with uniform
velocity, the parts of the molecule have a vibratory motion with respect to
the centre of mass of the whole molecule, and it is the disturbance of the
luminiferous medium communicated to it by the vibrating molecules which
constitutes the emitted light.

We may compare the vibrating molecule to a bell. When struck, the
bell is set in motion. This motion is compounded of harmonic vibrations of
many different periods, each of which acts on the air, producing notes of as
many different pitches. As the bell communicates its motion to the air, these
vibrations necessarily decay, some of them faster than others, so that the
sound contains fewer and fewer notes, till at last it is reduced to the funda-
mental note of the bell*. If we suppose that there are a great many bells
precisely similar to each other, and that they are struck, first one and then
another, in a perfectly irregular manner, yet so that, on an average, as many
bells are struck in one second of time as in another, and also in such a way

* Part of the energy of motion is, in the case of the bell, dissipated in the substance of the bell
in virtue of the viscosity of the metal, and assumes the form of heat, but it is not necessary, for the
purpose of illustration, to take this cause of the decay of vibrations into account.

ATOM.

on ,0 Average, any one bell is not again struck till it has ceased to
vibrmtr, then the audible result will appear a continuous sound, composed of
UM »ouud emitted by bells in all states of vibration, from the clang of the
stroke to the final hum of the dying fundamental tone.

Hut now let the number of bells be reduced while the same number of
stroke* are given in a second. Each bell will now be struck before it has
oeawd to vibrate, so that in the resulting sound there will be less of the
fundamental tone and more of the original clang, till at last, when the peal is
reduced to one bell, on which innumerable hammers are continually plying
their strokes all out of time, the sound will become a mere noise, in which
no inimical note can be distinguished.

In the case of a gas we have an immense number of molecules, each of
which is set in vibration when it encounters another molecule, and continues
to vibrate as it describes its free path. The molecule is a material system,
the parts of which are connected in some definite way, and from the fact that
the bright lines of the emitted light have always the same wave-lengths, we
learn that the vibrations corresponding to these lines are always executed in
the same periodic time, and therefore the force tending to restore any part of
the molecule to its position of equilibrium in the molecule must be propor-
tional to its displacement relative to that position.

From the mathematical theory of the motion of such a system, it appears
tliat the whole motion may be analysed into the following parts, which may
be considered each independently of the others : — In the first place, the centre
of mass of the system moves with uniform velocity in a straight line. This
velocity may have any value. In the second place, there may be a motion of
rotation, the angular momentum of the system about its centre of mass re-
maining during the free path constant in magnitude and direction. This angular
momentum may have any value whatever, and its axis may have any direction.
In the third place, the remainder of the motion is made up of a number of
component motions, each of which is an harmonic vibration of a given type.
In each type of vibration the periodic time of vibration is determined by the
nature of the system, and is invariable for the same system. The relative
amount of motion in different parts of the system is also determinate for each
type, but the absolute amount of motion and the phase of the vibration of
each type are determined by the particular circumstances of the last encounter,
and may vary in any manner from one encounter to another.

ATOM. 465

The values of the periodic times of the different types of vibration are
given by the roots of a certain equation, the form of which depends on the
nature of the connections of the system. In certain exceptionally simple cases,
as, for instance, in that of a uniform string stretched between two fixed points,
the roots of the equation are connected by simple arithmetical relations, and
if the internal structure of a molecule had an analogous kind of simplicity, we
might expect to find in the spectrum of the molecule a series of bright lines,
whose wave-lengths are in simple arithmetical ratios.

But if we suppose the molecule to be constituted according to some dif-
ferent type, as, for instance, if it is an elastic sphere, or if it consists of a
finite number of atoms kept in their places by attractive and repulsive forces,
the roots of the equation will not be connected with each other by any simple
relations, but each may be made to vary independently of the others by a
suitable change of the connections of the system. Hence, we have no right to
expect any definite numerical relations among the wave-lengths of the bright
lines of a gas.

The bright lines of the spectrum of an incandescent gas are therefore due
to the harmonic vibrations of the molecules of the gas during their free paths.
The only effect of the motion of the centre of mass of the molecule is to
alter the time of vibration of the light as received by a stationary observer.
When the molecule is coming towards the observer, each successive impulse
will have a shorter distance to travel before it reaches his eye, and therefore
the impulses will appear to succeed each other more rapidly than if the mole-
cule were at rest, and the contrary will be the case if the molecule is receding
from the observer. The bright line corresponding to the vibration will there-
fore be shifted in the spectrum towards the blue end when the molecule is
approaching, and towards the red end when it is receding from the observer.
By observations of the displacement of certain lines in the spectrum, Dr Huggins
and others have measured the rate of approach or of recession of certain stars
with respect to the earth, and Mr Lockyer has determined the rate of motion
of tornadoes in the sun. But Lord Rayleigh has pointed out that according
to the dynamical theory of gases the molecules are moving hither and thither
with so great velocity that, however narrow and sharply-defined any bright line
due to a single molecule may be, the displacement of the line towards the
blue by the approaching molecules, and towards the red by the receding mole-
cules, will produce a certain amount of widening and blurring of the line in

VOL. IL 59

. I

AT.'M.

the •pectrom, »o that there u ft limit to the sharpness of definition of the
Ifrrcg Q[ a -^ jhe widening of the lines due to this cause will be in pro-
portion to the Telocity of agitation of the molecules. It will be greatest for
the molecules of smallest mass, as those of hydrogen, and it will increase with
the temperature. Hence the measurement of the breadth of the hydrogen lines,
such M C or /' in tin- sj>ectrum of the solar prominences, may furnish evidence
that the temperature of the sun cannot exceed a certain value.

On the Tfteory of Vortex Atoms.

The equations which form the foundations of the mathematical theory of
fluid motion were fully laid down by Lagrange and the great mathematicians
of the end of last century, but the number of solutions of cases of fluid
motion which had been actually worked out remained very small, and almost
all of these belonged to a particular type of fluid motion, which has been
since named the irrotational type. It had been shewn, indeed, by Lagrange,
that a perfect fluid, if its motion is at any time irrotational, will continue in
all time coming to move in an irrotational manner, so that, by assuming that
the fluid was at one time at rest, the calculation of its subsequent motion may
be very much simplified.

It was reserved for Helmholtz to point out the very remarkable properties
of rotational motion in a homogeneous incompressible fluid devoid of all viscosity.
We must first define the physical properties of such a fluid. In the first place,
it is a material substance. Its motion is continuous in space and time, and
if we follow any portion of it as it moves, the mass of that portion remains
invariable. These properties it shares with all material substances. In the next
place, it is incompressible. The form of a given portion of the fluid may change,
but its volume remains invariable; in other words, the density of the fluid
remains the same during its motion. Besides this, the fluid is homogeneous, or
the density of all parts of the fluid is the same. It is also continuous, so
that the mass of the fluid contained within any closed surface is always r.r</<7/y
proportional to the volume contained within that surface. This is equivalent
to asserting that the fluid is not made up of molecules ; for, if it were, the
mass would vary in a discontinuous manner as the volume increases continuously.
because first one and then another molecule would be included within the closed

ATOM. 467

surface. Lastly, it is a perfect fluid, or, in other words, the stress between
one portion and a contiguous portion is always normal to the surface which
separates these portions, and this whether the fluid is at rest or in motion.

We have seen that in a molecular fluid the interdiffusion of the molecules
causes an interdiffusion of motion of different parts of the fluid, so that
the action between contiguous parts is no longer normal but in a direction
tending to diminish their relative motion. Hence the perfect fluid cannot be
molecular.

All that is necessary in order to form a correct mathematical theory of
a material system is that its properties shall be clearly defined and shall be
consistent with each other. This is essential ; but whether a substance having
such properties actually exists is a question which comes to be considered
only when we propose to make some practical application of the results of the
mathematical theory. The properties of our perfect liquid are clearly defined
and consistent with each other, and from the mathematical theory we can
deduce remarkable results, some of which may be illustrated in a rough way
by means of fluids which are by no means perfect in the sense of not being
viscous, such, for instance, as air and water.

The motion of a fluid is said to be irrotational when it is such that if a
spherical portion of the fluid were suddenly solidified, the solid sphere so formed
would not be rotating about any axis. When the motion of the fluid is rota-
tional the axis and angular velocity of the rotation of any small part of the
fluid are those of a small spherical portion suddenly solidified.

The mathematical expression of these definitions is as follows :— Let u, v, w
be the components of the velocity of the fluid at the point (x, y, z), and let

dv dw ,._dw du _du _dv^ , .

a = dz~Jy' P = ~dx~dz' 7~d^~dx-

then a, £, y are the components of the velocity of rotation of the fluid at the
point (x, y, z). The axis of rotation is in the direction of the resultant of
a, ft, and y, and the velocity of rotation, (a, is measured by this resultant.
A line drawn in the fluid, so that at every point of the line

__^=
a ds ~ ft ds ~ y ds <a

where s is the length of the line up to the point x, y, z, is called a vortex

59—2

line. IU direction coincides at every point with that of the axis of rotation
of the fluid.

We may now prove the theorem of Helmholtz, that the points of the
fluid which at any instant lie in the same vortex line continue to lie in the
^m vortex line during the whole motion of the fluid.

The equations of motion of a fluid are of the form

dV

when p is the density, which in the case of our homogeneous incompressible

g
fluid we may assume to be unity, the operator *- represents the rate of varia-

tion of the symbol to which it is prefixed at a point which is carried forward
with the fluid, so that

Sw du du du , du . .

•K7=j7 + u-T--fv-7- + M> -j- ......................... (4),

8t at ax ay dz

p is the pressure, and V is the potential of external forces. There are two
other equations of similar form in y and z. Differentiating the equation in //
with respect to z, and that in z with respect to y, and subtracting the second
from the first, we find

d 8v d 810 .

Performing the differentiations and remembering equations (1) and also the
condition of incompressibility,

du dv dw

we find

Now, let us suppose a vortex line drawn in the fluid so as always to
begin at the same particle of the fluid. The components of the velocity of this
point are «, v, u>. Let us find those of a point on the moving vortex line at
a distance ds from this point where

ds = otda- .................................... (8).

ATOM. 469

The co-ordinates of this point are

x + adcr, y + ftdv, z + ydar ........................... (9),

and the components of its velocity are

Consider the first of these components. In virtue of equation (7) we may
write it

du 7 . du _ 7 du j

du dx , du dy ^ du dz , /ir.v

or u + -j- -j- da-+ -j- -f- d<r + -j- -3- da- .................. (12),

dx dcr ay da dz do-

or u + (d<r ................................................... (13)'

But this represents the value of the component u of the velocity of the fluid
itself at the same point, and the same thing may be proved of the other
components.

Hence the velocity of the second point on the vortex line is identical
with that of the fluid at that point. In other words, the vortex line swims
along with the fluid, and is always formed of the same row of fluid particles.
The vortex line is therefore no mere mathematical symbol, but has a physical
existence continuous in time and space.

By differentiating equations (1) with respect to x, y, and z respectively,
and adding the results, we obtain the equation —

^ + ^ + ^ = 0 ................................ (14).

dx dy dz

This is an equation of the same form with (6), which expresses the con-
dition of flow of a fluid of invariable density. Hence, if we imagine a fluid,
quite independent of the original fluid, whose components of velocity are a, ft, y,
this imaginary fluid will flow without altering its density.

Now, consider a closed curve in space, and let vortex lines be drawn in
both directions from every point of this curve. These vortex lines will form
a tubular surface, which is called a vortex tube or a vortex filament. Since the
imaginary fluid flows along the vortex lines without change of density, the

470

ATOM.

quantity which in unit of time flows through any section of the same vort.-x
tube niuat be the same. Hence, at any section of a vortex tube the product
of the an* of the section into the mean velocity of rotation is the same.
Thia quantity is called the ftrrnyth of the vortex tube.

A vortex tube cannot begin or end within the fluid; for, if it did, the
imaginary fluid, whose velocity components are a, /?, y, would be generated from
nothing at the beginning of the tube, and reduced to nothing at the end of it.
fTnnm if the tube has a beginning and an end, they must lie on the surface
of the fluid mass. If the fluid is infinite the vortex tube must be infinite, or
else it must return into itself.

We have thus arrived at the following remarkable theorems relating to a
finite vortex tube in an infinite fluid: — (1) It returns into itself, forming a
ring. We may therefore describe it as a vortex ring. (2) It always
of the same portion of the fluid. Hence its volume is invariable. (3)

Its strength remains always the same. Hence the velocity of rotation at any
section varies inversely as the area of that section, and that of any segment
varies directly as the length of that segment. (4) No part of the fluid which
is not originally in a state of rotational motion can ever enter into that state,
and no part of the fluid whose motion is rotational can ever cease to move
rotationally. (5) No vortex tube can ever pass through any other vortex tube,
or through any of its own convolutions. Hence, if two vortex tubes are linked
together, they can never be separated, and if a single vortex tube is knotted
"ii itself, it can never become untied. (G) The motion at any instant of every
part of the fluid, including the vortex rings themselves, may be accurately
represented by conceiving an electric current to occupy the place of each vortex
ring, the strength of the current being proportional to that of the ring. The
magnetic force at any point of space will then represent in direction and mag-
nitude the velocity of the fluid at the corresponding point of the fluid.

These properties of vortex rings suggested to Sir William Thomson* the
possibility of founding on them a new form of the atomic theory. The con-
ditions which must be satisfied by an atom are — permanence in magnitude,
capability of internal motion or vibration, and a sufficient amount of possible
characteristics to account for the difference between atoms of different kinds.

The small hard body imagined by Lucretius, and adopted by Newton, was
invented for the express purpose of accounting for the permanence of the pro-

• "On Vortex Atoms," Proc. Hoy. Soc. Edin., 18th February, 1867.

ATOM. 471

perties of bodies. But it fails to account for the vibrations of a molecule as
revealed by the spectroscope. We may indeed suppose the atom elastic, but
this is to endow it with the very property for the explanation of which, as
exhibited in aggregate bodies, the atomic constitution was originally assumed.
The massive centres of force imagined by Boscovich may have more to recom-
mend them to the mathematician, who has no scruple in supposing them to be
invested with the power of attracting and repelling according to any law of the
distance which it may please him to assign. Such centres of force are no
doubt in their own nature indivisible, but then they are also, singly, incapable
of vibration. To obtain vibrations we must imagine molecules consisting of
many such centres, but, in so doing, the possibility of these centres being
separated altogether is again introduced. Besides, it is in questionable scientific
taste, after using atoms so freely to get rid of forces acting at sensible dis-
tances, to make the whole function of the atoms an action at insensible distances.

On the other hand, the vortex ring of Helmholtz, imagined as the true
form of the atom by Thomson, satisfies more of the conditions than any atom
hitherto imagined. In the first place, it is quantitatively permanent, as regards
its volume and its strength, — two independent quantities. It is also qualita-
tively permanent as regards its degree of implication, whether " knottedness "
on itself or " linkedness " with other vortex rings. At the same time, it is
capable of infinite changes of form, and may execute vibrations of different
periods, as we know that molecules do. And the number of essentially dif-
ferent implications of vortex rings may be very great without supposing the
degree of implication of any of them very high.

But the greatest recommendation of this theory, from a philosophical point
of view, is that its success in explaining phenomena does not depend on the
ingenuity with which its contrivers "save appearances," by introducing first one
hypothetical force and then another. When the vortex atom is once set in
motion, all its properties are absolutely fixed and determined by the laws of
motion of the primitive fluid, which are fully expressed in the fundamental
equations. The disciple of Lucretius may cut and carve his solid atoms in the
hope of getting them to combine into worlds; the follower of Boscovich may
imagine new laws of force to meet the requirements of each new phenomenon;
but he who dares to plant his feet in the path opened up by Helmholtz and
Thomson has no such resources. His primitive fluid has no other properties than
inertia, invariable density, and perfect mobility, and the method by which the

47*

i»HKm of this fluid is to be traced is pure mathematical analysis. The diffi-
culties of this method are enormous, but the glory of surmounting them would
be unique.

There Menu to be little doubt that an encounter between two vortex
*,»**»» would be in it* general character similar to those which we have already
described. Indeed, the encounter between two smoke rings in air gives a very
lively illustration of the elasticity of vortex rings.

But one of the first, if not the very first desideratum in a complete theory
of matter is to explain — first, mass, and second, gravitation. To explain mass
may seem an absurd achievement. We generally suppose that it is of the essence
of matter to be the receptacle of momentum and energy, and even Thomson,
in hia definition of his primitive fluid, attributes to it the possession of mass.
But according to Thomson, though the primitive fluid is the only true matter,
yet that which we call matter is not the primitive fluid itself, but a mode
of motion of that primitive fluid. It is the mode of motion which constitutes
the vortex rings, and which furnishes us with examples of that permanence and
continuity of existence which we are accustomed to attribute to matter itself.
The primitive fluid, the only true matter, entirely eludes our perceptions when
it is not endued with the mode of motion which converts certain portions of
it into vortex rings, and thus renders it molecular.

In Thomson's theory, therefore, the mass of bodies requires explanation. \\'<-
have to explain the inertia of what is only a mode of motion, and inertia is
a projx'rty of matter, not of modes of motion. It is true that a vortex ring
at any given instant has a definite momentum and a definite energy, but to
shew that bodies built up of vortex rings would have such momentum and
energy as we know them to have is, in the present state of the theory, a
very difficult task.

It may seem hard to say of an infant theory that it is bound to explain
gravitation. Since the time of Newton, the doctrine of gravitation has been
admitted and expounded, till it has gradually acquired the character rather of
an ultimate fact than of a fact to be explained.

It seems doubtful whether Lucretius considers gravitation to be an essential
property of matter, as he seems to assert in the very remarkable lines—

"Nam si tantundem est in lame glomere, quantum
Corporis in plumbo est, tantundem pendere par est :
Corpora officium est quoniam premere omnia deorsum."— De Rerun. Natura, i. 361.

ATOM. 473

If this is the true opinion of Lucretius, and if the downward flight of the
atoms arises, in his view, from their own gravity, it seems very doubtful whether
he attributed the weight of sensible bodies to the impact of the atoms. The
latter opinion is that of Le Sage, of Geneva, propounded in his Lucrece New-
tonien, and in his Traite de Physique Mecanique, published, along with a second
treatise of his own, by Pierre Prevost, of Geneva, in 1818*. The theory of
Le Sage is that the gravitation of bodies towards each other is caused by the
impact of streams of atoms flying in all directions through space. These atoms
he calls ultramundane corpuscules, because he conceives them to come in all
directions from regions far beyond that part of the system of the world which
is in any way known to us. He supposes each of them to be so small that
a collision with another ultramundane corpuscule is an event of very rare occur-
rence. It is by striking against the molecules of gross matter that they dis-
charge their function of drawing bodies towards each other. A body placed by
itself in free space and exposed to the impacts of these corpuscules would be
bandied about by them in all directions, but because, on the whole, it receives
as many blows on one side as on another, it cannot thereby acquire any sensible
velocity. But if there are two bodies in space, each of them will screen the
other from a certain proportion of the corpuscular bombardment, so that a
smaller number of corpuscules will strike either body on that side which is next
the other body, while the number of corpuscules which strike it in other direc-
tions remains the same.

Each body will therefore be urged towards the other by the effect of the
excess of the impacts it receives on the side furthest from the other. If we
take account of the impacts of those corpuscules only which come directly from
infinite space, and leave out of consideration those which have already struck
mundane bodies, it is easy to calculate the result on the two bodies, supposing
their dimensions small compared with the distance between them.

The force of attraction would vary directly as the product of the areas of
the sections of the bodies taken normal to the distance and inversely as the
square of the distance between them.

Now, the attraction of gravitation varies as the product of the masses of
the bodies between which it acts, and inversely as the square of the distance
between them. If, then, we can imagine a constitution of bodies such that

* See also Constitution de la Matiere, &c., par le P. Leray, Paris, 1869.

VOL. II. 60

474

the effective area* of the bodies are proportional to their masses, we shall make
tfce two laws coincide. Here, then, seems to be a path leading towards an
of tlie law of gravitation, which, if it can be shewn to be in
consistent with (acts, may turn out to be a royal road into the
aroana of science.

La Sage himself shews that, in order to make the effective area of a body,
in virtue of which it acts as a screen to the streams of ultramundane cor-
puscules, un>|x>rtional to the mass of the body, whether the body be large or
nn«H we must admit that the size of the solid atoms of the body is exceed-
ingly small compared with the distances between them, so that a very small
proportion of the corpuscules are stopped even by the densest and largest bodies.
W may picture to ourselves the streams of corpuscules coming in every direc-
tion, like light from a uniformly illuminated sky. We may imagine a material
body to consist of a congeries of atoms at considerable distances from each other,
and we may represent this by a swarm of insects flying in the air. To an
observer at a distance this swarm will be visible as a slight darkening of the
«ky in a certain quarter. This darkening will represent the action of the
material body in stopping the flight of the corpuscules. Now, if the proportion
• -I* light stopped by the swarm is very small, two such swarms will stop nearly
the same amount of light, whether they are in a line with the eye or not, but
if one of them stops an appreciable proportion of light, there will not be so
much left to be stopped by the other, and the effect of two swarms in a line
with the eye will be less than the sum of the two effects separately.

Now, we know that the effect of the attraction of the sun and earth on
the moon is not appreciably different when the moon is eclipsed than on other
occasions when full moon occurs without an eclipse. This shews that the number
«.f the corpuscules which are stopped by bodies of the size and mass of the
earth, and even the sun, is very small compared with the number which pass
straight through the earth or the sun without striking a single molecule. To
tlie streams of corpuscules the earth and the sun are mere systems of atoms
scattered in space, which present far more openings than obstacles to their recti-
linear flight.

Such is the ingenious doctrine of Le Sage, by which he endeavours to
explain universal gravitation. Let us try to form some estimate of this con-
tinual bombardment of ultramundane corpuscules which is being kept up on all
sides of us.

ATOM. 475

We have seen that the sun stops but a very small fraction of the cor-
puscules which enter it. The earth, being a smaller body, stops a still smaller
proportion of them. The proportion of those which are stopped by a small
body, say a 1 Ib. shot, must be smaller still in an enormous degree, because its
thickness is exceedingly small compared with that of the earth.

Now, the weight of the ball, or its tendency towards the earth, is produced,
according to this theory, by the excess of the impacts of the corpuscules which
come from above over the impacts of those which come from below, and have
passed through the earth. Either of these quantities is an exceedingly small
fraction of the momentum of the whole number of corpuscules which pass through
the ball in a second, and their difference is a small fraction of either, and yet
it is equivalent to the weight of a pound. The velocity of the corpuscules
must be enormously greater than that of any of the heavenly bodies, other-
wise, as may easily be shewn, they would act as a resisting medium opposing
the motion of the planets. Now, the energy of a moving system is half the
product of its momentum into its velocity. Hence the energy of the corpuscules,
which by their impacts on the ball during one second urge it towards the earth,
must be a number of foot-pounds equal to the number of feet over which a
corpuscule travels in a second, that is to say, not less than thousands of millions.
But this is only a small fraction of the energy of all the impacts which the
atoms of the ball receive from the innumerable streams of corpuscules which fall
upon it in all directions.

Hence the rate at which the energy of the corpuscules is spent in order
to maintain the gravitating property of a single pound, is at least millions of
millions of foot-pounds per second.

What becomes of this enormous quantity of energy ? If the corpuscules,
after striking the atoms, fly off with a velocity equal to that which they had
before, they will carry their energy away with them into the ultramundane
regions. But if this be the case, then the corpuscules rebounding from the body
in any given direction will be both in number and in velocity exactly equiva-
lent to those which are prevented from proceeding in that direction by being
deflected by the body, and it may be shewn that this will be the case what-
ever be the shape of the body, and however many bodies may be present in
the field. Thus, the rebounding corpuscules exactly make up for those which
are deflected by the body, and there will be no excess of the impacts on any
other body in one direction or another.

60—2

476

The explanation of gravitation, therefore, falls to the ground if the cor-
uosouW are like perfectly elastic spheres, and rebound with a velocity of
•Halation equal to that of approach. If, on the other hand, they rebound with
•nailer velocity, the effect of attraction between the bodies will no doubt
be produced, but then we have to find what becomes of the energy which
the molecules have brought with them but have not carried away.

If any appreciable fraction of this energy is communicated to the body in
the form of heat, the amount of heat so generated would in a few seconds
HUM it, and in like manner the whole material universe, to a white heat.

It lias been suggested by Sir W. Thomson that the corpuscules may be
m constructed as to carry off their energy with them, provided that part of
their kinetic energy is transformed, during impact, from energy of translation
to energy of rotation or vibration. For this purpose the corpuscules must be
material systems, not mere points. Thomson suggests that they are vortex
atoms, wliich are set into a state of vibration at impact, and go off with a
smaller velocity of translation, but in a state of violent vibration. He has also
suggested the possibility of the vortex corpuscule regaining its swiftness and
losing part of its vibratory agitation by communion with its kindred cor-
puscules in infinite space.

We have devoted more space to this theory than it seems to deserve,
because it is ingenious, and because it is the only theory of the cause of
gravitation which has been so far developed as to be capable of being attacked
and defended. It does not appear to us that it can account for the tem-
|H-niture of bodies remaining moderate while their atoms are exposed to the
tamhardment. The temperature of bodies must tend to approach that at
which the average kinetic energy of a molecule of the body would be equal
to the average kinetic energy of an ultramundane corpuscule.

Now, suppose a plane surface to exist which stops all the corpuscules.
The pressure on this plane will be p = NMiiP where M is the mass of a cor-
puscule, N the number in unit of volume, and u its velocity normal to the
plane. Now, we know that the very greatest pressure existing in the universe
must be much less than the pressure p, which would be exerted against a body
which stops all the corpuscules. We are also tolerably certain that N, the
number of corpuscules which are at any one time within unit of volume, is
small compared with the value of N for the molecules of ordinary bodies.
Hence, Jfua must be enormous compared with the corresponding quantity for

ATOM. 477

ordinary bodies, and it follows that the impact of the corpuscules would raise
all bodies to an enormous temperature. We may also observe that according
to this theory the habitable universe, which we are accustomed to regard as
the scene of a magnificent illustration of the conservation of energy as the
fundamental principle of all nature, is in reality maintained in working order
only by an enormous expenditure of external power, which would be nothing
less than ruinous if the supply were drawn from anywhere else than from the
infinitude of space, and which, if the contrivances of the most eminent mathe-
maticians should be found in any respect defective, might at any moment tear
the whole universe atom from atom.

We must now leave these speculations about the nature of molecules and
the cause of gravitation, and contemplate the material universe as made up of
molecules. Every molecule, so far as we know, belongs to one of a definite
number of species. The list of chemical elements may be taken as repre-
senting the known species which have been examined in the laboratories.
Several of these have been discovered by means of the spectroscope, and more
may yet remain to be discovered in the same way. The spectroscope has also
been applied to analyse the light of the sun, the brighter stars, and some of
the nebulae and comets, and has shewn that the character of the light
emitted by these bodies is similar in some cases to that emitted by terres-
trial molecules, and in others to light from which the molecules have absorbed
certain rays. In this way a considerable number of coincidences have been
traced between the systems of lines belonging to particular terrestrial sub-
stances and corresponding lines in the spectra of the heavenly bodies.

The value of the evidence furnished by such coincidences may be estimated
by considering the degree of accuracy with which one such coincidence may be
observed. The interval between the two lines which form Fraunhofer's line D
is about the five hundredth part of the interval between B and G on Kirch-
hoff's scale. A discordance between the positions of two lines amounting to
the tenth part of this interval, that is to say, the five thousandth part of
the length of the bright part of the spectrum, would be very perceptible in
a spectroscope of moderate power. We may define the power of the spectro-
scope to be the number of times which the smallest measurable interval is
contained in the length of the visible spectrum. Let us denote this by p. In
the case we have supposed p will be about 5000.

If the spectrum of the sun contains n lines of a certain degree of inten-

471

aity, U* probability that any one line of the spectrum of a gas will coincide
with oa* of these a lines is

ud when p «• large compared with n, this becomes nearly -. If there are
r lines in the spectrum of the gas, the probability that each and every one
•hall coincide with a line in the solar spectrum is approximately — . Hence,

in the case of a gas whose spectrum contains several lines, we have to com-
pare the results of two hypotheses. If a large amount of the gas exists in
the sun, we have the strongest reason for expecting to find all the r lines in
the solar spectrum. If it does not exist, the probability that r lines out of
tin* n observed lines shall coincide with the lines of the gas is exceedingly
small. If, then, we find all the r lines in their proper places in the solar
apectrum, we have very strong grounds for believing that the gas exists in the
sun. The probability that the gas exists in the sun is greatly strengthened
if the character of the lines as to relative intensity and breadth is found to
correspond in the two spectra.

The absence of one or more lines of the gas in the solar spectrum tends
of course to weaken the probability, but the amount to be deducted from the
probability must depend on what we know of the variation in the relative
intensity of the lines when the temperature and the pressure of the gas are
made to vary.

Coincidences observed, in the case of several terrestrial substances, with
several systems of lines in the spectra of the heavenly bodies, tend to increase
the evidence for the doctrine that terrestrial substances exist in the heavenly
bodies, while the discovery of particular lines in a celestial spectrum which do
not coincide with any line in a terrestrial spectrum does not much weaken
the general argument, but rather indicates either that a substance exists in the
heavenly body not yet detected by chemists on earth, or that the temperature
of the heavenly body is such that some substance, undecomposable by our
methods, is there split up into components unknown to us in their separate
state.

We are thus led to believe that in widely-separated parts of the visible
universe molecules exist of various kinds, the molecules of each kind having

ATOM. 479

their various periods of vibration either identical, or so nearly identical that our
spectroscopes cannot distinguish them. We might argue from this that these
molecules are alike in all other respects, as, for instance, in mass. But it is
sufficient for our present purpose to observe that the same kind of molecule,
say that of hydrogen, has the same set of periods of vibration, whether we
procure the hydrogen from water, from coal, or from meteoric iron, and that
light, having the same set of periods of vibration, comes to us from the sun,
from Sinus, and from Arcturus.

The same kind of reasoning which led us to believe that hydrogen exists
in the sun and stars, also leads us to believe that the molecules of hydrogen
in all these bodies had a common origin. For a material system capable of
vibration may have for its periods of vibration any set of values whatever.
The probability, therefore, that two material systems, quite independent of each
other, shall have, to the degree of accuracy of modern spectroscopic measure-
ments, the same set of periods of vibration, is so very small that we are
forced to believe that the two systems are not independent of each other.
When, instead of two such systems, we have innumerable multitudes all
having the same set of periods, the argument is immensely strengthened.

Admitting, then, that there is a real relation between any two molecules
of hydrogen, let us consider what this relation may be.

We may conceive of a mutual action between one body and another
tending to assimilate them. Two clocks, for instance, will keep time with each
other if connected by a wooden rod, though they have different rates if they
were disconnected. But even if the properties of a molecule were as capable
of modification as those of a clock, there is no physical connection of a suffi-
cient kind between Sirius and Arcturus.

There are also methods by which a large number of bodies differing from
each other may be sorted into sets, so that those in each set more or less
resemble each other. In the manufacture of small shot this is done by making
the shot roll down an inclined plane. The largest specimens acquire the greatest
velocities, and are projected farther than the smaller ones. In this way the
various pellets, which differ both in size and in roundness, are sorted into
different kinds, those belonging to each kind being nearly of the same size,
and those which are not tolerably spherical being rejected altogether.

If the molecules were originally as various as these leaden pellets, arid
were afterwards sorted into kinds, we should have to account for the dis-

ATOM.

of all the molecules which did not fall under one of the very limited
of kinds known to us; and to get rid of a number of indestructible
exceeding by fer the number of the molecules of all the recognised
kinds, would be one of tot seraest labours ever proposed to a cosmogonist.

* well known that living beings may be grouped into a certain number
of species, defined with more or less precision, and that it is difficult or im-
possible to find a tfltfr^ of individuals forming the links of a continuous chain
holms* one species and another. In the case of living beings, however, the
generation of individuals is always going on, each individual differing more or
less from its parents. Each individual during its whole life is undergoing
modification, and it either survives and propagates its species, or dies early,
accordingly as it is more or less adapted to the circumstances of its environ-
ment. Hence, it has been found possible to frame a theory of the distribution
of organisms into species by means of generation, variation, and discriminative
destruction. But a theory of evolution of this kind cannot be applied to the
case of molecules, for the individual molecules neither are born nor die, they
have neither parents nor offspring, and so far from being modified by their en-
vironment, we find that two molecules of the same kind, say of hydrogen,
have the same properties, though one has been compounded with carbon and
buried in the earth as coal for untold ages, while the other has been "oc-
cluded " in the iron of a meteorite, and after unknown wanderings in the
heavens has at last fallen into the hands of some terrestrial chemist.

The process by which the molecules become distributed into distinct species
is not one of which we know any instances going on at present, or of which
we have as yet been able to form any mental representation. If we suppose
that the molecules known to us are built up each of some moderate number
of atoms, these atoms being all of them exactly alike, then we may attribute
the limited number of molecular species to the limited number of ways in
which the primitive atoms may be combined so as to form a permanent system.

But though this hypothesis gets rid of the difficulty of accounting for the
independent origin of different species of molecules, it merely transfers the diffi-
culty from the known molecules to the primitive atoms. How did the atoms
come to be all alike in those properties which are in themselves capable of
assuming any value ?

If we adopt the theory of Boscovich, and assert that the primitive atom
is a mere centre of force, having a certain definite mass, we may get over the

ATOM. 481

difficulty about the equality of the mass of all atoms by laying it down as
a doctrine which cannot be disproved by experiment, that mass is not a
quantity capable of continuous increase or diminution, but that it is in its
own nature discontinuous, like number, the atom being the unit, and all masses
being multiples of that unit. We have no evidence that it is possible for the
ratio of two masses to be an incommensurable quantity, for the incommen-
surable quantities in geometry are supposed to be traced out in a continuous
medium. If matter is atomic, and therefore discontinuous, it is unfitted for
the construction of perfect geometrical models, but in other respects it may
fulfil its functions.

But even if we adopt a theory which makes the equality of the mass of
different atoms a result depending on the nature of mass rather than on any
quantitative adjustment, the correspondence of the periods of vibration of actual
molecules is a fact of a different order.

We know that radiations exist having periods of vibration of every value
between those corresponding to the limits of the visible spectrum, and probably
far beyond these limits on both sides. The most powerful spectroscope can
detect no gap or discontinuity in the spectrum of the light emitted by incan-
descent lime.

The period of vibration of a luminous particle is therefore a quantity which
in itself is capable of assuming any one of a series of values, which, if not
mathematically continuous, is such that consecutive observed values differ from
each other by less than the ten thousandth part of either. There is, therefore,
nothing in the nature of time itself to prevent the period of vibration of a
molecule from assuming any one of many thousand different observable values.
That which determines the period of any particular kind of vibration is the
relation which subsists between the corresponding type of displacement and the
force of restitution thereby called into play, a relation involving constants of
space and time as well as of mass.

It is the equality of these space- and time-constants for all molecules of
the same kind which we have next to consider. We have seen that the very
different circumstances in which different molecules of the same kind have been
placed have not, even in the course of many ages, produced any appreciable
difference in the values of these constants. If, then, the various processes of
nature to which these molecules have been subjected since the world began
have not been able in all that time to produce any appreciable difference

VOL. II. 61

ATOM.

lula^on the oocwUnU of one molecule and those of another, we are forced
•>tttrlu«lc that it u not to the operation of any of these processes that the
unifurmit v of the constant* is dua

The formation of the molecule is therefore an event not belonging to that
tinier of nature under which we live. It is an operation of a kind which is
not, *» fiu- at we are aware, going on on earth or in the sun or the stars,
cither now or since these bodies began to be formed. It must be referred to
the epoch, not of the formation of the earth or of the solar system, but of
the estnbluihment of the existing order of nature, and till not only these

n rj old r of nature tta It' i> dinolved, \\v haw
no reMon to expect the occurrence of any operation of a similar kind.

In the present state of science, therefore, we have strong reasons for be-
liering that in a molecule, or if not hi a molecule, in one of its component
atoms, we have something which has existed either from eternity or at least
from times anterior to the existing order of nature. But besides this atom,
there are immense numbers of other atoms of the same kind, and the constants
of each of these atoms are incapable of adjustment by any process now in
action. Each is physically independent of all the others.

Whether or not the conception of a multitude of beings existing from all
eternity is in itself self-contradictory, the conception becomes palpably absurd
when we attribute a relation of quantitative equality to all these beings. We
are then forced to look beyond them to some common cause or common origin
to explain why this singular relation of equality exists, rather than any one
of the infinite number of possible relations of inequality.

Science is incompetent to reason upon the creation of matter itself out
<>f nothing. We have reached the utmost limit of our thinking faculties when
we have admitted that, because matter cannot be eternal and self-existent, it
must have been created. It is only when we contemplate not matter in itself,
but the form in which it actually exists, that our mind finds something on
which it can lay hold.

That matter, as such, should have certain fundamental properties, that it
should have a continuous existence in space and time, that all action should
he between two portions of matter, and so on, are truths which may, for
aught we know, be of the kind which metaphysicians call necessary. We may
uw our knowledge of such truths for purposes of deduction, but we have no
data for speculating on their origin.

ATOM. 483

But the equality of the constants of the molecules is a fact of a very
different order. It arises from a particular distribution of matter, a collocation,
to use the expression of Dr Chalmers, of things which we have no difficulty in
imagining to have been arranged otherwise. But many of the ordinary instances
of collocation are adjustments of constants, which are not only arbitrary in
their own nature, but in which variations actually occur; and when it is
pointed out that these adjustments are beneficial to living beings, and are
therefore instances of benevolent design, it is replied that those variations
which are not conducive to the growth and multiplication of living beings
tend to their destruction, and to the removal thereby of the evidence of any
adjustment not beneficial.

The constitution of an atom, however, is such as to render it, so far as
we can judge, independent of all the dangers arising from the struggle for
existence. Plausible reasons may, no doubt, be assigned for believing that if
the constants had varied from atom to atom through any sensible range, the
bodies formed by aggregates of such atoms would not have been so well fitted
for the construction of the world as the bodies which actually exist. But as
we have no experience of bodies formed of such variable atoms this must remain
a bare conjecture.

Atoms have been compared by Sir J. Herschel to manufactured articles,
on account of their uniformity. The uniformity of manufactured articles may
be traced to very different motives on the part of the manufacturer. In
certain cases it is found to be less expensive as regards trouble, as well as
cost, to make a great many objects exactly alike than to adapt each to its
special requirements. Thus, shoes for soldiers are made in large numbers without
any designed adaptation to the feet of particular men. In another class of
cases the uniformity is intentional, and is designed to make the manufactured
article more valuable. Thus, Whitworth's bolts are made in a certain number
of sizes, so that if one bolt is lost, another may be got at once, and accurately
fitted to its place. The identity of the arrangement of the words in the
different copies of a document or book is a matter of great practical importance,
and it is more perfectly secured by the process of printing than by that of
manuscript copying.

In a third class not a part only but the whole of the value of the object
arises from its exact conformity to a given standard. Weights and measures
belong to this class, and the existence of many well-adjusted material standards

01—3

\I..\I.

of wright and measure in any country furnishes evidence of the existence of
• •jntera of Uw regulating the transactions of the inhabitants, and enjoining
in all profcoMd measures a conformity to the national standard.

There are thus three kinds of usefulness in manufactured articles — cheap-
ntm, eerrioeableneea, and quantitative accuracy. Which of these was present
to tho mind of Sir J. Herscliel we cannot now positively affirm, but it was
at leact as likely to have been the last as the first, though it seems more
probable that he meant to assert that a number of exactly similar things cannot
be each of them eternal and self-existent, and must therefore have been made,
and that he and the phrase "manufactured article" to suggest the idea of
their being made in great numbers.

[From the Encyclopaedia Britannica.]

LXXIV. Attraction.

THAT the different parts of a material system influence each other's mo-
tions is a matter of daily observation. In some cases we cannot discover any
material connection extending from the one body to the other. We call these
cases of action at a distance, to distinguish them from those in which we can
trace a continuous material bond of union between the bodies. The mutual
action between two bodies is called stress. When the mutual action tends to
bring the bodies nearer, or to prevent them from separating, it is called tension
or attraction. When it tends to separate the bodies, or to prevent them from
approaching, it is called pressure or repulsion. The names tension and pressure
are used when the action is seen to take place through a medium. Attraction
and repulsion are reserved for cases of action at a distance. The configuration
of a material system can always be defined in terms of the mutual distances
of the parts of the system. Any change of configuration must alter one or
more of these distances. Hence the force which produces or resists such a
change may be resolved into attractions or repulsions between those parts of
the system whose distance is altered.

There has been a great deal of speculation as to the cause of such forces,
one of them, namely, the pressure between bodies in contact, being supposed
to be more easily conceived than any other kind of stress. Many attempts
have therefore been made to resolve cases of apparent attraction and repulsion
at a distance into cases of pressure. At one time the possibility of attraction
at a distance was supposed to be refuted by asserting that a body cannot act
where it is not, and that therefore all action between different portions of
matter must be by direct contact. To this it was replied that we have no
evidence that real contact ever takes place between two bodies, and that, in
fact, when bodies are pressed against each other and in apparent contact, we

I ATTRACTION.

mar •onetime* actually measure the distance between them, as when one piece
of gbuB is hid on another, in which case a considerable pressure must be ap-
plied to bring the surfaces near enough to shew the block spot of Newton's
ring*, which indicate* a distance of about a ten thousandth of a millimetre. If,
in older to get rid of the idea of action at a distance, we imagine a ma-
terial medium through which the action is transmitted, all that we have done
is to substitute for a single action at a great distance a series of actions at
smaller distances between the parts of the medium, so that we cannot even
thus get rid of action at a distance.

The study of the mutual action between the parts of a material system
hat, in modem tiroes, been greatly simplified by the introduction of the idea
of the energy of the system. The energy of the system is measured by the
amount of work which it can do in overcoming external resistances. It depends
on the present configuration and motion of the system, and not on the manner
in which the system has acquired that configuration and motion. A complete
knowledge of the manner in which the energy of the system depends on its
configuration and motion, is sufficient to determine all the forces acting between
the parts of the system. For instance, if the system consists of two bodies,
and if the energy depends on the distance between them, then if the energy
increases when the distance increases, there must be attraction between the
bodies, and if the energy diminishes when the distance increases, there must
be repulsion between them. In the case of two gravitating masses m and m'

at a distance r, the part of the energy which depends on r is — • - . We

may therefore express the fact that there is attraction between the two bodies
by saying that the energy of the system consisting of the two bodies increases
when their distance increases. The question, therefore, Why do the two bodies
attract each other? may be expressed in a different form. Why does the
energy of the system increase when the distance increases ?

But we must bear in mind that the scientific or science-producing value of
the efforts made to answer these old standing questions is not to be measured
by the prospect they afford us of ultimately obtaining a solution, but by their
effect in stimulating men to a thorough investigation of nature. To propose a
scientific question presupposes scientific knowledge, and the questions which
exercise men's minds in the present state of science may very likely be such
that a little more knowledge would shew us that no answer is possible. The

ATTEACTION. 487

scientific value of the question, How do bodies act on one another at a dis-
tance ? is to be found in the stimulus it has given to investigations into the
properties of the intervening medium.

Newton, in his Principia, deduces from the observed motions of the hea-
venly bodies the fact that they attract one another according to a definite law.
This he gives as a result of strict dynamical reasoning, and by it he shews
how not only the more conspicuous phenomena, but all the apparent irregu-
larities of the celestial motions are the calculable results of a single principle.
In his Principia he confines himself to the demonstration and development of
this great step in the science of the mutual action of bodies. He says nothing
there about the means by which bodies gravitate towards each other. But his
mind did not rest at this point. We know that he did not believe in the
direct action of bodies at a distance.

" It is inconceivable that inanimate brute matter should, without the mediation of something else
which is not material, operate upon and affect other matter without mutual contact, as it must do
if gravitation in the sense of Epicurus be essential and inherent in it... That gravity should be innate,
inherent, and essential to matter, so that one body can act upon another at a distance, through a
vacuum, without the mediation of anything else, by and through which their action and force may be
conveyed from one to another, is to me so great an absurdity, that I believe no man, who has in
philosophical matters a competent faculty of thinking, can ever fall into it." — Letter to Bentley.

And we also know that he sought for the mechanism of gravitation in the
properties of an aethereal medium diffused over the universe.

" It appears, from his letters to Boyle, that this was his opinion early, and if he did not publish
it sooner it proceeded from hence only, that he found he was not able, from experiment and obser-
vation, to give a satisfactory account of this medium and the manner of its operation in producing
the chief phenomena of nature*."

In his Optical Queries, indeed, he shews that if the pressure of this
medium is less in the neighbourhood of dense bodies than at great distances
from them, dense bodies will be drawn towards each other, and that if the
diminution of pressure is inversely as the distance from the dense body the
law will be that of gravitation. The next step, as he points out, is to account
for this inequality of pressure in the medium ; and as he was not able to do
this, he left the explanation of the cause of gravity as a problem to succeeding
ages. As regards gravitation the progress made towards the solution of the
problem since the time of Newton has been almost imperceptible. Faraday

* Maclaurin's account of Sir Isaac Newton's discoveries.

. . , ATTRACTION.

•hewed that the transmission of electric and magnetic forces is accompanied by
lr1T|— occurring in every part of the intervening medium. He traced the
Unas of tmse through the medium; and he ascribed to them a tendency to
^^m themselves and to separate from their neighbours, thus introducing the
idea of ftnm in the medium in a different form from that suggested by Newton ;
for. whereas Newton's stress was a hydrostatic pressure in every direction,
Faradav's is a tension along the lines of force, combined with a pressure in
all normal directions. By shewing that the plane of polarisation of a ray of
light passing through a transparent medium in the direction of the magnetic
force is made to rotate, Faraday not only demonstrated the action of magnetism
on light, but by using light to reveal the state of magnetisation of the medium,
he " illuminated," to use his own phrase, " the lines of magnetic force."

From this phenomenon Thomson afterwards proved, by strict dynamical
reasoning, that the transmission of magnetic force is associated with a rotatory
motion of the small parts of the medium. He shewed, at the same time, how
the centrifugal force due to this motion would account for magnetic attraction.

A theory of this kind is worked out in greater detail in Clerk Maxwell's
Treatise on Electricity and Magnetism. It is there shewn that, if we assume
that the medium is in a state of stress, consisting of tension along the lines
of force and pressure in all directions at right angles to the lines of force,
the tension and the pressure being equal in numerical value and proportional
to the square of the intensity of the field at the given point, the observed
electrostatic and electromagnetic forces will be completely accounted for.

The next step is to account for this state of stress in the medium. In
the case of electromagnetic force we avail ourselves of Thomson's deduction
from Faraday's discovery stated above. We assume that the small parts of the
medium are rotating about axes parallel to the lines of force. The centrifugal
force due to this rotation produces the excess of pressure perpendicular to the
lines of force. The explanation of electrostatic stress is less satisfactory, but
there can be no doubt that a path is now open by which we may trace to
the action of a medium all forces which, like the electric and magnetic forces,
vary inversely as the square of the distance, and are attractive between bodies
of different names, and repulsive between bodies of the same names.

The force of gravitation is also inversely as the square of the distance,
but it differs from the electric and magnetic forces in this respect, that the bodies
between which it acts cannot be divided into two opposite kinds, one positive

ATTRACTION. 489

and the other negative, but are in respect of gravitation all of the same kind,
and that the force between them is in every case attractive. To account for
such a force by means of stress in an intervening medium, on the plan adopted
for electric and magnetic forces, we must assume a stress of an opposite kind
from that already mentioned. We must suppose that there is a pressure in
the direction of the lines of force, combined with a tension in all directions
at right angles to the lines of force. Such a state of stress would, no doubt,
account for the observed effects of gravitation. We have not, however, been
able hitherto to imagine any physical cause for such a state of stress. It is
easy to calculate the amount of this stress which would be required to account
for the actual effects of gravity at the surface of the earth. It would require
a pressure of 37,000 tons weight on the square inch in a vertical direction,
combined with a tension of the same numerical value in all horizontal directions.
The state of stress, therefore, which we must suppose to exist in the invisible
medium, is 3000 times greater than that which the strongest steel could sup-
port.

Another theory of the mechanism of gravitation, that of Le Sage, who
attributes it to the impact of "ultramundane corpuscles," has been already
discussed in the article Atom, supra, p. 473.

Sir William Thomson* has shewn that if we suppose all space filled with
a uniform incompressible fluid, and if we further suppose either that material
bodies are always generating and emitting this fluid at a constant rate, the
fluid flowing off to infinity, or that material bodies are always absorbing and
annihilating the fluid, the deficiency flowing in from infinite space, then, in
either of these cases, there would be an attraction between any two bodies
inversely as the square of the distance. If, however, one of the bodies were
a generator of the fluid and the other an absorber of it, the bodies would repel
each other.

Here, then, we have a hydrodynamical illustration of action at a distance,
which is so far promising that it shews how bodies of the same kind may
attract each other. But the conception of a fluid constantly flowing out of a
body without any supply from without, or flowing into it without any way
of escape, is so contradictory to all our experience, that an hypothesis, of
which it is an essential part, cannot be called an explanation of the phenomenon
of gravitation.

* Proceedings of the Royal Svciety of Edinburgh, 7th Feb. 1870.

VOL. II, 62

ATTRACTION*.

I* Robert Hooke, a man of singular inventive power, in 1671 entUa-
rouml to trace tht «*o«e of gravitation to waves propagated in a medium.
II' found thH bodiM floating on water agitated by waves were drawn towards
the centra of agitation*. He does not appear, however, to have followed up
that observation in such a way as to determine completely the action of waves
on an immened liody.

Professor Challis has investigated the mathematical theory of the effect of
waves of condensation and rarefaction in an elastic fluid on bodies immersed
in the fluid. He found the difficulties of the investigation to be so great that
he baa not been able to arrive at numerical results. He concludes, however,
^i the effect of such waves would be to attract the body towards the centre
..I" agitation, or to repel it from that centre, according as the wave's length
is very large or very small compared with the dimensions of the body. Prac-
tical illustrations of the effect of such waves have been given by Guyot, Schell-
bach. Guthrie, and Thomson t.

A tuning-fork is set in vibration, and brought near a delicately suspended
iiu'lit body. The body is immediately attracted towards the tuning-fork. If the
tuning-fork is itself suspended, it is seen to be attracted towards any body
placed near it

Sir W. Thomson has shewn that this action can in all cases be explained
by the general principle that in fluid motion the average pressure is least
where the average energy of motion is greatest. Now, the wave-motion is
greatest nearest the tuning-fork, the pressure is therefore least there ; and the
suspended body being pressed unequally on opposite sides, moves from the side
of greater pressure to the side of less pressure, that is towards the tuning-fork.
He has also succeeded in producing repulsion in the case of a small body
lighter than the surrounding medium.

It is remarkable that of the three hypotheses, which go some way towards
a physical explanation of gravitation, every one involves a constant expenditure
of work. Le Sage's hypothesis of ultramundane corpuscles does so, as we have
shewn in the article ATOM. That of the generation or absorption of fluid
requires, not only constant expenditure of work in emitting fluid under pressure,
but actual creation and destruction of matter. That of waves requires some
agent in a remote part of the universe capable of generating the waves.

* Po»thum<na Workt, edited by R Waller, pp. xiv. and 184
t l'Mosoi>kical Jfayatiru, June, 1871.

ATTRACTION. 491

According to such hypotheses we must regard the processes of nature not
as illustrations of the great principle of the conservation of energy, but as
instances in which, by a nice adjustment of powerful agencies not subject to
this principle, an apparent conservation of energy is maintained. Hence, we are
forced to conclude that the explanation of the cause of gravitation is not to
be found in any of these hypotheses.

62—2

Gi«»*rufy» PktlotojAioal Society t Proceedings, Vol. II. 1870.]

LXXV. On Bow's method of drawing diagrams in graphical statics with
illustrations from Peaucellier's linkage.

THE use of Diagrams is a particular instance of that method of symbols
which ia so powerful an aid in the advancement of science.

A diagram differs from a picture in this respect, that in a diagram no
attempt is made to represent those features of the actual material system which
are not the special objects of our study.

Thus when we are studying the internal equilibrium of a particular piece
of a structure or a machine, we require to know its shape and dimensions, and
the specification of these may often be made easier by means of a drawing of
the piece.

But when we are studying the equilibrium of a framework composed of
such pieces jointed together, in which each piece acts only by tension or by
pressure between its extremities, it is not necessary to know whether a particular
piece is straight or curved or what may be the form of its section. In order,
therefore, to exhibit the structure of the frame in the most elementary manner
we may draw it as a skeleton in which the different joints are connected by
straight lines. The tension or pressure of each piece may be indicated on such
a diagram by numbers attached to the line which represents that piece in the
diagram. The stresses in the frame would thus be indicated in a way which
is geometrical as regards the position and direction of the forces, but arithmetical
as regards their magnitude.

But a purely geometrical representation of a force has been made use of
from the earliest beginnings of mechanics as a science. The force is represented
by a straight line drawn from the point of application of the force, in the
direction of the force, and containing as many units of length as there are

BOW'S METHOD OF DRAWING DIAGRAMS IN GRAPHICAL STATICS. 493

units of force in the force. The end of the line is marked by an arrow-head
to shew in which direction the force acts.

According to this method each force is drawn in its proper position in
the diagram which represents the configuration of the system. Such a diagram
might be useful as a record of the results of calculation of the magnitude of
the forces, but it would be of no use in enabling us to test the correctness
of the calculation. It would be of less use than the diagram in which the
magnitudes of the forces were indicated by numbers. . ,

But we have a geometrical method of testing the equilibrium of any set
of forces acting at a point by drawing in series a set of lines parallel and
proportional to these forces. If these lines form a closed polygon the forces
are in equilibrium. We might thus form a set of polygons of forces, one for
each joint of the frame. But in so doing we give up the principle of always
drawing the line representing a force from its point of application, for all the
sides of a polygon cannot pass through the same point as the forces do.

We also represent every stress twice over, for it appears as a side of both
the polygons corresponding to the two joints between which it acts.

But if we can arrange the polygons in such a way that the sides of any
two polygons which represent the same force coincide with each other, we may
form a diagram in which every stress is represented in direction and magnitude,
though not in position, by a single line, which is the common boundary of the
two polygons which represent the points of concourse of the pieces of the frame.

Here we have a pure diagram of forces, in which no attempt is made to
represent the configuration of the material system, and in which every force
is not only represented in direction and magnitude by a straight line, but the
equilibrium of the forces is manifest by inspection, for we have only to examine
whether each polygon is closed or not.

The relations between the diagram of the frame and the diagram of stress
are as follows :

To every piece in the frame corresponds a line in the diagram of stress
which represents in magnitude and direction the stress acting on that piece.

To every joint of the frame corresponds a closed polygon in the diagram,
and the forces acting at that joint are represented by the sides of the polygon
taken in a certain cyclical order. The cyclical order of the sides of two adjacent
polygons is such that their common side is traced in opposite directions in
going round the two polygons.

'tt METHOD OF DRAWING I'lAi.KAMS IN URAPHICAL 8TATIC8.

When to every point of concourse of the lines in the diagram of stress
a closed polygon in the skeleton of the frame, the two diagrams

are aaki to be reciprocal.

The first extensions of the method of diagrams of forces to other cases
than that of the funicular polygon were given by Rankine in his Applied

Ifafcmtw (1857).

The method was indejwndently applied to a large number of cases by
Mr W. P. Taylor, a practical draughtsman in the office of the well-known con-
tractor Mr J. B. Cochrane. I pointed out the reciprocal properties of the
diagram in 1864, and in 1870 shewed the relations of this method to Airy's
function of stress and other mathematical methods.

Prof. Fleeming Jenkin has given a number of applications of the method
practice. Trans. R. S. K, Vol. xxv.

Cremona* has deduced the construction of the reciprocal figures from the
theory of the two linear components of a wrench.

Cul ma nn in his Graphiscfie Statik makes great use of diagrams of forces,
some of which, however, are not reciprocal

M. Maurice Levy in his Statique Graphiqw (Paris, 1874) has treated the
whole subject in an elementary and complete manner.

Mr R. H. Bow, C.E., F.R.S.E., in a recent work On the Economics of
Construction in relation to Framed Structures (Spon, 1873), has materially
simplified the process of drawing a diagram of stress reciprocal to a given frame
acted on by any system of equilibrating external forces.

Instead of lettering the joints of the frame as is generally done, or the
pieces of the frame as was my own custom, he places a letter in each of
the polygonal areas enclosed by the pieces of the frame, and also in each of
the divisions of the surrounding space as separated by the lines of action of
the external forces.

When one piece of the frame crosses another, the point of intersection is
treated as if it were a real joint, and the stresses of each of the intersecting
pieces are represented twice in the diagram of stress, as the opposite sides of the
parallelogram which represents the forces at the point of intersection. Thus
the point V in figures 1 and 3, p. 495, is represented by the parallelogram
BDCE in figure 2, and the point A in figure 2 is represented by the
parallelogram PRQS in figures 1 and 3.

* L» figure reciproche nella sfatica grafica (Milano, 1872).

BOWS METHOD OF DRAWING DIAGRAMS IN GRAPHICAL STATICS. 495

Peaucellier's linkage consists of the four equal pieces forming the jointed
rhombus PQRS together with two equal arms OS and OR.

Fig 1

When these arms are longer than the sides of the rhombus the linkage
is said to be positive ; when they are shorter the linkage is said to be negative.

When Peancellier's linkage is employed as a machine it is acted on by
three forces, applied respectively at the fulcrum 0, and the two tracing poles
(J and S.

496 BOW'S MOTHOD OP DRAWING DIAGRAMS IN GRAPHICAL STATICS.

These three forces, if in equilibrium, must meet in some point T. We
may therefore suppose them to be stresses in three new pieces OT, QT, ST,
which will complete the frame.

Lei us suppose that both 0 and T are outside the rhombus, and that
OS intersect* PT in the point V, and let us apply Bow's method to construct
the diagram of stress reciprocal to this frame.

If we letter the areas as follows, putting

A for the rhombus PRQS,
B for the triangle PSV,
C for the triangle OTV,
D for the quadrilateral ORPV,
E for the quadrilateral QSVT,
and F for the space outside the frame,

then, in the diagram of stress, the stresses of the four sides of the rhombus
will meet in A, and since the opposite sides of the rhombus are parallel, the
lines EA and AD will be in one straight line, and the lines BA and AF
will also be in a straight line.

Also since in the frame the pieces OR and OS are equal, the angles
ORP, PSV are equal, and the corresponding angles FDA, ABE must be equal,
and therefore the quadrilateral BEFD can be inscribed in a circle, and therefore
the angles FEA, DBA are equal, and the corresponding angles in the frame
TQS, SPY are equal, and therefore PT is equal to QT.

If; therefore, 0 is in one diagonal of the rhombus, T must be in the
other diagonal.

The diagram of stress is completed by drawing EC parallel to BD, and
DC parallel to BE, and joining FC.

This diagram therefore consists of a parallelogram BDCE, a diagonal ED,
a point F in the circle passing through FBD, and four lines drawn from F
to the angles of the parallelogram.

If we now begin with the diagram of stress, and proceed to construct a
frame reciprocal to it, the form of the frame will be different according to the
cyclical direction in which the sides of the rhombus PRQS are lettered. If
in the one case we have the points 0 and T both outside the rhombus as
in fig. 1, in the other 0 and T will both be within the rhombus as in fig. 3.

BOW'S METHOD OF DRAWING DIAGRAMS IN GRAPHICAL STATICS. 497

The stresses in the corresponding pieces of fig. 1 and fig. 3 are all equal if
they are equal in any pair of them.

If in the frames represented in fig. 1 and fig. 3, we consider that the
pieces OS and TP cross one another at V without intersecting, we have six
points 0, P, Q, R, S, T joined by nine lines. Now in general if p points in
a plane are joined by 2p — 3 lines the figure is simply stiff, that is to say
the form of the figure is determined by the lengths of the lines, and there are
no necessary relations between the lengths of the lines.

But in Peaucellier's linkage the length of any line, as OT, is determined
when those of the other eight are given. For if a is the length of a side
of the rhombus, b the length of either arm OR or OS, c the length of either
arm TP or TQ, then if OT=d,

Hence if any one of the nine pieces of the linkage be removed, the motion
of the remaining eight will be the same as before, and a given stress in any
one of the nine will produce stresses in each of the other eight which are
determinate in magnitude when the configuration of the linkage is given, though
they alter during the motion of the linkage.

VOL. II.

63

[From the Pnc*di*9» of the Cambridge Philotophical Society, Vol. II., 1876.]

LXXVI. On the Equilibrium of Heterogeneous Substances.

TUB thermodynamical problem of the equilibrium of heterogeneous substances
was first attacked by Kirchoff in 1855, who studied the properties of mixtures
of sulphuric acid with water, and the density of the vapour in equilibrium with
the mixture. His method has recently been adopted by C. Neumann in his
Vorksungen fiber die mechanische Theorie der Warme (Leipzig, 1875). Neither
of these writers, however, makes use of two of the most valuable concepts in
Thermodynamics, namely, the intrinsic energy and the entropy of the substance.

It is probably for this reason that their methods do not readily give ;m
explanation of those states of equilibrium which are stable in themselves, but
which the contact of certain substances may render unstable.

I therefore wish to point out to the Society the methods adopted by
Professor J. Willard Gibbs of Yale College, published in the Transactions of
the Academy of Sciences of Connecticut, which seem to me to throw a new-
light on Thermodynamics.

He considers the intrinsic energy (c) of a homogeneous mass consisting of
H kinds of component matter to be a function of n + 2, variables, namely, the
volume of the mass v, its entropy rj, and the n masses, w,, m,...wn, of its
component substances.

Each of these variables represents a physical quantity, the value of which,
for a material system, is the sum of its values for the parts of the system.

By differentiating the energy with respect to each of these variables (con-
sidered as independent), we obtain a set of n + 2 differential coefficients which
represent the intensity of various properties of the substance. Thus,

EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 499

gj = -p, where p is the pressure of the substance ;

j- = 0, where 0 is the temperature on the thermodynamic scale;

= /i1) where p., is the potential of the component (m,) with respect to the

compound mass.

Each of the component substances has therefore a potential with respect
to the whole mass.

The idea of the potential of a substance is, I believe, due to Prof. Gibbs.
His definition is as follows : —

If to any homogeneous mass we suppose an infinitesimal quantity of any
substance to be added, the mass remaining homogeneous, and its entropy and
volume remaining unchanged, the increase of the energy of the mass, divided
by the mass of the substance added, is the potential of that substance in the
mass considered.

The condition of the stable equilibrium of the mass is expressed by Prof.
Gibbs in either of the two following ways :

I. For the equilibrium of any isolated system it is necessary and sufficient
that in all possible variations of the state of the system which do not alter its
energy, the variation of its entropy shall either vanish or be negative.

II. For the equilibrium of any isolated system it is necessary and sufficient
that in all possible variations of the state of the system which do not alter its
entropy, the variation of the energy shall either vanish or be positive.

The variations here spoken of must not involve the transportation of any
matter through any finite distance.

It follows from this that the quantities 0, p, /v-/"^ must have the same
values in all parts of the mass. For if not, heat will flow from places of
higher to places of lower temperature, the mass as a whole will move from
places of higher to places of lower pressure, and each of the several component
substances will pass from places where its potential is higher to places where
it is lower, if it can do so continuously.

Hence Prof. Gibbs shews that if @, P, M^..Mn are the values of 0, p, ^...p,,
for a given phase of the compound, and if the quantity

K= e - @r? + Pv - M1m1 - &c. - Mnmnt

63—2

EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES.

M aero for the given fluid, and is positive for every other phase of the same
component*, the condition of the given fluid will be stable.

If this condition holds for all variations of the variables the fluid will be
absolutely stable, but if it holds only for small variations but not for certain
finite variations, then the fluid will be stable when not in contact with matter
in any of those phases for which K is positive, but if matter in any one of
these phases is in contact with it, its equilibrium will be destroyed, and a
portion will pass into the phase of the substance with which it is in contact.

Thus in Professor F. Guthrie's experiments, a solution of chloride of calcium
of 37 per cent, was cooled to a temperature somewhat below — 37* C. without
solidification.

In this state, however, the contact of three different solids determines three
different kinds of solidification. A piece of ice causes ice to separate from the
fluid. A piece of the cryohydrate of cMoride of calcium determines the formation
of cryohydrate from the fluid, and the anhydrous salt causes a precipitation of
anhydrous salt.

The phase of the fluid is such that K is positive for all phases differing
slightly from its own phase, and its equilibrium is therefore stable, but for
certain widely different phases, namely, ice, cryohydrate and anhydrous salt,
A' is negative.

If none of these substances are in contact with the fluid, the fluid cannot
alter in phase without a transport of matter through a finite distance, and is
therefore stable; but if any one of them is in contact with the fluid, part of
the fluid is enabled to pass into a phase in which K is negative. The con-
ditions of consistent phases are that the values of 0, p, ft,.../tn, and K are
equal for all phases which can coexist in equilibrium, the surface of contact
being plane.

This was illustrated by Mr Main's experiments on co-existent phases of
mixtures of chloroform, alcohol and water.

[From Nature, Vol. xiv.]

LXXVII. Diffusion of Gases through Absorbing Substances.

THE importance of the exact study of the motions of gases, not only as
a method of distinguishing one gas from another, but as likely to increase our
knowledge of the dynamical theory of gases, was pointed out by Thomas Graham.
Graham himself studied the most important phenomena, and distinguished from
each other those in which the principal effect is due to different properties of
gases.

The motion of large masses of the gas approximates to that of a perfect
fluid having the same density and pressure as the gas. This is the case with
the motion of a single gas when it flows through a large hole in a thin
plate from one vessel into another in which the pressure is less. The result
in this case is found to be in accordance with the principles of the dynamics
of fluids. This was approximately established by Graham, and the more accurate
formula, in which the thermodynamic properties of the gas are taken into
account, has been verified by the experiments of Joule and Thomson. (Proc.
R. S., May, 1856.)

When the orifice is exceedingly small, it appears from the molecular theory
of gases that the total discharge may be calculated by supposing that there
are two currents in opposite directions, the quantity flowing in each current
being the same as if it had been discharged into a vacuum.

For different gases the volume discharged in a given time, reduced to
standard pressure and temperature, is proportional to —

where p is the actual pressure, s is the specific gravity, and 6 the temperature
reckoned from — 274° C.

Dim-BIOS OF OAOi THBOUOH ABSORBING SUBSTANCES.

When the gM« in the two vends are different, each gas is discharged
according to this l»w independently of the other.

Tbew phenomena, however, can be observed only when the thickness of the
pkte and the diameter of the aperture are very small.

When this u the case, the distance is very small between a point in the
limt TOM til where the mixed gas has a certain composition, and a point in the
i^yprnl reaael where the mixed gas has a quite different composition, so that
the Telocity of diffusion through the hole between these two points is large
r..miiared with the velocity of flow of the mixed gas arising from the difference
uf tin- total pressures in the two vessels.

When the hole is of sensible magnitude this distance is larger, because
the region of mixed gases extends further from the hole, and the effects of
diffusion become completely masked by the effect of the current of the gas
in mass, arising from the difference of the total pressures in the two vessels.
In this latter case the discharge depends only on the nature of the gas in
the vessel of greater pressure, and on the resultant pressures in the two vessels.
It consists entirely of the gas of the first vessel, and there is no appreciable
o»unter current of the gas of the other vessel.

Hi -nee the experiments on the double current must be made either through
H single very small aperture, as in Graham's first experiment with a glass vessel
accidentally cracked, or through a great number of apertures, as in Graham's
later experiments with porous septa of plaster of Paris or of plumbago.

With such septa the following phenomena are observed :—

When the gases on the two sides of the septum are different, but have the
same pressure, the reduced volumes of the gases diffused in opposite directions
through the septum are inversely as the square roots of their specific gravities.

If one or both of the vessels is of invariable volume, the interchange of
gas will cause an inequality of pressure, the pressure becoming greater in the
vessel which contains the heavier gas.

If a vessel contains a mixture of gases, the gas diffused from the vessel through
a porous septum will contain a larger proportion of the lighter gas, and the pro-
portion of the heavier gas remaining in the vessel will increase during the process.

The rate of flow of a gas through a long capillary tube depends upon the
viscosity OB internal friction of the gas, a property quite independent of its
specific gravity.

DIFFUSION OF GASES THROUGH ABSORBING SUBSTANCES. 503

The phenomena of diffusion studied by Dr v. Wroblewski are quite distinct
from any of these. The septum through which the gas is observed to pass is
apparently quite free from pores, and is indeed quite impervious to certain
gases, while it allows others to pass.

It was the opinion of Graham that the substance of the septum is capable
of entering into a more or less intimate combination with the substance of the
gas ; that on the side where the gas has greatest pressure the process of
combination is always going on ; that at the other side, where the pressure of
the gas is smaller, the substance of the gas is always becoming dissociated from
that of the septum ; while in the interior of the septum those parts which
are richer in the substance of the gas are communicating it to those which
are poorer.

The rate at which this diffusion takes place depends therefore on the power
of the gas to combine with the substance of the septum. Thus if the septum
be a film of water or a soap-bubble, those gases will pass through it most
rapidly which are most readily absorbed by water, but if the septum be of
caoutchouc the order of the gases will be different. The fact discovered by
St Claire-Deville and Troost that certain gases can pass through plates of red-
hot metals, was explained by Graham in the same manner.

Franz Exner * has studied the diffusion of gases through soap-bubbles, and
finds the rate of diffusion is directly as the absorption-coefficient of the gas,
and inversely as the square root of the specific gravity.

Stefan t in his first paper on the diffusion of gases has shewn that a law
of this form is to be expected, but he says that he will not go further into
the problem of the motion of gases in absorbing medium, as it ought to form
the subject of a separate investigation.

Dr v. Wroblewski has confined himself to the investigation of the relation
between the rate of diffusion and the pressure of the diffusing gas on the two
sides of the membrane. The membrane was of caoutchouc, 0'0034 cm. thick.
It was almost completely impervious to air. The rate at which carbonic acid
diffused through the membrane was proportional to the pressure of that gas,
and was independent of the pressure of the air on the other side of the

* Pogg. Ann., Bd. 155.

f Ueber das Gkichgeuncht u. d. Diffusion von Gasgemengen. Sitzb. der k. Akad. (Wien). Jan. 5,

1871.

: niriTSION OF OABBB THROUGH ABSORBING SUBSTANCES.

provided thtt air was from carbonic acid. The connexion between
UIM rMok and Henry's law of absorption is pointed out.

The time of diffusion of hydrogen through caoutchouc is 3*6 times that of
an equal volume of carbonic acid The diffusion of a mixture of hydrogen and
carbonic acid takes place as if each gas diffused independently of the other
at a rate proportional to the part of the pressure which is due to that gas.

We hope that Dr v. Wroblewski will continue his researches, and make
a complete investigation of the phenomena of diffusion through absorbing sub-

[From the Kensington Museum Handbook, pp. 1 — 21.]

LXXVIII. General considerations concerning Scientific Apparatus.

1. EXPERIMENTS.

THE aim of Physical Science is to observe and interpret natural phenomena.

Of natural phenomena, some — as, for example, those of astronomy — are not
subject to our control, and in the study of these we can make use only of
the method of Observation. When, however, we can cause the phenomenon to
be repeated under various conditions, we are in possession of a much more
powerful method of investigation — that of Experiment.

An Experiment, like every other event which takes place, is a natural
phenomenon ; but in a Scientific Experiment the circumstances are so arranged
that the relations between a particular set of phenomena may be studied to
the best advantage.

In designing an Experiment the agents and phenomena to be studied are
marked off from all others and regarded as the Field of Investigation. All
agents and phenomena not included within this field are called Disturbing
Agents, and their effects Disturbances ; and the experiment must be so arranged
that the effects of these disturbing agents on the phenomena to be investigated
shall be as small as possible.

We may afterwards change the field of our investigation, and include
within it those phenomena which in our former investigation we regarded as
disturbances. The experiments must now be designed so as to bring into
prominence the phenomena which we formerly tried to get rid of. When we
have in this way ascertained the laws of the disturbances, we shall be better
prepared to make a more thorough investigation of what we began by regarding
as the principal phenomena.

VOL. II. 64

ThtM, in experiment* where we endeavour to detect or to measure a force
by irfrrr-'t the motion which it produces in a movable body, we regard
Friction M a disturbing agent, and we arrange the experiment so that the
motion to be ob»erred may be impeded as little as possible by friction.

2. APPARATUS.
Ewrthing which is required in order to make an experiment is called

A piece of apparatus constructed specially for the performance of experiments
is called an Instrument.

Apparatus may be designed to produce and exhibit a particular phenomenon,
to eliminate the effects of disturbing agents, to regulate the physical conditions
of the phenomenon, or to measure the magnitude of the phenomenon itself.

In many experiments, special apparatus is required for all these purposes,
but certain pieces of apparatus are used in a great variety of experiments, and
there are whole classes of instruments which have certain principles of COH-
Ktruction in common.

Thus, in all instruments in which motion is to be produced there must
be a prime mover or driving power, and a train of mechanism to connect the
prime mover with the body to be moved ; and in many cases additional apparatus

fceasary — such as a break to destroy the superfluous energy of the prime
mover, or a reservoir to store up its energy when not required; and we may
liave special apparatus to measure the force transmitted, the velocity produced,
or the work done, or to regulate them by automatic governors.

We may make a somewhat similar classification of the functions of apparatus
U-li.nging to other physical sciences — such as Electricity, Heat, Light, Sound, &c.

:*. GENERAL PRINCIPLE OF THE CONSTRUCTION OF APPARATUS.

There are certain primary requisites, however, which are common to all
uments, and which therefore are to be carefully considered in designing or
selecting them. The fundamental principle is, that the construction of the
iii-trument should be adapted to the use that is to be made of it, and in
(•articular, that the parts intended to be fixed should not be liable to become
displaced; that those which ought to be movable should not stick fast; that

CONCERNING SCIENTIFIC APPARATUS. 507

parts which have to be observed should not be covered up or kept in the
dark; and that pieces intended to have a definite form should not be disfigured
by warping, straining, or wearing.

It is therefore desirable, before we enter on the classification of instruments
according to the phenomena with which they are connected, to point out a
few of the principles which must be attended to in all instruments.

Each solid piece of an instrument is intended to be either fixed or movable,
and to have a certain definite shape. It is acted on by its own weight, and
other forces, but it ought not to be subjected to unnecessary stresses, for these
not only diminish its strength, but (what for scientific purposes may be much
more injurious) they alter its figure, and may, by their unexpected changes
during the course of an experiment, produce disturbance or confusion in the
observations we have to make.

We have, therefore, to consider the methods of relieving the pieces of an
instrument from unnecessary strain, of securing for the fixed parts a determinate
position, and of ensuring that the movable parts shall move freely, yet without
shake.

This we may do by attending to the well-known fact in kinematics — " A
EIGID BODY HAS Six DEGREES OF FREEDOM."

A rigid body is one whose form does not vary. The pieces of our instru-
ments are solid, but not rigid. They are liable to change of form under stress,
but such change of form is not desirable, except in certain special parts, such
as springs.

Hence, if a solid piece is constrained in more than six ways it will be
subject to internal stress, and will become strained or distorted, and this in
a manner which, without the most exact micrometrical measurements, it would
be impossible to specify.

In instruments which are exposed to rough usage it may sometimes be
advisable to secure a piece from becoming loose, even at the risk of straining
and jamming it ; but in apparatus for accurate work it is essential that the
bearings of every piece should be properly defined, both in number and in position.

4. METHODS OF PLACING AN INSTRUMENT IN A DEFINITE POSITION.

When an instrument is intended to stand in a definite position on a fixed
base it must have six bearings, so arranged that if one of the bearings were

64—2

GEXKRAL CONSIDERATIONS

direction in which the corresponding point of the instrument would
be left ftve to move by the other bearings must be as nearly as possible
normal to the tangent plane at the bearing.

(This condition implies that, of the normals to the tangent planes at the
bearings, no two coincide; no three are in one plane, and either meet in a
point or are parallel ; no four are in one plane, or meet in a point, or are
parallel, or, more generally, belong to the same system of generators of an
hyperboloid of one sheet. The conditions for five normals and for six are more
complicated.)*

These uomjiiinnt are satisfied by the well-known method of forming on the
fixed base three V grooves, whose sides are inclined 45* to the base, and whose
directions meet in a point at angles of 120°. The instrument has three feet ;
the end of each foot is, roughly speaking, conical, but so rounded off that it
bears against the two sides of the groove, and cannot reach the bottom. The
instrument has thus six solid bearings, and is kept in its place by its weight,
without being subjected to any unnecessary strain.

Sir William Thomson, who has bestowed much attention on this subject,
has adopted a somewhat different arrangement in some of his instruments. A
triangular hole, like that formed by pressing an angle of a cube into a mass
of clay, is formed in the base, and a V groove is cut in a direction passing
through the centre of the hole. The three feet of the instrument are all rounded,
but of different lengths. The longest stands in the triangular hole, and has
three bearings ; the second stands in the V groove, and has two bearings ; and
the third stands on the horizontal plane of the base, and has one bearing.
There are, thus six bearings in all. This method, though it does not give so
large a margin of stability as the method of three grooves, has this advantage,
that as each of the three feet is differently formed, it is impossible to put
any but the proper foot into the hole without detecting the mistake.

5. BEARINGS OF MIRRORS.

In mounting mirrors it is especially important to attend to the number
and position of their bearings, for any stress on the mirror spoils its figure,
and renders it useless for accurate work.

• See Ball on the Theory of Screwt.

CONCERNING SCIENTIFIC APPARATUS. 509

For small mirrors it is best to make one face of the mirror rest against
three solid bearings, and to keep it in contact with these by three spring
bearings placed exactly opposite to them against the other face of the mirror.
These will prevent any displacement of the mirror out of its proper plane. The
bearings against the edges of the mirror, by which it is prevented from shifting
in its own plane, are, in the case of small mirrors, of less importance.

When the mirror is large, as in the case of the speculum of a large
telescope, a greater number of bearings is required to prevent the mirror from
becoming strained by its own weight ; but in all cases the number of fixed
bearings at the back of the mirror must be three and only three, otherwise
any warping of the framework will entirely spoil the figure of the surface.

6. BEARINGS OP STANDARDS OF LENGTH.

It is of the greatest importance that the standard measure of length, by
which the national unit of length is defined, should not be exposed to strain.

The box in which the standard yard is kept in the Exchequer Chamber
is provided with bearings, the positions of which have been arranged so that
the bar may rest on them with as little strain throughout its substance as is
consistent with the fact that it is a heavy body.

7. ON THE BEARINGS OF MOVABLE PARTS.

The most important kinds of motion with one degree of freedom are,
(1) Rotation round an axis; (2) Motion of translation without rotation; and
(3) Screw motion, in which a definite rotation about an axis corresponds to a
definite motion of translation along that axis.

For one degree of freedom five solid bearings are required, the sixth con-
dition being supplied by that part of the instrument which regulates the motion

of the piece.

The construction of pieces capable of rotation about an axis is better
understood than any other department of mechanism.

In astronomical instruments, four of the bearings are supplied by the two
Y's on which the cylindrical end-pieces of the axle rest, and the fifth by the
longitudinal pressure of a bearing against one end of the axle, or a shoulder
formed upon it. The weight of the instrument is generally sufficient to keep

COH8IDKUATIOM8

• contact with iU bearing*; but when the weight is so great that the

MOTTO on the bearing* » Wcely to injure them, the greater part of the weight

» mppoctad by auxiliary bearings, the pressure of which is regulated by counter-

niBM w gpriogs, leaving only a moderate pressure to be borne by the true

brariflft

8. TRANSLATION.

Motion of translation in a fixed direction, without rotation.

This kind of motion is required for pieces which slide along straight fixed
pieces, as the verniers and microscopes of measuring apparatus, such as catheto-
m^nrn and micrometers, the slide-rests of lathes, the pistons of steam-engines

and pumps, Ac.

When a tripod stand is to have a motion of this kind in a horizontal
plane, two of its feet may be made to slide in a V groove, while the third
rests on the horizontal plane.

When a cylindrical rod is to have a longitudinal motion, it must be made
to bear against two fixed Y's, and must be prevented from rotating on its
axis by a bearing, connected either with the cylinder or the fixed piece, which
glides on a surface whose plane passes through the axis of the cylinder.

When, as in cathetometers and other measuring apparatus, a piece has to
hliilo along a bar, the five bearings of the piece may be arranged so that
three of them form a triangle on one face of the bar, while the two others
rest against an adjacent face of the bar, the line joining these two being in
the direction of motion. These bearings may be kept tight, without the possi-
liility of jamming, by means of spring bearings against the other sides of the bar.

9. PARALLEL MOTIOX BY LINKWORK.

In all these methods of guiding a piece by sliding contact, there is a
considerable waste of energy by friction. In many cases, however, this is of
little moment, compared with the errors depending on the necessary imperfection
«f the guiding surfaces, arising not only from original defects of workmanship
but from straining and wearing during use.

It is true that great advances have been made, and notably by Sir J.
Whitworth, in the art of forming truly plane and cylindric surfaces; but even

CONCERNING SCIENTIFIC APPARATUS. 511

these are liable to become altered, not only by wear but by strain and by
inequalities of temperature, so that it is never safe to depend upon the perfect
accuracy of the fitting of a large bearing surface, except when the pressure is
very great.

In linkwork, on the other hand, the relative motion of any two pieces
at their mutual bearing' is one of pure rotation about a well-turned axle. The
extent of the sliding surfaces is thus reduced to a minimum, so that less
power is lost by friction, and the workmanship .of such bearings can be brought
much nearer to perfection than that of any other kind of fittings. Hence, in
all prime movers and other machines, in which waste of power by friction is
to be avoided, and even in those in which great accuracy is required, it is
desirable, if it is possible, to guide the motion by linkwork.

The so-called " Parallel Motion " invented by James Watt was the first
attempt to guide a motion of translation by means of linkwork ; but though
the motion as thus guided is very nearly rectilinear, it is not exactly so.
Various other contrivances have been invented since the time of Watt, as, for
instance, that fitted to the engines of the Gorgon by Mr Seaward ; but all of
them involved either a deviation from true rectilinear motion, or a sliding
contact on a plane surface, and it was generally supposed by mathematicians
that a true rectilinear motion, guided by pure linkwork, was a geometrical
impossibility.

It was in the year 1864 that M. Peaucellier published his invention of an
exact parallel motion by pure linkwork, and thus opened up the path to a
very great extension of the science of mechanism, and its practical applications.
The linkwork motions constructed by M. Garcia, Mr Penrose, and others, and
the extensions of the theory of linkwork by Sylvester, Hart, and Kempe, are
now well known, but they could not be fully described within our present limits.

10. SCREW MOTION.

The adjustments of instruments are to a great extent made by means of
screws. In the case of levelling screws, which bear the weight of an instru-
ment, the thread of the screw is always in contact with its proper bearing
in the nut; but in micrometer screws it is necessary to secure this contact
by means of a spring. This spring is sometimes made to bear against the end
of the screw, or a shoulder turned upon it; but this arrangement causes a

I U

CON8I DERATIONS

variable praMure a* the screw moves forward. A much better arrangement is
to make the spring bear, not against the screw itself, but against a nut which
» fa* to move on the screw, but which is prevented from turning round by a
proper bearing. Tliis movable nut always remains at the same distance from
the fixed one, so that the pressure of the spring remains constant. This is
the arnogement of the micrometer screws in Sir W. Thomson's electrometers.

11. ON CONTRIVANCES FOR SECURING FREEDOM OF MOTION.

In many instruments there is a movable part or indicator, the position or
motion of which is to be observed in order to deduce therefrom some conclusion
with respect to the force which acts upon it. This force may be the weight
of a body, or an attractive or repulsive force of any kind ; but, besides the
force we are investigating, the resistance called Friction is always acting as a
disturbing force.

If the magnitude and direction of the force of friction were at all times
accurately known, this would be of less consequence ; but the amount of friction
Is liable to sudden alterations, owing to causes which we can often neither
suspect nor detect, so that the only way in which we can make any approach
to accuracy is by diminishing as much as possible the effect of friction. The
modes by which this is effected are of two kinds. Whenever there is sliding
contact, there is friction ; and wherever there is complete freedom of motion
tlu-re must be sliding contact ; but by making the extent of the sliding motion
small compared with the motion of the indicating part, we may reduce the effect
• •f friction to a very small part of the whole effect.

This is done in rotating parts by diminishing the size of the axle, and
by supporting it on friction-wheels ; and in toothed wheels by keeping the
bearings of the teeth as near as possible to the line of centres, or more perfectly
by cutting the teeth obliquely, as in Hooke's teeth.* A compass needle is
balanced on a fine point, and the extent of the bearing is so small that a
very small force applied to either end of the needle is sufficient to turn it round.

In all these instances the effect of friction is reduced by diminishing the
extent of the sliding motion.

In balances and other levers the bearing of the lever is in the form of
a prism, called a knife-edge, having an angle of about 120°; the edge of this

* Communicated to the Royal Society in 1666. See Willis's Principles of Mecftaiiism, 1870, p. 53.

CONCERNING SCIENTIFIC APPAEATUS. 513

prism is accurately ground to a straight line, and rests on a plane horizontal
surface of agate.

The relative motion in this case is one of rolling contact.

In another class of instruments sliding and rolling are entirely done away
with, and sufficient freedom of motion is secured by the pliability of certain
solid parts.

Thus many pendulums are hung, not on knife-edges, but on pieces of
watch-spring, and torsion balances are suspended by metallic wires or by silk
fibres. The motion of the piece is then affected by the elastic force of the
suspension apparatus, but this force is much more regular in its action than
friction, and its effects can be accurately taken account of, and a proper cor-
rection applied to the observed result.

12. THE TORSION ROD, OR BALANCE OF TORSION.

The balance of torsion has been of the greatest benefit to modern science
in the measurement of small forces. The first instrument of the kind was that
constructed by the Rev. John Michell, formerly Woodwardian Professor of Geology
at Cambridge, in order to observe the effect of the attraction of a pair of
large lead balls on a pair of smaller balls hung from the extremities of the
rod of the balance. Michell, however, died before he had opportunity to make
the experiment, and his apparatus came into the hands of Professor F. J. H.
Wollaston, and was transmitted by him to Henry Cavendish. Cavendish
greatly improved the apparatus,* and successfully measured the attraction of the
balls, and thus determined the density of the earth. -f-

The experiment has since been repeated by Reich and Baily. In the mean-
time, however, independently of Michell, and before Cavendish had actually
used the instrument, Coulomb J had invented a torsion balance, by which he
established the laws of the attraction and repulsion of electrified and magnetic
bodies.

* Cavendish's apparatus now belongs to the Royal Institution,
t Philosophical Transactions, 1798.

* Mem. de I' Academic, 1784, tfcc.

VOL. IL

514 pf*"***- CONSIDERATIONS

13. Hi FILAR SUSPENSION.

elastic force of torsion of a wire, though much more regular than the
force of friction, is subject to alterations arising from hitherto unknown causes,
but probably depending on facts in the previous history of the wire, such as
iu baring been subjected to twists and other strains before it was hung up.
it is sometimes better to employ another mode of suspension, in which

force of restitution depends principally on the weight of the suspended parts.

The body is suspended by two wires or fibres, which are close together
and nearly vertical, and are so connected by a pulley that their tensions are
equal. The body is in equilibrium when the two fibres are in the same plane.
When the body is turned about a vertical axis, the tension of the fibres produces
a force tending to turn the body back towards its position of equilibrium ; and
this force is very regular in its action, and may be accurately determined by
proper experiments.

This arrangement, which is called the Bifilar suspension, was invented by
Gauss and Weber for their magnetic apparatus. It was afterwards used by
Baily in his experiments on the attraction of balls.

14. METHODS OF READING.

The observed position of the indicating part of an instrument is recorded
as the "Reading." To ascertain the position of the indicating part of the
instrument various methods have been adopted. The commonest method is to
make the indicating part in the form of a light needle, the point of which
moves near a graduated circle. The position of the needle is estimated by
observing the position of its point with respect to the divisions of the scale.
By giving the needle two points at opposite extremities of a diameter, and
observing the position of both points, we may eliminate the errors arising from
the want of coincidence between the centre of the graduated circle and the
axis of motion of the needle.

This is the method adopted in ordinary magnetic compasses. As it is

necessary for freedom of motion that the point of the needle should not be in

ctual contact with the graduated limb, the reading will be affected by any

change in the position of the eye of the observer. The error thus introduced

CONCERNING SCIENTIFIC APPARATUS. 515

is called the error of Parallax. In some instruments, therefore, the observation
is made through an eye-hole in a definite position. A better plan, however, is
to place a plane mirror under the needle, and in taking the reading to place
the eye so that the needle appears to cover its own reflexion in the mirror.

15. SPIEGEL- ABLESUNG, OR MIRROR-READING.

A still more accurate method is that invented by Poggendorff, and used
by Gauss and Weber in their magnetic observations. A small plane mirror is
attached to the indicating piece, so as to turn with it about its axis, which
we may suppose vertical. A divided scale is placed so as to be perpendicular
to this axis, and so that a normal to the scale at its middle point passes
through the axis. The image of this scale by reflexion in the mirror is observed
by means of a telescope having a vertical wire in the plane of distinct vision.
As the indicating piece turns about its axis, the image of the scale passes
across the field of view of the telescope, and the coincidence of the image of
any division of the scale with the vertical wire of the telescope may be observed.

The error of parallax is entirely got rid of by this method, for the two
optical images whose relative position is observed are in the same plane.

Another method of using the mirror is to reverse the direction of the rays
of light by removing the eye-piece of the telescope, and putting the flame of
a lamp in its place. The light emerging from the object-glass falls on the
mirror, and is reflected so as to form on the scale a somewhat confused image
of the flame, with a distinct image of the vertical wire crossing it. The reading
is made by observing the position on the scale of the image of the vertical
wire. In many instruments the telescope is dispensed with, and the mirror is
a concave one, as in Thomson's reflecting galvanometer.

Some German writers distinguish this method of using the mirror and scale
with a lamp as the objective method, the method in which the observer looks
through the telescope being called the subjective method. The objective method
is the only one adapted for the photographic registration of the readings.

16. RAMSDEN'S GHOST.

To ascertain the exact position of an instrument with respect to a plumb-
line without touching the line, Ramsden fixed a convex lens to a part of his

65—2

i|(J OKXKRAL CONSIDERATIONS

.iirtniiiMjnt, and J*~-* a wire so that when the instrument was in its proper
pOMtion the image of the plumb-line formed by the lens exactly coincided with
the (bud wire. By moving the instrument till this coincidence was observed,
^ ingtrumrnt wan m«t«d to its proper position. This contrivance was long
known a* "lUnuden's Ghost." It is, in fact, a simple form of the reading
niiiiiiOOtlpn; »nd there is no better method of ascertaining that a delicately-
oliject w exactly in its proper place.

17. COLLIMATINO TELESCOPE.

If two telescopes are made to face one another, and if the cross-wires of
the ftflti M M0n through the second, appear to coincide with the cross-wires
of the second, the optic axes of the two instruments are parallel. This mode
of ascertaining parallelism is used in practical astronomy, and is called the method
of colliniating telescopes, or of collimators.

It is also used in the Kew portable magnetometer. The magnet is hollow,
and carries a lens at one end and a scale at the other, at the principal focus
of the lens. The magnet is thus a collimating telescope, and is observed by
means of a telescope mounted on a divided circle. The disadvantage of this
method is, that when the magnet is deflected, the scale soon passes out of the
field of view, and the observer has to shift his telescope, in order to get a
new reading.

18. THREE CLASSES OF READINGS.

We- may, in fact, arrange instruments in three classes, according to the
method of reading them.

In the self-recording class the observer leaves the instrument to itself, and
examines the record at his own convenience.

In those which depend on eye observations alone, the observer must be
there to look at the indicator of the instrument, but he does not touch it.

In the third class, which depend on eye and hand, the observer, before
taking the reading, must make some adjustment of the instrument.

19. FUNCTIONS OF INSTRUMENTS.

The foregoing remarks apply to the constituent parts of instruments, without
reference to the special department of science to which they belong.

CONCERNING SCIENTIFIC APPARATUS. 517

The classification of special instruments will be best understood if we arrange
those belonging to each department of science according to their respective
functions ; some of these functions having instruments corresponding to them
in several departments of science, while others are peculiar to one department.

All the physical sciences relate to the passage of energy, under its various
forms, from one body to another ; but Optics and Acoustics are often represented
as relating to the sensations of sight and hearing. These two sciences, in fact,
have a physiological as well as a physical aspect, and therefore some parts of
them have less analogy with the purely dynamical sciences.

The most important functions belonging to instruments, or elements of
instruments, are as follows : —

1. The Source of energy. The energy involved in the phenomenon we

are studying is not, of course, produced from nothing, but it enters
the apparatus at a particular place, which we may call the Source.

2. The channels or distributors of energy, which carry it to the places

where it is required to do work.

3. The restraints, which prevent it from doing work when it is not

required.

4. The reservoirs in which energy is stored up till it is required.

5. Apparatus for allowing superfluous energy to escape.

6. Regulators for equalising the rate at which work is done.

7. Indicators, or movable pieces, which are acted on by the forces under

investigation.

8. Fixed scales on which the position of the indicator is read off.

Thus in solid machinery we have —

1. Prime movers.

2. Trains of mechanism.

3. Fixed framework.

4. Fly-wheels, springs, weights raised.

5. Friction breaks.

6. Governors, pendulums, balance springs in watches.

7. Dynamometers, Strophometers, Watt's indicator, chronographs, &c.

8. Scales for these indicators. Standards of length and mass. Astro-

nomical standard of time.

-

L CONSIDERATIONS

tt» ph«0»«ft depending on fluid pressure we have—

1. Pump*, condensing and rarifying syringes, Orsted's Piezometer, Andrews's

apparatus for high pressure.

2. Pipes and tubes.

8. Packing, washers, caoutchouc tubes, paraffin joints; fusion and other
methods of making joints tight.

4. Air chambers, water reservoirs, vacuum chambers.

5. Safety valves.

6. Governors by Siemens and others, Cavailld-Col's regulator for organ

blast

7. Pressure-gauges, barometers, manometers, sphygmographs, &c. ; areometers,

and specific gravity bottles ; current meters, gas meters.
a Scales for these gauges.

For thermal phenomena —

1. Furnaces, blow-pipe flames, freezing mixtures, solar and electric heat.

2. Hot water pipes, copper conductors.

3. Non-conducting packing, cements, clothing, &c.; steam-jackets, and

ice-jackets.

4. Regenerators, heaters, <fec.

5. Condensers and safety valves.

6. Thermostats, (1) by regulation of gas; (2) by boiling a liquid of known

composition.

7. Thermometers. Pyrometers, Thermoelectric Pile, Siemens' resistance

thermometer; Calorimeters.

8. Standard temperatures : as those of melting ice, boiling water, &c.

For electric phenomena —

1. Electric machines, frictional machine, electrophorus, Holtz' machine;

voltaic batteries, thermo-electric batteries, magneto-electric machines.

2. Wires and other metallic conductors ; armatures of magnets.

3. Insulators.

4. Leyden jars and other "condensers;" secondary batteries or cells of

polarization ; magnets, and electro- magnets.

5. Rheostats, lightning conductors, &c.

6. Guthrie's voltastat, regulators of electric lamps, &c.

CONCERNING SCIENTIFIC APPARATUS. 519

7. Electroscopes and electrometers, Coulomb's torsion balance, voltameters,

galvanometers and electrodynamometers, magnetometers.

8. Standards of resistance, capacity, electro-motive force, &c., as the Ohm,

the microfarad, L. Clark's voltaic constant cell.

From the physical, as distinguished from the physiological point of view,
the science of Acoustics relates to the excitement of vibrations and the propa-
gation of waves in solids, liquids, and gases, and that of Optics to the excitement
of vibrations, and the propagation of radiation, in the luminiferous medium.

From the physiological point of view, only those waves in ordinary matter
are considered which excite in us the sensation of Sound, though waves which
do not excite this sensation can be detected and studied by appropriate methods.

In the physiological treatment of Optics, only those radiations are considered
which excite in us the sensation of Light, though other radiations can be
detected and studied by their thermal, chemical, and even mechanical effects.

VIBRATIONS AND WAVES.
PHYSICAL ASPECT OF ACOUSTICS.

1. Sources. Vibrations of various bodies.

Air — Organ pipes, resonators and other wind instruments.

Reed instruments.

The Siren.

Strings . . . Harp, &c.

Membranes . . . Drum, &c.

Plates . . . Gong, &c.

Bods . . . . Tuning-fork, &c.

2. Distributors. Air . . Speaking tubes, stethoscopes, &c.

Wood, . Sounding rods.
Metal, . Wires.

3. Pugging of floors, &c.

4. Reservoirs. Resonators, Organ Pipes, Sounding-boards.

5. Dampers of pianofortes.

6. Regulators. Organ Swell.

7. Detectors, the ear; Sensitive Flames, Membranes, Phonautographs, &c.

8. Tuning-forks, pitch-pipes, and musical scales.

GKVBRAL CONSIDERATIONS

HEARING.
PHYSIOLOGICAL ASPECT OP ACOUSTICS.

A|>|Mtr*tua for determining the conditions —

I. Of the audibility of sounds.

'2. Of the perception of the distinction of sounds.

M. Of the harmony or discord of simultaneous sounds.

4. Of the melodious succession of sounds.

5. Of the timbre of sounds, and of the distinction of vowel sounds.

6. Of the time required for the perception of the sensation of sound.

RADIATION.
PHYSICAL ASPECT OP OPTICS.

1. Sources of Radiation. Heated bodies, solid, liquid, and gaseous.
Solid. Heated by a blow-pipe as in the oxy-hydrogen limelight.

Heated by their own combustion, as in the magnesium light

and glowing coals.
Heated by an electric current, as the carbon electrodes of

the electric lamp.
Heated by concentrated radiation from other sources, as in

the phenomenon called Calescence.
Liquid, as in hot fused metals and other bodies.
Gaseous. Heated by their own combustion, as in flames.

Heated by a Bunsen burner, as the sodium light.
Heated by the voltaic arc.
Heated by the induction spark.

•J. Distributors. Burning mirrors and lenses, condensing lenses for solar
microscopes, magic lanterns, &c., lighthouse apparatus ; telescopes,
microscopes, &c.

M. Selectors. Absorbing media and coloured bodies in general, prisms
and spectroscopes, ruled gratings, &c. : tourmalines, Nicol's prisms,
and other polarizers.

CONCERNING SCIENTIFIC APPARATUS. 521

4. Phosphorescent, fluorescent, and calescent bodies.

5. Opaque screens, diaphragms, and slits.

6. Regulators. The iris of the eye.

7. Photometers, photographic apparatus, actinometer, thermopile, Bunsen

and Roscoe's photometer, selenium photometer, Crookes' radiometer,
and other instruments.

8. Fraunhofer's lines of reference, and maps of the spectrum. Standard

sperm candle, burning 120 grains of sperm per hour.

SIGHT.
PHYSIOLOGICAL ASPECT OF OPTICS.

Apparatus for determining the conditions of —

1. The visibility of objects, with respect to size, illumination, &c.

2. The perception of the distinction of objects.

3. The perception of colour, as depending on the composition of the

light coming from the object.

Apparatus for comparing the intensity of luminous impressions, as depending
on the intensity of the exciting cause and on the time during which it has
acted, and for tracing the course of the development of a luminous impression
from its first excitation to its final decay and extinction.

Ophthalmometers, for measuring the dimensions of the eye and determining
its motions ; and for ascertaining the two limits of distinct vision.

Ophthalmoscopes, for illuminating and observing the interior of the eye.

BIOLOGICAL APPARATUS.

1. For measuring the ingesta, egesta, and weight of living beings.

2. For testing the strength of animals and measuring the work done

by them.

3. For measuring the heat which they generate.

4. For determining the conditions of fatigue of muscles.

5. For investigating the phenomena of the propagation of impulses through

the nerves, and of the excitement of muscular action.
VOL. II. 66

CONCERNING SCIENTIFIC AITAHATUS.

6. For tracing and registering the rhythmic action of the circulatory and

VMpiratory operation* — cardiographs, sphygmographs, stethoscopes, &c.

7. Marey's apparatus for registering the paces of men, horses, &c., and

the actions of birds and insects during flight.

8. Instrument* for illuminating and rendering visible parts within the

living body— ophthalmoscopes, laryngoscopes, &c.— and arrangements
for transmitting light through parts of the living body.

9. Instruments for varying the electrical state of the body— induction

coils, Ac.

10. Instruments for determining the electric state of living organs—
galvanometers with proper electrodes, electrometers, &c.

[From the Kensington Museum Hand Booh]

LXXIX. Instruments connected with Fluids.

IN countries where the fertility of the soil is capable of being greatly
increased by artificial irrigation, the attention of all ingenious persons is naturally
directed to devising means whereby the labour of raising water may be diminished.
Hence we find that in China and in India, but especially in Egypt, great
progress was made in the art of producing and guiding the motion of water.

The first pump worked by a piston of which we have any account seems
to be that invented by Ctesibius of Alexandria, about 130 B.C.

The construction of this pump, as described by Vitruvius, resembles that
of the modern fire-engine. It had two barrels, which discharged the water
alternately into a closed vessel, the upper part of which contained air. This
air-chamber acted as a reservoir of energy, and equalised the pressure under
which the water was emitted from the discharge pipe.

Hero, a scholar of Ctesibius, invented a number of ingenious machines. He
delighted in curious combinations of siphons, by which fountains were made to
play under unusual circumstances. We should call such machines toys, but though
to us they have no longer any scientific value, we must regard them as among
the first instances of apparatus constructed not in order to minister directly
to man's necessities or luxuries, but to excite or to satisfy his curiosity with
respect to the more unusual phenomena of nature.

From the time of Ctesibius and Hero to that of Galileo (1600) pumps were
constructed chiefly for useful, as distinguished from scientific, purposes, and con-
siderable skill was developed in the art of forming the barrel and piston so
as to work with a certain degree of accuracy.

Galileo shewed that the reason why water ascends in a sucking pump is
not that Nature abhors a vacuum, but that the pressure of the atmosphere
acts on the free surface of the water, and that this pressure will force the

66—2

t IX8TBUM*XTS OOKNKTE1) WITH H.t ll>s.

water only to a height of 34 feet; for when this height is reached, the water

i DO further, and thU shews either that the pressure of the atmosphere is
tabooed by that of the column of water, or else thut Nature has become
reconciled to a vacuum.

From this time the production of a vacuum became the scientific object
^jin^d at by a greet number of inventors. Torricelli, the pupil of Galileo, in
Irtl'i made the greatest step in this direction, by filling with mercury a tube
cloud at one end and then inverting the tube with its open end in a vessel
ef mercury. If the tube was long enough the mercury fell, leaving an empty
1PQQ* at the top of the tube. The vacuum obtained by filling a vessel with
^JJLMIIJ jfiA removing the mercury without admitting any other matter is hence
called the Torricellian vacuum. Torricelli, therefore, gave us at once the mercury
pump and the barometer, though the subsequent history of these two instruments
w very different.

A little before 1654 Otto Von Guericke, of Magdeburg, first applied the
principle of the common pump to the production of a vacuum. The fitting of
the pistons of the pumps of those days, however, though sufficiently water-tight,
was by no means air-tight. He therefore began by filling a vessel with WUUT
and then removing the water by a water-pump. In his experiments he met
with many failures, but he continued to improve his apparatus till he could
not only exhibit most of the phenomena now shewn in exhausted receivers,
but till he had discovered the reason of the imperfection of the vacuum when
water was used to keep the pump air-tight.

"In the year 1658 Hooke finished an air-pump for Boyle, in whose laboratory
he was . an assistant ; it was more convenient than Guericke's, but the vacuum
was not so perfect ; yet Boyle's numerous and judicious experiments gave to
the exliausted receiver of the air-pump the name of the Boy lean vacuum, by
which it was long known in the greatest part of Europe. Hooke's air-pump
had two barrels, and with some improvements by Hauksbee it remained in use
until the introduction of Smeaton's pump, which, however, has not wholly
superseded it*."

The history of the air-pump after this time relates chiefly to the con-
trivances for insuring the working of the valves when the pressure of the
remaining air is no longer sufficient to effect it, and to methods of rendering

* Thomas Young's Lecture* on Nat. Phil (1807). Lecture xxx

INSTRUMENTS CONNECTED WITH FLUIDS. 525

the working parts air-tight without introducing substances the vapour of which
would continue to fill the otherwise empty space.

There is one form of air-pump, however, which we must notice, as in it
all packing and lubricating substances are dispensed with. This is the air-pump
constructed by M. Deleuil, of Paris*, in which the pistons are solid cylinders
of considerable length, and are not made to fit tightly in the barrels of the
pump. No grease or lubricating substance is used, and the pistons work easily
and smoothly in. the barrels. The space between the piston and the barrel
contains air, but the internal friction of the air in this narrow space is so great
that the rate at which it leaks into the exhausted part of the barrel is not
comparable with the rate at which the pump is exhausting the air from the
receiver. It has been shewn by the present writer that the internal friction
of air is not diminished even when its density is greatly reduced. It is for
this reason that this pump works satisfactorily up to a very considerable degree
of exhaustion.

Pumps of the type already described are still used for the rapid exhaustion
of large vessels, but since the physical properties of extremely rarefied gases
have become the object of scientific research, the original method of Torricelli
has been revived under various forms.

Thus we have one set of mercury-pumps in which the mercury is alternately
made to fill a certain chamber completely and to drive out whatever gas may
be hi it, and then to flow back leaving the chamber empty.

Sprengel's pump is the type of the other set. The working part is a vertical
glass tube longer than the height of the barometer, and so narrow that a
small portion of mercury placed in it is compelled by its surface-tension to fill
the whole section of the tube. The mercury is introduced into this tube from
a funnel at the top through a small India-rubber tube regulated by a pinch-
cock, so that the mercury falls in small detached portions, each of which drives
before it any air which may be in the tube till it escapes into a mercury-
trough, into which the bottom of the tube dips.

The vessel to be exhausted is connected to a tube which enters at the
side of the vertical tube near the top. Any air or other gas which may be
in the vessel expands into the vacuum left in the vertical tube between suc-
cessive portions of the falling mercury, and is driven down the tube by the
next portion of mercury into the mercury-trough, where it may be collected.
* Comptes Rendus, t. Ix. p. 571. Carl's Repertorium.

CONNECTED WITH FLUIDS.

A* long as the mrefection of the air is not very great the quantity of
air which ia included between successive portions of mercury is sufficient to act
M a sort of buffer, but as the rarefaction increases the portions of mercury
ooinQ together more abruptly, and produce a sound which becomes sharper as
the vacuum becomes more perfect.

After the mercury-pump has been in action for some hours the quantity
of matter remaining in the vessel is very small. If the tubes have been joined
by means of caoutchouc connections there is a trace of gaseous matter emitted
by the caoutchouc. It is therefore necessary, when a very perfect vacuum is
desired, to moke all the joints " hermetical " by fusing the glass. Still, however,
there remains a trace of matter. The vapour of mercury is, of course, present,
and the sides of the glass vessel retain water very strongly, and part with
it very slowly, when all other matter is removed.

By passing a little strong sulphuric acid through the pump along with the
mercury, vapours both of mercury and of water may be in great measure
removed.

MM. Kundt and Warburg have got rid of an additional quantity of water-
substance by heating the vessel to as high a temperature as the glass will
bear while the pump was kept in action.

A method which has been long in use for getting a good vacuum is to
place in the vessel a stick of fused potash, and to fill it with carbonic acid,
and, after exhausting as much as possible, to seal up the vessel. The potash
is then heated, and when it has again become cold, most of the remaining
carbonic acid has combined with the potash.

Another method, employed by Professor Dewar, is to place in a compart-
ment of the vessel a piece of freshly heated cocoa-nut charcoal, and to heat
it strongly during the last stages of the exhaustion by the mercury-pump.
The vessel is then sealed up, and as the charcoal cools it absorbs a very large
proportion of the gases remaining in the vessel.

The interior of the vessel, after exhaustion, is found to be possessed of
very remarkable properties.

One of these properties furnishes a convenient test of the completeness of
the exhaustion. The vessel is provided with two metallic electrodes, the ends
of which within the vessel are within a quarter of an inch of each other. When
the vessel contains air at the ordinary pressure a considerable electromotive
force is required to produce an electric discharge across this interval. As the

INSTRUMENTS CONNECTED WITH FLUIDS. 527

exhaustion proceeds the resistance to the discharge diminishes till the pressure
is reduced to that of about a millimetre of mercury. When, however, the
exhaustion is made very perfect the discharge cannot be made to take place
between the electrodes within the vessel, and the spark actually passes through
several inches of air outside the vessel before it will leap the small interval
in the empty vessel. A vacuum, therefore, is a stronger insulator of electricity
than any other medium.

MM. Kundt and Warburg have experimented on the viscosity of the air
remaining after exhaustion, and on its conductivity for heat. They find that
it is only when the exhaustion is very perfect that the viscosity and con-
ductivity begin sensibly to diminish, even when the stratum of the medium
experimented on is very thin.

But the most remarkable phenomenon hitherto observed in an empty space
is that discovered by Mr Crookes. A light body is delicately suspended in an
exhausted vessel, and the radiation from the sun, or any other source of light
or heat, is allowed to fall on it. The body is apparently repelled and moves
away from the side on which the radiation falls.

This action is the more energetic the greater the perfection of the vacuum.
When the pressure amounts to a millimetre or two the repulsion becomes very
feeble, and at greater pressures an apparent attraction takes place, which, how-
ever, cannot be compared either in regularity or in intensity to the repulsion
in a good vacuum.

From these instances we may see what important scientific discoveries may
be looked for in consequence of improvements in the methods of obtaining a
vacuum.

[From Nature, VoL xnr.]

LXXX. nTietpflFs Writings and Correspondence.

WE frequently hear the complaint that as the boundaries of science are
widened its cultivators become less of philosophers and more of specialists, each
confining himself with increasing exclusiveness to the area with which he is
familiar. This is probably an inevitable result of the development of science,
which has made it impossible for any one man to acquire a thorough knowledge
of the whole, while each of its sub-divisions is now large enough to afford
occupation for the useful work of a lifetime. The ablest cultivators of science
are agreed that the student, in order to make the most of his powers, should
ascertain in what field of science these powers are most available, and that he
should then confine his investigations to this field, making use of other parts
of science only in so far as they bear upon his special subject.

Accordingly we find that Dr Whewell, in his article in the Encyclopedia
.\f'-tii>jnJit<tmt, on "Archimedes and Greek Mathematics," says of Eratosthenes,
who, like himself, was philologer, geometer, astronomer, poet, and antiquary :
"It is seldom that one person attempts to master so many subjects without
incurring the charge and perhaps the danger of being superficial."

It is probably on account of the number and diversity of the kinds of
intellectual work in which Dr Whewell attained eminence that his name is most
widely known. Of his actual performances the "History" and the "Philosophy
nf the Inductive Sciences" are the most characteristic, and this because his
practical acquaintance with a certain part of his great subject enabled him the
better to deal with those parts which he had studied only in books, and to
describe their relations in a more intelligent manner than those authors who
devoted themselves entirely to the general aspect of human knowledge
without being actual workers in any particular department of it.

WHEWELL'S WRITINGS AND CORRESPONDENCE. 529

But the chief characteristic of Dr Whewell's intellectual life seems to have
been the energy and perseverance with which he pursued the development of
each of the great ideas which had in the course of his life presented itself to
him. Of these ideas some might be greater than others, but all were large.

The special pursuit, therefore, to which he devoted himself was the elabora-
tion and the expression of the ideas appropriate to different branches of
knowledge. The discovery of a new fact, the invention of a theory, the solution
of a problem, the filling up of a gap in an existing science, were interesting
to him not so much for their own sake as additions to the general stock of
knowledge, as for their illustrative value as characteristic instances of the
processes by which all human knowledge is developed.

To watch the first germ of an appropriate idea as it was developed either
in his own mind or in the writings of the founders of the sciences, to frame
appropriate and scientific words in which the idea might be expressed, and then
to construct a treatise in which the idea should be largely developed and the
appropriate words copiously exemplified — such seems to have been the natural
channel of his intellectual activity in whatever direction it overflowed. When
any of his great works had reached this stage he prepared himself for some
other labour, and if new editions of his work were called for, the alterations
which he introduced often rather tended to destroy than to complete the unity
of the original plan.

Mr Todhunter has given us an exhaustive account of Dr "Whewell's writings
and scientific work, and in this we may easily trace the leading ideas which
he successively inculcated as a writer. We can only share Mr Todhunter's regret
that it is only as a writer that he appears in this book, and it is to be
hoped that the promised account of his complete life as a man may enable us
to form a fuller conception of the individuality and unity of his character, which
it is hard to gather from the multifarious collection of his books.

Dr Whewell first appears before us as the author of a long series of
text-books on Mechanics. His position as a tutor of his College, and the
interest which he took in University education, may have induced him to spend
more time in the composition of elementary treatises than would otherwise have
been congenial to him, but in the prefaces to the different editions, as well
as in the introductory chapters of each treatise, he shews that sense of the
intellectual and educational value of the study of first principles which dis-
tinguishes all his writings. It is manifest from his other writings, that the

= VOL. II. 67

WHEWKLL'S WRITINGS AND CORRESPONDENCE.

oompoaiUon of the** t«fc-book«, involving as it did a thorough study of the
fundamental science of Dynamics, was a most appropriate training for his
wbamuent labour* in the survey of the sciences in their widest extent.

• It KM alway* appeared to me," my* Mr Todhunter, " that Mr Whewell would have been of
— ^4 Lmnflt to rtiidonu if b« had undertaken a critical revision of the technical language of Mechanics.
Ttk UMTUJT WM formed to a great extent by the early writers at an epoch when the subject was
nndcratood, and many terms were used without well-defined meanings. Gradually the
JH> bran improved, but it is still open to objection."

In after years, when his authority in scientific terminology was widely
recognised, we find Faraday, Lyell, and others applying to him for appropriate
expressions for the subject-matter of their discoveries, and receiving in reply
systems of scientific terms which have not only held their place in technical
treatises, but are gradually becoming familiar to the ordinary reader.

"Is it not true," Dr Whewell asks in his Address to the Geological Society,
" in our science as in all others, that a technical phraseology is real wealth,
because it puts in our hands a vast treasure of foregone generalisations ? "

Perhaps, however, he felt it less difficult to induce scientific men to adopt
a new term for a new idea than to persuade the students and teachers of a
University to alter the phraseology of a time-honoured study.

But even in the elementary treatment of Dynamics, if we compare the
text-books of different dates, we cannot fail to recognise a marked progress.
Those by Dr Whewell were far in advance of any former text-books as regards
logical coherence and scientific accuracy, and if many of those which have been
published since have fallen behind in these respects, most of them have intro-
duced some slight improvement in terminology which has not been allowed to
be lost.

Dr Whewell's opinion with respect to the evidence of the fundamental
doctrines of mechanics is repeatedly inculcated in his writings. He considered
that experiment was necessary in order to suggest these truths to the mind,
but that the doctrine when once fairly set before the mind is apprehended by
it as strictly true, the accuracy of the doctrine being in no way dependent
on the accuracy of observation of the result of the experiment.

He therefore regarded experiments on the laws of motion as illustrative
experiments, meant to make us familiar with the general aspect of certain phe-
nomena, and not as experiments of research from which the results are to be
deduced by careful measurement and calculation.

WHEWELL'S WRITINGS AND CORRESPONDENCE. 531

Thus experiments on the fall of bodies may be regarded as experiments of
research into the laws of gravity. We find by careful measurements of times
and distances that the intensity of the force of gravity is the same whatever
be the motion of the body on which it acts. We also ascertain the direction
and magnitude of this force on different bodies and in different places. All
this can only be done by careful measurement, and the results are affected by
all the errors of observation to which we are liable.

The same experiments may be also taken as illustrations of the laws of
motion. The performance of the experiments tends to make us familiar with
these laws, and to impress them on our minds. But the laws of motion cannot
be proved to be accurate by a comparison of the observations which we make,
for it is only by taking the laws for granted that we have any basis for our
calculations. We may ascertain, no doubt, by experiment, that the acceleration
of a body acted on by gravity is the same whatever be the motion of that
body, but this does not prove that a constant force produces a constant accele-
ration, but only that gravity is a force, the intensity of which does not depend
on the velocity of the body on which it acts.

The truth of Dr Whewell's principle is curiously illustrated by a case in
which he persistently contradicted it. In a paper communicated to the Philo-
sophical Society of Cambridge, and reprinted at the end of his Philosophy of
the Inductive Sciences, Dr Whewell conceived that he had proved, a priori, that
all matter must be heavy. He was well acquainted with the history of the
establishment of the law of gravitation, and knew that it was only by careful
experiments and observations that Newton ascertained that the effect of gravi-
tation on two equal masses is the same whatever be the chemical nature of
the bodies, but in spite of this he maintained that it is contrary not only to
observation but to reason, that any body should be repelled instead of attracted
by another, whereas it is a matter of daily experience, that any two bodies
when they are brought near enough, repel each other.

The fact seems to be that, finding the word weight employed in ordinary
language to denote the quantity of matter in a body, though in scientific language
it denotes the tendency of that body to move downwards, and at the same
time supposing that the word mass in its scientific sense was not yet sufficiently
established to be used without danger in ordinary language, Dr Whewell en-
deavoured to make the word weight carry the meaning of the word mass. Thus
he tells us that "the weight of the whole compound must be equal to the

weights of the separate elements."

' 67—2

WUEWELL'S WRITINGS AND CORRESPONDENCE.
On this Mr Todhuntar very properly observes :

"Of wm Uwr* to »• practical uncertainty a* to this principle; but Dr \VTicwell seems to allow
lu» nadm to ioMCtM IfcM it ia of the came nature as the axiom that 'two straight lines cannot
iaeloip a mea,' TU*e u, however, a wide difference between them, depending on a fact which
Dr Wtowvil baa Mm**1* raooynued in another place (nee vol. i. p. 224). The truth is, that strictly
•i-miidf UM weight of the whole compound is not equal to the weight of the separate elements;
for UM weight ittfW** upon the poaition of the compound particles, and in general by altering
UM poeitiaa of UM particle*, the resultant effect which we call weight is altered, though it may
he to an inappreciable extent"

It U evident that what Dr Whowell should have said was : " The mass of
the whole compound must be equal to the sum of the masses of the separate
elements." This statement all would admit to be strictly true, and yet not a
single experiment has ever been made in order to verify it. All chemical
measurements are made by comparing the weights of bodies, and not by com-
paring the forces required to produce given changes of motion in the bodies ;
and as we have just been reminded by Mr Todhunter, the method of comparing
quantities of matter by weighing them is not strictly correct.

Thus, then, we are led by experiments which are not only liable to error,
hut which are to a certain extent erroneous in principle, to a statement which
is universally acknowledged to be strictly true. Our conviction of its truth must
therefore rest on some deeper foundation than the experiments which suggested
it to our minds. The belief in and the search for such foundations is, I think,
the most characteristic feature of all Dr Whewell's work.

[From the British Association Report, 1876.]

LXXXI. On Ohm's Laio.

THE service rendered to electrical science by Dr G. S. Ohm can only be
rightly estimated when we compare the language of those writers on electricity
who were ignorant of Ohm's law with that of those who have understood
and adopted it.

By the former, electric currents are said to vary as regards both their
"quantity" and their "intensity," two qualities the nature of which was very
imperfectly explained by tedious and vague expositions.

In the writings of the latter, after the elementary terms "Electromotive
Force," " Strength of Current," and " Electric Resistance " have been defined,
the whole doctrine of currents becomes distinct and plain.

Ohm's law may be stated thus :

The electromotive force which must act on a homogeneous conductor in
order to maintain a given steady current through it, is numerically equal to
the product of the resistance of the conductor into the strength of the current
through it. If, therefore, we define the resistance of a conductor as the ratio
of the numerical value of the electromotive force to the numerical value of
the strength of the current, Ohm's law asserts that this ratio is constant—
that is, that its value does not depend on that of the electromotive force or
of the current.

The resistance, as thus defined, depends on the nature and form of the
conductor, and on its physical condition as regards temperature, strain, &c. ;
but if Ohm's law is true, it does not depend on the strength of the current.

Ohm's law must, at least at present, be considered a purely empirical
one. No attempt to deduce it from pure dynamical principles has as yet
been successful ; indeed Weber's latest theoretical investigations* on this subject
have led him to suspect that Ohm's law is not true, but that, as the electro-

* Pogg. Ann. 1875.

.. ON OHM'S LAW.

motive ferae increase* without limit, the current increases slower and slower,
so »H«» the "resistance," M defined by Ohm's law, would increase with the
electromotive force. On the other hand, Schuster* has described experiments
which lead him to suspect a deviation from Ohm's law, but in the opposite
direction, the resistance being smaller for great currents than for small ones.

Lorenti*, of Leyden, has also proposed a theory according to which Ohm's
law would cease to be true for rapidly varying currents. The rapidity of
variation, however, which, as he supposes, would cause a perceptible deviation
from Ohm's law, must be comparable with the rate of vibration of light, so
that it would be impossible by any experiments other than optical ones to
test this theory.

The conduction of electricity through a resisting medium is a process in
which part of the energy of an electric current, flowing in a definite direc-
tion, is spent in imparting to the molecules of the medium that irregular
agitation which we call heat. To calculate from any hypothesis as to the
molecular constitution of the medium at what rate the energy of a given
current would be spent in this way, would require a far more perfect know-
ledge of the dynamical theory of bodies than we at present possess. It is
only by experiment that we can ascertain the laws of processes of which we
do not understand the dynamical theory.

We therefore define, as the resistance of a conductor, the ratio of the
numerical value of the electromotive force to that of the strength of the
current, and we have to determine by experiment the conditions which affect
the value of this ratio.

Thus if E denotes the electromotive force acting from one electrode of the
conductor to the other, C the strength of the current flowing through the
conductor, and R the resistance of the current, we have by definition

B-c-

and if H is the heat generated in the time t, and if J is the dynamical
equivalent of heat, we have by the principle of conservation of energy

* Report of British Auociation, 1874.

t Over de Terugkaatging en Brdcing van het Lie/a. Leiden, 1875.

ON OHM'S LAW. 535

The quantity R, which we have defined as the resistance of the conductor,
can be determined only by experiment. Its value may therefore, for any thing
we know, be affected by each and all of the physical conditions to which the
conductor may be subjected.

Thus we know that the resistance is altered by a change of the tempe-
rature of the conductor, and also by mechanical strain and by magnetization.

The question which is now before us is whether the current itself is or
is not one of the physical conditions which may affect the value of the
resistance ; and this question we cannot decide except by experiment.

Let us therefore assume that the resistance of a given conductor at a
given temperature is a function of the strength of the current. Since the
resistance of a conductor is the same for the same current in whichever
direction the current flows, the expression for the resistance can contain only
even powers of the current.

Let us suppose, therefore, that the resistance of a conductor of unit length
and unit section is

where r is the resistance corresponding to an infinitely small current, and
c is the current through unit of section, and s, s', &c. are small coefficients to
be determined by experiment. The coefficients s, s', &c. represent the deviations
from Ohm's law. If Ohm's law is accurate, these coefficients are zero ; also if
e is the electromotive force acting on this conductor,

e = re (1 + so* + s'c4 + &c.).

Now let us consider another conductor of the same substance whose length
is L and whose section is A ; then if E is the electromotive force on this
conductor, and e that on unit of length,

E = Le.
Also if C be the current through the conductor and c that through unit of

area,

C=Ac.

Hence the resistance of this conductor will be

E Lrf *<?*'<?

ON ODM'S LAW.

Now let u« suppose two conductors of the same material but of different
A-~—i~~ arranged in series and the same current passed through both :

where the suffixes indicate to which conductor the quantities belong. The
ratio of the resistances is

Hence if Ohm's law is not true, and if, therefore, any of the quantities s, s',
Ac. have sensible values, the ratio of the resistances will depend on the strength
of the current

Now the ratio of two resistances may be measured with great accuracy
by means of Wheatstone's bridge.

We therefore arrange the bridge so that one branch of the current passes
first through a very fine wire a few centimetres long, and then through a
much longer and thicker wire of about the same resistance. The other branch
of the current passes through two resistances, equal to each other, but much
greater than the other two, so that very little of the heating-effect of the
current is produced in these auxiliary resistances.

The bridge is formed by connecting the electrodes of a galvanometer, one
to the junction of the fine wire and the thick one, and the other to a point
between the other two resistances.

We have thus a method of testing the ratio of the resistances of the
fine wire to that of the thick one; and by passing through the bridge some-
times a feeble current and sometimes a powerful one, we might ascertain if
the ratio differed in the two cases.

But this direct method is rendered useless by the fact that the current
generates heat, which raises the temperature of both wires, but that of the
thin wire most rapidly ; and this makes it impossible to compare the effects
of strong and weak currents through a conductor at one and the same
temperature.

ON OHM'S LAW. 537

It is also useless to work with weak currents, as the effect depends on
the square of the current, and is so small as to have escaped observation in
all ordinary experiments.

Again, if we were to use a single very strong current acting for a very
short time, we should not be able to observe the galvanometer in a satis-
factory manner. In fact it was found in the experiment that currents which
lasted for a sixtieth part of a second produced a heating-effect which interfered
with the measurements. The experiment was therefore arranged so that a
strong current and a weak one were passed through the bridge alternately ;
and when the bridge was so arranged that the galvanometer was in equili-
brium, the direction of the weaker current was reversed. If Ohm's law were
not true, the condition of equilibrium for strong currents would be different
from that for weaker ones, so that when the weak currents were reversed
there would be no longer equilibrium. Since, in point of fact, the reversal
of the weaker currents did not affect the equilibrium, it follows that the
bridge was in equilibrium for the weaker currents as well as for the stronger
ones, and therefore the conditions were the same for both, and Ohm's law
is true to within the limits of error of the experiment4'".

[* The mode in which the actual strength of the currents was measured and the limits of
error ascertained are described in the Report which follows the above by Mr Chrystal, now Pro-
fessor of Mathematics at Edinburgh University.]

VOL. II. 68

[From Nat*rt, Vol. xiv.]

I. XXXII. On the protection of luiUlings from lightning.

MOST of those who have given directions for the construction of lightning-
conductors have paid great attention to the upper and lower extremities of
the conductor. They recommend that the upper extremity of the conductor
should extend somewhat above the highest part of the building to be protected,
and that it should terminate in a sharp point, and that the lower extremity
should be carried as far as possible into the conducting strata of the ground,
so as to "make" what telegraph engineers call "a good earth."

The electrical effect of such an arrangement is to tap, as it were, the
gathering charge by facilitating a quiet discharge between the atmospheric
accumulation and the earth. The erection of the conductor will cause a some-
what greater number of discharges to occur at the place than would have
occurred if it had not been erected ; but each of these discharges will be
smaller than those which would have occurred without the conductor. It is
probable, also, that fewer discharges will occur in the region surrounding the
conductor.

It appears to me that these arrangements are calculated rather for the
benefit of the surrounding country and for the relief of clouds labouring under
an accumulation of electricity, than for the protection of the building on which
the conductor is erected.

What we really wish is to prevent the possibility of an electric discharge
taking place within a certain region, say in the inside of a gunpowder manu-
factory. If this is clearly laid down as our object, the method of securing it
is equally clear.

An electric discharge cannot occur between two bodies, unless the difference
of their potentials is sufficiently great, compared with the distance between them.
If, therefore, we can keep the potentials of all bodies within a certain region
equal, or nearly equal, no discharge will take place between them. We may

ON THE PROTECTION OF BUILDINGS FROM LIGHTNING. 539

secure this by connecting all these bodies by means of good conductors, such
as copper wire ropes, but it is not necessary to do so, for it may be shewn
by experiment that if eveiy part of the surface surrounding a certain region
is at the same potential, every point within that region must be at the same
potential, provided no charged body is placed within the region.

It would, therefore, be sufficient to surround our powder-mill with a con-
ducting material, to sheathe its roof, walls, and ground-floor with thick sheet
copper, and then no electrical effect could occur within it on account of any
thunderstorm outside. There would be no need of any earth connection. We
might even place a layer of asphalte between the copper floor and the ground,
so as to insulate the building. If the mill were then struck with lightning,
it would remain charged for some time, and a person standing on the ground
outside and touching the wall might receive a shock, but no electrical effect
would be perceived inside, even on the most delicate electrometer. The poten-
tial of everything inside with respect to the earth would be suddenly raised
or lowered as the case might be, but electric potential is not a physical con-
dition, but only a mathematical conception, so that no physical effect would be
perceived.

It is therefore not necessary to connect large masses of metal such as
engines, tanks, &c., to the walls, if they are entirely within the building. If,
however, any conductor, such as a telegraph wire or a metallic supply-pipe for
water or gas comes into the building from without, the potential of this con-
ductor may be different from that of the building, unless it is connected with
the conducting-shell of the building. Hence the water or gas supply pipes, if
any enter the building, must be connected to the system of lightning con-
ductors, and since to connect a telegraph wire with the conductor would render
the telegraph useless, no telegraph from without should be allowed to enter
a powder-mill, though there may be electric bells and other telegraphic appa-
ratus entirely within the building.

I have supposed the powder-mill to be entirely sheathed in thick sheet
copper. This, however, is by no means necessary in order to prevent any
sensible electrical effect taking place within it, supposing it struck by lightning.
It is quite sufficient to inclose the building with a network of a good conduct-
ing substance. For instance, if a copper wire, say No. 4, B.W.G. (0'238 inches
diameter), were carried round the foundation of the house, up each of the
corners and gables and along the ridges, this would probably be a sufficient

68—2

540 9f TOM I'BOTBCTION OF BUILDINGS FROM LIGHTNING.

ptutujtiuii for an ordinary building against any thunderstorm in this climate.
The copper wire may be built into the wall to prevent theft, but should be
^mmH^j»^ to any outside metal such as lead or zinc on the roof, and to metal
nun-water pipes. In the case of a powder-mill it might be advisable to make
(he network closer by carrying one or two additional wires over the roof and
down the walla to the wire at the foundation. If there are water or gas-
I.IJKW which enter the building from without, these must be connected with
the system of conducting-wires, but if there are no such metallic connections
with distant points, it is not necessary to take any pains to facilitate the
escape of the electricity into the earth.

Still less is it advisable to erect a tall conductor with a sharp point in
order to relieve the thunder-clouds of their charge.

It is hardly necessary to add, that it is not advisable, during a thunder-
storm, to stand on the roof of a house so protected, or to stand on the ground
outside and lean against the wall.

[From the Encyclopaedia Britannica.]

LXXXIII. Capillary Action.

A TUBE, the bore of which is so small that it will only admit a hair
(capilla), is called a capillary tube. When such a tube of glass, open at both
ends, is placed vertically with its lower end immersed in water, the water is
observed to rise in the tube, and to stand within the tube at a higher level
than the water outside. The action between the capillary tube and the water
has been called Capillary Action, and the name has been extended to many
other phenomena which have been found to depend on properties of liquids
and solids similar to those which cause water to rise in capillary tubes.

The forces which are concerned in these phenomena are those which act
between neighbouring parts of the same substance, and which are called forces
of cohesion, and those which act between portions of matter of different kinds,
which are called forces of adhesion. These forces are quite insensible between
two portions of matter separated by any distance which we can directly measure.
It is only when the distance becomes exceedingly small that these forces be-
come perceptible. Quincke* has made experiments to determine the greatest
distance at which the effect of these forces is sensible, and he finds for various
substances distances about the twenty-thousandth part of a millimetre.

Poggendorfft tells us that Leonardo da Vinci J must be considered as the
discoverer of capillary phenomena.

The first accurate observations of the capillary action of tubes and glass
plates were made by Hauksbee§. He ascribes the action to an attraction be-
tween the glass and the liquid. He observed that the effect was the same in
thick tubes as in thin, and concluded that only those particles of the glass
which are very near the surface have any influence on the phenomenon.

* Fogg. Ann., cxxxvn. p. 402. t Fogg. Ann., ci. p. 551. J Died 1519.

§ Physico- Mechanical Experiments, London, 1709, pp. 139—169; and Phil. Trans., 1711 and 1712.

, CAPILLARY ACTION.

Dr Jurin* shewed that the height at which the liquid is suspended
depend* on the section of the tube at the surface of the liquid, and is inde-
ndent of the form of the lower part of the tube. He considered that the
^upp^nn of the liquid is due to "the attraction of the periphery or section
of the mfr"* of the tube to which the upper surface of the water is con-
tiguous and coheres." From this he shews that the rise of the liquid in tubes
of the same substance is inversely proportional to their radii.

Newton devotes the 31st query in the last edition of his Opticla to
molecular forces, and instances several examples of the cohesion of liquids, such
M the suspension of mercury in a barometer tube at more than double the
height at which it usually stands. This arises from its adhesion to the tube,
and the upper part of the mercury sustains a considerable tension, or negative
pressure, without the separation of its parts. He considers the capillary phe-
nomena to be of the same kind, but his explanation is not sufficiently explicit
with respect to the nature and the limits of the action of the attractive force.

It is to be observed that, while these early speculators ascribe the phe-
nomena to attraction, they do not distinctly assert that this attraction is
sensible only at insensible distances, and that for all distances which we can
directly measure the force is altogether insensible. The idea of such forces,
however, had been distinctly formed by Newton, who gave the first example
of the calculation of the effect of such forces in his theorem on the alteration
of the path of a light-corpuscle when it enters or leaves a dense body.

Clairautf appears to have been the first to shew the necessity of taking
account of the attraction between the parts of the fluid itself in order to
explain the phenomena. He does not, however, recognise the fact that the
distance at which the attraction is sensible is not only small but altogether
insensible.

Segner J introduced the very important idea of the surface-tension of liquids,
which he ascribed to attractive forces, the sphere of whose action is so small
"ut nullo adhuc sensu percipi potuerit." In attempting to calculate the effect
of this surface-tension in determining the form of a drop of the liquid, Segner
took account of the curvature of a meridian section of the drop, but neglected
the effect of the curvature in a plane at right angles to this section.

* Phil. Trans., 1718, No. 355, p. 739, and 1719, No. 363, p. 1083.
t Clairaut, Thcone de la Figure de la Terre, Paris, 1808, pp. 105, 128.
J Segner, Comment. Soc. Reg. Gotting., i. (1751), p. 301.

CAPILLARY ACTION. 543

But the idea of surface-tension introduced by Segner had a most important
effect on the subsequent development of the theory. We may regard it as a
physical fact established by experiment in the same way as the laws of the
elasticity of solid bodies. We may investigate the forces which act between
finite portions of a liquid in the same way as we investigate the forces which
act between finite portions of a solid. The experiments on solids lead to certain
laws of elasticity expressed in terms of coefficients, the values of which can
be determined only by experiments on each particular substance. Various
attempts have also been made to deduce these laws from particular hypotheses
as to the action between the molecules of the elastic substance. We may
therefore regard the theory of elasticity as consisting of two parts. The first
part establishes the laws of the elasticity of a finite portion of the solid sub-
jected to a homogeneous strain, and deduces from these laws the equations of
the 'equilibrium and motion of a body subjected to any forces and displace-
ments. The second part endeavours to deduce the facts of the elasticity of a
finite portion of the substance from hypotheses as to the motion of its con-
stituent molecules and the forces acting between them.

In like manner we may by experiment ascertain the general fact that the
surface of a liquid is in a state of tension similar to that of a membrane
stretched equally in all directions, and prove that this tension depends only
on the nature and temperature of the liquid and not on its form, and from
this as a secondary physical principle we may deduce all the phenomena
of capillary action. This is one step of the investigation. The next step
is to deduce this surface-tension from an hypothesis as to the molecular
constitution of the liquid and of the bodies that surround it. The scientific
importance of this step is to be measured by the degree of insight which
it affords or promises into the molecular constitution of real bodies by the
suggestion of experiments by which we may discriminate between rival molecular

theories.

In 1756 Leidenfrost * shewed that a soap-bubble tends to contract, so that
if the tube with which it was blown is left open the bubble will diminish
in size and will expel through the tube the air which it contains. He attri-
buted this force, however, not to any general property of the surfaces of liquids,
but to the fatty part of the soap which he supposed to separate itself from

* De aquae communis nonnullis qualitatibus tractalus, Duisburg.

I . CAPILLABY ACTION*.

the other constituents of the solution, and to form a thin skin on the outer

free of the bubble.

In 1787 Monge* ••!•>! 1 that "by supposing the adherence of the par-
tfpjj, ^ a fluy to have a sensible effect only at the surface itself and in the
direction of the surface it would be easy to determine the curvature of the
of fluids in the neighbourhood of the solid boundaries which contain

r that these surfaces would be lintearice of which the tension, constant
in all directions, would be everywhere equal to the adherence of two particles,
and the phenomena of capillary tubes would then present nothing which could
not be determined by analysis." He applied this principle of surface-tension
to the explanation of the apparent attractions and repulsions between bodies
floating on a liquid.

In 1802 Leslie t gave the first correct explanation of the rise of a liquid
in a tube by considering the effect of the attraction of the solid on the very
thin stratum of the liquid in contact with it. He does not, like the earlier
speculators, suppose this attraction to act in an upward direction so as to
support the fluid directly. He shews that the attraction is everywhere normal
to the surface of the solid. The direct effect of the attraction is to increase
the pressure of the stratum of the fluid in contact with the solid, so as to
make it greater than the pressure in the interior of the fluid. The result of
this pressure if unopposed is to cause this stratum to spread itself over the
surface of the solid as a drop of water is observed to do when placed on a
clean horizontal glass plate, and this even when gravity opposes the action, as
when the drop is placed on the under surface of the plate. Hence a glass
tube plunged into water would become wet all over were it not that the
ascending liquid film carries up a quantity of other liquid which coheres to it,
so that when it has ascended to a certain height the weight of the column
balances the force by which the film spreads itself over the glass. This ex-
planation of the action of the solid is equivalent to that by which Gauss
afterwards supplied the defect of the theory of Laplace, except that, not being
expressed in terms of mathematical symbols, it does not indicate the mathe-
matical relation between the attraction of individual particles and the final
result Leslie's theory was afterwards treated according to Laplace's mathe-
matical methods by James Ivory in the article on capillary action, under the

* Memovres de FAcad. des Sciences, 1787, p. 506.
t Philosophical Magazine, 1802, Vol. xiv. p. 193.

CAPILLARY ACTION. 545

heading " Fluids, Elevation of," in the supplement to the fourth edition of the
Encyclopedia Britannica, published in 1819.

In 1804 Thomas Young* founded the theory of capillary phenomena on
the principle of surface-tension. He also observed the constancy of the angle
of contact of a liquid surface with a solid, and shewed how from these two
principles to deduce the phenomena of capillary action. His essay contains the
solution of a great number of cases, including most of those afterwards solved
by Laplace, but his methods of demonstration, though always correct, and often
extremely elegant, are sometimes rendered obscure by his scrupulous avoidance
of mathematical symbols. Having applied the secondary principle of surface-
tension to the various particular cases of capillary action, Young proceeds to
deduce this surface-tension from ulterior principles. He supposes the particles
to act on one another with two different kinds of forces, one of which, the
attractive force of cohesion, extends to particles at a greater distance than
those to which the repulsive force is confined. He further supposes that the
attractive force is constant throughout the minute distance to which it extends,
but that the repulsive force increases rapidly as the distance diminishes. He
thus shews that at a curved part of the surface, a superficial particle would
be, urged towards the centre of curvature of the surface, and he gives reasons
for concluding that this force is proportional to the sum of the curvatures of
the surface in two normal planes at right angles to each other.

The subject was next taken up by Laplace t. His results are in many
respects identical with those of Young, but his methods of arriving at them
are very different, being conducted entirely by mathematical calculations. The
form into which he has thrown his investigation seems to have deterred many
able physicists from the inquiry into the ulterior cause of capillary phenomena,
and induced them to rest content with deriving them from the fact of surface-
tension. But for those who wish to study the molecular constitution of bodies
it is necessary to study the effect of forces which are sensible only at insen-
sible distances ; and Laplace has furnished us with an example of the method
of this study which has never been surpassed. Laplace investigates the force
acting on the fluid contained in an infinitely slender canal normal to the
surface of the fluid arising from the attraction of the parts of the fluid out-

* Essay on the "Cohesion of Fluids," Philosophical Transactions, 1805, p. 65.
t Mecanique Celeste, supplement to the tenth book, published in 1806.
vol.. II. 69

CAPILLARY ACTION.

aid* the canal. He thus finds for the pressure at a point in the interior of
the fluid an expression of the form

when A" u * con»Unt pressure, probably very large, which, however, does not
influence capillary phenomena, and therefore cannot be determined from observa-
tion of such phenomena ; // is another constant on which all capillary phenomena
depend; and R and K are the radii of curvature of any two normal sections
,/ the surface at right angles to each other.

In the first part of our own investigation we shall adhere to the symbols
myrjl by Laplace, as we shall find that an accurate knowledge of the physical
intiTprvtation of these symbols is necessary for the further investigation of the
xubject. In the Supplement to the Theory of Capillary Action, Laplace deduces
the equation of the surface of the fluid from the condition that the resultant
force on a particle at the surface must be normal to the surface. His expla-
nation, however, of the rise of a liquid in a tube is based on the assumption
• •I" the constancy of the angle of contact for the same solid and fluid, and of
this he has nowhere given a satisfactory proof. In this supplement Laplace
gives many important applications of the theory, and compares the results with
the experiments of Gay-Lussac.

The next great step in the treatment of the subject was made by Gauss*.
The principle which lie adopts is that of virtual velocities, a principle which
under liis hands was gradually transforming itself into what is now known as
the principle of the conservation of energy. Instead of calculating the direction
and magnitude of the resultant force on each particle arising from the action
of neighbouring particles, he forms a single expression which is the aggregate
of all the potentials arising from the mutual action between pairs of particles.
This expression lias been called the force-function. With its sign reversed it
is now called the potential energy of the system. It consists of three parts,
the first depending on the action of gravity, the second on the mutual action

een the particles of the fluid, and the third on the action between the
I "articles of the fluid and the particles of a solid or fluid in contact with it.

The condition of equilibrium is that this expression (which we may for
the sake of distinctness call the potential energy) shall be a minimum. This

• Priitcipia gewmlia Theoria Figune Fluidorum in flatu jEquilibrii (Gottingen. 1830), or Werke,
T. 29 (Gottingw,, 1867).

CAPILLARY ACTION. 547

condition when worked out gives not only the equation of the free surface in
the form already established by Laplace, but the conditions of the angle of
contact of this surface with the surface of a solid.

Gauss thus supplied the principal defect in the great work of Laplace.
He also pointed out more distinctly the nature of the assumptions which we
must make with respect to the law of action of the particles in order to be
consistent with observed phenomena. He did not, however, enter into the ex-
planation of particular phenomena, as this had been done already by Laplace.
He points out, however, to physicists the advantages of the method of Segner
and Gay-Lussac, afterwards carried out by Quincke, of measuring the dimen-
sions of large drops of mercuiy on a horizontal or slightly concave surface,
and those of large bubbles of air in transparent liquids resting against the
under side of a horizontal plate of a substance wetted by the liquid.

In 1831 Poisson published his Noiivelle Theorie de I' Action Capillaire.
He maintains that there is a rapid variation of density near the surface of a
liquid, and he gives very strong reasons, which have been only strengthened
by subsequent discoveries, for believing that this is the case. He then pro-
ceeds to an investigation of the equilibrium of a fluid on the hypothesis of
uniform density, and he arrives at the conclusion that on this hypothesis none
of the observed capillary phenomena would take place, and that, therefore,
Laplace's theory, in which the density is supposed uniform, is not only insuffi-
cient but erroneous. In particular he maintains that the constant pressure K,
which occurs in Laplace's theory, and which on that theory is very large,
must be in point of fact very small, but the equation of equilibrium from
which he concludes this is itself defective. Laplace assumes that the liquid
has uniform density, and that the attraction of its molecules extends to a
finite though insensible distance. On these assumptions his results are certainly
right, and are confirmed by the independent method of Gauss, so that the
objections raised against them by Poisson fall to the ground. But whether
the assumption of uniform density be physically correct is a very different
question, and Poisson has done good service to science in shewing how to
carry on the investigation on the hypothesis that the density very near the
surface is different from that in the interior of the fluid.

The result, however, of Poisson's investigation is practically equivalent to
that already obtained by Laplace. In both theories the equation of the liquid
surface is the same, involving a constant H, which can be determined only by

69—2

CAPILLARY ACTION.

experiment. The only difference is in the manner in which this quantity //
depend* on the law of the molecular forces and the law of density near the
mr&ee of the fluid, and as these laws are unknown to us we cannot obtain
any te*t to discriminate between the two theories.

Wa haw now described the principal forms of the theory of capillary
action during its earlier development. In more recent times the method of
GAUM has been modified so as to take account of the variation of density
near the surface, and its language has been translated in terms of the modern
doctrine of the conservation of energy *.

M. Plateau t, who has himself made the most elaborate study of the
phenomena of surface-tension, has adopted the following method of getting rid
of the effects of gravity : He forms a mixture of alcohol and water of the
same density as olive oil. He then introduces a quantity of oil into the
mixture. It assumes the form of a sphere under the action of surface-tension
alone. He then, by means of rings of iron-wire, disks, and other contrivances,
alters the form of certain parts of the surface of the oil. The free portions
of the surface then assume new forms depending on the equilibrium of surface-
tension. In this way he has produced a great many of the forms of equilibrium

a liquid under the action of surface-tension alone, and compared them with
the results of mathematical investigation. He has also greatly facilitated the

idy of liquid films by shewing how to form a liquid, the films of which
will last for twelve or even for twenty-four hours. The debt which science
owes to M. Plateau is not diminished by the fact that, while investigating
these beautiful phenomena, he has never himself seen them. He lost his sight
long ago1 in the pursuit of science, and has ever since been obliged to depend
on the eyes and the hands of others.

M. Van der Mensbrugghe J has also devised a great number of beautiful
illustrations of the phenomena of surface-tension, and has shewn their connection
with the experiments of Mr Tomlinson on the figures formed by oils dropped
on the clean surface of water.

* See Prof. Betti, Teoria delta CapillaritA: Nuavo Cimento, 1867; a memoir by M. Stahl,
"Ueber einige Punckto in der Theorie der Capillarerscheimmgen," Pogg. Ann., cxxxix. p. 239 (1870);
and M. Van der Waal's Over da Continuiteit van den Gcu- en Vloeistoftoesland. The student will
find a good account of the subject from a mathematical point of view in Professor Challis's
"Report on the Theory of Capillary Attraction," Brit. Ass. Report, iv. p. 253 (1834).

t M. Plateau, Statique experimental* et theorique des liquides.

J J/em. d* FAcad. Roy. de Belyiqu*, xxxvn. (1873).

CAPILLARY ACTION. 549

M. Duprd in his 5th, 6th, and 7th Memoirs on the Mechanical Theory
of Heat (Ann. de Chimie et de Physique, 1866 to 1868) has done much to-
wards applying the principles of thermodynamics to capillary phenomena, and
the experiments of his son are exceedingly ingenious and well devised, tracing
the influence of surface-tension in a great number of very different circumstances,
and deducing from independent methods the numerical value of the surface-
tension. The experimental evidence which M. Dupre" has obtained bearing on
the molecular structure of liquids must be very valuable, even if many of our
present opinions on this subject should turn out to be erroneous.

M. Quincke* has made a most elaborate series of experiments on the tension
of the surfaces separating one liquid from another and from air.

M. Liidtget has experimented on liquid films, and has shewn how a film
of a liquid of high surface-tension is replaced by a film of lower surface-tension.
He has also experimented on the effects of the thickness of the film, and has
come to the conclusion that the thinner a film is, the greater is its tension.
This result, however, has been tested by M. Van der Mensbrugghe, who finds
that the tension is the same for the same liquid whatever be the thickness,
as long as the film does not burst. The phenomena of very thin liquid films
deserve the most careful study, for it is in this way that we are most likely
to obtain evidence by which we may test the theories of the molecular structure
of liquids.

Sir W. Thomson J has investigated the effect of the curvature of the
surface of a liquid on the thermal equilibrium between the liquid and the
vapour in contact with it. He has also calculated the effect of surface-tension
on the propagation of waves on the surface of a liquid, and has determined
the minimum velocity of a wave, and the velocity of the wind when it is just
sufficient to disturb the surface of still water §.

THEORY OF CAPILLARY ACTION.

When two different fluids are placed in contact, they may either diffuse
into each other or remain separate. In some cases diffusion takes place to a
limited extent, after which the resulting mixtures do not mix with each other.

* Pogg. Ann., cxxxix. (1870), p. 1. t Ibid. p. 620.

J Proceedings R. S., Edinburgh, February 7, 1870.
§ Philosophical Magazine, November, 1871.

CAPILLARY ACTION.

Tha nmf substance may be able to exist in two different states at the same
temperature and pressure, as when water and its saturated vapour are con-
tained in the same vooool The conditions under which the thermal and
iMffihan^l equilibrium of two fluids, two mixtures, or the same substance in
two physical states in contact with each other, is possible belong to thermo-
dynamics. All that we have to observe at present is that, in the cases in
which the fluids do not mix of themselves, the potential energy of the system
mu*t be greater when the fluids are mixed than when they are separate.

It is found by experiment that it is only very close to the bounding
Atirfkce of a liquid that the forces arising from the mutual action of its parts
have any resultant effect on one of its particles. The experiments of Quincke
and others seem to shew that the extreme range of the forces which produce
capillary action lies between a thousandth and a twenty thousandth part of
a millimetre.

We shall use the symbol e to denote this extreme range, beyond which
the action of these forces may be regarded as insensible. If x denotes tl it-
potential energy of unit of mass of the substance, we may treat x as sensibly
constant except within a distance e of the bounding surface of the fluid. In
the interior of the fluid it has the uniform value x<>- In l^e manner the
density, p, is sensibly equal to the constant quantity p,, which is its value
in the interior of the liquid, except within a distance e of the bounding surface.
Hence if V is the volume of a mass M of liquid bounded by a surface whose
area is S, the integral

(1),

where the integration is to be extended throughout the volume V, may be
divided into two parts by considering separately the thin shell or skin extend-
ing from the outer surface to a depth c, within which the density and other
properties of the liquid vary with the depth, and the interior portion of the
liquid within which its properties are constant.

Since e is a line of insensible magnitude compared with the dimensions of
the mass of liquid and the principal radii of curvature of its surface, the
volume of the shell whose surface is S and thickness e will be Se, and that
of the interior space will be V '— Se.

If we suppose a normal v less than e to be drawn from the surface S
into the liquid, we may divide the shell into elementary shells whose thick-

CAPILLARY ACTION. 551

ness is dv, in each of which the density and other properties of the liquid
will be constant.

The volume of one of these shells will be Sdv. Its mass will be Spdv.
The mass of the whole shell will therefore be S\ pdv, and that of the interior

Jo

part of the liquid (F-Se)/30. We thus find for the whole mass of the liquid

M=Vpt-st'(pt-p)dv ........................... (2).

J o

To find the potential energy we have to integrate

E = I \xpdxdydz ....................... ....... (3).

Substituting yjp for p in the process we have just gone through, we find

E=VXP*-S \(x°p»-xp)dv ........................ (4)-

J 0

Multiplying equation (2) by x*> and subtracting it from (4),

......................... (5).

In this expression M and ^0 are both constant, so that the variation of
the right hand side of the equation is the same as that of the energy E,
and expresses that part of the energy which depends on the area of the
bounding surface of the liquid. We may call this the surface energy.

The symbol x expresses the energy of unit of mass of the liquid at a
depth v within the bounding surface. When the liquid is in contact with a
rare medium, such as its own vapour or any other gas, ^ is greater than ^0,
and the surface energy is positive. By the principle of the conservation of
energy, any displacement of the liquid by which its energy is diminished will
tend to take place of itself. Hence if the energy is the greater, the greater
the area of the exposed surface, the liquid will tend to move in such a way
as to diminish the area of the exposed surface, or in other words, the exposed
surface will tend to diminish if it can do so consistently with the other
conditions. This tendency of the surface to contract itself is called the surface-
tension of liquids.

M. Dupre' has described an arrangement by which the surface-tension of
a liquid film may be illustrated.

551 ' VPILLARY ACTION.

A piece of sheet metal is cut out in the form A A (fig. 1). A very fine
idip of metal U laid on it in the position BB, and the whole is dipped into

a solution of soap, or M. Plateau's glycerine mixture.
When it is taken out the rectangle AACC is filled up
by a liquid film. This film, however, tends to contract
on iteelf, and the loose strip of metal BB will, if it
is let go, be drawn up towards A A, provided it is
sufficiently light and smooth.

Let T be the surface-energy per unit of area; then the energy of a
sur&ce of area S will be ST. If, in the rectangle AACC, AA=a, and (7(7=6,
it* area is S=al, and its energy Tab. Hence if .F is the force by which
the alip BB is pulled towards A A,

F-^Tab-Ta (6),

..r the force arising from the surface-tension acting on a length a of the strip
is Tti, so that T represents the surface-tension acting transversely on every
unit of length of the periphery of the liquid surface. Hence if we write

r = J4(x-X.)/**" (7),

we may define T either as the surface-energy per unit of area, or as the
surface- tension per unit of contour, for the numerical values of these two
quantities are equal.

If the liquid is bounded by a dense substance, whether liquid or solid, the
value of x ma7 be different from its value when the liquid has a free surface.
If the liquid is in contact with another liquid, let us distinguish quantities
belonging to the two liquids by suffixes. We shall then have

(8),
(9).

Adding these expressions, and dividing the second member by S, we obtain
for the tension of the surface of contact of the two liquids

f'i f't

(x.-X".)/>A+ (x»~x-)/vki (10).

|"

/«

o

CAPILLARY ACTION. 553

If this quantity is positive, the surface of contact will tend to contract,
and the liquids will remain distinct. If, however, it were negative, the dis-
placement of the liquids which tends to enlarge the surface of contact would
be aided by the molecular forces, so that the liquids, if not kept separate by
gravity, would at length become thoroughly mixed. No instance, however, of
a phenomenon of this kind has been discovered, for those liquids which mix
of themselves do so by the process of diffusion, which is a molecular motion,
and not by the spontaneous puckering and replication of the bounding surface,
as would be the case if T were negative.

It is probable, however, that there are many cases in which the integral
belonging to the less dense fluid is negative. If the denser body be solid we
can often demonstrate this; for the liquid tends to spread itself over the
surface of the solid, so as to increase the area of the surface of contact, even
although in so doing it is obliged to increase the free surface in opposition
to the surface-tension. Thus water spreads itself out on a clean surface of glass.

This shews that (x~Xo) P^v must be negative for water in contact with glass.

J D

ON THE TENSION OF LIQUID FILMS.

The method already given for the investigation of the surface-tension of
a liquid, all whose dimensions are sensible, fails in the case of a liquid film,
such as a soap-bubble. In such a film it is possible that no part of the liquid
may be so far from the surface as to have the potential and density corre-
sponding to what we have called the interior of a liquid mass, and measurements
of the tension of the film when drawn out to different degrees of thinness
may possibly lead to an • estimate of the range of the molecular forces, or at
least of the depth within a liquid mass, at which its properties become sensibly
uniform. We shall therefore indicate a method of investigating the tension of
such films.

Let S be the area of the film, M its mass, and E its energy; a- the
mass, and e the energy of unit of area ; then

M=S<r (11),

E = Se (12).

VOL. 1L 70

CAPILLARY ACTION.

Let us now suppose that by some change in the form of the boundary
of UM film iU area is changed from S to S+dS. If its tension is T the
work required to eflect this increase of surface will be TdS, and the energy
of the film will be increased by this amount. Hence

(13).
But since J/ is constant,

Q ............................ (14).

Eliminating dS from equations (13) and (14), and dividing by S, we find

In this expression a- denotes the mass of unit of area of the film, and
• the energy of unit of area.

If we take the axis of z normal to either surface of the film, the radius
of curvature of which we suppose to be very great compared with its thick-
ness c, and if p is the density, and \ *ne energj °f un^ °f mass a^ depth z,

=t' pdz

j

(16),
and

(17).

Both p and \ aie functions of z, the values of which remain the same
when z — c is substituted for z. If the thickness of the film is greater than
2€, there will be a stratum of thickness c — 2e in the middle of the film,
within which the values of p and \ w^ ^ P« an(^ X°' ^n ^e ^wo strata
on either side of this the law, according to which p and x depend on the
depth, will be the same as in a liquid mass of large dimensions. Hence in
this case

(7= 1C —

(18),

CAPILLARY ACTION. 555

da- _ de _ de

Tc~p" dc~x°p<" •'•^• = Xo>

pdv,

(20).

Hence the tension of a thick film is equal to the sum of the tensions of

its two surfaces as already calculated (equation 7). On the hypothesis of

uniform density we shall find that this is true for films whose thickness
exceeds e.

The symbol x *s defined as the energy of unit of mass of the substance.
A knowledge of the absolute value of this energy is not required, since in
every expression in which it occurs it is under the form x~Xo> that is to
say, the difference between the energy in two different states. The only cases,
however, in which we have experimental values of this quantity are when the
substance is either liquid and surrounded by similar liquid, or gaseous and sur-
rounded by similar gas. It is impossible to make direct measurements of the
properties of particles of the substance within the insensible distance e of the
bounding surface.

When a liquid is in thermal and dynamical equilibrium with its vapour,
then if p and x' are the values of p and x for the vapour, and pa and x»
those for the liquid,

where J is the dynamical equivalent of heat, L is the latent heat of unit of
mass of the vapour, and p is the pressure. At points in the liquid very near
its surface it is probable that x is greater than XD, and at points in the gas
very near the surface of the liquid it is probable that x is less than x'. but
this has not as yet been ascertained experimentally. We shall therefore en-
deavour to apply to this subject the methods used in Thermodynamics, and
where these fail us we shall have recourse to the hypotheses of molecular

physics.

70—2

CAPILLARY ACTION.

We hare next to determine the value of x in terms of the action between
one particle and another. Let ua suppose that the force between two particles
M and m' at the distance / is

(22),

being reckoned positive when the force is attractive. The actual force between

the particles arises in part from their mutual gravitation, which is inversely

n
as the square of the distance. This force is expressed by mm' ^. It is easy

to shew that a force subject to this law would not account for capillary action.
We shall, therefore, in what follows, consider only that part of the force which
depends on $(f), where <f>(f) is a function of f which is insensible for all
sensible values of f, but which becomes sensible and even enormously great
when / is exceedingly small.

If we next introduce a new function of/ and write

(23),

then utin'll (f) will represent (1) the work done by the attractive force on
the particle m, while it is brought from an infinite distance from m' to the
distance f from m'; or (2) the attraction of a particle m on a narrow straight
rod resolved in the direction of the length of the rod, one extremity of the
rod being at a distance / from m, and the other at an infinite distance, the
mass of. unit of length of the rod being m'. The function IT (f ) is also in-
sensible for sensible values of /, but for insensible values of / it may become
sensible and even very great.

If we next write

»A(z) (24),

then 2irm<rV»(z) will represent (l) the work done by the attractive force while
a particle m is brought from an infinite distance to a distance z from an
infinitely thin stratum of the substance whose mass per unit of area is a- ;
(2) the attraction of a particle m placed at a distance z from the plane surface
of an infinite solid whose density is cr.

CAPILLARY ACTION.

Let us examine the case in which the particle m is
placed at a distance z from a curved stratum of the sub-
stance, whose principal radii of curvature are R1 and Rt.
Let P (fig. 2) be the particle and PB a normal to the
surface. Let the plane of the paper be a normal section
of the surface of the stratum at the point B, making an
angle at with the section whose radius of curvature is Rr
Then if O is the centre of curvature in the plane of the
paper, and B0 = u,

1 cos2 to sina to

557

Fig. 2.

u

Let

= r, PQ=f,

The element of the stratum at Q may be expressed by

<n? sin 9 dO dco,
or expressing dO in terms of df by (26),

(25).

(26).

<r - fdfdoi.

Multiplying this by m and by Ilf, we obtain for the work done by the
attraction of this element when m is brought from an infinite distance to P,

Integrating with respect to f from f=z to f=a, where a is a line very
great compared with the extreme range- of the molecular force, but very small
compared with either of the radii of curvature, we obtain for the work

mcr - {\fi (z) — •$ (a)} do),
and since «/>(«) is an insensible quantity we may omit it. We may also write

since z is very small compared with u, and expressing u in terras of <a by
(25), we find

CAPILLARY ACTION.

This then expresses the work done by the attractive forces when a particle
M M brought from an infinite distance to the point P at a distance z from
a stratum whose suHkce-density is <r, and whose principal radii of curvature
mn Rt and J(.

To find the work done when m is brought to the point P in the neigh-
liourhood of a solid body, the density of which is a function of the depth v
Mow the surface, we have only to write instead of <r pdz, and to integrate

2irm j pj (:) dz + •* hr+ ft) I f^ (z) dz>

where, in general, we must suppose p a function of z. This expression, when
integrated, gives (1) the work done on a particle m while it is brought from
an infinite distance to the point P, or (2) the attraction on a long slender
column normal to the surface and terminating at P, the mass of unit of
length of the column being m. In the form of the theory given by Laplace,
the density of the liquid was supposed to be uniform. Hence if we write

K=2ir\ ^i(z)dz, H=2irt zt/»(z)efe,
the pressure of a column of the fluid itself terminating at the surface will be

mid the work done by the attractive forces when a particle m is brought to
the surface of the fluid from an infinite distance will be

If we write

"*(*)&-*(*),

then 2irmp0(z) will express the work done by the attractive forces, while a
particle m is brought from an infinite distance to a distance z from the plane
surface of a mass of the substance of density p and infinitely thick. The
function 9(z) is insensible for all sensible values of z. For insensible values it
may become sensible, but it must remain finite even when z = 0, in which case
6(0) = K.

CAPILLARY ACTION. 559

If x is the potential energy of unit of mass of the substance in vapour,
then at a distance z from the plane surface of the liquid

x=x'

At the surface

X = x'
At a distance z within the surface

If the liquid forms a stratum of thickness c, then

(z) + 2irp0 (z - c).

The surface-density of this stratum is <r = cp. The energy per unit of
area is

e = xPdz>

J 0

(c fc

= c/j{x'-4irp0(0)} + 27r/9a d(z)dz + 2Trp*\ 0(c-z)dz.

Jo Jo

Since the two sides of the stratum are similar the last two terms are
equal, and

e = cp \x - 4irpO (0)} + 477y>2 (e 6 (z) dz.

J o

Differentiating with respect to c, we find

da-
dc=P>

Hence the surface-tension

„ de

T=e-(rcfc>

= 47r/)2|fC e(z)dz-c6(c)\.
Integrating the first term within brackets by parts, it becomes

CAPILLARY ACTION.

d6
Remembering that 0(0) ia a finite quantity, and that ^ = -$(z), we find

|

When c is greater than e this ia equivalent to 2// in the equation of Laplace.
Hence the tension is the same for all films thicker than c, the range of the
forces. For thinner films

IT

Hence if ^(c) is positive, the tension and the thickness will increase together.
Now 1wmp^(c) represents the attraction between a particle m and the plane
tfiir&ce of an infinite mass of the liquid, when the distance of the particle out-
side the surface is c. Now, the force between the particle and the liquid is
certainly, on the whole, attractive ; but if between any two small values of
<• it should be repulsive, then for films whose thickness lies between these
values the tension will increase as the thickness diminishes, but for all other
cases the tension will diminish as the thickness diminishes.

We have given several examples in which the density is assumed to be
uniform, because Poisson has asserted that capillary phenomena would not take
place unless the density varied rapidly near the surface. In this assertion we
think he was mathematically wrong, though in his own hypothesis that the
density does actually vary, he was probably right. In fact, the quantity 4wp-K,
which we may call with Van der Waals the molecular pressure, is so great
for most liquids (5000 atmospheres for water), that in the parts near the sur-
face, where the molecular pressure varies rapidly, we may expect considerable
variation of density, even when we take into account the smallness of the
compressibility of liquids.

The pressure at any point of the liquid arises from two causes, the external
pressure P to which the liquid is subjected, and the pressure arising from the
mutual attraction of its molecules. If we suppose that the number of molecules
within the range of the attraction of a given molecule is very large, the part
of the pressure arising from attraction will be proportional to the square of
the number of molecules in unit of volume, that is, to the square of the
density. Hence we may write

(1),

CAPILLARY ACTION. 561

where A is a constant. But by the equations of equilibrium of the liquid

dp= -pdx (2).

Hence — pdx = 2Apdp (3),

and x'-X = 2^p-2£ (4),

where B is another constant.

Near the plane surface of a liquid we may assume p a function of z. We
have then for the value of x a^ the point where z = c,

r'p(z)t(z-c)dz (5),

Jc-e

where c is the range beyond which the attraction of a mass of liquid bounded
by a plane surface becomes insensible. The value of x depends, therefore, on
those values only of p which correspond to strata for which z is nearly equal
to c. We may, therefore, expand p in terms of z — c, or writing * for z — c,

-) + +&c

where the suffix (c) denotes that in the quantity to which it is applied after
differentiation, z is to be made equal to c. We may now write

[+e

The function »// (x) has equal values for + x and - x. Hence afi/> (x) dx

J -•

vanishes if n is odd.

But if we write

f+t i [+•

K=tr\ $(x)dx, L = -TT\ y?ty(x}dx,

1 f+'

M = . TT x4^ (x) dx, &c.

1 . / . • ) . 4 J — «

This is the expression for x on ^ne hypothesis that the value of p can
be expanded in a series of powers of z — c within the limits z — c and z + e. It

VOL. II. 71

CAPILLARY ACTION.

M onlj when the point /' b within the distance c of the surface of the liquid
thk COMOO to be possible.
If we now substitute. for x »te value from equation (4), we obtain

2Ap — 2B = 2A'/o + 2L-r? + 2 J/ T~ + Ac.,
linear differential equation in p, the solution of which is

p - jj-
where «„ n,, «,, «4 are the roots of the equation

The coefficient J/ is less than JL, where « is the range of the attractive
force. Hence we may consider M very small compared with L. If we neglect
.!/ altogether,

n,= -

If we assume a quantity a such that a'K=2L, we may call a the cur/"</<
range of the molecular forces. If we also take b, so that 6n=l, we may call
b the modulus of the variation of the density near the surface.

Our calculation hitherto has been made on the hypothesis that a is small
when compared with b, and in that case we have found that a' : 6' :: A — K : K.

But it appears from experiments on liquids that A — K is in general large
when compared with AT, and sometimes very large. Hence we conclude, first,
that the hypothesis of our calculation is incorrect, and, secondly, that the
phenomena of capillary action do not in any very great degree depend on the
variation of density near the surface, but that the principal part of the force
depends on the finite range of the molecular action. .

In the following table, Ap is half the cubical elasticity of the liquid, and
Kp the molecular pressure, both expressed in atmospheres (the absolute value
of an atmosphere being one million in centimetre-gramme-second measure, see
below, p. 589). p is the density, T the surface-tension, and a the average range
of the molecular action, as calculated by Van der Waals from the values of
T and K.

CAPILLARY ACTION.

563

The unit in which a is expressed is 1cm. xlO~"; a is therefore the twenty-
millionth part of a centimetre for mercury, the thirty-millionth for water, and
the forty-millionth part for alcohol. Quincke, however, found by direct experi-
ment that certain molecular actions were sensible at a distance of a two-
hundred-thousandth part of a centimetre, so that we cannot regard any of these
numbers as accurate.

Ap

KP

P

rp

.'

Ether

4600

1300

•73

18

29

Alcohol

5500

2100

•79

25-5

25

Bisulphide of Carbon . . .
Water

16000
22200

2900
5000

1-27
1

32-1
81

23
31 '

Mercury

542000

22500

13-54

540

49

ON SURFACE-TENSION.

Definition. — The tension of a liquid surface across any line drawn on the
surface is normal to the line, and is the same for all directions of the line,
and is measured by the force across an element of the line divided by the
length of that element.

Experimental Laws of Surface-tension.

1. For any given liquid surface, as the Surface which separates water from
air, or oil from water, the surface-tension is the same at every point of the
surface and in every direction. It is also practically independent of the curva-
ture of the surface, although it appears from the mathematical theory that
there is a slight increase of tension where the mean curvature of the surface
is concave, and a slight diminution where it is convex. The amount of this
increase and diminution is too small to be directly measured, though it has
a certain theoretical importance in the explanation of the equilibrium of the
superficial layer of the liquid where it is inclined to the horizon.

2. The surface-tension diminishes as the temperature rises, and when the
temperature reaches that of the critical point at which the distinction between
the liquid and its vapour ceases, it has been observed by Andrews that the

71—2

• :

CAPILLARY ACTION.

capillary action also vanishes. The early writers on capillary action supposed
that the diminution of capillary action was due simply to the change of density
oomcpooding to the rise of temperature, and, therefore, assuming the surface-
tension to vary as the square of the density, they deduced its variations from
the observed dilatation of the liquid by heat. This assumption, however, does
not appear to be verified by the experiments of Brunner and Wolff on the
rue of water in tubes at different temperatures.

3. The tension of the surface separating two liquids which do not mix
cannot be deduced by any known method from the tensions of the surfaces
of the liquids when separately in contact with air.

When the surface is curved, the effect of the surface-tension is to make
the pressure on the concave side exceed the pressure on the convex side by

T(-K+-n), where T is the intensity of the surface-tension and JRlt 72, are
the radii of curvature of any two sections normal to the surface and to each

If three fluids which do not mix are in contact with each other, the three
surfaces of separation meet in a line, straight or curved.
Let 0 (fig. 3) be a point in this line, and let the
plane of the paper be supposed to be normal to the
line at the point 0. The three angles between the
tangent planes to the three surfaces of separation at
the point 0 are completely determined by the tensions
of the three surfaces. For if in the triangle abc the
side ab is taken so as to represent on a given scale
the tension of the surface of contact of the fluids a
and 6, and if the other sides be and ca are taken so
as to represent on the same scale the tensions of the surfaces between b and
,md between c and a respectively, then the condition of equilibrium at O
for the corresponding tensions R, P, and Q is that the angle ROP shall be
the supplement of abc, POQ of bca, and, therefore, QOR of cab. Thus the
angles at which the surfaces of separation meet are the same at all parts of
the line of concourse of the three fluids. When three films of the same liquid
meet, their tensions are equal, and, therefore, they make angles of 120° with
each other. The froth of soap-suds or beat-up eggs consists of a multitude
of small films which meet each other at angles of 120°.

Fig. 3.

CAPILLARY ACTION. 565

If four fluids, a, b, c, d, meet in a point 0, and if a tetrahedron ABCD
is formed so that its edge AB represents the tension of the surface of contact
of the liquids a and b, BC that of b and c, and so on; then if we place
this tetrahedron so that the face ABC is normal to the tangent at O to the
line of concourse of the fluids abc, and turn it so that the edge AB is normal
to the tangent plane at O to the surface of contact of the fluids a and b,
then the other three faces of the tetrahedron will be normal to the tangents
at O to the other three lines of concourse of the liquids, and the other five
edges of the tetrahedron will be normal to the tangent planes at 0 to the
other five surfaces of contact.

If six films of the same liquid meet in a point the corresponding tetra-
hedron is a regular tetrahedron, and each film, where it meets the others, has
an angle whose cosine is — f. Hence if we take two nets of wire with
hexagonal meshes, and place one on the other so that the point of concourse
of three hexagons of one net coincides with the middle of a hexagon of the
other, and if we then, after dipping them in Plateau's liquid, place them
horizontally, and gently raise the upper one, we shall develop a system of
plane laminae arranged as the walls and floors of the cells are arranged in a
honeycomb. We must not, however, raise the upper net too much, or the
system of films will become unstable.

When a drop of one liquid, B, is placed on the surface of another, A,
the phenomena which take place depend on the relative magnitude of the three
surface-tensions corresponding to the surface between A and air, between B
and air, and between A and B. If no one of these tensions is greater than
the sum of the other two, the drop will assume the form of a lens, the
angles which the upper and lower surfaces of the lens make with the free
surface of A and with each other being equal to the external angles of the
triangle of forces. Such lenses are often seen formed by drops of fat floating
on the surface of hot water, soup, or gravy. But when the surface-tension of
A exceeds the sum of the tensions of the surfaces of contact of B with air
and with A, it is impossible to construct the triangle of forces, so that equi-
librium becomes impossible. The edge of the drop is drawn out by the surface-
tension of A with a force greater than the sum of the tensions of the two
surfaces of the drop. The drop, therefore, spreads itself out, with great velocity,
over the surface of A till it covers an enormous area, and is reduced to such
extreme tenuity that it is ' not probable that it retains the same properties of

••'

CAPILLARY ACTION.

which it has in a large mass. Thus a drop of train oil will

OTer Uw surface of the sea till it shews the colours of tliin plates.

Thaw rapidly dwcend in Newton's scale and at last disappear, shewing that
the thidtMM of the film is less than the tenth part of the length of a wave
nf light But even when thus attenuated, the film may be proved to be pre-
iwnt, since the surface-tension of the liquid is considerably less than that of
pore wmter. This may be shewn by placing another drop of oil on the surface.
Thk drop will not spread out like the first drop, but will take the form of
a flat lens with a distinct circular edge, shewing that the surface-tension of
what is still apparently pure water is now less than the sum of the tensions
nf the surfaces separating oil from air and water.

The spreading of drops on the surface of a liquid has formed the subject
of a very extensive series of experiments by Mr Tomlinson. M. Van der
Mensbragghe has also written a very complete memoir on this subject*.

When a solid body is in contact with two fluids, the surface
of the solid cannot alter its form, but the angle at which the

\> I surface of contact of the two fluids meets the surface of the solid
s^ depends on the values of the three surface-tensions. If a and l>
are the two fluids and c the solid then the equilibrium of the
tensions at the point 0 depends only on that of thin components
parallel to the surface, because the surface-tensions normal to the
surface are balanced by the resistance of the solid. Hence if the
angle ROQ (fig. 4) at which the surface of contact OP meets the solid is
denoted by a,

^-2^-2

whence

-'oft

As an experiment on the angle of contact only gives us the difference of the
surface-tensions at the solid surface, we cannot determine their actual value.
It is theoretically probable that they are often negative, and may be called
surface-pressures.

The constancy of the angle of contact between the surface of a fluid and
a solid was first pointed out by Dr Young, who states that the angle of
contact between mercury and glass is about 140°. Quincke makes it 128° 52'.

* Sur la Tension StiperficieUe des lAquldes, Bruxclles, 1873.

CAPILLARY ACTION.

567

If the tension of the surface between the solid and one of the fluids ex-
ceeds the sum of the other two tensions, the point of contact will not be in
equilibrium, but will be dragged towards the side on which the tension is
greatest. If the quantity of the first fluid is small it will stand in a drop
on the surface of the solid without wetting it. If the quantity of the second
fluid is small it will spread itself over the surface and wet the solid. The
angle of contact of the first fluid is 180° and that of the second is zero.

If a drop of alcohol be made to touch one side of a drop of oil on a
glass plate, the alcohol will appear to chase the oil over the plate, and if a
drop of water and a drop of bisulphide of carbon be placed in contact in a
horizontal capillary tube, the bisulphide of carbon will chase the water along
the tube. In both cases the liquids move in the direction in which the surface-
pressure at the solid is least.

ON THE RISE OF A LIQUID IN A TUBE.

Let a tube (fig. 5) whose internal radius is r, made of a solid substance c,
be dipped into a liquid a. Let us suppose that the angle of contact for this
liquid with the solid c is an acute angle. This
implies that the tension of the free surface of
the solid c is greater than that of the surface
of contact of the solid with the liquid a. Now
consider the tension of the free surface of the
liquid a. All round its edge there is a tension
T acting at an angle a with the vertical. The
circumference of the edge is 2nr, so that the
resultant of this tension is a force ZirrTco&a
acting vertically upwards on the liquid. Hence
the liquid will rise in the tube till the weight of the vertical column between
the free surface and the level of the liquid in the vessel balances the resultant
of the surface-tension. The upper surface of this column is not level, so that
the height of the column cannot be directly measured, but let us assume that
h is the mean height of the column, that is to say, the height of a column
of equal weight, but with a flat top. Then if r is the radius of the tube at
the top of the column, the volume of the suspended column is irr'h, and its

Fig. 5.

CAPILLARY ACTION.

..

i-

, when p i« it* density and 0 the intensity of gravity.
Equating thia force with the resultant of the tension

ZTcoaa

" = '

Heooe the mean height to which the fluid rises is inversely as the radius of
the tube. For water in a clean glass tube the angle of contact is zero, and

ZT
n = — .

For mercury in a glass tube the angle of contact is 128' 52', the cosine of
which is negative. Hence when a glass tube is dipped into a vessel of mercury,
the mercury within the tube stands at a lower level than outside it.

RISE OF A LIQUID BETWEEN Two PLATES.

When two parallel plates are placed vertically in a liquid the liquid rises
between them. If we now suppose fig. 5 to represent a vertical section per-
pendicular to the plates, we may calculate the rise of the liquid. Let / be
the breadth of the plates measured perpendicularly to the plane of the paper,
then the length of the line which bounds the wet and the dry parts of the
plates inside is / for each surface, and on this the tension T acts at an angle
a to the- vertical. Hence the resultant of the surface-tension is ZITcosa. If
the distance between the inner surfaces of the plates is a, and if the mean
height of the film of fluid which rises between them is h, the weight of fluid
raised is pyhla. Equating the forces —

, ZTcosa

whence h = -

pga

This expression is the same as that for the rise of a liquid in a tube, except
that instead of r, the radius of the tube, we have a the distance of the plates.

CAPILLARY ACTION. 569

FORM OF THE CAPILLARY SURFACE.

The form of the surface of a liquid acted on by gravity is easily deter-
mined if we assume that near the part considered the line of contact of the
surface of the liquid with that of the solid bounding it is straight and hori-
zontal, as it is when the solids which constrain the liquid are bounded by
surfaces formed by horizontal and parallel generating lines. This will be the
case, for instance, near a flat plate dipped into the liquid. If we suppose these
generating lines to be normal to the plane of the paper then all sections of
the solids parallel to this plane will be equal and similar to each other, and
the section of the surface of the liquid will be of the same form for all such
sections.

Let us consider the portion of the liquid between two parallel sections
distant one unit of length. Let Plt P2 (fig. 6) be
two points of the surface ; 6it 0a, the inclination of
the surface to the horizon at Px and P2; ylt ?/2 the
heights of P1 and P2 above the level of the liquid
at a distance from all solid bodies. The pressure at
any point of the liquid which is above this level is
negative unless another fluid as, for instance, the air,
presses on the upper surface, but it is only the differ-
ence of pressures with which we have to do, because Fi«- 6-
two equal pressures on opposite sides of the surface produce no effect.

We may, therefore, write for the pressure at a height y

where p is the density of the liquid, or if there are two fluids the excess of
the density of the lower fluid over that of the upper one.

The forces acting on the portion of liquid PJP^A^ are — first, the hori-
zontal pressures, -\pgy? and fogy? ; second, the surface-tension T acting at
P! and P, in directions inclined 01 and 02 to the horizon. Resolving horizontally

we find —

T (cos 0, - cos 0,) + \g? (y? - y?) = 0,

whence cos 02 = cos 6l — fajpy? + 3 -jr y*>

VOL. II. ?2

7 CAPILLARY ACTION.

or if we suppoM P, fixed and P, variable, we may write

ooa 0 - i y\[- +constant.

This equation gives a relation between the inclination of the curve to the
horuun and the height above the level of the liquid.

Resolving vertically we find that the weight of the liquid raised above the
level most be equal to T (sin 0, — sin 0,), and this is therefore equal to the area
/',/Vff^t multiplied by gp. The form of the capillary surface is identical with
that of the "elastic curve," or the curve formed by a uniform spring originally
straight, when its ends are acted on by equal and opposite forces applied either
to the ends themselves or to solid pieces attached to them. Drawings of

the different forms of the curve may be found in
Thomson and Tait's Natural Philosophy, Vol. i.
p. 455.

We shall next consider the rise of a liquid
between two plates of different materials for which
the angles of contact are a, and a,, the distance
between the plates being a, a small quantity. Since

the plates are very near one another we may use the following equation of

the surface as an approximation: —

whence

y = h1+Ax+JBxt
h^h + Aa+Ea',

cot 04 = — A
cot a, = A + 2Ba
=pga (h,

whence we obtain

• - (2 cot cu, — cot a,).

Let X be the force which must be applied in a horizontal direction to either
keep from approaching the other, then the forces acting on the first

CAPILLARY ACTION. 571

plate are T+X in the negative direction, and T sin ^ + $gph* in the positive
direction. Hence

For the second plate

Hence X = £ gp (h? + hf) - T (1 - 1 (sin a, + sin a,)},

or, substituting the values of hr and hu

T-

X = % — j (cos c^ + cos a^f — T{1 — £ (sin 04 + sin a,) — ^ (cos c^ + cos cu.) (cot a^ + cot a,)},

the remaining terms being negligible when a is small. The force, therefore, with
which the two plates are drawn together consists first of a positive part, or
in other words an attraction, varying inversely as the square of the distance,
and second, of a negative part or repulsion independent of the distance. Hence
in all cases except that in which the angles at and o^ are supplementary to
each other, the force is attractive when a is small enough, but when cos a,
and cos a, are of different signs, as when the liquid is raised by one plate,
and depressed by the other, the first term may be so small that the repulsion
indicated by the second term comes into play. The fact that a pair of plates
which repel one another at a certain distance may attract one another at a
smaller distance was deduced by Laplace from theory, and verified by the
observations of the Abb£ Hauy.

A DROP BETWEEN TWO PLATES.

If a small quantity of a liquid which wets glass be introduced between
two glass plates slightly inclined to each other, it will run towards that part
where the glass plates are nearest together. When the liquid is in equilibrium
it forms a thin film, the outer edge of which is all of the same thickness.
If d is the distance between the plates at the edge of the film and n the

atmospheric pressure, the pressure of the liquid in the film is II -=. — ,

and if A is the area of the film between the plates and B its circumference,
the plates will be pressed together with a force

ZATcosa j,™ .

- + BTsma,
a

72—2

CAPIIXABY ACTION.

and this, whether the atmosphere exerts any pressure or not. The force thus
produced by tbe introduction of a drop of water between two plates is enormous,
and i. often wfficient to press certain parts of the plates together so power-
fbOy M to bruise them or break them. When two blocks of ice are placed
loosely together so that the superfluous water which melts from them may
drain away, the remaining water draws the blocks together with a force suffi-
cMOt to cause the blocks to adhere by the process called Regelation.

In many experiments bodies are floated on the surface of water in order
that they may be free to move under the action of slight horizontal forces.
Thus Newton placed a magnet in a floating vessel and a piece of iron in
another in order to observe their mutual action, and Ampere floated a voltaic
battery with a coil of wire in its circuit in order to observe the effects of
the earth's magnetism on the electric circuit. When such floating bodies come
near the edge of the vessel they are drawn up to it, and are apt to stick
fast to it. There are two ways of avoiding this inconvenience. One is to grease
the float round its water-line so that the water is depressed round it. This,
however, often produces a worse disturbing effect, because a thin film of grease
spreads over the water and increases its surface-viscosity. The other method
is to fill the vessel with water till the level of the water stands a little
higher than the rim of the vessel. The float will then be repelled from the
edge of the vessel. Such floats, however, should always be made so that the
section taken at the level of the water is as small as possible.

PHENOMENA ARISING FROM THE VARIATION OF THE SURFACE-TENSION.

Pure water has a higher surface-tension than that of any other substance
liquid at ordinary temperatures except mercury. Hence any other liquid if
mixed with water diminishes its surface-tension. For example, if a drop of
alcohol be placed on the surface of water, the surface-tension will be diminished
from 80, the value for pure water, to 25, the value for pure alcohol. The
surface of the liquid will therefore no longer be in equilibrium, and a current
will be formed at and near the surface from the alcohol to the surrounding
water, and this current will go on as long as there is more alcohol at one
j»art of the surface than at another. If the vessel is deep, these currents will
be balanced by counter currents below them, but if the depth of the water

CAPILLARY ACTION. 573

is only two or three millimetres, the surface-current will sweep away the whole
of the water, leaving a dry spot where the alcohol was dropped in. This
phenomenon was first described and explained by Professor James Thomson,
who also explained a phenomenon, the converse of this, called the "tears of
strong wine."

If a wine glass be half-filled with port wine the liquid rises a little up
the side of the glass as other liquids do. The wine, however, contains alcohol
and water, both of which evaporate, but the alcohol faster than the water, so
that the superficial layer becomes more watery. In the middle of the vessel
the superficial layer recovers its strength by diffusion from below, but the film
adhering to the side of the glass becomes more watery, and therefore has a
higher surface-tension than the surface of the stronger wine. It therefore creeps
up the side of the glass dragging the strong wine after it, and this goes on
till the quantity of fluid dragged up collects into a drop and runs down the
side of the glass.

The motion of small pieces of camphor floating on water arises from the
gradual solution of the camphor. If this takes place more rapidly on one side
of the piece of camphor than on the other side, the surface-tension becomes
weaker where there is most camphor in solution, and the lump, being pulled
unequally by the surface-tensions, moves off in the direction of the strongest
tension, namely, towards the side on which least camphor is dissolved.

If a drop of ether is held near the surface of water the vapour of ether
condenses on the surface of the water, and surface-currents are formed flowing
in every direction away from under the drop of ether.

If we place a small floating body in a shallow vessel of water and wet
one side of it with alcohol or ether, it will move off with great velocity and
skim about on the surface of the water, the part wet with alcohol being always
the stern.

The surface-tension of mercury is greatly altered by slight changes in the
state of the surface. The surface-tension of pure mercury is so great that it
is very difficult to keep it clean, for every kind of oil or grease spreads over
it at once.

But the most remarkable effects of change of surface-tension are those pro-
duced by what is called the electric polarization of the surface. The tension
of the surface of contact of mercury and dilute sulphuric acid depends on the
electromotive force acting between the mercury and the acid. If the electro-

N

t'APILLARY ACTION.

motive fcwe M from tlie acid to the mercury the surface-tension increases; if
it H from the roereoiy to the acid, it diminishes. Faraday observed that a
Urge drop of mercury, resting on the flat bottom of a vessel containing dilute
acid, changes it* form in a remarkable way when connected with one of the
uloBtrodm of •> battery, the other electrode being placed in the acid. When
the mercury is made positive it becomes dull and spreads itself out; when it
» made negative it gathers itself together and becomes bright again. M. Lipp-
raaim. who has made a careful investigation of the subject, finds that exceed-
ingly small variations of the electromotive force produce sensible changes in
the surface-tension. The effect of one Darnell's cell is to increase the tension
from 30-4 to 40'6. He has constructed a capillary electrometer by which
differences of electric potential less than O'Ol of that of a Daniell's cell can
be detected by the difference of the pressure required to force the mercury to
a given point of a fine capillary tube. He has also constructed an apparatus
in which this variation in the surface-tension is made to do work and drive
a machine. He has also found that this action is reversible, for when the
ana of the surface of contact of the acid and mercury is made to increase,
an electric current passes from the mercury to the acid, the amount of elec-
tricity which passes while the surface increases by one square centimetre being
sufficient to decompose '000013 grammes of water.

Ux THE FORMS OF LIQUID FILMS WHICH ARE FIGURES OF REVOLUTION.

A Spherical Soap-bubble.

A soap-bubble is simply a small quantity of soap-suds spread out so as
to expose a large surface to the air. The bubble, in fact, has two surfaces,
an outer and an inner surface, both exposed to air. It has, therefore, a certain
amount of surface-energy depending on the area of these two surfaces. Since
in the case of thin films the outer and inner surfaces are approximately equal,
we shall consider the area of the film as representing either of them, and
shall use the symbol T to denote the energy of unit of area of the film, both
surfaces being taken together. If T is the energy of a single surface of the
liquid, T the energy of the film is 27". When by means of a tube we blow
air into the inside of the bubble we increase its volume and therefore its

CAPILLARY ACTION. 575

surface, and at the same time we do work in forcing air into it, and thus
increase the energy of the bubble.

That the bubble has energy may be shewn by leaving the end of the tube
open. The bubble will contract, forcing the air out, and the current of air
blown through the tube may be made to deflect the flame of a candle. If the
bubble is in the form of a sphere of radius r this material surface will have
an area

3=4^ ..................................... (1).

If T be the energy corresponding to unit of area of the film the surface-energy
of the whole bubble will be

ST=±irr>T ......................... . .......... (2).

The increment of this energy corresponding to an increase of the radius from
r to r + dr is therefore

TdS=8irrTdr ................................. (3).

Now this increase of energy was obtained by forcing in air at a pressure
greater than the atmospheric pressure, and thus increasing the volume of the
bubble.

Let II be the atmospheric pressure and H+p the pressure of the air
within the bubble. The volume of the sphere is

and the increment of volume is

cZF=4irrtfr .................................... (5).

Now if we suppose a quantity of air already at the pressure II +p, the work
done in forcing it into the bubble is pdV. Hence the equation of work and
energy is

pdV=Tds .................................... (6),

or 4wpr>dr = SirrdrT ................................. (7),

or p = 2T ..................................... (8).

This, therefore, is the excess of the pressure of the air within the bubble over
that of the external air, and it is due to the action of the inner and outer
surfaces of the bubble. "We may conceive this pressure to arise from the ten-
dency which the bubble has to contract, or in other words from the surface-
tension of the bubble.

If to increase the area of the surface requires the expenditure of work,
the Mir&ee roust reewt extension, and if the bubble in contracting can do work,
the Kirface roost tend to contract The surface must therefore act like a sheet
of india-rubber when extended both in length and breadth, that is, it must
exert •uriace-tension. The tension of the sheet of india-rubber, however, de-
pend* on the extent to which it is stretched, and may be different in different
directions, whereas the tension of the surface of a liquid remains the same
however much the film is extended, and the tension at any point is the same
in all directions.

The intensity of this surface-tension is measured by the stress which it
exerts across a line of unit length. Let us measure it in the case of the
spherical soap-bubble by considering the stress exerted by one hemisphere of
the bubble on the other, across the circumference of a great circle. This stress
is balanced by the pressure p acting over the area of the same great circle :
it is therefore equal to irr*p. To determine the intensity of the surface-tension
we have to divide this quantity by the length of the line across which it acts,
which is in this case the circumference of a great circle Zirr. Dividing irfp
by this length we obtain %pr as the value of the intensity of the surface-
tension, and it is plain from equation 8 that this is equal to T. Hence the
numerical value of the intensity of the surface-tension is equal to the numerical
value of the surface-energy per unit of surface. We must remember that since
the film has two surfaces the surface-tension of the film is double the tension
of the surface of the liquid of which it is formed.

To determine the relation between the surface-tension and
the pressure which balances it when the form of the surface
is not spherical, let us consider the following case : —

Let fig. 8 represent a section through the axis Cc of a
soap-bubble in the form of a figure of revolution bounded by
two circular disks AB and ab, and having the meridian section
APa. Let PQ be an imaginary section normal to the axis.
Let the radius of this section PR be y, and let PT, the
tangent at P, make an angle a with the axis.

Let us consider the stresses which are exerted across this
imaginary section by the lower part on the upper part. If
the internal pressure exceeds the external pressure by p,
there is in the first place a force infp acting upwards arising

CAPILLARY ACTION. 577

from the pressure p over the area of the section. In the next place, there is
the surface-tension acting downwards, but at an angle a with the vertical,
across the circular section of the bubble itself, whose circumference is 2iry, and
the downward force is therefore

Now these forces are balanced by the external force which acts on the
disk ACB, which we may call F. Hence equating the forces which act on
the portion included between ACB and PRQ

Try*p-2TryTcosa = -F ............................. (9).

If we make CR — z, and suppose z to vary, the shape of the bubble of course
remaining the same, the values of y and of a will change, but the other
quantities will be constant. In studying these variations we may if we please
take as our independent variable the length s of the meridian section AP
reckoned from A. Differentiating equation 9 with respect to s we obtain, after
dividing by 2ir as a common factor

Now -^ = sina .................................... (11).

The radius of curvature of the meridian section is

The radius of curvature of a normal section of the surface at right angles to
the meridian section is equal to the part of the normal cut off by the axis,
which is

(13).

cos a
Hence dividing equation 10 by 7/sina, we find

This equation, which gives the pressure in terms of the principal radii of curva-
ture, though here proved only in the case of a surface of revolution, must be
true of all surfaces. For the curvature of any surface at a given point may

73

VOL. II.

378

i \PILLABY ACTION.

be completely defined in terms of the positions of its principal normal sections
and their radii of curvature.

Before going farther we may deduce from equation 9 the nature of all
the figures of revolution which a liquid film can assume. Let us first deter-
^fc^ ^|je nature of a curve, such that if it is rolled on the axis its origin
will trace out the meridian section of the bubble. Since at any instant the
rolling curve is rotating about the point of contact with the axis, the line
drawn from this point of contact to the tracing point must be normal to the
direction of motion of the tracing point. Hence if N is the point of contact,
NP must be normal to the traced curve. Also, since the axis is a tangent
to the rolling curve, the ordinate PR is the perpendicular from the tracing
point P on the tangent. Hence the relation between the radius vector and
the perpendicular on the tangent of the rolling curve must be identical with
the relation between the normal PN and the ordinate PR of the traced curve.
If we write r for PN, then y = r cos a, and equation 9 becomes

pr

This relation between y and r is identical with the relation between the per-
pendicular from the focus of a conic section on the tangent at a given point
and the focal distance of that point, provided the transverse and conjugate axes
<>f the conic are 2a and 2b respectively, where

T F

a = — , and 6 = — .
p Trp

Hence the meridian section of the film may be traced by the focus of such
a conic, if the conic is made to roll on the axis.

ON THE DIFFERENT FORMS OF THE MERIDIAN LINE.

(1) When the conic is an ellipse the meridian line is in the form of a
series of waves, and the film itself has a series of alternate swellings and con-
tractions as represented in figs. 8 and 9. This form of the film is called the
unduloid.

CAPILLARY ACTION. 579

(1 a.) When the ellipse becomes a circle, the meridian line becomes a
straight line parallel to the axis, and the film passes into the form of a
cylinder of revolution.

(1 6.) As the eUipse degenerates into the straight line joining its foci, the
contracted parts of the unduloid become narrower, till at last, the figure becomes
a series of spheres in contact.

In all these cases the internal pressure exceeds the external by — where

J a

a is the semi-transverse axis of the conic. The resultant of the internal pres-
sure and the surface-tension is equivalent to a tension along the axis, and the
numerical value of this tension is equal to the force due to the action of this
pressure on a circle whose diameter is equal to the conjugate axis of the
ellipse.

(2) When the conic is a parabola the meridian line is a catenary (fig. 10),
the internal pressure is equal to the external pressure, and the tension along
the axis is equal to 2irTm where m is the distance of the vertex from the
focus.

Fig. 9.— Unduloid. Fig. 10.— Catenoid. Fig. 11.— Nodoid.

(3) When the conic is a hyperbola the meridian line is in the form of
a looped curve (fig. 11). The corresponding figure of the film is called the
nodoid. The resultant of the internal pressure and the surface-tension is equiva-
lent to a pressure along the axis equal to that due to a pressure p acting
on a circle whose diameter is the conjugate axis of the hyperbola.

When the conjugate axis of the hyperbola is made smaller and smaller,
the nodoid approximates more and more to the series of spheres touching each
other along the axis. When the conjugate axis of the hyperbola increases with-
out limit, the loops of the nodoid are crowded on one another, and each
becomes more nearly a ring of circular section, without, however, ever reaching
this form. The only closed surface belonging to the series is the sphere.

73—2

CAPILLARY ACTION.

These nguws of revolution have been studied mathematically by Poisson*.
Goldschmidtt. Lindeldf and MoignoJ, Delaunayll, Lamarle§, Beerl, and Mann-
»•*, and have been produced experimentally by Plateautt in the two different
ways already described.

The limiting conditions of the stability of these figures have been studied
both mathematically and experimentally. We shall notice only two of them,
the cylinder and the catenoid.

STABILITY OF THE CYLINDER.

The cylinder is the limiting form of the unduloid when the rolling ellipse
a circle. When the ellipse differs infinitely little from a circle, the

x

c is

equation of the meridian line becomes approximately y = a + c sin - where

mull. This is a simple harmonic wave-line, whose mean distance from the axis
is a, whose wave-length is 2ira, and whose amplitude is c. The internal pres-

T
sure corresponding to this unduloid is as before p = - . Now consider a

portion of a cylindric film of length x terminated by two equal disks of radius ;•
und containing a certain volume of air. Let one of these disks be made to

approach the other by a small quantity dx. The
film will swell out into the convex part of an
unduloid, having its largest section midway between
the disks, and we have to determine whether the
internal pressure will be greater or less than before.
If A and C (fig. 12) are the disks, and if x the

Ffe. 13.

• NouvMe Uieorie de Faction capillaire (1831).

f Ddwrnimitio ttiperficiei minima rolatione curves data duo puncta jungentis circa datum axem

(Oottingen, 1831).

} Ltymt da calcttl de* variations (Paris, 1861).

'8ur U surface de revolution dont la courbure moyenne eat constant*," Liouvitte's Journal, vi.
{ "TbforiB gfomftrique dea rayons et centres de courbure," Bullet, de CAcad. de Belgigue, 1857.

* Tractate* de Theoria Mathematica Pfuenomenorum in Liqnidix actione gravitate detractis obter-

(Bonn, 1857).
** Journal F/nttitut, No. 1260.
ft Statique exjxrimentale et theorique dea liquidea.

CAPILLARY ACTION. 581

distance between the disks is equal to TTT half the wave-length of the harmonic
curve, the disks will be at the points where the curve is at its mean distance

rrt

from the axis, and the pressure will therefore be - as before. If A C, are

T

the disks, so that the distance between them is less than irr, the curve must
be produced beyond the disks before it is at its mean distance from the axis.
Hence in this case the mean distance is less than r, and the pressure will be

T

greater than - . If, on the other hand, the disks are at 42 and C2, so that the

distance between them is greater than TTT, the curve will reach its mean dis-
tance from the axis before it reaches the disks. The mean distance will there-

71

fore be greater than r, and the pressure will be less than — . Hence if one

r

of the disks be made to approach the other, the internal pressure will be
increased if the distance between the disks is less than half the circumference
of either, and the pressure will be diminished if the distance is greater than
this quantity. In the same way we may shew that if the distance between
the disks is increased, the pressure will be diminished or increased according
as the distance is less or more than half the circumference of either.

Now let us consider a cylindric film contained between two equal fixed disks
A and B, and let a third disk, C, be placed midway between. Let C be slightly
displaced towards A. If AC and OS are each less than half the circumference
of a disk the pressure on C will increase on the side of A and diminish on
the side of B. The resultant force on C will therefore tend to oppose the
displacement and to bring C back to its original position. The equilibrium of
C is therefore stable. It is easy to shew that if C had been placed in any
other position than the middle, its equilibrium would have been stable. Hence
the film is stable as regards longitudinal displacements. It is also stable as
regards displacements transverse to the axis, for the film is in a state of
tension, and any lateral displacement of its middle parts would produce a re-
sultant force tending to restore the film to its original position. Hence if the
length of the cylindric film is less than its circumference, it is in stable equi-
librium. But if the length of the cylindric film is greater than its circumference,
and if we suppose the disk C to be placed midway between A and B, and
to be moved towards A, the pressure on the side next A will dimmish, and
that on the side next B will increase, so that the resultant force will tend to

CAPILLAKY ACTION.

ii,.l,0<MBlL «* the equUibrium of the disk C is therefore mi-
Hem* tho equUibrium of a cylindric film whose length is greater than
unstable. Such a film, if ever so little disturbed, will

l it x'll 'x f

begin to contract at one section and to expand at another, till its form ceases
to iwetnble a cylinder, if it does not break up into two parts which become
ultimately portions of spheres.

INSTABILITY OF A JET OF LIQUID.

When a liquid flows out of a vessel through a circular opening in the
bottom of the vessel, the form of the stream is at first nearly cylindrical though
its diameter gradually diminishes from the orifice downwards on account of the
ring velocity of the liquid. But the liquid after it leaves the vessel is

subject to no forces except gravity, the pressure of the air, and its own surface-
tension. Of these gravity has no effect on the form of the stream except in
drawing asunder its parts in a vertical direction, because the lower parts are
moving faster than the upper parts. The resistance of the air produces little
disturbance until the velocity becomes very great. But the surface-tension,
acting on a cylindric column of liquid whose length exceeds the limit of stability,
begins to produce enlargements and contractions in the stream as soon as the
liquid has left the orifice, and these inequalities in the figure of the column
go on increasing till it is broken up into elongated fragments. These fragments
aa they are felling through the air continue to be acted on by surface-tension.
They therefore shorten themselves, and after a series of oscillations in which
they become alternately elongated and flattened, settle down into the form of
spherical drops.

This process, which we have followed as it takes place on an individual
portion of the falling liquid, goes through its several phases at different dis-
tances from the orifice, so that if we examine different portions of the stream
as it descends, we shall find next the orifice the unbroken column, then a
aeries of contractions and enlargements, then elongated drops, then flattened
drops, and so on till the drops become spherical.

CAPILLARY ACTION. 583

STABILITY OF THE CATENOID.

When the internal pressure is equal to the external, the film forms a
surface of which the mean curvature at every point is zero. The only surface
of revolution having this property is the catenoid formed by the revolution of
a catenary about its directrix. This catenoid, however, is in stable equilibrium
only when the portion considered is such that the tangents to the catenary
at its extremities intersect before they reach the directrix.

To prove this, let us consider the catenary as the form of equilibrium of
a chain suspended between two fixed points A and B. Suppose the chain
hanging between A and B to be of very great length, then the tension at
A or B will be very great. Let the chain be hauled in over a peg at A.
At first the tension will diminish, but if the process be continued the tension
will reach a minimum value and will afterwards increase to infinity as the
chain between A and B approaches to the form of a straight line. Hence for
every tension greater than the minimum tension there are two catenaries pass-
ing through A and B. Since the tension is measured by the height above
the directrix these two catenaries have the same directrix. Every catenary
lying between them has its directrix higher, and every catenary lying beyond
them has its directrix lower than that of the two catenaries.

Now let us consider the surfaces of revolution formed by this system of
catenaries revolving about the directrix of the two catenaries of equal tension.
We know that the radius of curvature of a surface of revolution in the plane
normal to the meridian plane is the portion of the normal intercepted by the
axis of revolution.

The radius of curvature of a catenary is equal and opposite to the por-
tion of the normal intercepted by the directrix of the catenary. Hence a
catenoid whose directrix coincides with the axis of revolution has at every point
its principal radii of curvature equal and opposite, so that the mean curvature
of the surface is zero.

The catenaries which lie between the two whose direction coincides with
the axis of revolution generate surfaces whose radius of curvature convex
towards the axis in the meridian plane is less than the radius of concave
curvature. The mean curvature of these surfaces is therefore convex towards the
axis. The catenaries which lie beyond the two generate surfaces whose radius

CAPILLARY ACTION.

of curvature convex towards the axis in the meridian plane is greater than
the radios of concave curvature. The mean curvature of these surfaces is,
therefore, concave towards the axis.

Now if the pressure is equal on both sides of a liquid film, if its mean
curvature is aero, it will be in equilibrium. This is the case with the two
If the mean curvature is convex towards the axis the film will

from the axis. Hence if a film in the form of the catenoid which is
the axis is ever so slightly displaced from the axis it will move further
from the axis till it reaches the other catenoid.

If the mean curvature is concave towards the axis the film will tend to
approach the axis. Hence if a film in the form of the catenoid which is
nearest the axis be displaced towards the axis, it will tend to move further
towards the axis and will collapse. Hence the film in the form of the cateuoid
which is nearest the axis is in unstable equilibrium under the condition that
it is exposed to equal pressures within and without. If, however, the circular
ends of the catenoid are closed with solid disks, so that the volume of air
contained between these disks and the film is determinate, the film will be
in stable equilibrium however large a portion of the catenary it may consist o£
The criterion as to whether any given catenoid is stable or not may be
obtained as follows.

Let PABQ and ApqB (fig. 13) be two catenaries having the same direc-

trix and intersecting in A and B. Draw Pp and
Qq touching both catenaries, Pp and Qq will
intersect at T, a point in the directrix ; for since
any catenary with its directrix is a similar figure
to any other catenary with its directrix, if the
directrix of the one coincides with that of the
other the centre of similitude must lie on the
common directrix. Also, since the curves at P
and p are equally inclined to the directrix, P
and p are corresponding points and the line Pp

must pass through the centre of similitude. Similarly Qq must pass through
the centre of similitude. Hence T, the point of intersection of Pp and Qq,
must be the centre of similitude and must be on the common directrix. Hence
tlie tangents at A and B to the upper catenary must intersect above the
directrix, and the tangents at A and B to the lower catenary must intersect

CAPILLARY ACTION. 585

below the directrix. The condition of stability of a catenoid is therefore that
the tangents at the extremities of its generating catenary must intersect before
they reach the directrix.

STABILITY OF A PLANE SURFACE.

We shall next consider the limiting conditions of stability of the horizontal
surface which separates a heavier fluid above from a lighter fluid below. Thus,
in an experiment of M. Duprez*, a vessel containing olive oil is placed with
its mouth downwards in a vessel containing a mixture of alcohol and water,
the mixture being denser than the oil. The surface of separation is in this
case horizontal and stable, so that the equilibrium is established of itself.
Alcohol is then added very gradually to the mixture till it becomes lighter
than the oil. The equilibrium of the fluids would now be unstable if it were
not for the tension of the surface which separates them, and which, when the
orifice of the vessel is not too large, continues to preserve the stability of the
equilibrium.

When the equilibrium at last becomes unstable, the destruction of equili-
brium takes place by the lighter fluid ascending in one part of the orifice and
the heavier descending in the other. Hence the displacement of the surface
to which we must direct our attention is one which does not alter the volume
of the liquid in the vessel, and which therefore is upward in one part of the
surface and downward in another. The simplest case is that of a rectangular
orifice in a horizontal plane, the sides being a and b.

Let the surface of separation be originally in the plane of the orifice, and
let the co-ordinates x and y be measured from one corner parallel to the sides
a and b respectively, and let z be measured upwards. Then if p be the
density of the upper liquid, and a- that of the lower liquid, and P the original
pressure at the surface of separation, then when the surface receives an upward
displacement z, the pressure above it will be P-pgz, and that below it will
be P-a-gz, so that the surface will be acted on by an upward pressure (p-<r)gz.
Now if the displacement z be everywhere very small, the curvature in the

planes parallel to xz and yz will be ^4 and ^-2 respectively; and if T is

* "Sur un caa particulier de 1'equilibre des liquides," par F. Duprez, Nouveaux Mem. de I'Acad.
de Belgique, 1851 et 1854.

VOL. II. 74

CAPILLABY ACTION.

the snAce-tension the whole upward force will be

If thi* quantity is of the aame sign as z, the displacement will be increased,
and UM equilibrium will be unstable. If it is of the opposite sign from z,
the equilibrium will lie stable. The limiting condition may be found by put-
ting it equal to *ero. One form of the solution of the equation, and that
which is applicable to the case of a rectangular orifice, is

Substituting in the equation we find the condition

!+" stable.
0 neutral.
— "• unstable.

Tl»at the surface may coincide with the edge of the orifice, which is a
rectangle, whose sides are a and b, we must have

pa = mir, qb = HIT,

when m and n are integral numbers. Also, if in and n are both unity, the
displacement will be entirely positive, and the volume of the liquid will not
be constant. That the volume may be constant, either n or m must be an
number. We have, therefore, to consider the conditions under which

cannot be made negative. Under these conditions the equilibrium is stable for

all small displacements of the surface. The smallest admissible value of ^+^

a o

is — , + ^ , where a is the longer side of the rectangle. Hence the condition of
stability is that

_

is a positive quantity. When the breadth 6 is less than £-2fp the length
a may be unlimited.

CAPILLARY ACTION. 587

7T~
- Z,

where z is the least root of the equation

2 Tl Z2 Z4 Z6

z (*>~ ~274 + 2.42.6~2.42.62.8~l" *C"~
The least root of this equation is

z~ 3-831 71.

If h is the height to which the liquid will rise in a capillary tube of
unit radius, then the diameter of the largest orifice is

= 5-4188^.
M. Duprez found from his experiments

EFFECT OF SURFACE-TENSION ON THE VELOCITY OF WAVES*.

When a series of waves are propagated on the surface of a liquid, the
surface-tension has the effect of increasing the pressure at the crests of the
waves and diminishing it in the troughs. If the wave-length is X, the equa-

tion of the surface is

, • x
y = b sm ZTT r- .

A

The pressure due to the surface-tension T is

This pressure must be added to the pressure due to gravity gpy. Hence the
waves will be propagated as if the intensity of gravity had been

477* T7

instead of g. Now it is shewn in hydrodynamics that the velocity of propa-
gation of waves in deep water is that acquired by a heavy body falling through
half the radius of the circle whose circumference is the wave-length, or

277 A./J

* See Sir W. Thomson, " Hydrokinetio Solutions and Observations," Phil. Mag., Nov. 1871.

74—2

CAFILLABY ACTION.

velocity is a minimum when
and the minimum value is

For waves whoee length from crest to crest is greater than X, the prin-
cipal force concerned in the motion is that of gravitation. For waves whose
length is less than X the principal force concerned is that of surface-tension.
Sir William Thomson proposes to distinguish the latter kind of waves by the
of ripples.

When a small body is partly immersed in a liquid originally at rest, and
horizontally with constant velocity F, waves are propagated through the
liquid with various velocities according to their respective wave-lengths. In front
of the body the relative velocity of the fluid and the body varies from F
where the fluid is at rest, to zero at the cutwater on the front surface of the
body. The waves produced by the body will travel forwards faster than the
body till they reach a distance from it at which the relative velocity of the
body and the fluid is equal to the velocity of propagation corresponding to the
wave-length. The waves then travel along with the body at a constant dis-
tance in front of it. Hence at a certain distance in front of the body there
is a series of waves which are stationary with respect to the body. Of these,
the waves of minimum velocity form a stationary wave nearest to the front
of the body. Between the body and this first wave the surface is compara-
tively smooth. Then comes the stationary wave of minimum velocity, which is
the most marked of the series. In front of this is a double series of stationary
waves, the gravitation waves forming a series increasing in wave-length with
their distance in front of the body, and the surface-tension waves or ripples
diminishing in wave-length with their distance from the body, and both sets
of waves rapidly diminishing in amplitude with their distance from the body.

If the current-function of the water referred to the body considered as
origin is $, then the equation of the form of the crest of a wave of velocity
tr, the crest of which travels along with the body, is

d\(> = wds
where da is an element of the length of the crest. To integrate this equation

CAPILLARY ACTION.

589

for a solid of given form is probably difficult, but it is easy to see that at
some distance on either side of the body, where the liquid is sensibly at rest,
the crest of the wave will approximate to an asymptote inclined to the path

ntj

of the body at an angle whose sine is -^., where w is the velocity of the
wave and V is that of the body.

The crests of the different kinds of waves will therefore appear to diverge
as they get further from the body, and the waves themselves will be less and
less perceptible. But those whose wave-length is near to that of the wave of
minimum velocity will diverge less than any of the others, so that the most
marked feature at a distance from the body will be the two long lines of
ripples of minimum velocity. If the angle between these is 2,0, the velocity
of the body is wsecO, where iv for water is about 23 centimetres per second.

TABLES OF SURFACE-TENSION.

In the following tables the units of length, mass, and time are the centi-
metre, the gramme, and the second, and the unit of force is that which if
it acted on one gramme for one second would communicate to it a velocity of
one centimetre per second : —

Table of Surface-Tension at 20° C. (Quincke).

Liquid.

Specific
Gravity.

Tension of surface sepa-
rating the liquid from

Angle of contact with glass
in presence of

Air.

Water.

Mercury.

Air.

Water.

Mercury.

"Water

1
13-5432
1-2687
1-4878
0-7906
0-9136
0-8867
0-7977
1-1
1-1248

81
540
32-1
30-6
25-5
36-9
29-7
31-7
70-1
77-5

418
41-75
29-5

20-56
11-55
27-8

418

372-5
399
399
335

250-5
284
377
442-5

25° 32'
51° 8'
32° 16'

25° 12'
21° 50'
37° 44'
36° 20'

23" 20'

26°8'
13° 8'

17°'
37° 44'
42° 46'

26° 8'

47'°"2'
47° 2'

10" 42'

Bisulphide of Carbon

Olive Oil

Solution of Hyposulphite of Soda

Olive oil and alcohol, 12-2.

Olive oil and aqueous alcohol (sp. g. '9231, tension of free surface 25-5), 6-8, angle 87° 48'.

OAP1LLABY ACTION.

guincke has datennincd the surface-tension of a great many substances near
their point of fusion or solidification. His method was that of observing the
form of » large drop Bt*ry*ing on a plane surface. If A" is the height of the
lUt miriaoe of the drop, and t that of the point where its tangent plane is

s 'faff-Tenaotu cf Liquids at tfieir Point of Solidification. From, Quinc.k<.

«_.

Temperature of
Solidification.

Surface-Tension.

Flfttinain

2000° C.

1658

Gold

1200"

983

Zinc .

360"

860

Tin

230°

587

Mercurv..

- 40°

577

Lead •.

330°

448

SilYw

1000°

419

Bumath

265°

382

58°

364

Sodium

90°

253

Antimony

432°

244

Borax ... ,,, ......

1000°

212

Carbonate of Soda

1000°

206

Chloride of Sodium

114

Water

0"

86 -2

Solmiuin ...

217°

70-4

Sulphur

111"

41-3

43°

41-1

Wax

68°

Q-J.J.

Quincke finds that for several series of substances the surface-tension is

nearly proportional to the density, so that if we call (K-Kf=— the specific

9P
cohesion, we may state the general results of his experiments as follows :

The bromides and iodides have a specific cohesion about half that of mercury.
The nitrates, chlorides, sugars, and fats, as also the metals, lead, bismuth, and
antimony, have a specific cohesion nearly equal to that of mercury. Water,
tl»- carbonates and sulphates, and probably phosphates, and the metals, platinum,
p.l.l. silver, cadmium, tin, and copper have a specific cohesion double that of
•ury. Zinc, iron, and palladium, three times that of mercury, and sodium,
six times that of mercury.

CAPILLARY ACTION. 591

RELATION OF SURFACE-TENSION TO TEMPERATURE.

It appears from the experiments of Brunner and of Wolff on the ascent
of water in tubes that at the temperature t" centigrade

r= 75-20 (i-o-ooi 87*);

= 76-08 (1-0'002« + 0'00000415£2), for a tube '02346cm. diameter (Wolff) ;
= 77-34(1-0-001810, for a tube '03098cm. diameter (Wolff).

Sir W. Thomson has applied the principles of Thermodynamics to deter-
mine the thermal effects of increasing or diminishing the area of the free surface
of a liquid, and has shewn that in order to keep the temperature constant
while the area of the surface increases by unity, an amount of heat must be
supplied to the liquid which is dynamically equivalent to the product of the
absolute temperature into the decrement of the surface-tension per degree of
temperature. We may call this the latent heat of surface-extension.

It appears from the experiments of Brunner and Wolff that at ordinary
temperatures the latent heat of extension of the surface of water is dynamically
equivalent to about half the mechanical work done in producing the surface-
extension.

[From NcUurt, Vol. xv.]

LXXXIV // l.'idwig Ferdinand Helmholtz.

THE contributions made by Helmholtz to mathematics, physics, physiology,
psychology, and aesthetics, are well known to all cultivators of these various
subject*. Most of those who have risen to eminence in any one of these
have done so by devoting their whole attention to that science ex-

clusively, so that it is only rarely that the cultivators of different branches can
be of service to each other by contributing to one science the skill they have
acquired by the study of another.

Il'iice the ordinary growth of human knowledge is by accumulation round
a number of distinct centres. The time, however, must sooner or later arrive
when two or more departments of knowledge can no longer remain independent
of each other, but must be fused into a consistent whole. But though men
of science may be profoundly convinced of the necessity of such a fusion, the
operation itself is a most arduous one. For though the phenomena of nature
are all consistent with each other, we have to deal not only with these, but
with tin- hyjKitheses which have been invented to systematise them ; and it
by no means follows that because one set of observers have laboured with all
sincerity to reduce to order one group of phenomena, the hypotheses which
they have formed will be consistent with those by which a second set of
observers have explained a different set of phenomena. Each science may appear
tolerably consistent within itself, but before they can be combined into one,
each must be stripped of the daubing of untempered mortar by which its parts
have been prematurely made to cohere.

Hence the operation of fusing two sciences into one generally involves much
criticism of established methods, and the explosion of many pieces of fancied
knowledge which may have been long held in scientific reputation.

Most of those physical sciences which deal with things without life IKIVC
either undergone this fusion or are in a fair state of preparation for it, and
the form which each finally assumes is that of a branch of dynamics.

; II HUMANS LUDWIO FERDINAND HELMHOLTZ.

by experiment, calculation, or speculation, to the establishment of the principle
• >f the conservation of energy; but there can be no doubt that a very great
impulse was communicated to this research by the publication in 1847, of
Hchuholu'a essay Ueber die Erhaltung der Kraft, which we must now (and
correctly, as a matter of science) translate Conservation of Energy, though in
the translation which appeared in Taylor's Scientific Memoirs, the word AV.;/'c
was translated Force in accordance with the ordinary literary usage of that

time.

In this essay Helmholtz shewed that if the forces acting between material
bodies were equivalent to attractions or repulsions between the particles of these
bodies, the intensity of which depends only on the distance, then the con-
figuration and motion of any material system would be subject to a certain
equation, which, when expressed in words, is the principle of the conservation
uf energy.

Whether this equation applies to actual material systems is a matter which
experiment alone can decide, but the search for what was called the perpetual
motion has been carried on for so long, and always in vain, that we may now
appeal to the united experience of a large number of most ingenious men, any
one of whom, if he had once discovered a violation of the principle, would
have turned it to most profitable account.

Besides this, if the principle were in any degree incorrect, the ordinary
processes of nature, carried on as they are incessantly and in all possible com-
binations, would be certain now and then to produce observable and even
startling phenomena, arising from the accumulated effects of any slight diver-
gence from the principle of conservation.

But the scientific importance of the principle of the conservation of energy
does not depend merely on its accuracy as a statement of fact, nor even on
the remarkable conclusions which may be deduced from it, but on the fertility
of the methods founded on this principle.

Whether our work is to form a science by the colligation of known facts,
or to seek for an explanation of obscure phenomena by devising a course of
experiments, the principle of the conservation of energy is our unfailing guide.
It gives us a scheme by which we may arrange the facts of any physical
science as instances of the transformation of energy from one form to another.
It also indicates that in the study of any new phenomenon our first inquiry
must be, How can this phenomenon be explained as a transformation of energy?

HERMANN LUDWIG FERDINAND HELMHOLTZ. 595

What is the original form of the energy? What is its final form? and What
are the conditions of transformation ?

To appreciate the full scientific value of Helmholtz's little essay on this
subject, we should have to ask those to whom we owe the greatest discoveries
in thermodynamics and other branches of modern physics, how many times they
have read it over, and how often during their researches they felt the weighty
statements of Helmholtz acting on their minds like an irresistible driving-power.

We come next to his researches on the eye and on vision, as they are
given in his book on Physiological Optics. Every modern oculist will admit
that the ophthalmoscope, the original form of which was invented by Helmholtz,
has substituted observation for conjecture in the diagnosis of diseases of the
inner parts of the eye, and has enabled operations on the eye to be made
with greater certainty.

But though the ophthalmoscope is an indispensable aid to the oculist, a
knowledge of optical principles is of still greater importance. Whatever optical
information he had was formerly obtained from text-books, the only practical
object of which seemed to be to explain the construction of telescopes. They
were full of very inelegant mathematics, and most of the results were quite
inapplicable to the eye.

The importance to the physiologist and the physician of a thorough know-
ledge of physical principles has often been insisted on, but unless the physical
principles are presented in a form which can be directly applied to the com-
plex structures of the living body, they are of very little use to him ; but
Helmholtz, Bonders, and Listing, by the application to the eye of Gauss's
theory of the cardinal points of an instrument, have made it possible to acquire
a competent knowledge of the optical effects of the eye by a few direct
observations.

But perhaps the most important service conferred on science by this great
work consists in the way in which the study of the eye and vision is made
to illustrate the conditions of sensation and of voluntary motion. In no depart-
ment of research is the combined and concentrated light of all the sciences
more necessary than in the investigation of sensation. The purely subjective
school of psychologists used to assert that for the analysis of sensation no
apparatus was required except what every man carries within himself, for, since
a sensation can exist nowhere except in our own consciousness, the only possible
method for the study of sensations must be an unbiased contemplation of our

75—2

HERMANN l.l'BWIG FKK1MNANU HKLMHOLTZ.

own frame of mind Othera might study the conditions under which an impulse
it propagated along a nerve, and might suppose that while doing so they were
studying actuations, but though such a procedure leaves out of account the
very oaymrin of the phenomenon, and treats a fact of consciousness as if it
were an electric current, the methods which it has suggested have been more
f.Ttilf in results than the method of self-contemplation has ever been.

But the best results are obtained when we employ all the resources of
physical science so as to vary the nature and intensity of the external stimulus,
and then consult consciousness as to the variation of the resulting sensation.
It was by this im-thod that Johannes Muller established the great principle
that the difference in the sensations due to different senses does not depend
upon the actions which excite them, but upon the various nervous arrange-
ments which receive them. Hence the sensation due to a particular nerve may
\arv in intensity, but not in quality, and therefore the analysis of the infinitely
lious states of sensation of which we are conscious must consist in ascertain-
ing the number and nature of those simple sensations which, by entering into
consciousness each in its own degree, constitute the actual state of feeling at
any instant.

If, after this analysis of sensation itself, we should find by anatomy an
apparatus of nerves arranged in natural groups corresponding in number to the
»-lementa of sensation, this would be a strong confirmation of the correctness
of our analysis, and if we could devise the means of stimulating or deadening
each particular nerve in our own bodies, we might even make the investigation
physiologically complete.

The two great works of Helmholtz on Physiological Optics and on the
Sensations of Tone, form a splendid example of this method of analysis applied
to the two kinds of sensation which furnish the largest proportion of the raw
materials for thought.

In the first of these works the colour-sensation is investigated and shewn
to depend upon three variables or elementary sensations. Another investigation,
in which exceedingly refined methods are employed, is that of the motions of
the eyes. Each eye has six muscles by the combined action of which its
angular position may be varied in each of its three components, namely, in
altitude and azimuth as regards the optic axis, and rotation about that axis.
There is no material connection between these muscles or their nerves which
would cause the motion of one to be accompanied by the motion of any other,

HERMANN LUDWIG FERDINAND HELMHOLTZ. 597

so that the three motions of one eye are mechanically independent of the three
motions of the other eye. Yet it is well known that the motions of the axis
of one eye are always accompanied by corresponding motions of the other.
This takes place even when we cover one eye with the fingers. We feel the
cornea of the shut eye rolling under our fingers as we roll the open eye up
or down, or to left or right ; and indeed we are quite unable to move one
eye without a corresponding motion of the other.

Now though the upward and downward motions are effected by corresponding
muscles for both eyes, the motions to right and left are not so, being pro-
duced by the inner muscle of one eye along with the outer muscle of the
other, and yet the combined motion is so regular, that we can move our eyes
quite freely while maintaining during the whole motion the condition that the
optic axes shall intersect at some point of the object whose motions we are
following. Besides this, the motion of each eye about its optic axis is found
to be connected in a remarkable way with the motion of the axis itself.

The mode in which Helmholtz discusses these phenomena, and illustrates
the conditions of our command over the motions of our bodies, is well worth
the attention of those who are conscious of no limitation of their power of
moving in a given manner any organ which is capable of that kind of motion.

In his other great work on the Sensation of Tone as a Physiological Basis
far the Theory of Music, he illustrates the conditions under which our senses
are trained in a yet clearer manner. We quote from Mr Ellis's translation,
p. 95 :—

" Now practice and experience play a far greater part in the use of our senses than we are
usually inclined to assume, and since, as just remarked, our sensations derived from the senses
are primarily of importance only for enabling us to form a correct conception of the world with-
out us, our practice in the observation of these sensations usually does not extend in the slightest
degree beyond what is necessary for this purpose. We are certainly only far too much disposed
to believe that we must be immediately conscious of all that we feel and of all that enters into
our sensations. This natural belief, however, is founded only on the fact that we are always
immediately conscious, without taking any special trouble, of everything necessary for the practical
purpose of forming a correct acquaintance with external nature, because during our whole life we
have been daily and hourly using our organs of sense and collecting results of experience for this
precise object."

Want of space compels us to leave out of consideration that paper 011
Vortex Motion, in which he establishes principles in pure hydrodynamics which
had escaped the penetrative power of all the mathematicians who preceded him,
including Lagrange himself; and those papers on electrodynamics where he

HERMANN LUDWIO FEKI'1N\M> II KI.MHOLTZ.

reduce* to an intelligible and systematic form the laborious and intricate in-
vestigations of several independent theorists, so as to compare them with each
other and with experiment.

Hut we must not dwell on isolated papers, each of which might have been
taken for the work of a specialist, though few, if any, specialists could have
treated them in so able a manner. We prefer to regard Helmholtz as the
author of the two great books on Vision and Hearing, and now that we are
no longer under the sway of that irresistible power which has been bearing
us along through the depths of mathematics, anatomy, and music, we may
venture to observe from a safe distance the whole figure of the intellectual
giant as he sits on some lofty cliff watching the waves, great and small, as
each pursues its independent course on the surface of the sea below.

•• I imut own," he says, " that whenever I attentively observe this spectacle, it awakens in
•M a peculiar kind of intellectual pleasure, because here is laid open before the bodily eye what,
in the cue of the wave* of the invisible atmospheric ocean, can be rendered intelligible only to
tfca ey« of the understanding, and by the help of a long aeries of complicated propositions." _
p. 42.)

Helmholtz is now in Berlin, directing the labours of able men of science
in his splendid laboratory. Let us hope that from his present position he will
again take a comprehensive view of the waves and ripples of our intellectual
progress, and give us from time to time his idea of the meaning of it all.

[From the Proceedings of the Cambridge Philosophical Society, Vol. III. 1877.]

LXXXV. On a Paradox in the Theory of Attraction.

LET A^AZ be a straight line, P a point in the same, Xlt X3 corresponding points
in the segments PAU PAy

Let the distances of these points from the origin 0 measured in the positive

0 A S P X A

a

direction be alt a2, p, xlf x.2, respectively, and let the equation of correspondence
between a;, and xt be

JL 1 J_ J_ m

— V1/'

If xl and xa vary simultaneously,

ClX-t CtJCq , \

T— — \i = — 7— — s (2).

i 'Y* .I. i /n l" / <n ^— o" i

I ut-j ^^ /y I i /_/ iX^o /

Hence o^ and «j move in opposite directions, and the lengths of the
corresponding elements dxl and dx^ (considered both positive) are as the squares
of their respective distances from the point P.

If therefore AB is a uniform rod of matter attracting inversely as the
square of the distance, the attractions of the corresponding elements on a
particle at the point P will be equal and opposite.

Now by giving values to xlt varying continuously from a± to p, we may obtain
a corresponding series of values of x.2, varying from «„ to p, and since every
corresponding pair of elements dxl and dxa exert equal and opposite attractions
on a particle at P, we might conclude that the attraction of the whole segment
AJ* on a particle at P is equal and opposite to that of the segment Af on the
same particle.

OK A PARADOX IN THE THEORY OF ATTRACTION.

Bat it is still more evident that if A,P is the greater of the two segments,
and if we cut off /*a-/M, the attractions of Pa and PAl on the particle at
P will be equal and opposite. But the attraction of PA, exceeds that of Pa
by the attraction of the part o.-!,, therefore the attraction of PA, exceeds
that of PAt by a finite quantity, contrary to our first conclusion.

BP»M* our first conclusion is wrong, and for this reason. The attractions
of any two corresponding segments yl,A', and J,A', are exactly equal, but
however near the corresponding points A', and A', approach to P, the attraction
of each of the parts X,P and A',/' on P is infinite, but that of X3P exceeds
that of A',/* by a constant quantity, equal to the attraction of Ajt on P.

This method of corresponding elements leads to a very simple investigation
of the distribution on straight lines, circular and elliptic disks and solid spheres
and ellipsoids of fluids repelling according to any power of the distance.

The problem has been already solved by Green* in a far more general
manner, but at the same time by a far more intricate method

We have, as before, for corresponding values of xl and #,

-1 --- L = _I ___ L. ........................ (I).

Z,-p 0^-p p-X, Jp-0,

Transposing -- 1 -- = -- f- - (o)

x,-p p-a, p-x, a,-p

Multiplying

If we write -xx-a = - (4),

(5),

we find from equation (3) ^£ = P~X* (6]

yi y,

Let />„ p, be the densities and .<?„ *, the sections of the rod at the
corresponding points A', and X9 and let the repulsion of the matter of the

Ooorge Green. " Mathematical Investigations concerning the laws of the Equilibrium of Fluids
•nalogotu to the electric fluid, with other similar researches." Transactions of the Cambridge Philosophical
(Read Nov. 12, 1832.) Ferrers' Edition of Green's Papers, p. 119.

ON A PARADOX IN THE THEORY OF ATTRACTION. 601

rod vary inversely as the wth power of the distance, then the condition of
equilibrium of a particle at P under the action of the elements dxl and — dx3 is

p1sldx1(x1-p)~n = -p^jlx^p-x,)-* ..................... (7).

Eliminating dx1 and cfax, by means of equation (2), we find

p1s1(x1-p)*-» = pA(p-x,)*-» ........................ (8),

and from this by means of equation (6) we obtain

as the condition of equilibrium between the elements.

The condition of equilibrium is therefore satisfied for every pair of elements

by making

psy-~n — constant = C ........................... (10).

In a uniform rod s is constant, so that the distribution of density is
given by the equation

P=Cy»-"- .................................... (11).

If m = 2, as in the case of electricity, the density is uniform.

We have already shewn that when the density is uniform a particle not
at the middle of the rod cannot be in equilibrium, but on the other hand
any finite deviation from uniformity of density would be inconsistent with
equilibrium. We may therefore assert that the distribution of the fluid when
in equilibrium is not absolutely uniform, but is least at the middle of the
rod, while at the same time the deviation from uniformity is less than any
assignable quantity.

If the force is independent of the distance, n = 0 and

P=0r* .................................... (12),

or if r is the distance from the middle of the rod, 2l being the length of the rod,

If C were finite, the whole mass would be infinite. Hence if the mass of
fluid in the rod is finite it must be concentrated into two equal masses and placed
at the two ends of the rod.

VOL. n. 76

A PARADOX IN THK THEORY. OF ATTRACTION.

Let us neat consider a disk on which two chords are drawn intersecting
•t th* point P at a small angle 0, and let corresponding elements be taken
of the two sectors so formed.

In this case the section of either sector is proportional to the distance of

••lenient from the point of intersection, and therefore the two sections are
proportional to the values of y at the two elements. Hence if pif~n is constant.

particle at the point of intersection will be in equilibrium.

If the edge of the disk is the ellipse whose equation is

and if at any point within it

l-^-£=|>' (15),

rt ft

iui«l if the length of a diameter parallel to the given chord is '•Zd, then the
value of y for any point of the chord is

y=pd... .'.... (IG).

Henre if p=Cp"-i (17),

u particle placed at any point of the disk will be in equilibrium under the
.u-tion of any pair of sectors formed by chords intersecting at that point, and
then-fore it will be absolutely in equilibrium.

When as in the case of electricity, n = 2,

P = Cp~*. (18),

the known law of distribution of density.

If the repulsion were inversely as the distance, the fluid would be accu-
mulated in the circumference of the disk, leaving the rest entirely empty.

If the force were inversely as the cube of the distance, the density would
l>e uniform over the surface of the disk.

Lastly, let us consider a solid ellipsoid, the equation of the surface being

-*1 v £-0
0§

and at any point within it let

a

ON A PARADOX IN THE THEORY OP ATTRACTION. 603

At any point of a chord drawn parallel to a diameter whose length is Id
the value of y is pd.

If we consider a double cone of small angular aperture whose vertex is
at a given point, and whose axis is this chord, the sections at two correspond-
ing elements are in the ratio of the squares of the distances of the elements
from the given point, and therefore in the ratio of the values of p* at these
elements. Hence the condition to be satisfied is

pp*~n = C, a constant.

If this condition be fulfilled the fluid will he in equilibrium at every point
of the ellipsoid.

Ifn = 2, P=Cp->

is the condition of equilibrium. But if C is finite the whole mass of the fluid
in the ellipsoid if distributed according to this law of density would be infinite.
Hence if the whole quantity of fluid is finite it must be accumulated entirely
on the surface, and the interior will be entirely empty, as we know already.

If the force is inversely as the fourth power of the distance the density
within the ellipsoid will be uniform.

76—2

[fnn Uw Pnemtutgt o/ the Cambridge Philosophical Society, Vol. in. 1877.]

LXXXVI. OH Approximate Multiple Integration between Limits by Summation.

IT is often desirable to obtain the approximate value of an integral taken
between limits in cases in which, though we can ascertain the value of the
quantity to be integrated for any given values of the variables, we are not
able to express the integral as a mathematical function of the variables.

A method of deducing the result of a single integration between limits
fr.ua the values of the quantity corresponding to a series of equidistant values
• •»' the independent variable was invented by Cotes in 1707, and given in his
Lectures in 1709. Newton's tract Methodius Different ialia (see Horsley's edition
.•f Newton's Works (1779), Vol. I. p. 521) was published in 1711.

<'"tes* rules are given in his Opera Miscellanea, edited by Dr Uobert Smith,
and placed at the end of his Harmonia Mensurarum. He gives the proper
multipliers for the ordinates up to eleven ordinates, but he gives no details
of the method by which he ascertained the values of these multipliers.

Gauss, in his McthoJits nova Integralium Values per Approximationem
Inofiiietuli (Gottingische gelehrte Anzeigen, 1814, Sept. 26, or Werke, m. 202)
shews how to calculate Cotes' multipliers, and goes on to investigate the case
in which the values of the independent variable are not supposed to be equi-
distant, but are chosen so as with a given number of values to obtain the
highest degree of approximation.

He finds that by a proper choice of the values of the variable the value
of the integral may be calculated to the same degree of approximation as would
be obtained by means of doable the number of equidistant values.

The equation, the roots of which give the proper values of the variable,
is identical in form with that which gives the zero values of a zonal spherical
harmonic.

ON APPROXIMATE MULTIPLE INTEGRATION BETWEEN LIMITS BY SUMMATION. 605

DOUBLE INTEGRATION.

in

There is a particular kind of double integration which can be treated
a somewhat similar manner, namely, when the quantity to be integrated is a
function of a linear function of the two independent variables.

Thus if I=t'\udxdy ............ ....... ..m

J *i J yi
where u is a function of r, and

r = a + bx + cy .................................. (2),

let x = x x -x1) ........... ...... .......... (3),

!&) ........................... (4),

-. .................. (5).

If we write

t + yl) ........................ (6),

. ..................... (7),

................. . ............ (8),

• ............ • .......... (9),

we may consider u as a function of £ of the form

. ..... v ........ (10),

Now let «„ be the value of u corresponding to £ = 0,

tt, and u\ C= ±4.

u, and < £= ±4>

and if we assume

! = (**- «,) (ft - ft) Rw» + A («i + «',) + R, (u, + u',) + &c.} ...... (12),

OK APPROXIMATE MULTIPLE INTEGRATION

Uteo «nce the form of the function 11, and therefore the values of the coefficients
Ac. must be considered entirely arbitrary, we may equate the
coefficient* of A., Ac. in equations (11) and (13) as follows:

. •j/tt'+Ac.-ifl'+iy =Blt

'

X
2^c,*+

Ac.

If we write St for the sum of all the values of £*,

S, for the sum of all products such as £,*, &*,
S, ................................................ f, f,

then for r terms

BtSr - BjS,_ , + 5A. , - &c. ( - Y Br+l = 0,
B& - BJS^ + BtS,_, - &c. ( - )' Br,t = 0,

a aet of r equations, from which the quantities JR., .R, have been eliminated,
ami from which we may determine the r quantities S, . . . Sr, and the values
<«f £ are then given as the roots of the equation

r-s,r-'+s8r-4-&c. (-)rsr=o.

Thus if we have three values of £ they should be

'

4=0,

When the quantity to be integrated is a perfectly general function of the
variables we must proceed in a different manner.

BETWEEN LIMITS BY SUMMATION. (J07

We may begin as before by transforming the double integral into one
between the limits +1 for both variables, so that

=1 rudxdy = ±(x.i-x1)(y,-y1)\ i udpdq .............. (l).

J *J Vi J -IJ -1

Let £(«„) denote the sum of the eight values of u corresponding to the
following eight systems of values of p and q,

(an, &„), (an, - bn), ( - an, bn), (-an, - bn) ;
(bn,an), (bn,-an), (-bn, an), (-bn> -an),
and let us assume that the value of the integral is of the form

/=£(*,- x,) (y, - yO^K) + R^(Ul) + &c. + J2.2K)} ......... (2).

The values of the coefficients JR, a and b are to be deduced from equations
formed by equating the sum of the terms in p°-<f in this expression with the
integral

Only those terms in which both a and /3 are even will require to be
considered, for the symmetrical distribution of the values of p and q ensures
that the terms in which either a or /3 is odd must disappear.

Also since the expression is symmetrical with respect to p and q, the
term in yf(f will give an equation identical with that in p*qft.

We may therefore write down the equations at once, leaving out the factor

p"qf common to each term, but writing it at the side to indicate how the

equation was obtained. There are a + 1 equations in the first group, in which
0-0,

=4,

] =f,

=f,

OX ArWUMCIMATS MULTIPLE INTEGRATION

There are «-l equations in the second group, in which 0 = 2,

4

(21)73 '

There will be a- 3 equations in which 0 = 4, and so on. Hence if a is
the whole number of equations is

(H-

If a is odd, the number is

4

To satisfy these equations we have in general, for each group of values
of ii, three disposable quantities, R, p and q.

If, however, the central ordinate be selected it will constitute the first
group, and will introduce only one disposable quantity, namely JRt.

Also, if ordinates lying on the axes of p or of q be chosen, the groups
so formed contain only two disposable quantities, one of the ordinates being
zero.

Also, for ordinates lying on the diagonals, q=p, so that for these also
there are only two disposable quantities.

Thus if a = 3, the number of equations is - - = 6 ; and if we select the

central ordinate, giving one disposable quantity, a group of four points on the
axes, giving two disposable quantities, and a group of eight points giving
three disposable quantities, we shall be able to satisfy the six equations, and
to form an expression for the integral which will be correct for any function
not exceeding the seventh degree.

We assume

n n

br) W

•

BETWEEN LIMITS BY SUMMATION.

609

The equations are

Op

The solution of these equations gives

p= ±0-5855571,
g= ±0-9294971,
r= ±0-57969554,

The positions of the thirteen points are given in the annexed figure.

R R

Q R

R

R

TRIPLE INTEGRATION.

In extending this method to triple integration we meet with the curious
result, that in certain cases the solution indicates that we are to employ
values of the function some of which correspond to values of the variables
outside the limits of integration.

VOL. II. 77

Of HJaUJLimTir MULTIPLE rNTEGRATION.

Thwi if we endeavour to determine twenty-seven sets of values of x, y
mad i, with a uunMpnnrtinff «* of multipliers, so as to express the value of
U* triple integral in the form

P f f

......... (l),

when* ». denotes the value of u when x = y = z=Q;

S(M ) iluootoi the sum of the six values of u for which

x= ±pa, 0, 0,

y=o, ±pd, o,

z=0, 0, ±pc-;

2(«f) denotes the sum of the twelve values of u for which

x = 0, ±qa, ±qa,
y=±qb, 0, ±qb,

* = ±qc, ±qc, 0;

and £(«,) denotes the sum of the eight values of u for which

x= ±ra,

y= ±rb,

z= ±rc,
where the signs may be taken in any order.

The equations to be satisfied in order that equation (1) may be satisfied
for any function of x, y and z of not more than seven dimensions are

0 + 6P +12Q+8R =8,

BETWEEN LIMITS BY SUMMATION. 611

The solution of these equations gives two systems of values :

First System. Second System.

^, = 0-969773, jp2 = 2-638430.
fc= 1-119224, ^ = 0-602526.

r, = 0-652817, r, = 0-855044.

0,= -2-856446, 02= -7'999462.

P,= 1-106789, P2= 0-000513.

Q,= 0-032304, Q,= 1-238514.

12,= 0-478508, R,= 0-094777.

In the first system ql is greater than unity, and in the second system
pt is greater than unity, so that in either case one set of the values of u
corresponds to values of the variables outside of the limits of integration.

This, of course, renders the method useless in determining the integral
from the measured values of the quantity u, as when we wish to determine
the weight of a brick from the specific gravities of samples taken from 27
selected places in the brick, for we are directed by the method to take some
of the samples from places outside the brick.

But this is not the case contemplated in the mathematical enunciation.
All that we have proved is that if u be a function of x, y, z of not more
than seven dimensions, our method will lead to a correct value, and of course
we can determine the value of such a function for any values of the variables,
whether they lie within the limits of integration or not.

77—2

[From U» Pnendutgt of the Cambridge Philosophical Society. Vol. HI. Pt. in.]

LXXXVII. On the Unpublished Electrical Papers of the Hon. HENRY CAVENDISH.

CAVKKDISH published only two papers relating to electricity, "An attempt
to explain some of the principal Phenomena of Electricity by Means of an
Ebtftic Fluid" (Phil. Trans. 1771, pp. 584 — 677), and "An account of some
Attempts to imitate the Effects of the Torpedo by Electricity" (Phil. Trans.
for 1776, pp. 196 — 225). He left behind him, however, some twenty packets
of manuscript on mathematical and experimental electricity. These were
placed by the then Earl of Burlington, now Duke of Devonshire, in the
h^iwJa of the late Sir William Snow Harris, who appears to have made "an
abstract of them with a commentary of great value on their contents." This
was sent to Dr George Wilson when he was preparing his Life of Cavendish
i I I'.'rtfo of the Cavendish Society, Vol. I. London, 1851). It was afterwards
returned to Sir W. S. Harris, but I have not been able to learn whether
it ia still in existence. The Cavendish manuscripts, however, were placed in
my hands by the Duke of Devonshire in 1874, and they are now almost
ready for publication.

They may be divided into three classes :

(A) Mathematical propositions, intended to follow those in the paper <>t
1771, and numbered accordingly. Some of these are important as shewing the
clear ideas of Cavendish with respect to what we now call charge, potential,
and the capacity of a conductor ; but the great improvements in the mathe-
matical treatment of electricity since the time of Cavendish have rendered
others superfluous.

We come next to an account of the experiments on which the mathe-
matical theory was founded. This is a manuscript fully prepared for the press,
and since it refers to the second part of the published paper of 1771 as "the

THE ELECTRICAL EXPERIMENTS OF HENRY CAVENDISH. 613

second part of this Work," it must have been intended to be published as a
book, along with a reprint of that paper. It contains no dates, but as it
refers to experiments which we know were made in 1773, it must have been
written after that time, but I do not think later than 1775.

It forms a scientifically arranged treatise on electricity. A manuscript entitled
" Thoughts concerning electricity" seems to form a kind of introduction to
this treatise, for it contains several important definitions and hypotheses which
are not afterwards repeated.

Next comes the fundamental experiment, in which it is proved that a
conducting sphere insulated within a hollow conducting sphere does not become
charged when the hollow sphere is charged and the inner sphere is made to
communicate with it.

Cavendish proves that if this is the case, the law of force must be that
of the inverse square, and also that if the index instead of being 2 had
been 2 + ^g-, his method would have detected the charge on the inner sphere.

The experiment has been repeated this summer by Mr MacAlister of St
John's College with a delicate quadrant electrometer capable of detecting a
charge many thousand times smaller than Cavendish could detect by his straw
electrometer, so that we may now assert that the index cannot exceed or
fall short of 2 by the millionth of a unit.

The second experiment is a repetition of this, using one parallelepiped within
another instead of the two spheres.

He then describes his apparatus for comparing the charges of different
bodies, or, as we should say, their capacities.

He first shews (Exp. 3) that the charge, communicated to a body con-
nected to another body at a great distance by a fine wire, does not depend
on the form of the wire, or on the point where it touches the body.

Exp. 4 is on the capacities of bodies of the same shape and size but of
different substances.

Exp. 5 compares the capacity of two circles with that of another of twice
the diameter.

Exp. 6 compares the capacity of two short wires with that of a long one.

Exp. 7 compares the capacities of bodies of different forms, the most im-
portant of which are a disk and a sphere.

Exp. 8 compares the charge of the middle of three parallel plates with
that of the outer plates

TIIK ELECTRICAL EXPERIMENTS OF HENRY CAVENDISH.

In the next part of his researches he investigates the capacities of con-
turned of plate* of different kinds of glass, rosin, wax, shellac, &c.
with di*k« of tinfoil, and also of plates of air between two flat con-
ductor* He fioda that the electricity spreads on the surface of the plate
beyond the tinfoil coatings, and he investigates most carefully the extent of
thU spreading, and how it depends on the strength of the electrification.

After correcting for the spreading, he finds that for coated plates of the
mmri mbctance the observed capacity is proportional to the computed capacity,
but it in always several times greater than the computed capacity, except in
the rttt of plates of air. Cavendish thus anticipated Faraday in the discovery
of the specific inductive capacity of dielectrics, and in the measurement of
this quantity for different substances.

For these experiments Cavendish constructed a large number of coated
platea with capacities so arranged that by combining them he could measure
the capacity of any conductor from a sphere 12*1 inches diameter to his large
battery of 49 Leyden jars. He expressed the capacity of any conductor in
what he calls "inches of electricity," that is to say the diameter of a sphere
of equal capacity expressed in inches.

The details and dates of the experiments referred to in this work are
contained in three volumes of experiments in the years 1771, 1772 and 1773,
in a separate collection of "Measurements" and in a paper entitled " Itesults,"
in which the experiments of different days are compared together. Besides these
there are experiments of other kinds which are not described in the treatise.
The most important of these experiments are those on the electric re-
sistance of different substances, which were continued to the year 1781.

He compares the resistance of solutions of sea salt of various strengths
from saturation to 1 in 20000, and measures the diminution of resistance as
the temperature rises. He also compares the resistance of solutions of sea salt
with that of solutions containing chemical equivalents of other salts in the
aame quantity of water. He finds the resistance of distilled water to be very
great, and much greater for fresh distilled water than for distilled water kept
for some time in a glass bottle.

I have compared Cavendish's results with those recently obtained by
Kohlrausch, and find them all within 10 per cent, and many much nearer.

Cavendish also investigates the relation between the resistance and the
velocity of the current, and finds the power of the velocity to be by dif-

THE ELECTEICAL EXPERIMENTS OF HENRY CAVENDISH. 615

ferent experiments 1-08, 1'03, 0'976 and 1, and he finally concludes that
the resistance is as the first power of the velocity, thus anticipating Ohm's
Law.

The general accuracy of these results is the more remarkable when we
consider the method by which they were obtained, forty years before the
invention of the galvanometer.

Every comparison of two resistances was made by Cavendish by connecting
one end of each resistance-tube with the external coatings of a set of equally
charged Leyden jars and touching the jars in succession with a piece of metal
held in one hand, while with a piece of metal in the other hand he touched
alternately the ends of the two resistances. He thus compared the sensation
of the shock felt when the one or the other resistance in addition to the
resistance of his body was placed in the path of the discharge. His results
therefore are derived from the comparison of the sensations produced by an
enormous number of shocks passed through his own body.

The skill which he thus acquired in the discrimination of shocks was so
great that he is probably accurate even when he tells us that the shock
when taken through a long thin copper wire wound on a large reel was
sensibly greater than when taken direct. The experiment is certainly worth
repeating, to determine whether the intensification of the physiological effect
on account of the oscillatory character of the discharge through a coil would
in any case compensate for the weakening effect of the resistance of the coil.
But I have not hitherto succeeded in obtaining this result. Indeed on com-
paring the shock through two coils of equal resistance, one of which had far
more self-induction than the other, I found the shock sensibly feebler through
the coil of large self-induction.

[From the Encydopetdia Britannica.]

I A X X V 1 1 1. Constitution of Bodies.

IIIK question whether the smallest parts of which bodies are composed are
finite in number, or whether, on the other hand, bodies are infinitely divisible,
rvl.iUw to the ultimate constitution of bodies, and is treated of in the article
ATOM.

The mode in which elementary substances combine to form compound sub-
stance* is called the chemical constitution of bodies, and is treated of in

• 'lIKMISTBY.

The mode in which sensible quantities of matter, whether elementary or
compound, are aggregated together so as to form a mass having certain observed
properties, is called the physical constitution of bodies.

Bodies may be classed in relation to their physical constitution by con-
sidering the effects of internal stress in changing their dimensions. When a
body can exist in equilibrium under the action of a stress which is not uniform
in all directions it is said to be solid.

When a body is such that it cannot be in equilibrium unless the stress
«t every point is uniform in all directions, it is said to be fluid.

There are certain fluids, any portion of which, however small, is capable
«»f expanding indefinitely, so as to fill any vessel, however large. These are
called gases. There are other fluids, a small portion of which, when placed in
a large vessel, does not at once expand so as to fill the vessel uniformly, but
remains in a collected mass at the bottom, even when the pressure is removed.
These fluids are called liquids.

When a liquid is placed in a vessel so large that it only occupies a
part of it, part of the liquid begins to evaporate, or in other words it passes
int.. the state of a gas, and this process goes on either till the whole of the
liquid is evaporated, or till the density of the gaseous part of the substance

CONSTITUTION OF BODIES. 617

has reached a certain limit. The liquid and the gaseous portions of the sub-
stance are then in equilibrium. If the volume of the vessel be now made
smaller, part of the gas will be condensed as a liquid, and if it be made larger,
part of the liquid will be evaporated as a gas.

The processes of evaporation and condensation, by which the substance
passes from the liquid to the gaseous, and from the gaseous to the liquid state,
are discontinuous processes, that is to say, the properties of the substance are
very different just before and just after the change has been effected. But this
difference is less in all respects the higher the temperature at which the change
takes place, and Cagniard de la Tour in 1822* first shewed that several sub-
stances, such as ether, alcohol, bisulphide of carbon, and water, when heated
to a temperature sufficiently high, pass into a state which differs from the
ordinary gaseous state as much as from the • liquid state. Dr Andrews has
since t made a complete investigation of the properties of carbonic acid both
below and above the temperature at which the phenomena of condensation and
evaporation cease to take place, and has thus explored as well as established
the continuity of the liquid and gaseous states of matter.

For carbonic acid at a temperature, say of 0° C., and at the ordinary pressure
of the atmosphere, is a gas. If the gas be compressed till the pressure rises
to about 40 atmospheres, condensation takes place, that is to say, the substance
passes in successive portions from the gaseous to the liquid condition.

If we examine the substance when part of it is condensed, we find that
the liquid carbonic acid at the bottom of the vessel has all the properties of
a liquid, and is separated by a distinct surface from the gaseous carbonic acid
which occupies the upper part of the vessel.

But we may transform gaseous carbonic acid at 0" C. into liquid carbonic
acid at 0° C. without any abrupt change, by first raising the temperature of
the gas above 30°.92 C. which is the critical temperature, then raising the
pressure to about 80 atmospheres, and then cooling the substance, still at high
pressure, to zero.

During the whole of this process the substance remains perfectly homo-
geneous. There is no surface of separation between two forms of the substance,
nor can any sudden change be observed like that which takes place when the
gas is condensed into a liquid at low temperatures ; but at the end of the

* Annales de Chimie, 2me s^rie, xxi. et xxn.
t Phil. Trans. 1869, p. 575.

VOL. II. 78

OM8TITI-TION OF

the suUrtanoe is undoubtedly in the liquid state, f..r if we now diminish
pnwurr to sjsjlimlllf If" than 40 atmospheres the substance will exhibit
ordinary distinction between the liquid and the gaseous state, that is t<.
mr, part of it will evaporate, leaving the rest at the bottom of the vessel,
ith a dUtinct surface of separation between the gaseous and the liquid parts.

The MUMUgn of a substance between the liquid and the solid state takes place
with Tarioua degrees of abruptness. Some substances, such as some of the more
crystalline metals, sesm to pass from a completely fluid to a completely solid
lit* very sudd«Milv. In some cases the melted matter appears to 'become
thicker before it solidifies, but this may arise from a multitude of solid crystals
being formed in the still liquid mass, so that the consistency of the mass be-
comes like that of a mixture of sand and water, till the melted matter in
which the crystals are swimming becomes all solid.

There are other substances, most of them colloidal, such that when the
! i it- 1 tod substance cools it becomes more and more viscous, passing into tlu-
•olid state with hardly any discontinuity. This is the case with pitch.

The theory of the consistency of solid bodies will be discussed in the article
-rii-iTY, but the manner in which a solid behaves when acted on by
furnishes us with a system of names of different degrees and kinds uf

A fluid, as we have seen, can support a stress only when it is uniform in
itll directions, that is to say, when it is of the nature of a hydrostatic pressure.

There are a great many substances which so far correspond to this defini-
te «n of (i fluid that they cannot remain in permanent equilibrium if the stress
within them is not uniform in all directions.

In all existing fluids, however, when their motion is such that the shape
of any small portion is continually changing, the internal stress is not uniform
in all directions, but is of such a kind as to tend to check the relative motion
of the part* of the fluid.

This capacity of having inequality of stress called into play by inequality
• •!' motion is called viscosity. All real fluids are viscous, from treacle and tar
to water and ether and air and hydrogen.

When the viscosity is very small the fluid is said to be mobile, like water
and ether.

When the viscosity is so great that a considerable inequality of str
though it produces a continuously increasing displacement, produces it so slowly

CONSTITUTION OP BODIES. G19

that we can hardly see it, we are often inclined to call the substance a solid,
and even a hard solid. Thus the viscosity of cold pitch or of asphalt is so
great that the substance will break rather than yield to any sudden blow, and
yet if it is left for a sufficient time it will be found unable to remain in
equilibrium under the slight inequality of stress produced by its own weight,
but will flow like a fluid till its surface becomes level.

If, therefore, we define a fluid as a substance which cannot remain in
permanent equilibrium under a stress not equal in all directions, we must call
these substances fluids, though they are so viscous that we can walk on them
without leaving any footprints.

If a body, after having its form altered by the application of stress, tends
to recover its original form when the stress is removed, the body is said to
be elastic.

The ratio of the numerical value of the stress to the numerical value of
the strain produced by it is called the coefficient of elasticity, and the ratio of
the strain to the stress is called the coefficient of pliability.

There are as many kinds -of these coefficients as there are kinds of stress
and of strains or components of strains produced by them.

If, then, the values of the coefficients of elasticity were to increase without
limit, the body would approximate to the condition of a rigid body.

We may form an elastic body of great pliability by dissolving gelatine or
isinglass in hot water and allowing the solution to cool into a jelly. By
diminishing the proportion of gelatine the coefficient of elasticity of the jelly
may be diminished, so that a very small force is required to produce a large
change of form in the substance.

When the deformation of an elastic body is pushed beyond certain limits
depending on the nature of the substance, it is found that when the stress
is removed it does not return exactly to its original shape, but remains per-
manently deformed. These limits of the different kinds of strain are called the
limits of perfect elasticity.

There are other limits which may be called the limits of cohesion or of
tenacity, such that when the deformation of the body reaches these limits the
body breaks, tears asunder, or otherwise gives way, and the continuity of its
substance is destroyed.

A body which can have its form permanently changed without any flaw
or break taking place is called mild. When the force required is small the

78—2

CONSTITUTION OF BODIES.

body » said to be «^J; when it is great the body is said to be touyh. A
body which becomes 8awed or broken before it can be permanently defoniu-,1
w called brittle. When the force required is great the body is said to be

The stiffness of a body is measured by the force required to product a
amount of deformation.

It* strength is measured by the force required to break or crush it.

We may conceive a solid body to approximate to the condition of a fluid
in several different ways.

If we knead fine clay with water, the more water we add the softer does
the mixture become till at last we have water with particles of clay slowly
subsiding through it This is an instance of a mechanical mixture the r«n-
stttoents of which separate of themselves. But if we mix bees-wax with oil,
or rosin with turpentine, we may form permanent mixtures of all degrees of
softness, and so pass from the solid to the fluid state through all degrees of
viscosity.

We may also begin with an elastic and somewhat brittle substance like
palatine, and add more and more water till we form a very weak jelly which
opposes a very feeble resistance to the motion of a solid body, such as a spoon,
through it. But even such a weak jelly may not be a true fluid, for it mav
be able to withstand a very small force, such as the weight of a small mote.
If a small mote or seed is enclosed in the jelly, and if its specific gravity is
different from that of the jelly, it will tend to rise to the top or sink to the
bottom. If it does not do so we conclude that the jelly is not a fluid but
a solid body, very weak, indeed, but able to sustain the force with which the
mote tends to move.

It appears, therefore, that the passage from the solid to the fluid state
may be conceived to take place by the diminution without limit either of the
coefficient of rigidity, or of the ultimate strength against rupture, as well as
by the diminution of the viscosity. But whereas the body is not a true fluid
till the ultimate strength, or the coefficient of rigidity, is reduced to zero, it
in not a true solid as long as the viscosity is not infinite.

Solids, however, which are not viscous in the sense of being capul>li
iui unlimited amount of change of form, are yet subject to alterations depend-
ing on the time during which stress has acted on them. In other words, th«-
at any given instant depends, not only on the strain at that instant.

CONSTITUTION OP BODIES. 621

but on the previous history of the body. Thus the stress is somewhat greater
when the strain is increasing than when it is diminishing, and if the strain
is continued for a long time, the body, when left to itself, does not at once
return to its original shape, but appears to have taken a set, which, however,
is not a permanent set, for the body slowly creeps back towards its original
shape with a motion which may be observed to go on for hours and even
weeks after the body is left to itself.

Phenomena of this kind were pointed out by Weber and Kohlrausch (Pogy.
Ann. Bd. 54, 119 and 128), and have been described by O. E. Meyer (Poyy.
Ann. Bd. 131, 108), and by Maxwell (Phil. Trans. 1866, p. 249), and a theory
of the phenomena has been proposed by Dr L. Boltzmann (Wiener Sitzunyx-
berichte, 8th October 1874).

The German writers refer to the phenomena by the name of "elastische
Nachwirkung," which might be translated "elastic reaction" if the word reaction
were not already used in a different sense. Sir W. Thomson speaks of the
viscosity of elastic bodies.

The phenomena are most easily observed by twisting a fine wire suspended
from a fixed support, and having a small mirror suspended from the lower end,
the position of which can be observed in the usual way by means of a tele-
scope and scale. If the lower end of the wire is turned round through an
angle not too great, and then left to itself, the mirror makes oscillations, the
extent of which may be read off on the scale. These oscillations decay much
more rapidly than if the only retarding force were the resistance of the air,
shewing that the force of torsion in the wire must be greater when the twist
is increasing than when it is diminishing. This is the phenomenon described
by Sir W. Thomson under the name of the viscosity of elastic solids. But
we may also ascertain the middle point of these oscillations, or the point of
temporary equilibrium when the oscillations have subsided, and trace the varia-
tions of its position.

If we begin by keeping the wire twisted, say for a minute or an hour,
and then leave it to itself, we find that the point of temporary equilibrium is
displaced in the direction of twisting, and that this displacement is greater the
longer the wire has been kept twisted. But this displacement of the point of
equilibrium is not of the nature of a permanent set, for the wire, if left to
itself, creeps back towards its original position, but always slower and slower.
This slow motion has been observed by the writer going on for more than a

COXSTITtTIOX OF BODIES.

week and lie al*> found that if the wire was set in vibration the motion of
the point «.f equilibrium was more rapid than when the wire was not in vibration.

We mav produce a very complicated series of motions of the lower end
of tbe win by previously subjecting the wire to a series of twists. For in-
rtanf*. we KAY frst twist it in the positive direction, and keep it twisted for
a <j-V( then in tbe negative direction for an hour, and then in the positive
direction for a minute. When the wire is left to itself the displacement, at
fin* positive, becomes negative in a few seconds, and this negative displacement
increases for some time. It then diminishes, and the displacement becomes
utive, and lasts a longer time, till it too finally dies away.

Tbe phenomena are in some respects analogous to the variations of the
surface temperature of a very large ball of iron which has been heated in a
furnace for a day, then placed in melting ice for an hour, then in boiling
water for a minute, and then exposed to the air ; but a still more perfect
analogy may be found in the variations of potential of a Leyden jar which
bat been charged positively for a day, negatively for an hour, and positively
again for a minute*.

The effects of successive magnetization on iron and steel are also in many
respects analogous to those of strain and electrification f.

The method proposed by Boltzmann for representing such phenomena mathe-
matically is to express the actual stress, L(f), in terms not only of the actual
Htruin, 0(l, but of the strains to which the body has been subjected during
all previous time.

His equation is of the form

-f

Jo

where <a is the interval of time reckoned backwards from the actual time t to
the time t-u, when the strain Ot_m existed, and $ (o>) is some t function of
that intcrv.il.

\Ve may describe this method of deducing the actual state from the pre-

us states as the historical method, because it involves a knowledge of the

previous history of the body. But this method may be transformed into another,

1 tT ^ "°Pkin8on' '<0n the R^™l Charge of the Loyden Jar," Proc. R. S. «iv. 408,
.Mmreb 30, 18 1 '•.

t See Wicdctn»nn'« Galvanitmtu, vol. n. p. 567.

CONSTITUTION OF BODIES. 623

in which the present state is not regarded as influenced by any state which
has ceased to exist. For if we expand 6t_u by Taylor's theorem,

,, a dd w2 d-0

^**^»* ITS *"*•**

and if we also write

f f30 f* 2

A = $ (w) dca' B=\ «4 (») dot, C= -2- 1/, («) dw, &c.

Jo Jo J o 1 . 2

then equation (l) becomes

where no symbols of time are subscribed, because all the quantities refer to
the present tune.

This expression of Boltzmann's, however, is not in any sense a physical
theory of the phenomena ; it is merely a mathematical formula which, though
it represents some of the observed phenomena, fails to express the phenomenon
of permanent deformation. Now we know that several substances, such as
gutta-percha, India-rubber, &c., may be permanently stretched when cold, and
yet when afterwards heated to a certain temperature they recover their original
form. Gelatine also may be dried when in a state of strain, and may recover
its form by absorbing water.

We know that the molecules of all bodies are in motion. In gases and
liquids the motion is such that there is nothing to prevent any molecule from
passing from any part of the mass to any other part ; but in solids we must
suppose that some, at least, of the molecules merely oscillate about a certain
mean position, so that, if we consider a certain group of molecules, its con-
figuration is never very different from a certain stable configuration, about which
it oscillates.

This will be the case even when the solid is in a state of strain, pro-
vided the amplitude of the oscillations does not exceed a certain limit, but if
it exceeds this limit the group does not tend to return to its former configu-
ration, but begins to oscillate about a new configuration of stability, the strain
in which is either zero, or at least less than in the original configuration.

The condition of this breaking up of a configuration must depend partly
on the amplitude of the oscillations, and partly on the amount of strain in
the original configuration ; and we may suppose that different groups of mole-

CONSTITUTION* OF BODIES.

in a homogeneous solid, are not in similar circumstances in this

Thus we may suppose that in a certain number of groups the ordinary
agitation of the molecules is liable to accumulate so much that every now and
then the configuration of one of the groups breaks up, and this whether it is
in a state of strain or not. We may in this case assume that in every second
a certain projwrtion of these groups break up, and assume configurations cor-
responding to a strain uniform in all directions.

If all the groups were of this kind, the medium would be a viscous fluid.

But we may suppose that there are other groups, the configuration of which
is so stable that they will not break up under the ordinary agitation of the
molecules unless the average strain exceeds a certain limit, and this limit may
be different for different systems of these groups.

Now if such groups of greater stability are disseminated through the sub-
stance in such abundance as to build up a solid framework, the substance will
be a solid, which will not be permanently deformed except by a stress greater
than a certain given stress.

But if the solid also contains groups of smaller stability and also groups
of the first kind which break up of themselves, then when a strain is applied
the resistance to it will gradually diminish as the groups of the first kind
break up, and this will go on till the stress is reduced to that due to the
more permanent groups. If the body is now left to itself, it will not at once
return to its original form, but will only do so when the groups of the first
kind have broken up so often as to get back to their original state of strain.

This view of the constitution of a solid, as consisting of groups of mole-
cules some of which are in different circumstances from others, also helps to
explain the state of the solid after a permanent deformation has been given
t-i it. In this case some of the less stable groups have broken up and assumed
new configurations, but it is quite possible that others, more stable, may still
retain then* original configurations, so that the form of the body is determined
by the equilibrium between these two sets of groups ; but if, on account of
rise of temperature, increase of moisture, violent vibration, or any other cause,
the breaking up of the less stable groups is facilitated, the more stable groups
may again assert their sway, and tend to restore the body to the shape it
had before its deformation.

[From the Encyclopaedia Britannica.]

LXXXIX. Diffusion.

SOME liquids, such as mercury and water, when placed in contact with
each other do not mix at all, but the surface of separation remains distinct,
and exhibits the phenomena described under CAPILLARY ACTION. Other pairs
of liquids, such as chloroform and water, mix, but only in certain proportions.
The chloroform takes up a little water, and the water a little chloroform;
but the two mixed liquids will not mix with each other, but remain in con-
tact separated by a surface shewing capillary phenomena. The two liquids are
then in a state of equilibrium with each other. The conditions of the equi-
librium of heterogeneous substances have been investigated by Professor J.
Willard Gibbs in a series of papers published in the Transactions of the Con-
necticut Academy of Arts and Sciences, Vol. in. part I. p. 108. Other pairs
of liquids, and all gases, mix in all proportions.

When two fluids are capable of being mixed, they cannot remain in equi-
librium with each other; if they are placed in contact with each other the
process of mixture begins of itself, and goes on till the state of equilibrium
is attained, which, in the case of fluids which mix in all proportions, is a
state of uniform mixture.

This process of mixture is called diffusion. It may be easily observed by
taking a glass jar half full of water and pouring a strong solution of a
coloured salt, such as sulphate of copper, through a long-stemmed funnel, so
as to occupy the lower part of the jar. If the jar is not disturbed we may
trace the process of diffusion for weeks, months, or years, by the gradual rise
of the colour into the upper part of the jar, and the weakening of the colour
in the lower part.

This, however, is not a method capable of giving accurate measurements
of the composition of the liquid at difierent depths in the vessel. For more

VOL. ii. 79

.

MKFU8IOX.

determination* w« may draw off a portion from a given stratum of the
mixed liquid, and di-tennine it* composition either by chemical methods or by
its specific gtmrity, or any other property from which its composition may be

deduced.

But as the act of removing a portion of the fluid interferes with the
ttrocce* of diffusion, it is desirable to be able to ascertain the composition of
any at rat inn of the mixture without removing it from the vessel. For this
propose Sir W. Thomson places in the jar a number of glass beads of different
dimtfrW, which indicate the densities of the strata in which they are observed
to float. The principal objection to this method is, that if the liquids contain
air or any other gas, bubbles are apt to form on the glass beads, so as to
make them float in a stratum of less density than that marked on them.

M. Voit has observed the diffusion of cane-sugar in water by passing a
r»y of plane-polarized light horizontally through the vessel, and determining
the angle through which the plane of polarization is turned by the solution
of sugar. This method is of course applicable only to those substances which
cause rotation of the plane of polarized light.

Another method is to place the diffusing liquids in a hollow glass prism,
with its refracting edge vertical, and to determine the deviation of a ray of
light passing through the prism at different depths. The ray is bent doAvn-
wards on account of the variable density of the mixture, as well as towards
the thicker part of the prism; but by making it pass as near the edge of
the prism as possible, the vertical component of the refraction may be made
vety small; and by placing the prism within a vessel of water having parallel
aides of glass, we can get rid of the constant part of the deviation, and are
able to use a prism of large angle, so as to increase the part due to the
diffusing substance. At the same time we can more easily control and register
the temperature.

The laws of diffusion were first investigated by Graham. The diffusion of
gases has recently been observed with great accuracy by Loschmidt, and that
• >f liquids by Fick and by Voit.

Diffusion as a Molecular Motion.

If we observe the process of diffusion with our most powerful microscopes,
we cannot follow the motion of any individual portions of the fluids. We

DIFFUSION. 627

cannot point out one place in which the .lower fluid is ascending, and another
in which the upper fluid is descending. There are no currents visible to us,
and the motion of the material substances goes on as imperceptibly as the
conduction of heat or of electricity. Hence the motion which constitutes dif-
fusion must be distinguished from those motions of fluids which we can trace
by means of floating motes. It may be described as a motion of the fluids,
not in mass but by molecules.

When we reason upon the hypothesis that a fluid is a continuous homo-
geneous substance, it is comparatively easy to define its density and velocity;
but when we admit that it may consist of molecules of different kinds, we
must revise our definitions. We therefore define these quantities by considering
that part of the medium which at a given instant is within a certain small
region surrounding a given point. This region must be so small that the
properties of the medium as a whole are sensibly the same throughout the
region, and yet it must be so large as to include a large number of mole-
cules. We then define the density of the medium at the given point as the
mass of the medium within this region divided by its volume, and the
velocity of the medium as the momentum of this portion of the medium
divided by its mass.

If we consider the motion of the medium relative to an imaginary surface
supposed to exist within the region occupied by the medium, and if we define
the flow of the medium through the surface as the mass of the medium
which in unit of time passes through unit of area of the surface, then it
follows from the above definitions that the velocity of the medium resolved
in the direction of the normal to the surface is equal to the flow divided by
the density. If we suppose the surface itself to move with the same velo-
city as the fluid, and in the same direction, there will be no flow through it.

Having thus defined the density, velocity, and flow of the medium as a
whole, or, as it is sometimes expressed, "in mass," we may now consider one
of the fluids which constitute the medium, and define its density, velocity,
and flow in the same way. The velocity of this fluid may be different from
that of the medium in mass, and its velocity relative to that of the medium
is the velocity of diffusion which we have to study.

79—2

DIFFUSION.

o/ Ghoet acconling to the Kinetic Theory.

80 many of the phenomena of gases are found to be explained in a c<>n
MM- by the kinetic theory of gases, that we may describe \\ith
probability of correctness the kind of motion which constitute!
in gave*. We shall therefore consider gaseous diffusion in the light
of the kinetic theory before we consider diffusion in liquids.

A gat, according to the kinetic theory, is a collection of particles or mole-
cule* which are in rapid motion, and which, when they encounter each other,
before pretty much as elastic bodies, such as billiard balls, would do if no
energy were lost in their collisions. Each molecule travels but a very small
distance between one encounter and another, so that it is every now and then
Altering its velocity both in direction and magnitude, and that in an exceed-
ingly irregular manner.

The result is that the velocity of any molecule may be considered as com-
pounded of two velocities, one of which, called the velocity of the medium,
in the same for all the molecules, while the other, called the velocity of agi-
tation, is irregular both in magnitude and in direction, though the average
magnitude of the velocity may be calculated, and any one direction is just as
likely as any other.

The result of this motion is, that if in any part of the medium the
molecules are more numerous than in a neighbouring region, more molecules
will pass from the first region to the second than in the reverse direction,
and for this reason the density of the gas will tend to become equal in all
puts 6f the vessel containing it, except in so far as the molecules may be
crowded towards one direction by the action of an external force such as
gravity. Since the motion of the molecules is very swift, the process of
equalization of density in a gas is a very rapid one, its velocity of propaga-
tion through the gas being that of sound.

Let us now consider two gases in the same vessel, the proportion of the
gases being different in different parts of the vessel, but the pressure being
everywhere the same. The agitation of the molecules will still cause more
molecules of the first gas to pass from places where that gas is dense to
places where it is rare than in the opposite direction, but since the second
gas is dense where the first one is rare, its molecules will be for the most
part travelling in the opposite direction. Hence the molecules of the two

DIFFUSION. 629

gases will encounter each other, and every encounter will act as a check to
the process of equalization of the density of each gas throughout the mixture.

The interdiffusion of two gases in a vessel is therefore a much slower
process than that by which the density of a single gas becomes equalized,
though it appears from the theory that the final result is the same, and that
each gas is distributed through the vessel in precisely the same way as if no
other gas had been present, and this even when we take into account the
effect of gravity.

If we apply the ordinary language about fluids to a single gas of the
mixture, we may distinguish the forces which act on an element of volume
as follows : —

1st. Any external force, such as gravity or electricity.

2nd. The difference of the pressure of the particular gas on opposite sides
of the element of volume. [The pressure due to other gases is to be con-
sidered of no account.]

3rd. The resistance arising from the percolation of the gas through the
other gases which are moving with different velocity.

The resistance due to encounters with the molecules of any other gas is
proportional to the velocity of the first gas relative to the second, to the
product of their densities, and to a coefficient which depends on the nature
of the gases and on the temperature. The equations of motion of one gas of
a mixture are therefore of the form

- u.3) + &c. = 0,

where the symbol of operation ^ prefixed to any quantity denotes the time-

variation of that quantity at a point which moves along with that medium
which is distinguished by the suffix (,), or more explicitly

8, d d d d
cr-= -r + u,-j- + v1-j- + w1-j-.
8t at dx dy ldz

In the state of ultimate equilibrium itl = ut = &c. = Q, and the equation is
reduced to

which is the ordinary form of the equations of equilibrium of a single fluid.

DHTFLT I

Hence, when the process of diffusion is complete, the density of each gas at
any point of the vessel is the same as if no other gas were present.

If I" is the potential of the force which acts on the gas, and if in tin-
equation !»,-£,/>,, i\ is constant, as it is when the temperature is uniform,
then the equation of equilibrium becomes

the solation of which is

Hence if, as in the case of gravity, V is the same for all gases, but /• is
different for different gases, the composition of the mixture will be different
in different parts of the vessel, the proportion of the heavier gases, for which
k is smaller, being greater at the bottom of the vessel than at the top. It
\\.nild be difficult, however, to obtain experimental evidence of this difference
of composition except in a vessel more than 100 metres high, and it would
be necessary to keep the vessel free from inequalities of temperature for more
than a year, in order to allow the process of diffusion to advance to a state
even half-way towards that of ultimate equilibrium. The experiment might,
however, be made in a few minutes by placing a tube, say 10 centimetres
long, on a whirling apparatus, so that one end shall be close to the axis,
while the other is moving at the rate, say, of 50 metres per second. Thus
if equal volumes of hydrogen and carbonic acid were used, the proportion of
hydrogen to carbonic acid would be about xl^ greater at the end of the tube
nearest the axis. The experimental verification of the result is important, as
it establishes a method of effecting the partial separation of gases without the
selective action of chemical agents.

Let us next consider the case of diffusion in a vertical cylinder. Let TO,
U- the mass of the first gas in a column of unit area extending from the
bottom of the vessel to the height x, and let v, be the volume which this
mass would occupy at unit pressure, then

«,,

dm.

-~ar'

dv,

'

DIFFUSION. G31

and the equation of motion becomes

=p fd\ djj> _ dX~ciy 2| d-v, _Xdo^
l (dx* dt \ ~ d? dx j + ~da? ^ dx

dxl

dx dt dx

If we add the corresponding equations together for all the gases, we find
that the terms in (712 destroy each other, and that if the medium is not
affected with sensible currents the first term of each equation may be neg-
lected. In ordinary experiments we may also neglect the effect of gravity, so
that we get

or

where p is the uniform pressure of the mixed medium. Hence

dv2_ dv1 , dv2_ dvt

di= ~^t dx~p~~dx'

and the equation becomes

d\ _ Cu dVi
~

an equation, the form of which is identical with the well-known equation for
the conduction of heat. We may write it

D is called the coefficient of diffusion. It is equal to

It therefore varies inversely as the total pressure of the medium, and if the
coefficient of resistance, <712, is independent of the temperature, it varies directly
as the product kjc.,, i.e., as the square of the absolute temperature. It is
probable, however, that the effect of temperature is not so great as this would
make it.

DIFFUSION.

In liquids D probably depends on the proportion of the ingredients of the
mixed medium u well as on the temperature. The dimensions of D are DT~\
when L b the unit of length and T the unit of time.

Tbd vduet of the coefficients of diffusion of several pairs of gases have
fopu jafcennined by Loechmidt*. They are referred in the following table to
the centimetre and the second as units, for the temperature 0°C. and the
of 76 centimetres of mercury.

D

Carbonic acid and air . . 0'1423

Carbonic acid and hydrogen . . . 0'5558

Oxygen and hydrogen . . . 07214

Carbonic acid and oxygen .... 0'1409

Carbonic acid and carbonic oxide . . 0'1406

Carbonic acid and marsh gas . . . 0*1586

Carbonic acid and nitrous oxide . . . 0'0983

Sulphurous acid and hydrogen . . . 0'4800

Oxygen and carbonic oxide . . . 0'1802

Carbonic oxide and hydrogen . . . 0'6422

Diffusion in Liquids.

The nature of the motion of the molecules in liquids is less understood
than in' gases, but it is easy to see that if there is any irregular displacement
among the molecules in a mixed liquid, it must, on the whole, tend to cause
each component to pass from places where it forms a large proportion of the
mixture to places where it is less abundant. It is also manifest that any
relative motion of two constituents of the mixture will be opposed by a
resistance arising from the encounters between the molecules of these com-
|M>nents. The value of this resistance, however, depends, in liquids, on more
complicated conditions than in gases, and for the present we must regard it
as a function of all the physical properties of the mixture at the given place,
that is to say, its temperature and pressure, and the proportions of the
different components of the mixture.

* Imperial Academy of Vienna, 10th March, 1870.

DIFFUSION. 633

The coefficient of Intel-diffusion of two liquids must therefore be considered
as depending on all the physical properties of the mxiture according to laws
which can be ascertained only by experiment.

Thus Fick has determined the coefficient of diffusion for common salt in
water to be O'OOOOOllB, and Voit has found that of cane-sugar to be
0-00000365.

It appears from these numbers that in a vessel of the same size the
process of diffusion of liquids requires a greater number of days to reach a
given stage than the process of diffusion of gases in the same vessel requires
seconds.

When we wish to mix two liquids, it is not sufficient to place them in
the same vessel, for if the vessel is, say, a metre in depth, the lighter liquid
will lie above the denser, and it will be many years before the mixture
becomes even sensibly uniform. We therefore stir the two liquids together,
that is to say, we move a solid body through the vessel, first one way, then
another, so as to make the liquid contents eddy about in as complicated a
manner as possible. The effect of this is that the two liquids, which originally
formed two thick horizontal layers, one above the other, are now disposed in
thin and excessively convoluted strata, which, if they could be spread . out,
would cover an immense area. The effect of the stirring is thus to increase
the area over which the process of diffusion can go on, and to diminish the
distance between the diffusing liquids ; and since the time required for diffusion
varies as the square of the thickness of the layers, it is evident that by a
moderate amount of stirring the process of mixture which would otherwise
require years may be completed in a few seconds. That the process is not
instantaneous is easily ascertained by observing that for some time after the
stirring the mixture appears full of streaks, which cause it to lose its trans-
parency. This arises from the different indices of refraction of different portions
of the mixture which have been brought near each other by stirring. The
surfaces of separation are so drawn out and convoluted, that the whole mass
has a woolly appearance, for no ray of light can pass through it without being
turned many times out of its path.

Graham observed that the diffusion both of liquids and gases takes place
through porous solid bodies, such as plugs of plaster of Paris or plates of
pressed plumbago, at a rate not very much less than when no such body is
interposed, and this even when the solid partition is amply sufficient to check

VOL. n. SO

.

PIFFUSIOX.

all ordinary current*, and even to sustain a considerable difference of pressure

on iu oppoaite sides.

But then? is another class of cases in which a liquid or a gas can pass
through a diaphragm, which is not, in the ordinary sense, porous. For instance,
when carbonic acid gas is confined in a soap bubble it rapidly escapes. The

i> absorbed at the inner surface of the bubble, and forms a solution of

carbonic acid in water. This solution diffuses from the inner surface of the
babble, where it is strongest, to the outer surface, where it is in contact
with air, and the carbonic acid evaporates and diffuses out into the atmosphere.
It w also found that hydrogen and other gases can pass through a layer of
caoutchouc. Graham shewed that it is not through pores, in the ordinary
tftim, that the motion takes place, for the ratios are determined by the
chemical relations between the gases and the caoutchouc, or the liquid film.

According to Graham's theory, the caoutchouc is a colloid substance, — that
is, one which is capable of combining, in a temporary and very loose manner,
with indeterminate proportions of certain other substances, just as glue will
form a jelly with various proportions of water. Another class of substances,
which Graham called crystalloid, are distinguished from these by being always
• •f definite composition, and not admitting of these temporary associations.
When a colloid body has in different parts of its mass different proportions
i >f water, alcohol, or solutions of crystalloid bodies, diffusion takes place through
the colloid body, though no part of it can be shewn to be in the liquid state.

On the other hand, a solution of a colloid substance is almost incapable
of diffusion through a porous solid, or another colloid body. Thus, if a solu-
tion of gum and salt in water is placed in contact with a solid jelly of gelatine
and alcohol, alcohol will be diffused into the gum, and salt and water will
be diffused into the gelatine, but the gum and the gelatine will not diffuse
into each other.

There are certain metals whose relations to certain gases Graham explained
by this theory. For instance, hydrogen can be made to pass through iron and
palladium at a high temperature, and carbonic oxide can be made to pass
through iron. The gases form colloidal unions with the metals, and are diffused
through them as water is diffused through a jelly. Root has lately found
that hydrogen can pass through platinum, even at ordinary temperatures.

By taking advantage of the different velocities with which different liquids
and gases pass through parchment-paper and other solid bodies, Graham was

DIFFUSION. 635

enabled to effect many remarkable analyses. He called this method the method
of Dialysis.

Diffusion and Evaporation, Condensation, Solution, and Absorption.

The rate of evaporation of liquids is determined principally by the rate
of diffusion of the vapour through the air or other gas which lies above the
liquid. Indeed, the coefficient of diffusion of the vapour of a- liquid through
air can be determined in a rough but easy manner by placing a little of the
liquid in a test tube, and observing the rate at which its weight diminishes
by evaporation day by day. For at the surface of the liquid the density of
the vapour is that corresponding to the temperature, whereas at the mouth
of the test tube the air is nearly pure. Hence, if p be the pressure of the
vapour corresponding to the temperature, and p = Jcp, and if ra be the mass
evaporated in time t, and diffused into the air through a distance h *, then

j^ _ Jchm
' pt'

This method is not, of course, applicable to vapours which are rarer than
the superincumbent gas.

The solution of a salt in a liquid goes on in the same way, and so does
the absorption of a gas by a liquid.

These processes are all accelerated by currents, for the reason already ex-
plained.

The processes of evaporation and condensation go on much more rapidly when
no air or other non-condensible gas is present. Hence the importance of the
air-pump in the steam engine.

Relation between Diffusion of Matter and Diffusion of Heat.

The same motion of agitation of the molecules of gases which causes two
gases to diffuse through each other also causes two portions of the same gas
to diffuse through each other, although we cannot observe this kind of diffusion,
because we cannot distinguish the molecules of one portion from those of the

* h should be taken equal to the height of the tube above the surface of the liquid, together
with about f of the diameter of the tube. — See Clerk Maxwell's Electricity, Art. 309.

80—2

DIFFUSION.

other whan they are once mixed If, however, the molecules of one portion
hare any property whereby they can be distinguished from those of the other,
then that property will be communicated from one part of the medium to an
adjoining part, and that either by convection — that is by the molecules them-
selves patting out of one part into the other, carrying the property with
then— or by tranamisaion— that is by the property being communicated from
one molecule to another during their encounters. The chemical properties by
which different substances are recognized are inseparable from their molecules,
•o that the diffusion of such properties can take place only by the trans-
ference of the molecules themselves, but the momentum of a molecule in any
given direction and its energy are also properties which may be different in
different molecules, but which may be communicated from one molecule to
another. Hence the diffusion of momentum and that of energy through the
medium can take place in two different ways, whereas the diffusion of matter
can take place only in one of these ways.

In gases the great majority of the particles, at any instant, are describing
free paths, and it is therefore possible to shew that there is a simple numerical
relation between the coefficients of the three kinds of diffusion.-^the diffusion
of matter, the lateral diffusion of velocity (which is the phenomenon known as
the internal friction or viscosity of fluids), and the diffusion of energy (which
is called the conduction of heat). But in liquids the majority of the molecules
are engaged at close quarters with one or more other molecules, so that the
transmission of momentum and of energy takes place in a far greater degree
by communication from one molecule to another, than by convection by the
molecules' themselves. Hence the ratios of the coefficient of diffusion to those
of viscosity and thermal conductivity ard much smaller in liquids than in gases.

Theory of the Wet Bulb Thermometer.

The temperature indicated by the wet bulb thermometer is determined in
great part by the relation between the coefficients of diffusion and thermal
conductivity. As the water evaporates from the wet bulb heat must be sup-
plied to it by convection, conduction, or radiation. This supply of heat will
not be sufficient to maintain the temperature constant till the temperature of
the wet bulb has sunk so far below that of the surrounding air and other

DIFFUSION. 637

bodies that the flow of heat due to the difference of temperature is equal to
the latent heat of the vapour which leaves the bulb.

The use of the wet bulb thermometer as a means of estimating the
humidity of the atmosphere was employed by Hutton* and Leslie f, but the
formula by which the dew-point is commonly deduced from the readings of the
wet and dry thermometers was first given by Dr ApjohnJ.

Dr Apjohn assumes that, when the temperature of the wet bulb is sta-
tionary, the heat required to convert the water into vapour is given out by
portions of the surrounding air in cooling from the temperature of the atmo-
sphere to that of the wet bulb, and that the air thus cooled becomes saturated
with the vapour which it receives from the bulb.

Let m be the mass of a portion of air at a distance from the wet bulb,
#„ its temperature, p, the pressure due to the aqueous vapour in it, and P the
whole pressure.

If cr is the specific gravity of aqueous vapour (referred to air), then the

mass of water in this portion of air is -p

Let this portion of air communicate with the wet bulb till its temperature
sinks to 0l} that of the wet bulb, and the pressure of the aqueous vapour in
it rises to p1} that corresponding to the temperature 0^

The quantity of vapour which has been communicated to the air is

, x a-m

(Pi-l>.)-p »

and if L is the latent heat of vapour at the temperature 0,, the quantity of
heat required to produce this vapour is

crm T
-- L-

According to Apjohn's theory, this heat is supplied by the mixed air and
vapour in cooling from 00 to Qv

If S is the specific heat of the air (which will not be sensibly different
from that of dry air), this quantity of heat is

(0,-ejmS.

* Playfair's "Life of Hutton," Edinburgh Transactions, Vol. v. p. 67, note.
t Eneyc. Brit., 8th ed. Vol. i. "Dissertation Fifth," p. 764.
J Trans. Royal Irish Academy, 1834.

DIFFUSION.

(vitiating the two values we obtain

Here A i* l^e pressure °f ^e vapour in the atmosphere. The tempi- ra-
turr for which this is the maximum pressure — is the dew-point, and p^ is the
maximum pressure corresponding to the temperature 0, of the wet bulb. Hence
thin formula, combined with tables of the pressure of aqueous vapour, enables
«• to find the dew-point from observations of the wet and dry bulb thermo-
metan.

We may call this the convection theory of the wet bulb, because \\c
consider the temperature and humidity of a portion of air brought from a
distance to be affected directly by the wet bulb without communication either
• >t' heat or of vapour with other portions of air.

Dr Everett has pointed out as a defect in this theory, that it does not
explain how the air can either sink in temperature or increase in humidiu
unless it comes into absolute contact with the wet bulb. Let us, therefore,
consider what we may call the conduction and diffusion theory in calm air,
taking into account the effects of radiation.

The steady conduction of heat is determined by the conditions —
0=6, at a great distance from the bulb,
6 = 6l at the surface of the bulb,
V'# = 0 at any point of the medium.

The steady diffusion of vapour is determined by the conditions—
P=pt at a great distance from the bulb,
p=pi at the surface of the bulb,
V'p = Q at any point of the medium.

Now, if the bulb had been an electrified conductor, the conditions with
respect to the potential would have been

F=0 at a great distance,
V—Vl at the surface,
V*F=0 at any point outside the bulb.

DIFFUSION. 639

Hence the solution of the electrical problem leads to that of the other
two. For if V is the potential at any point,

If E is the electric charge of the conductor,

where the double integral is extended over the surface of the bulb, and dv
is an element of a normal to the surface.

If H is the flow of heat in unit of time from the bulb,

and if Q is the flow of aqueous vapour from the bulb,

_

kjdv

where k is the ratio of the pressure of aqueous vapour to its density.
If C is the electrical capacity of the bulb, E=CViy

The heat which leaves the bulb by radiation to external objects at tem-
perature 00 may be written

h=AB(01-ett),

where A is the surface of the bulb and R the coefficient of radiation of unit of
surface.

When the temperature becomes constant

AR

This formula gives the result of the theory of diffusion, conduction, and
radiation in a still atmosphere. It differs from the formula of the convection
theory only by the factor in the last term.

, • j

DIFFUSION.

The firet part of this factor is certainly less than unity, and probably

•boat 77.

If the bulb is spherical and of radius r, A = 4trr* and C=r, so that the

i _. •
•wood part is

Hence, the larger the wet bulb, the greater will be the ratio of the effect
of radiation to that of conduction. If, on the other hand, the air is in motion,
this will increase both conduction and diffusion, so as to increase the ratio of
the first part to the second. By comparing actual observations of the dew-
point with Apjohn's formula, it has been found that the factor should l>e
somewhat greater than unity. According to our theory it oiight to be greater
if the bulb is larger, and smaller if there is much wind.

Relation beluxen Diffusion and Electrolytic Conduction.

Electrolysis (see separate article) is a molecular movement of the con-
stituents of a compound liquid in which, under the action of electromotive force,
one of the components travels in the positive and the other in the negative
direction, the flow of each component, when reckoned in electrochemical equiva-
lents, being in all cases numerically equal to the flow of electricity.

Electrolysis resembles diffusion in being a molecular movement of two
currents in opposite directions through the same liquid ; but since the liquid
is of the same composition throughout, we cannot ascribe the currents to the
molecular agitation of a medium whose composition varies from one part to
another as in ordinary diffusion, but we must ascribe it to the action of the
electromotive force on particles having definite charges of electricity.

The force, therefore, urging an electro-chemical equivalent of either com-
ponent, or ton, as it is called, in a given direction is numerically equal to the
electromotive force at a given point of the electrolyte, and is therefore com-
parable with any ordinary force. The resistance which prevents the current
from rising above a certain value is that arising from the encounters of the
molecules of the ion with other molecules as they struggle forward through
the liquid, and this depends on their relative velocity, and also on the nature
of the ion, and of the liquid through which it has to flow.

DIFFUSION. 641

The average velocity of the ions will therefore increase, till the resistance
they meet with is equal to the force which urges them forward, and they will
thus acquire a definite velocity proportional to the electric force at the point,
but depending also on the nature of the liquid.

If the resistance of the liquid to the passage of the ion is the same for
different strengths of solution, the velocity of the ion will be the same for
different strengths, but the quantity of it, and therefore the quantity of elec-
tricity which passes in a given time, will be proportional to the strength of
the solution.

Now, Kohlrauseh has determined the conductivity of the solutions of many
electrolytes in water, and he finds that for very weak solutions the conductivity
is proportional to the strength. When the solution is strong the liquid through
which the ions struggle can no longer be considered sensibly the same as pure
water, and consequently this proportionality does not hold good for strong
solutions.

Kohlrauseh has determined the actual velocity in centimetres per second
of various ions in weak solutions under an electro-motive force of unit value.
From these velocities he has calculated the conductivities of weak solutions of
electrolytes different from those of which he made use in calculating the velocity
of the ions, and he finds the results consistent with direct experiments on
those electrolytes.

It is manifest that we have here important information as to the resist-
ance which the ion meets with in travelling through the liquid. It is not
easy, however, to make a numerical comparison between this resistance and any
results of ordinary diffusion, for, in the first place, we cannot make experiments
on the diffusion of ions. Many electrolytes, indeed, are decomposed by the
current into components, one or both of which are capable of diffusion, but
these components, when once separated out of the electrolyte, are no longer
ions — they are no longer acted on by electric force, or charged with definite
quantities of electricity. Some of them, as the metals, are insoluble, and there-
fore incapable of diffusion ; others, like the gases, though soluble in the liquid
electrolyte, are not, when in solution, acted on by the current.

Besides this, if we accept the theory of electrolysis proposed by Clausius,
the molecules acted on by the electro-motive force are not the whole of the
molecules which form the constituents of the electrolyte, but only those which
at a given instant are in a state of dissociation from molecules of the other

VOL. II. 81

DIFFUSION.

kind. Wfg forced away from them temporarily by the violence of the molecular
• If these diMOciated molecules form a small proportion of the whole,

the velocitv of their paoage through the medium must be much greater than
the mean Telocity of the whole, which is the quantity calculated by Kohlrausch.

0* Proce*** by trA/cA the M'urt'nv and Sejxtration of Fluids can be
effected in a Reversible Manner.

A physical process is said to be reversible when the material system can
be made to return from the final state to the original state under conditions
which at every stage of the reverse process differ only infinitesimally from the
conditions at the corresponding stage of the direct process.
All other processes are called irreversible.

Thus the passage of heat from one body to another is a reversible process
if the temperature of the first body exceeds that of the second only by an
infinitesimal quantity, because by changing the temperature of either of the
bodies by an infinitesimal quantity, the heat may be made to flow back again
from the second body to the first

But if the temperature of the first body is higher than that of the second
by a finite quantity, the passage of heat from the first body to the second
is not a reversible process, for the temperature of one or both of the bodies
must be altered by a finite quantity before the heat can be made to flow
back again.

In like manner the interdiffusion of two gases is in general an irreversible
process, for in order to separate the two gases the conditions must be very
considerably changed. For instance, if carbonic acid is one of the gases, we
can separate it from the other by means of quicklime ; but the absorption of
carbonic acid by quicklime at ordinary temperatures and pressures is an irre-
versible process, for in order to separate the carbonic acid from the lime it
must be raised to a high temperature.

In all reversible processes the substances which are in contact must be
in complete equilibrium throughout the process ; and Professor Gibbs has shewn
the condition of equilibrium to be that not only the temperature and the
pressure of the two substances must be the same, but also that the potential
of each of the component substances must be the same in both compounds,
and that there is an additional condition which we need not here specify.

DIFFUSION. 643

Now, we may obtain complete equilibrium between quicklime and the
mixture containing carbonic acid if we raise the whole to a temperature at
which the pressure of dissociation of the carbonic acid in carbonate of lime is
equal to the pressure of the carbonic acid in the mixed gases. By altering
the temperature or the pressure very slowly we may cause carbonic acid to
pass from the mixture to the lime, or from the lime to the mixture, in such
a manner that the conditions of the system differ only by infinitesimal quan-
tities at the corresponding stages of the direct and the inverse processes. The
same thing may be done at lower temperatures by means of potash or soda.

If one of the gases can be condensed into a liquid, and if during the
condensation the pressure is increased or the temperature diminished so slowly
that the liquid and the mixed gases are always very nearly in equilibrium,
the separation and mixture of the gases can be effected in a reversible
manner.

The same thing can be done by means of a liquid which absorbs the
gases in different proportions, provided that we can maintain such conditions as
to temperature and pressure as shall keep the system in equilibrium during the
whole process.

If the densities of the two gases are different, we can effect their partial
separation by a reversible process which does not involve any of the actions
commonly called chemical. We place the mixed gases in a very long horizontal
tube, and we raise one end of the tube till the tube is vertical. If this is
done so slowly that at every stage of the process the distribution of the two
gases is sensibly the same as it would be at the same stage of the reverse
process, the process will be reversible, and if the tube is long enough the
separation of the gases may be carried to any extent.

In the Philosophical Magazine for 1876, Lord Rayleigh has investigated
the thermodynamics of diffusion, and has shewn that if two portions of different
gases are given at the same pressure and temperature, it is possible, by
mixing them by a reversible process, to obtain a certain quantity of work.
At the end of the process the two gases are uniformly mixed, and occupy a
volume equal to the sum of the volumes they occupied when separate, but
the temperature and pressure of the mixture are lower than before.

The work which can be gained during the mixture is equal to that which
would be gained by allowing first one gas and then the other to expand
from its original volume to the sum of the volumes ; and the fall of temperature

81—2

; ; DIFFUSION.

and pfOMuro is equal to that which would be produced in the mixture by
taking away a quantity of heat equivalent to this work.

i: the diffusion takes place by an irreversible process, such as goes on
when the gates are placed together in a vessel, no external work is done,
and there is BO fall of temperature or of pressure during the process.

We may arrive at this result by a method whkh, if not so instructive
a* that of Lord Rayleigh, is more general, by the use of a physical quantity
called by Clausius the Entropy of the system.

Hie entropy of a body in equilibrium is a quantity such that it remains
constant if no heat enters or leaves the body, and such that in general the
quantity of heat which enters the body is

where <f> is the entropy, and 6 the absolute temperature.

The entropy of a material system is the sum of the entropy of its parts.

In reversible processes the entropy of the system remains unchanged, but
in all irreversible processes the entropy of the system increases.

The increase of entropy involves a diminution of the available energy of
the system, that is to say, the total quantity of work which can be obtained
from the system. This is expressed by Sir W. Thomson by saying that a
certain amount of energy is dissipated.

The quantity of energy which is dissipated in a given process is equal to

W.-6),

where & is the entropy at the beginning, and <£, that at the end of the
process, 'and 0. is the temperature of the system in its ultimate state, when
1 1' i more work can be got out of it.

When we can determine the ultimate temperature we can calculate the
amount of energy dissipated by any process; but it is sometimes difficult to
do this, whereas the increase of entropy is determined by the known states
of the system at the beginning and end of the process.

The entfopy of a volume vl of a gas at pressure pl and temperature 0l
exceeds its entropy where its volume is vt and its temperature 0, by the quantity

Hence if volumes r, and i\ of two gases at the same temperature and pressure

DIFFUSION. 645

are mixed so as to occupy a volume Vi + va at the same temperature and
pressure, the entropy of the system increases during the process by the quantity

Since in this case the temperature does not change during the process, we
may calculate the quantity of energy dissipated by multiplying the gain of
entropy by the temperature, and we thus find for the dissipation

or the sum of the work which would be done by the two portions of gas if
each expanded under constant temperature to the volume v1 + vi.

It is greatest when the two volumes are equal, in which case it is
l'386jw, where p is the pressure and v the volume of one of the portions.

Let us now suppose that we have in a vessel two separate portions of
gas of equal volume, and at the same pressure and temperature, with a
movable partition between them. If we remove the partition the agitation of
the molecules will carry them from one side of the partition to the other in
an irregular manner, till ultimately the two portions of gas will be thoroughly and
uniformly mixed together. This motion of the molecules will take place whether
the two gases are the same or different, that is to say, whether we can
distinguish between the properties of the two gases or not.

If the two gases are such that we can separate them by a reversible
process, then, as we have just shewn, we might gain a definite amount of
work by allowing them to mix under certain conditions ; and if we allow them
to mix by ordinary diffusion, this amount of Work is no longer available, but
is dissipated for ever. If, on the other hand, the two portions of gas are
the same, then no work can be gained by mixing them, and no work is
dissipated by allowing them to diffuse into each other.

It appears, therefore, that the process of diffusion does not involve dis-
sipation of energy if the two gases are the same, but that it does if they can
be separated from each other by a reversible process.

Now, when we say that two gases are the same, we mean that we cannot
distinguish the one from the other by any known reaction. It is not probable,
but it is possible, that two gases derived from different sources, but hitherto
supposed to be the same, may hereafter be found to be different, and that a

(J4G DIFFUSION.

method mar be discovered of separating them by a reversible process. If this
•hould happen, the process of iuterdiffusion which we had formerly supposed
not to be an tiurtw"** of dissipation of energy would now be recognized as
snob an instance,

It follows from this that the idea of dissipation of energy depends on
th« extent of our knowledge. Available energy is energy which we can direct
int.. any desired channel Dissipated energy is energy which we cannot lay
hold of and direct at pleasure, such as the energy of the confused agitation
of molecules which we call heat. Now, confusion, like the correlative term
order, u not a property of material things in themselves, but only in relation
to the mind which perceives them. A memorandum-book does not, provided it
w neatly written, appear confused to an illiterate person, or to the owner who
understands it thoroughly, but to any other person able to read it appears to
be inextricably confused. Similarly the notion of dissipated energy could not
occur to a being who could not turn any of the energies of nature to his
own account, or to one who could trace the motion of every molecule and
seiae it at the right moment. It is only to a being in the intermediate si
who can lay hold of some forms of energy while others elude his grasp, that
energy appears to be passing inevitably from the available to the dissipated

[From the Encyclopedia Britanmca.]

XC. Diagrams.

A DIAGRAM is a figure drawn in such a manner that the geometrical
relations between the parts of the figure help us to understand relations between
other objects. A few have been selected for description in this article on account
of their greater geometrical significance.

Diagrams may be classed according to the manner in which they are
intended to be used, and also according to the kind of analogy which we
recognize between the diagram and the thing represented.

Diagrams of Illustration.

The diagrams in mathematical treatises are intended to help the reader
to follow the mathematical reasoning. The construction of the figure is defined
in words so that even if no figure were drawn the reader could draw one
for himself. The diagram is a good one if those features which form the
subject of the proposition are clearly represented. The accuracy of the drawing
is therefore of smaller importance than its distinctness.

Metrical Diagrams.

Diagrams are also employed in an entirely different way — namely, for
purposes of measurement. The plans and designs drawn by architects and
engineers are used to determine the value of certain real magnitudes by
measuring certain distances on the diagram. For such purposes it is essential
that the drawing be as accurate as possible.

We therefore class diagrams as diagrams of illustration, which merely
suggest certain relations to the mind of the spectator, and diagrams drawn to
scale, from which measurements are intended to be made.

•48

DIAGRAMS.

Methods in which diagrams are used for purposes of measurement are called

(Jraphkml method*.

of illustration, if sufficiently accurate, may be used for purjxxsea

of measurement; and diagrams for measurement, if sufficiently clear, may be
used for purposes of demonstration.

There are some diagrams or schemes, however, in which the form of the
parts is of no importance, provided their connections are properly shewn. Of
tfrfr fcjnd are the diagrams of electrical connections, and those belonging to
that department of geometry which treats of the degrees of cyclosis, periphraxy,
linkednen, and knottedne

Diagrams purely Graphic and mixed Symbolic and Graphic.

Diagrams may also be classed either as purely graphical diagrams, in which
DO symbols are employed except letters or other marks to distinguish particular
points of the diagrams, and mixed diagrams, in which certain magnitudes are
represented, not by the magnitudes of parts of the diagram, but by symbols,
such as numbers written on the diagram.

Thus in a map the height of places above the level of the sea is often
indicated by marking the number of feet above the sea at the corresponding
places on the map.

There is another method in which a line called a contour line is drawn
through all the places in the map whose height above the sea is a certain
number of feet, and the number of feet is written at some point or points
of this line.

By the use of a series of contour lines, the height of a great number of
places can be indicated on a map by means of a small number of written
symbols. Still this method is not a purely graphical method, but a partly
symbolical method of expressing the third dimension of objects on a diagram
in two dimensions.

Diagrams in Pairs.

In order to express completely by a purely graphical method the relations
of magnitudes involving more than two variables, we must use more than one
diagram. Thus in the arts of construction we use plans and elevations and

DIAGRAMS. 649

sections through different planes, to specify the form of objects having three
dimensions.

In such systems of diagrams we have to indicate that a point in one
diagram corresponds to a point in another diagram. This is generally done by
marking the corresponding points in the different diagrams with the same letter.
If the diagrams are drawn on the same piece of paper we may indicate
corresponding points by drawing a line from one to the other, taking care
that this line of correspondence is so drawn that it cannot be mistaken for
a real line in either diagram.

In the stereoscope the two diagrams, by the combined use of which the
form of bodies in three dimensions is recognized, are projections of the bodies
taken from two points so near each other that, by viewing the two diagrams
simultaneously, one with each eye, we identify the corresponding points intuitively.

The method in which we simultaneously contemplate two figures, and
recognize a correspondence between certain points in the one figure and certain
points in the other, is one of the most powerful and fertile methods hitherto
known in science. Thus in pure geometry the theories of similar, reciprocal,
and inverse figures have led to many extensions of the science. It is sometimes
spoken of as the method or principle of Duality.

DIAGRAMS IN KINEMATICS.

The study of the motion of a material system is much assisted by the
use of a series of diagrams representing the configuration, displacement, and
acceleration of the parts of the system.

Diagram of Configuration.

In considering a material system it is often convenient to suppose that

we have a record of its position at any given instant in the form of a diagram
of configuration.

The position of any particle of the system is defined by drawing a straight

line or vector from the origin, or point of reference, to the given particle.

The position of the particle with respect to the origin is determined by the
magnitude and direction of this vector.

VOL. n. 82

I'l \

If in the dkgnun we drew from the origin (which need not be the same
point of «pM» W the origin for the material system) a vector equal :m<l
llrl to the vector which determines the position of the particle, the t-nd
thw twctor will indicate the position of the particle in the digram ..f

If this » done for all the particles, we shall have a system of points in
the diagram of configumtion, each of which corresponds to a particle of the
material system, and the relative positions of any pair of these points will In-
the mate as the relative positions of the material particles which correspond

to them.

We have hitherto spoken of two origins or points from which the vectoti
are supposed to be drawn — one for the material system, the other for tin-
diagram. These point*, however, and the vectors drawn from them, may now
be omitted, so that we have on the one hand the material system ami <>n
the other a set of points, each point corresponding to a particle of the system.
and the whole representing the configuration of the system at a given instant.

This is called a diagram of configuration.

Diagram of Displacement.

Let us next consider two diagrams of configuration of the same system,
corresponding to two different instants.

We call the first the initial configuration and the second the final con-
figuration, and the passage from the one configuration to the other we call the
displacement of the system. We do not at present consider the length of
time during which the displacement was effected, nor the intermediate stages
through which it passed, but only the final result — a change of configuration.
To study this change we construct a diagram of displacement.

Let A, B, C be the points in the initial diagram of configuration, and
.V. It'. C1 be the corresponding points in the final diagram of configuration.

From o, the origin of the diagram of displacement, draw a vector on
fjual and parallel to A A', oh equal and parallel to BB', oc to CC', and so on.

The ]».ints. o, h, c, &c., will be such that the vector ab indicates tin-
displacement of I* relative to a, and so on. The diagram containing the points
«, 6, c, Ac., is therefore called the diagram of displacement.

In constructing the diagram of displacement we have hitherto assumed
that we know the absolute displacements of the points of the system. For

DIAGKAMS. 651

we are required to draw a line equal and parallel to AA', which we cannot
do unless we know the absolute final position of A, with respect to its initial
position. In this diagram of displacement there is therefore, besides the points
a, I), c, &c., an origin, o, which represents a point absolutely fixed in space.
This is necessary because the two configurations do not exist at the same
time ; and therefore to express their relative position we require to know a
point which remains the same at the beginning and end of the time.

But we may construct the diagram in another way which does not assume
a knowledge of absolute displacement or of a point fixed in space.

Assuming any point and calling it a, draw ak parallel and equal to BlAl
in the initial configuration, and from If draw kb parallel and equal to AnB,,
in the final configuration. It is easy to see that the position of the point 6
relative to a will be the same by this construction as by the former construction,
only we must observe that in this second construction we use only vectors
such as A^,, A.2B.,, which represent the relative position of points both of
which exist simultaneously, instead of vectors such as A^At, B^B^, which express
the position of a point at one instant relative to its position at a former
instant, and which therefore cannot be determined by observation, because the
two ends of the vector do not exist simultaneously.

It appears therefore that the diagram of displacements, when drawn by
the first construction includes an origin o, which indicates that we have assumed
a knowledge of absolute displacements. But no such point occurs in the second
construction, because we use such vectors only as we can actually observe.
Hence the diagram of displacements ivithout an origin represents neither more
nor less than all we can ever know about the displacement of the material
system.

Diagram of Velocity.

If the relative velocities of the points of the system are constant, then
the diagram of displacement corresponding to an interval of a unit of time
between the initial and the final configuration is called a diagram of relative
velocity.

If the relative velocities are not constant, we suppose another system in
which the velocities are equal to the velocities of the given system at the
given instant and continue constant for a unit of tune. The diagram of dis-

82—2

DIAGRAMS.

i4arenM>nu (or this imaginary system is the required diagram of relative velocities
of the actual system at the given instant.

It is easy to see that the diagram gives the velocity of any one point
relative to any other, but cannot give the absolute velocity of any of them.

Diagram of Acceleration.

By the same process by which we formed the diagram of displacements
from the two diagrams of initial and final configuration, we may form a diagram
of changes of relative velocity from the two diagrams of initial and final velocities.
ThU diagram may be called that of total accelerations in a finite interval of time.

By the same process by which we deduced the diagram of velocities from
tliat of displacements we may deduce the diagram of rates of acceleration from
that of total acceleration.

We have mentioned this system of diagrams in elementary kinematics
because they are found to be of use especially when we have to deal with
material systems containing a great number of parts, as in the kinetic theory
of gases. The diagram of configuration then appeal's as a region of space
swarming with points representing molecules, and the only way in which w
can investigate it is by considering the number of such points in unit of
volume in different parts of that region, and calling this the density of the gas.

In like manner the diagram of velocities appears as a region containing
(loints equal in number but distributed in a different manner, and the number
• •f ]x>inta in any given portion of the region expresses the number of molecules
whose velocities lie within given limits. We may speak of this as the velocity-
density.

Path and Hodograph.

When the number of bodies in the system is not so great, we may
construct diagrams each of which represents some property of the whole course
tif the motion.

Thus if we are considering the motion of one particle relative to another,
the point on the diagram of configuration which corresponds to the moving
{•article will trace out a continuous line called the path of the particle.

On the diagram of velocity the point corresponding to the moving particle
will trace another continuous line called the hodograph of the particle.

DIAGRAMS. 653

The hodograph was invented and used with great success by Sir W. K.
Hamilton as a method of studying the motions of bodies.

DIAGRAMS OF STRESS.

Graphical methods are peculiarly applicable to statical questions, because
the state of the system is constant, so that we do not need to construct a
series of diagrams corresponding to the successive states of the system.

The most useful of these applications relates to the equilibrium of plane
framed structures. Two diagrams are used, one called the diagram of the
frame and the other called the diagram of stress,

The structure itself consists of a number of separable pieces or links jointed
together at their extremities. In practice these joints have friction, or may
be made purposely stiff, so that the force acting at the extremity of a piece
may not pass exactly through the axis of the joint ; but as it is unsafe to
make the stability of the structure depend in any degree upon the stiffness
of joints, we assume in our calculations that all the joints are perfectly smooth,
and therefore that the force acting on the end of any link passes through the
axis of the joint.

The axes of the joints of the structure are represented by points in the
diagram of the frame.

The link which connects two joints in the actual structure may be of
any shape, but in the diagram of the frame it is represented by a straight
line joining the points representing the two joints.

If no force acts on the link except the two forces acting through the
centres of the joints, these two forces must be equal and opposite, and their
direction must coincide with the straight line joining the centres of the joints.

If the force acting on either extremity of the link is directed towards
the other extremity, the stress on the link is called pressure and the link is
called a strut. If it is directed away from the other extremity, the stress
on the link is called tension and the link is called a tie.

In this case, therefore, the only stress acting in a link is a pressure or
a tension in the direction of the straight line which represents it in the
diagram of the frame, and all that we have to do is to find the magnitude
of this stress.

In the actual structure, gravity acts on every part of the link, but in

DIAGRAMS.

the diagram we substitute for the actual weight of the different parts of the
Imk. two weights which have the same resultant acting at the extremities of

thf link.

We mar now treat the diagram of the frame as composed of links without
but loaded at each joint with a weight made up of portions of the
of all the links which meet in that joint.

If any link has more than two joints we may substitute for it in tlu>
diagram an imaginary stiff frame, consisting of links, each of which has only

two joints.

The diagram of the frame is now reduced to a system of points, certain
pairs of which are joined by straight lines, and each point is in general acted
on by a weight or other force acting between it and some point external to

the system.

To complete the diagram we may represent these external forces as links,
that is to say, straight lines joining the points of the frame to points external

lie frame. Thus each weight may be represented by a link joining the
point of application of the weight with the centre of the earth.

Hut we can always construct an imaginary frame having its joints in the
lines of action of these external forces, and this frame, together with the real
frame and the links representing external forces, which join points in the one
frame to points in the other frame, make up together a complete self-strained
system in equilibrium, consisting of points connected by links acting by pressure
or tension. We may in this way reduce any real structure to the case of a

in of points with attractive or repulsive forces acting between certain pairs
• •t' these points, and keeping them in equilibrium.

The direction of each of these forces is sufficiently indicated by that of
the line joining the points, so that we have only to determine its magnitude.

We might do this by calculation, and then write down on each link the
pressure or the tension which acts in it.

We should in this way obtain a mixed diagram in which the stresses
are represented graphically as regards direction and position, but symbolically
as regards magnitude.

But we know that a force may be represented in a purely graphical
manner by a straight line in the direction of the force containing as many
units of length as there are units of force in the force. The end of this line
is marked with an arrow head to shew in which direction the force acts.

DIAGRAMS. G55

According to this method each force is drawn in its proper position in
the diagram of configuration of the frame. Such a diagram might be useful
as a record of the result of calculation of the magnitude of the forces, but
it would be of no use in enabling us to test the correctness of the calculation.

But we have a graphical method of testing the equilibrium of any set of
forces acting at a point. We draw in series a set of lines parallel and pro-
portional to these forces. If these lines form a closed polygon the forces are in
equilibrium. We might in this way form a series of polygons of forces, one for
each joint of the frame. But in so doing we give up the principle of drawing
the line representing a force from the point of application of the force, for all
the sides of the polygon cannot pass through the same point, as the forces do.

We also represent every stress twice over, for it appears as a side of both
the polygons corresponding to the two joints between which it acts.

But if we can arrange the polygons in such a way that the sides of any
two polygons which represent the same stress coincide with each other, we
may form a diagram in which every stress is represented in direction and
magnitude, though not in position, by a single line which is the common
boundary of the two polygons which represent the joints at the extremities
of the corresponding piece of the frame.

We have thus obtained a pure diagram of stress in which no attempt is
made to represent the configuration of the material system, and in which every
force is not only represented in direction and magnitude by a straight line,
but the equilibrium of the forces at any joint is manifest by inspection, for
we have only to examine whether the corresponding polygon is closed or not.

The relations between the diagram of the frame and the diagram of stress
are as follows : —

To every link in the frame corresponds a straight line in the diagram of
stress which represents in magnitude and direction the stress acting in that link.

To every joint in the frame corresponds a closed polygon in the diagram,
and the forces acting at that joint are represented by the sides of the polygon
taken in a certain cyclical order. The cyclical order of the sides of the two
adjacent polygons is such that their common side is traced in opposite directions
in going round the two polygons.

The direction in which any side of a polygon is traced is the direction
of the force acting on that joint of the frame which corresponds to the polygon,
and due to that link of the frame which corresponds to the side.

DIAGRAMS.

This determine* whether the stress of the link is a pressure or a tension.

If w« know whether the stress of any one link is a pressure or a tension,
this determines the cyclical order of the sides of the two polygons corresponding
to the end* of the links, and therefore the cyclical order of all the polygons,
and the nature of the stress in every link of the frame.

Definition of Reciprocal Diagrams.

When to every point of concourse of the lines in the diagram of stress
corresponds a dosed polygon in the skeleton of the frame, the two diagrams
are said to be reciprocal.

The first extensions of the method of diagrams of forces to other cases
than that of the funicular polygon were given by Rankine in his Ap^i«l
Mechanics (1857). The method was independently applied to a large number

rises by Mr W. P. Taylor, a practical draughtsman in the office of the
well-known contractor Mr J. B. Cochrane, and by Professor Clerk Maxwell in
his lectures in King's College, London. In the Phil. Mag. for 1864 the latter
[x'inted out the reciprocal properties of the two diagrams, and in a paper on
"Reciprocal Figures, Frames, and Diagrams of Forces," Trans. R. S. Edinburgh,
Vol. xxvi. (1870), he shewed the relation of the method to Airy's function of
stress and to other mathematical methods.

Professor Fleeming Jenkin has given a number of applications of the method
to practice (Trans. R. S. Edin., Vol. xxv.).

Cremona (Le figure reciproche nella statica grafica, Milan, 1872) has deduced
the construction of reciprocal figures from the theory of the two components
of a wrench as developed by Mobius.

Culmann, in his Graphische Statik, makes great use of diagrams of forces,
some of which, however, are not reciprocal.

M. Maurice Levy in his Statique Graphique (Paris, 1874) has treated the
whole subject in an elementary but copious manner.

Mr R. H. Bow, C.E., F.R.S.E., in his work on The Economics of Constiiiction
in relation to Framed Structures, 1873, has materially simplified the process of
drawing a diagram of stress reciprocal to a given frame acted on by a system
• •f equilibrating external forces.

Instead of lettering the joints of the frame, as is usually done, or the
links of the frame, as was the writer's custom, he places a letter in each of

DIAGRAMS.

657

the polygonal areas inclosed by the links of the frame, and also in each of
the divisions of surrounding space as separated by the lines of action of the
external forces.

When one link of the frame crosses another, the point of apparent
intersection of the links is treated as if it were a real joint, and the stresses
of each of the intersecting links are represented twice in the diagram of
stress, as the opposite sides of the parallelogram which corresponds to the
point of intersection.

M

FIG. 1. Diagram of Configuration.

F

Fio. 2. Diagram of Stress.

This method is followed in the lettering of the diagram of configuration
(fig. 1), and the diagram of stress (fig. 2) of the linkwork which Professor
Sylvester has called a quadruplane.

In fig. 1 the real joints are distinguished from the places where one link
appears to cross another by the little circles 0, P, Q, R, S, T, V.

The four links RSTV form a " contraparallelogram " in which RS=TV and
RV=ST.

The triangles ROS, RPV, TQS are similar to each other. A fourth triangle
(TNV), not drawn in the figure, would complete the quadruplane. The four

VOL. II. 83

658

OIAURAM&

points O, P, N. Q form a parallelogram whose angle POQ is constant and
aqUlj to9-SOR, The product of the distances OP and OQ is constant.

Th« linkwork may be fixed at O. If any figure is traced by P, Q will
trace the inverse figure, but turned round O through the constant angle POQ.

In the diagram forces Pp, Qq are balanced by the force Oo at the fixed
point The forces Pp and Q? are necessarily inversely as OP and OQ, and
make equal angles with those lines.

Every closed area formed by the links or the external forces in the diagram
of configuration is marked by a letter which corresponds to a point of concourse
of lines in the diagram of stress.

The stress in the link which is the common boundary of two areas is
represented in the diagram of stress by the line joining the points corresponding
to those areas.

When a link is divided into two or more parts by lines crossing it, the
stress in each part is represented by a different line for each part, but as the
stress is the same throughout the link these lines are all equal and parallel.
Thus in the figure the stress in RV is represented by the four equal and
parallel lines ///, FG, DE, and AB.

If two areas have no part of their boundary in common the letters cor-
responding to them in the diagram of stress are not joined by a straight
line. If, however, a straight line were drawn between them, it would represent
in direction and magnitude the resultant of all the stresses in the links which
are cut by any line, straight or curved, joining the two areas.

For instance the areas F and C in fig. 1 have no common boundary, and
the points F and C in fig. 2 are not joined by a straight line. But every
path from the area F to the area C in fig. 1 passes through a series of other
areas, and each passage from one area into a contiguous area corresponds to
a line drawn in the diagram of stress. Hence the whole path from F to C
in fig. 1 corresponds to a path formed of lines in fig. 2 and extending from
F to C, and the resultant of all the stresses in the links cut by the path is
represented by FC in fig. 2.

Automatic Description of Diagrams.

There are many other kinds of diagrams in which the two co-ordinates of
a point in a plane are employed to indicate the simultaneous values of two
related quantities.

DIAGRAMS. 659

If a sheet of paper is made to move, say horizontally, with a constant
known velocity, while a tracing point is made to move in a vertical straight
line, the height varying as the value of any given physical quantity, the point
will trace out a curve on the paper from which the value of that quantity
at any given time may be determined.

This principle is applied to the automatic registration of phenomena of all
kinds, from those of meteorology and terrestrial magnetism to the velocity of
cannon-shot, the vibrations of sounding bodies, the motions of animals, voluntary
and involuntary, and the currents in electric telegraphs.

Indicator-Diagram.

In Watt's indicator for steam-engines the paper does not move with a
constant velocity, but its displacement is proportional to that of the piston of
the engine, while that of the tracing point is proportional to the pressure of
the steam. Hence the co-ordinates of a point of the curve traced on the
diagram represent the volume and the pressure of the steam in the cylinder.
The indicator-diagram not only supplies a record of the pressure of the steam
at each stage of the stroke of the engine, but indicates the work done by the
steam in each stroke by the area inclosed by the curve traced on the diagram.

The indicator-diagram was invented by James Watt as a method of estimating
the work done by an engine. It was afterwards used by Clapeyron to illustrate
the theory of heat, and this use of it was greatly developed by Kankine in
his work on the steam-engine.

The use of diagrams in thermodynamics has been very completely illustrated
by Prof. J. Willard Gibbs (Connecticut Acad. Sci., Vol. in.), but though his
methods throw much light on the general theory of diagrams as a method of
study, they belong rather to thermodynamics than to the present subject.

83—2

[From Naturt, Vol. xvn.]

XCI. Tail's "Thermodynamics."

THIS book, aa we are told in the preface, has grown out of two articles
contributed in 1864 by Prof. Tait to the North British Review. This journal,
about that time, inserted a good many articles in which scientific subjects were
discussed in scientific language, and in which, instead of the usual attempts
to conciliate the unscientific reader by a series of relapses into irrelevant and
incoherent writing, his attention was maintained by awakening a genuine interest
in the subject.

The attempt was so far successful that the publishers of the Review were
urged by men of science, especially engineers, to reprint these essays of Prof.
Tait, but the Review itself soon afterwards became extinct.

Prof. Tait added to the two essays a mathematical sketch of the funda-
mental principles of thermodynamics, and in this form the book was published
in 1868. In the present edition, though there are many additions and improve-
ments, the form of the book is essentially the same.

Whether on account of these external circumstances, or from internal
causes, it is impossible to compare this book either with so-called popular
treatises or with those of a more technical kind.

In the popular treatise, whatever shreds of the science are allowed to
appear, are exhibited in an exceedingly diffuse and attenuated form, apparently
with the hope that the mental faculties of the reader, though they would
reject any stronger food, may insensibly become saturated with scientific phraseo-
logy, provided it is diluted with a sufficient quantity of more familiar language.
In this way, by simple reading, the student may become possessed of the
phrases of the science without having been put to the trouble of thinking a
single thought about it. The loss implied in such an acquisition can be estimated

C61

only by those who have been compelled to unlearn a science that they might
at length begin to learn it.

The technical treatises do less harm, for no one ever reads them except
under compulsion. From the establishment of the general equations to the end
of the book, every page is full of symbols with indices and suffixes, so that
there is not a paragraph of plain English on which the eye may rest.

Prof. Tait has not adopted either of these methods. He serves up his
strong meat for grown men at the beginning of the book, without thinking
it necessary to employ the language either of the nursery or of the school ;
while for younger students he has carefully boiled down the mathematical
elements into the most concentrated form, and has placed the result at the
end as a bonne bouche, so that the beginner may take it in all at once, and
ruminate upon it at his leisure.

A considerable part of the book is devoted to the history of thermo-
dynamics, and here it is evident that with Prof. Tait the names of the founders
of his science call up the ideas, not so much of the scientific documents they
have left behind them in our libraries, as of the men themselves, whether he
recommends them to our reverence as masters in science, or bids us beware
of them as tainted with error. There is no need of a garnish of anecdotes
to enliven the dryness of science, for science has enough to do to restrain the
strong human nature of the author, who is at no pains to conceal his own
idiosyncrasies, or to smooth down the obtrusive antinomies of a vigorous mind
into the featureless consistency of a conventional philosopher.

Thus, in the very first page of the book, he denounces all metaphysical
methods of constructing physical science, and especially any d priori decisions
as to what may have been or ought to have been. In the second page he
does not indeed give us Aristotle's ten categories, but he lays down four of
his own : — matter, force, position, and motion, to one of which he tells us,
" it is evident that every distinct physical conception must be referred," and
then before we have finished the page we are assured that heat does not
belong to any of these four categories, but to a fifth, called energy.

This sort of writing, however unlike what we might expect from the con-
ventional man of science, is the very thing to rouse the placid reader, and
startle his thinking powers into action.

Prof. Tait next handles the caloric theory, but instead of merely shewing
up its weak points and then dismissing it with contempt, he puts fresh life

TJUT'S "THERMODYNAMICS."

into it by giving (in the new edition) a characteristic extract from Dr Black's
and proceeds to help the calorists out of some of their difficulties, by
ikinir over to them some excellent hints of his own.

H •— • — *j iiiia^»a^ ^

The hiatory of thermodynamics has an especial interest as the development
of a acienee, within a short time and by a small number of men, from the
condition of a vague anticipation of nature to that of a science with secure
famdatiwE, clear definitions, and distinct boundaries.

The earlier part of the history has already provoked a sufficient amount
of dincuauon. We shall therefore confine our remarks to the methods employed
for the advancement of the science by the three men who brought the theory

to maturity.

Of the three founders of theoretical thermodynamics, Rankine availed him-
•elf to the greatest extent of the scientific use of the imagination. His
imagination, however, though amply luxuriant, was strictly scientific. Whatever
he imagined about molecular vortices, with their nuclei and atmospheres, was
M> clearly imaged in his mind's eye, that he, as a practical engineer, could
see how it would work.

However intricate, therefore, the machinery might be which he imagined
to exist in the minute parts of bodies, there was no danger of his going
on to explain natural phenomena by any mode of action of this machinery
which was not consistent with the general laws of mechanism. Hence, though
the construction and distribution of his vortices may seem to us as complicated
and arbitrary as the Cartesian system, his final deductions are simple, necessary,
and consistent with facts.

Certain phenomena were to be explained. Rankine set himself to imagine
the mechanism by which they might be produced. Being an accomplished
engineer, he succeeded in specifying a particular arrangement of mechanism
competent to do the work, and also in predicting other properties of the
mechanism which were afterwards found to be consistent with observed facts.

As long as the training of the naturalist enables him to trace the action
only of particular material systems without giving him the power of dealing
with the general properties of all such systems, he must proceed by the method
so often described in histories of science — he must imagine model after model
of hypothetical apparatus till he finds one which will do the required work.
If this apparatus should afterwards be found capable of accounting for many
of the known phenomena, and not demonstrably inconsistent with any of them,

TAITS "THERMODYNAMICS. 663

he is strongly tempted to conclude that his hypothesis is a fact, at least
until an equally good rival hypothesis has been invented. Thus Rankine*,
long after an explanation of the properties of gases had been founded on the
theory of the collisions of molecules, published what he supposed to be a
proof that the phenomena of heat were invariably due to steady closed streams
of continuous fluid matter.

The scientific career of Rankine was marked by the gradual development
of a singular power of bringing the most difficult investigations within the range
of elementary methods. In his earlier papers, indeed, he appears as if battling
with chaos, as he swims, or sinks, or wades, or creeps, or flies,

" And through the palpable obscure finds out
His uncouth way ;"

but he soon begins to pave a broad and beaten way over the dark abyss, and
his latest writings shew such a power of bridging over the difficulties of science,
that his premature death must have been almost as great a loss to the diffusion
of science as it was to its advancement.

The chapter on thermodynamics in his book on the steam-engine was the
first published treatise on the subject, and is the only expression of his views
addressed directly to students.

In this book he has disencumbered himself to a great extent of the hypothesis
of molecular vortices, and builds principally on observed facts, though he, in
common with Clausius, makes several assumptions, some expressed as axioms,
others implied in definitions, which seem to us anything but self-evident. As
an example of Rankine's best style we may take the following definition :—

" A PERFECT GAS is a substance in such a condition that the total pressure exerted by any
number of portions of it, at a given temperature, against the sides of a vessel in which they
are enclosed, is the sum of the pressures which each portion would exert if enclosed in the vessel
separately at the same temperature."

Here we can form a distinct conception of every clause of the definition,
but when we come to Rankine's Second Law of Thermodynamics we find
that though, as to literary form, it seems cast in the same mould, its actual
meaning is inscrutable.

« "On the Second Law of Thermodynamics," Phil. Mag. Oct. 1865, § 12, p. 244; but in his
paper on the Thermal Energy of Molecular Vortices, Trans. E.S. Edin. xxv. p. 657 [1869], he
admits that the explanation of gaseous pressure by the impacts of molecules has been proved to
be possible.

TAIT'8 " THERMODYNAMICS."

*f n»rmadynamit». — If the total actual heat of a homogeneous and uniformly
1x4 mUUBM Iw aoactit^J *o b« divided into any number of equal parts, the effects of those
iwru in caving work to be performed are equal."

We find it difficult enough, even in 1878, to attach any distinct meaning
to the total actual heat of a body, and still more to conceive this heat
divided into equal parts, and to study the action of each of these parts ; but
•a if our powers of deglutition were not yet sufficiently strained, Hankino follows
this up with another statement of the same law, in which we have to assert
«>ur intuitive belief that —

••If the absolute temperature of any uniformly hot substance be divided into any number of
+^1 part*, the effect* of those parts in causing work to be performed are equal."

The student who thinks that he can form any idea of the meaning of this
sentence is quite capable of explaining on thermodynamical principles what
Mr Tennyson says of the great Duke : —

"Whose eighty winters freeze with one rebuke
All great self-seekers trampling on the right"

Prof. Clausius does not ask us to believe quite so much about the heat
in hot bodies. In his first memoir, indeed, he boldly dismisses one supposed
variety of heat from the science. Latent heat, he tells us, "is not only, as
its name imports, hidden from our perceptions, but has actually no existence;"
" it has been converted into work."

But though Clausius thus gets rid of all the heat which, after entering
a body, is expended in doing work, either exterior or interior, he allows a
certain quantity to remain in the body as heat, and this remnant of what
should have been utterly destroyed lives on in a sort of smouldering existence,
breaking out now and then with just enough vigour to mar the scientific coherence
of what might have been a well compacted system of thermodynamics.

Prof. Tait tells us :—

"The source of all this sort of speculation, which is as old as the time of Crawford and
Irvine — and which was countenanced to a certain extent even by Rankine — is the assumption that
bodies must contain a certain quantity of actual, or thermometric, heat. We are quite ignorant
of the condition of energy in bodies generally. We know how much goes in, and how much
comes out, and we know whether at entrance or exit it is in the form of heat or of work. But
that u all."

If we define thermodynamics, as I think we may now do, as the investi-
gation of the dynamical and thermal properties of bodies, deduced entirely

TAITS "THERMODYNAMICS." 665

from what are called the First and Second laws of Thermodynamics, without
anJ hypotheses as to the molecular constitution of bodies, all speculations as
to how much of the energy in a body is in the form of heat are quite out
of place.

Prof. Tait, however, does not seem to have noticed that Prof. Clausius, in
a footnote to his sixth memoir *, tells us what he means by the heat in a
body. In the middle of a sentence we read : —

" the heat actually present in a unit weight of the substance in question — in other words,

the vis viva of its molecular motions"

Thvis the doctrine that heat consists of the vis viva of molecular motions,
and that it does not include the potential energy of molecular configuration —
the most important doctrine, if true, in molecular science — is introduced in a
footnote under cover of the unpretending German abbreviation "d.h."

Prof. Clausius is himself the principal founder of the kinetic theory of
gases. The theory of the exchanges of the energy of collections of molecules
was afterwards developed by Boltzmann to a much greater extent than had
been done by Clausius, and it appears from his investigations that whether we
suppose the molecules to be acted on by forces towards fixed centres or not,
the condition of equilibrium of exchange of energy, or in other words the
condition of equality of temperature of two bodies, is that the average kinetic
energy of translation of a single molecule is the same in both bodies.

We may therefore define the temperature of a body as the average kinetic
energy of translation of one of its molecules multiplied into a constant which
is the same for all bodies. If we also define the total heat of the body as
the sum of the whole kinetic energy of its molecules, then the total heat
must be equal to the temperature multiplied into the number of molecules,
and by the ratio of the whole kinetic energy to the energy of translation,
and divided by the above constant.

The kinetic theory of gases has therefore a great deal to say about what
Rankine and Clausius call the actual heat of a body, and if we suppose that
molecules never coalesce or split up, but remain constant in number, then we
may also assert, all experiments notwithstanding, that the real capacity for
heat (as defined by Clausius) is constant for the same substance in all conditions.

* Hirst's translation, p. 230, German edition, 1864, p. 258, "wirklich vorhandene Warme, d.h.
die lebendige Kraft seiner Molecularbewegungen."

VOL. IL 84

" THKBMODVNAMIC8.

lUnkine, indeed, probably biassed by the results of experiments, allowed that
real specific heat of a substance might be different in different states of
Mtpth". kut Clausius has clearly shewn that this admission is illogical,
n.i tl»t if we admit any such changes, we had better give up real specific

heat altogether.

Statements of this kind have their legitimate place in molecular science,
where it is essential to specify the dynamical condition of the system, and to
distinguish the kinetic energy of the molecules from the potential energy of
their configuration; but they have no place in thermodynamics proper, in which
we deal only with sensible masses and their sensible motions.

Both Rankine and Clausius have pointed out the importance of a certain
function, the increase or diminution of which indicates whether heat is entering
or leaving the body. Rankine calls it the thermodynamic function, and Clausius
the entropy. Clausius, however, besides inventing the most convenient name
for this function, has made the most valuable developments of the idea of
entropy, and in particular has established the most important theorem in the
whole science, — that when heat passes from one body to another at a lower
temperature, there is always an increase of the sum of the entropy of the two
bodies, from which it follows that the entropy of the universe must always
be increasing.

He has also shewn that if the energy of a body is expressed as a
function of the volume and the entropy, then its pressure (with sign reversed)
and its temperature are the differential coefficients of the energy with respect
to the volume and the entropy respectively, thus indicating the symmetrical
relations of the five principal quantities in thermodynamics.

But Clausius, having begun by breaking up the energy of the body into
its thermal and ergonal content, has gone on to break up its entropy into
the transformational value of its thermal content and the disgregation.

Thus both the energy and the entropy, two quantities capable of direct
measurement, are broken up into four quantities, all of them quite beyond
the reach of experiment, and all this is owing to the actual heat which
Clausius, after getting rid of the latent heat, suffered to remain in the body.

Sir William Thomson, the last but not the least of the three great founders,
does not even consecrate a symbol to denote the entropy, but he was the
first to clearly define the intrinsic energy of a body, and to him alone are due
the ideas and the definitions of the available energy and the dissipation of

TAIT'S " THERMODYNAMICS." 667

energy. He has always been most careful to point out the exact extent of
the assumptions and experimental observations on which each of his statements
is based, and he avoids the introduction of quantities which are not capable
of experimental measurement. It is therefore greatly to be regretted that his
memoirs on the dynamical theory of heat have not been collected and reprinted
in an accessible form, and completed by a formal treatise, in which his method
of building up the science should be exhibited in the light of his present
knowledge.

The touchstone of a treatise on thermodynamics is what is called the second
law.

Rankine, as we have seen, founds it on statements which may or may not
be true, but which cannot be considered as established in the present state
of science.

The second law is introduced by Clausius and Thomson as an axiom on which
to found Carnot's theorem that the efficiency of a reversible engine is at least as
great as that of any other engine working between the same limits of temperature.

If an engine of greater efficiency exists, then, by coupling this engine with
Carnot's engine reversed, it is possible to restore to the hot body as much heat
as is taken from it, and at the same tune to do a certain amount of work.

If with Carnot we suppose heat to be a substance, then this work would
be performed in direct violation of the first law — the principle of the conser-
vation of energy. But if we regard heat as a form of energy, we cannot apply
this method of reductio ad absurdum, for the work may. be derived from the
heat taken from the colder body.

Clausius supposes all the work gained by the first engine to be expended
in driving the second. There is then no loss or gain of heat on the whole,
but heat is taken from the cold body, and an equal quantity communicated to
the hot body, and this process might be carried on to an indefinite extent.

In order to assert the impossibility of such a process in a form of words
having sufficient verisimilitude to be received as an axiom, Clausius, in his
first memoir, simply says that this process "contradicts the general deportment
of heat, which everywhere exhibits the tendency to equalize differences of
temperature, and therefore to pass from the warmer to the colder body *."

* Und das widerspricht dem sonstigen Verhalten der Warme, indem sie uberall das Bestreben
zeigt, vorkommende Temperaturdifferensen auszugleichen und also aus den warmeren Korpern in die
kalteren uberzugehen.

84—2

TAIT'S " THEHMODYXAMIC8. '

In iU obvious and «triot sense no axiom can be more irrefragable. Even
in the hypothetical process, the impossibility of which it was intended to
avert, erery communication of heat is from a warmer to a colder body.
When the beat is taken from the cold body it flows into the working substance
which is at that time still colder. The working substance afterwards becomes
hot, not by communication of heat to it, but by change of volume, and when
it communicates heat to the hot body it is itself still hotter.

It is therefore hardly correct to assert that heat has been transmitted or
transferred from the colder to the hotter body. There is undoubtedly a transfer
of energy, but in what form this energy existed during its middle passage is
a question for molecular science, not for pure thermodynamics.

In a note added in 1864 Clausius states the principle in a modified form,
" that heat cannot of itself pass from a colder to a warmer body " * and finally,
in the new edition of his Tlieory of Heat (1876) he substitutes for the words
"of itself" the expression "without compensation f."

With respect to the first of these emendations we must remember that
the words "of itself" are not intended to exclude the intervention of any kind
of self-acting machinery, and it is easy, by means of an engine which takes
in heat from a body at 200* C., and gives it out at 100° to drive a freezing
machine so as to take heat from water at 0°, and so freeze it, and also a
friction break so as to generate heat in a body at 500°. It would therefore
be necessary to exclude all bodies except the hot body, the cold body, and
the working substance, in order to exclude exceptions to the principle.

By the introduction of the second expression, " without compensation," com-
bined with a full interpretation of this phrase, the statement of the principle
becomes complete and exact ; but in order to understand it we must have a
previous knowledge of the theory of transformation-equivalents, or in other
words of entropy, and it is to be feared that we shall have to be taught
thermodynamics for several generations before we can expect beginners to receive
as axiomatic the theory of entropy.

Thomson, in his "Third Paper on the Dynamical Theory of Heat" (Trcuis.
R. S. Edin. xx. p. 265, read March 17, 1851), has stated the axiom as follows : —

1 DIM die Warrae nicht von selbst aus einem kalteren in einem warmeren Korper tiber-
gehen bum.

\ Ein Warmedbergang aus einem kalteren in einem warmeren Korper kann nicht ohne Com-
penmtion Suit finden.

TAITS "THERMODYNAMICS. 669

" It is impossible, by means of inanimate material agency, to derive mechanical effect from any
portion of matter by cooling it below the temperature of the coldest of surrounding objects."

Without some further restriction this axiom cannot be considered as true,
for by allowing air to expand we may derive mechanical effect from it by
cooling it below the temperature of the coldest of surrounding objects.

If we make it a condition that the material agency is to be left in the
same state at the end of the process as it was at first, and also that the
mechanical effect is not to be derived from the pressure of the hot or of the
cold body, the axiom will be rendered strictly true, but this brings us back
to a simple re-assertion of Carnot's principle, except that it is extended from
heat engines to all other kinds of inanimate material agency.

It is probably impossible to reduce the second law of thermodynamics to
a form as axiomatic as that of the first law, for we have reason to believe
that though true, its truth is not of the same order as that of the first law.

The first law is an extension to the theory of heat of the principle of
conservation of energy, which can be proved mathematically true if real bodies
consist of matter "as per definition," acted on by forces having potentials.

The second law relates to that kind of communication of energy which
we call the transfer of heat as distinguished from another kind of communication
of energy which we call work. According to the molecular theory the only
difference between these two kinds of communication of energy is that the
motions and displacements which are concerned in the communication of heat
are those of molecules, and are so numerous, so small individually, and so
irregular in their distribution, that they quite escape all our methods of obser-
vation ; whereas when the motions and displacements are those of visible bodies
consisting of great numbers of molecules moving altogether, the communication
of energy is called work.

Hence we have only to suppose our senses sharpened to such a degree
that we could trace the motions of molecules as easily as we now trace those
of large bodies, and the distinction between work and heat would vanish, for
the communication of heat would be seen to be a communication of energy of
the same kind as that which we call work.

The second law must either be founded on our actual experience in dealing
with real bodies of sensible magnitude, or else deduced from the molecular theory
of these bodies, on the hypothesis that the behaviour of bodies consisting of
millions of molecules may be deduced from the theory of the encounters of

TAIT'i " THBK1IODYNAMIC8."

•„ ^ mfrh^u,, by supposing the relative frequency of different kinds of
enotmntan to be distributed according to the laws of probability.

The troth of the second law is therefore a statistical, not a mathematical,
troth, for it depends on the feet that the bodies we deal with consist of
million* of molecules, and that we never can get hold of single molecules.

Sir Will**1" Thomson* has shewn how to calculate the probability of the
ooourreace within a given time of a given amount of deviation from the most
probable distribution of a finite number of molecules of two different kinds
in a vessel, and has given a numerical example of a particular case of the
diffusion of gases.

The same method might be extended to the diffusion of heat by conduction,
and the diffusion of motion by internal friction, which are also processes by
which energy is dissipated in consequence of the motions and encounters of
the molecules of the system.

The tendency of these motions and encounters is in general towards a
definite state, in which there is an equilibrium of exchanges of the molecules
and their momenta and energies between the different parts of the system.

If we restrict our attention to any one molecule of the system, we shall
find its motion changing at every encounter in a most irregular manner.

If we go on to consider a finite number of molecules, even if the system
to which they belong contains an infinite number, the average properties of this
group, though subject to smaller variations than those of a single molecule, are
rtill every now and then deviating very considerably from the theoretical mean
of the whole system, because the molecules which form the group do not submit
their procedure as individuals to the laws which prescribe the behaviour of
the average or mean molecule.

Hence the second law of thermodynamics is continually being violated, and
that to a considerable extent, in any sufficiently small group of molecules
belonging to a real body. As the number of molecules in the group is increased,
the deviations from the mean of the whole become smaller and less frequent;
and when the number is increased till the group includes a sensible portion
of the body, the probability of a measurable variation from the mean occurring
in a finite number of years becomes so small that it may be regarded as
practically an impossibility.

• "On the Kinetic Theory of the Dissipation of Energy," Proe. K.S. Edin., February 16, 1874,
Vol. via p. 323, ako iu Tafcwe, Vol. UL p. 441.

TAITS "THERMODYNAMICS. 671

This calculation belongs of course to molecular theory and not to pure
thermodynamics, but it shews that we have reason for believing the truth of
the second law to be of the nature of a strong probability, which, though it
falls short of certainty by less than any assignable quantity, is not an absolute
certainty.

Several attempts have been made to deduce the second law from purely
dynamical principles, such as Hamilton's principle, and without the introduction
of any element of probability. If we are right in what has been said above,
no deduction of this kind, however apparently satisfactory, can be a sufficient
explanation of the second law. Indeed some of them have already indicated
their unsoundness by leading to determinations of physical quantities which
have no existence, such as the periodic time of the alternations of the volume
of particular gases'".

* Szily, Phil. Mag., October, 1876 ; Clausius, Pogg. Ann. CXLII. p. 433 ; Fogg. Ann. CXLVI.
p. 585, May, 1872; J. J. Miiller, Pogg. Ann. CLII. p. 105.

[From tb« Proc*di*gt of tht London Mathematical Society, Vol. ix.]

X«'II On the Electrical Capacity of a long narrow Cylinder, and of a Disk of

sensible Thickness.

THK distribution of electricity in equilibrium on a straight line without
breadth is a uniform one. We may expect, therefore, that the distribution on
* cylinder will approximate to uniformity as the radius of the cylinder diminishes.

Let 2/ be the length of the cylinder, and 6 its radius.

Let x be measured along the axis from the middle point of the axis, and
let y be the distance of any point from the axis.

Let X be the linear density on the curved surface of the cylinder ; that
is, let Xrfor be the charge on the annular element dx.
Let <r be the surface-density on the flat ends.

Then, at a point on the axis for which £=x, the potential (i/») is

the first integral being extended over the curved surface, and the other two
over the positive and negative flat ends respectively.

If the electricity is in equilibrium in the cylinder, i/» must be constant
for all point* within the cylinder, and therefore for all points on the axis.

Also, by Art. 144 of Electricity and Magnetism, if </» is constant for all
point* on the axis, it is constant for all points within the surface of the
cylinder.

If we suppose X and o- constant,

ELECTRICAL CAPACITY OF A LONG NARROW CYLINDER, &C. 673

where / and /3 are the distances of the point (f ) on the axis from the positive
and negative edges of the curved surface.

At the middle of the axis,

£=0, and /,=/,=

1^0) = 2X log " + 47TO- (/- I).

At either end of the axis, £ = Z, / = &, /2 = 2/, nearly,

i/»w = X log y + 2-rra-b.

Just within the cylinder, when £ is just less than I,
d\jt /I l\

= ~ ~ + 27nr = ° by

Hence 2ir&cr = X,

or the density on the end must be equal to that on the curved surface.
The whole charge is therefore E=2rrba- (2l + b).

The greatest potential is i/»{0) = 2Tr6cr ( 2 log -,- + ^).

V b I]

The smallest potential is that at the curved edge, and is approximately

and the capacity must lie between

E 2l + b E

i~ = Xi — r and r~ = ri — ^ •

y/(0) i 21 U Y(e) « 4e 2

3 b 7 ° b TT

These are the limits between which Cavendish shews that the capacity
must lie. When the cylinder is very narrow, the upper limit is nearly double
the lower, so that we cannot obtain in this way any approximation to the
true value.

To obtain an approximation, we may make use of the following method,

in which we neglect the efiect of the flat ends, and consider the cylinder as
a hollow tube :—

VOL. II. 85

: : HJCCTRICAL CAPAOITT OF A LONO NARROW CYLINDER,

Lei Q be the potential enetfy of any arbitrary distribution of electricity

on the cylinder.

The charge m &'•

Let us now suppose this charge to distribute itself so as to pass into
the state of equilibrium ; then the potential will become uniform, and equal,
•ay, to *. and <?. = W.#

If AT is the capacity of the conductor,

£= and A'='

Since Q, the potential energy due to any arbitrary distribution of the
charge, may be greater, but cannot be less, than Q0, the energy of the same
charge in equilibrium, the capacity may be greater, but cannot be less, than

2Q

r

j -i

This inferior limit of the capacity is greater than that derived from the
maximum value of the potential, and, as we shall see, often gives a very close
approximation to the truth. Thus, if we suppose, in the case of the cylinder,
A to b» uniform,

where </"f«4/' + o>. For a long narrow cylinder,

I

To obtain a closer approximation, let us suppose the distribution to be of
any form, and to be expressed in the form of a series of harmonics.

The potential due to any such distribution at a given point may be
expressed in terms of spherical harmonics of the second kind. See
'•ical Harmonics, Chap. v.

AND OF A DISK OF SENSIBLE THICKNESS. 675

If we write «4^' H^P'

where r, and r2 are the distances of a given point from the ends of the line,
and if the linear density is expressed by

,, ,

w

where P< is the zonal harmonic of degree i, then the potential at the given

point (a, ft) is V = £4<&(«)pi08),

where Qt is the zonal harmonic of the second kind, and is of the form

where -R.-(a) is a rational function of a of (* — 1) degrees, and is such that
Q{ (a.) vanishes when a is infinite, thus :

5.7 3.5 t , 3\, a+1 5.7 , 5.11

«4 «2 J \ \f\rf ._ fi~ _L /I

-1 4 12

At a point at a very small distance & from the line, if we write

_L = log — — j +log— — ^ ,

the potential due to the distribution whose linear density is

is approximately
*

I

2.3.3 2.3.4.4

85—2

KUKTMCAL CAPACITY OF A LONG NARROW CYLINDER,
if . A.-A,

Sat 3

then approximately

^.7f 3.5? 3W 25

A

If we write | for kg ±5, or approxi.ately S=log£ we find, to

the same degree of approximation,

J

AND OF A DISK OF SENSIBLE THICKNESS. 677

Determining At so as to make I (X0 + Xj (^0 + ^2) dg a minimum, and re-
membering that E = ZlA<l, we find

A -A5
*'

101'

30.

and we obtain as a second approximation

I

36 g_101

Unless the length of the cylinder considerably exceeds 7 '245 times its
diameter, this approximation is of little use, for this ratio makes At infinite.
It shews, however, that when the ratio of the length to the diameter increases
without limit, the electric density becomes more nearly uniform, and the
expression for Ka approximates to the true capacity.

We may go on to a third approximation by determining _42 and A, so that
(X,, + A, + X4) (\fj0 + \ji, + </»4) dg shall be a minimum ; whence

g_ 3373

5 " f''™

A1 = A<>-

--

30 /\ 1260/ 196

,
A -A 9 21°

^,-^o

/ 101W 6989X 45'
\ " 30/V 1260/ 196

and

01 A_J ? V 210/

36«_101 400 / 101\r/o IglWo 6989\ 451

^- «/x I * f\f\ I I * AA I t ' io/»r\/ in/?l

_
30 \ 30 /\ 30 A 1260/ 196J

ST8 McnucAL OAPAcmr or A LONG NARROW CYLINDER,

When « b T«iy great, the distribution of electricity is expressed by the

X-/

which »hew« that, as the ratio of the length to the diameter increases, the
deMtty become* more nearly uniform, and the deviation from uniformity becomes
more confined to the parts near the ends of the cylinder.

To indicate the character of the approximation, I have calculated ? and
the three terms of the denominator of A'4 for different values of the ratio of

/.. When this ratio is less than 100, the third term is unavailable.

9 lit term. 2nd term. 3rd term.

10 3-68888 . 2-68888-0-43151

20 4-38203.3-38203-0-13680

30 478749 . 378749 - 0'09775

50 5-29832 . 4'29832-0'07191

100 5-99146. 4-99146-0-05291-0-13566

1000 8-29405 . 7'29405-0-02818-0'00892.

Examples of the application of the method to the calculation of the
capacities of a cylinder in presence of a plane conducting surface, and in
presence of another equal cylinder, will be given in the notes to the forth-
coming edition of Cavendish's Electrical Researclies, as illustrations of measure-
ments made by Cavendish in 1771.

Electric Capacity of a Disk of sensible Thickness.

We may apply the same method to determine the capacity of a disk of
radius a and thickness b, b being very small compared with a.

We may begin by supposing that the density on the flat surfaces is the
name as when the disk is infinitely thin.

Let a and ft be the elliptic co-ordinates of a given point with respect to
the lower disk, or in other words let the greatest and least distances of the
point from the edge of the disk be a (a +£) and a(a-yS).

AND OF A DISK OF SENSIBLE THICKNESS. G79

The distance of the given point from the axis is

r = aap (l),

and its distance from the plane of the lower disk is

z = a(a2-l)i(l-^)> (2).

If we write ajp2 = a2 — r2 (3);

then, if A2 is the charge of the upper disk, distributed as when undisturbed
by the lower disk, the density at any point is

A,

cr =

.(4).

If At is the charge of the lower disk, also undisturbed, the potential at
the given point due to it is

\l> = A1a~1 cosec"1 a .............................. (5),

or, if we write a2 = y2+l .................................... (6),

........................... (7).

We have next to find the relation between p and y when the given
point is in the upper disk, and therefore z = b.

Equation (2) becomes b" = ay (1 -/32) ................................. (8),

and p=l-a*p ....................................... (9),

Since the given point is on the upper disk, and since & is small, p must
be between 1 and 0, and y between- and (-) ; and between those limits we

Cb \ /

may write, as a sufficient approximation for our purpose,

We have now to find the value of the surface integral of the product of
the density into the potential taken over the upper disk, or

{

J

(12).

AL CAPACITY OF A U>NO NARROW CYLINDER, &C.

Substituting the value of UnMy from (11), the integral in (12) becomes

The corresponding quantity for the action of the upper disk on itself is
got by putting A^A, and 6 = 0, and is

In the actual case, A, = At-\Et when E is the whole charge; and the
lower limit of the capacity is therefore

but since we have assumed that b is very small compared with a, we may
express our result with sufficient accuracy in the form

or, the capacity of two equal disks is equal to that of a single disk whose
circumference exceeds that of either disk by b log y .

If the space between the disks is filled up, so as to form a single disk
uf sensible thickness, there will be a certain charge on the cylindric surface;
but, at the same time, the charge on the inner sides of the disks will vanish,
and that on the outer sides of the disks will be diminished, so that the capacity
of a disk of sensible thickness is very little greater than that given by (16).

[From the Philosophical Transactions of the Royal Society, Part I. 1879.]

XCIII. On Stresses in Rarified Gases arising from Inequalities of Temperature.

1. IN this paper I have followed the method given in my paper " On
the Dynamical Theory of Gases" (Phil. Trans., 1867, p. 49). I have shewn
that when inequalities of temperature exist in a gas, the pressure at a given
point is not the same in all directions, and that the difference between the
maximum and the minimum pressure at a point may be of considerable magnitude
when the density of the gas is small enough, and when the inequalities of
temperature are produced by small* solid bodies at a higher or lower temperature
than the vessel containing the gas.

2. The nature of this stress may be thus defined : — Let the distance from
a given point, measured in a given direction, be denoted by h ; then the space-

* The dimensions of the bodies must be of the same order of magnitude as a certain length
X, which may be defined as the distance travelled by a molecule with its mean velocity during
the time of relaxation of the medium.

The time of relaxation is the time in which inequalities of stress would disappear if the rate
at which they diminish were to continue constant. Hence

\*7>/ P

On the hypothesis that the encounters between the molecules resemble those between " rigid
elastic" spheres, the free path of a molecule between two successive encounters has a definite
meaning, and if I is its mean value,

; 3 /7r\i Sir ,ft

f=oMo — ) = -Q-A=lll8A.

2 r \2ppJ

So that the mean path of a molecule may be taken as representing what we mean by "small".

If the force between the molecules is supposed to be a continuous function of the distance,

the free path of a molecule has no longer a definite meaning, and we must fall back on the
quantity X, as defined above.

VOL. II. 86

8TR080 IN RAIUFIEO GASES

-

variation of the temperature for a point moving along this line will be denoted
U ?, and the ipeia whtinn of this quantity along the same line by

There will in general, be a particular direction of the line h for which

u a maximum, another for which it is a minimum, and a third for which it is
a maximum-minimunx These three directions are at right angles to each other,
and are the axes of principal stress at the given point; and the part of the
•trees arising from inequalities of temperature is, in each of these principal axes,

tl* (fv

pO <//<' '

where it. is the coefficient of viscosity, p the density, and 6 the absolute
temperature.

3. Now for dry air at 15'C., /t=r9xlO~4 in centimetre-gramme-second

measure, and *'*,= 0'315, where p is the pressure, the unit of pressure being
p6 p

one dyne per square centimetre, or nearly one-millionth part of an atmosphere.

If a sphere of 2a centimetres in diameter is T degrees centigrade hotter
than the air at large distances from it, then, when there is a steady flow
of heat, the temperature at a distance of r centimetres from the centre will be

Ta , d*0 2Ta
e=0.+ - , and-T3=— j-.
r dir r

Ht-nce, at a distance of r centimetres from the centre of the sphere, the
pressure in the direction of the radius arising from inequality of temperature
will be

Ta

—^ 0'63 dynes per square centimetre.

4. In Mr Crookes' experiments the pressure, p, was often so small that
thi« stress would be capable, if it existed alone, of producing rapid motion
in a radiometer.

Indeed, if we were to consider only the normal part of the stress exerted
on solid bodies immersed in the gas, most of the phenomena observed by
Mr Crookes could be readily explained.

5. Let us take the case of two small bodies symmetrical with respect
to the axis joining their centres of figure. If both bodies are wanner than

ARISING FROM INEQUALITIES OP TEMPERATURE. 683

the air at a distance from them, then, in any section perpendicular to the
axis joining their centres, the point where it cuts this line will have the
highest temperature, and there will be a flow of heat outwards from this
axis in all directions.

d'O
Hence ^ will be positive for the axis, and it will be a line of maximum

pressure, so that the bodies will repel each other.

If both bodies are colder than the air at a distance, everything will be
reversed; the axis will be a line of minimum pressure, and the bodies will
attract each other.

If one body is hotter and the other colder than the air at a distance,
the effect will be smaller, and it will depend on the relative sizes of the
bodies, and on their exact temperatures, whether the action is attractive or
repulsive.

6. If the bodies are two parallel disks very near to each other, the
central parts will produce very little effect, because between the disks the

d~9
temperature varies uniformly, and -^ = 0. Only near the edges will there be

any stress arising from inequality of temperature in the gas.

7. If the bodies are encircled by a ring having its axis in the line
joining the bodies, then the repulsion between the two bodies, when they are
warmer than the air in general, may be converted into attraction by heating
the ring so as to produce a flow of heat inwards towards the axis.

8. If a body in the form of a cup or bowl is warmer than the air, the
distribution of temperature in the surrounding gas is similar to the distribution
of electric potential near a body of the same form, which has been investigated

d~0
by Sir W. Thomson. Near the convex surface the value of -^ is nearly the

same as if the body had been a complete sphere, namely 2T— , where T is

Cv

the excess of temperature, and a is the radius of the sphere. Near the
concave surface the variation of temperature is exceedingly small.

Hence the normal pressure will be greater on the convex surface than
on the concave surface, and if we were to neglect the tangential pressures we
might think this an explanation of the motion of Mr Crookes' cups.

86—2

IN KAUIFIED OASES

Since the expNMOM for the stress are linear as regards the temperature,
everything will be reversed when the cup is colder than the surrounding air.

9. In a spherical vessel, if the two polar regions are made hotter than the
equatorial wroe, the pressure in the direction of the axis will be greater than
that parallel to the equatorial plane, and the reverse will be the case if the
polar regions are made colder than the equatorial zone.

10. All such explanations of the observed phenomena must be subjected
to careful criticism. They have been obtained by considering the normal

alone, to the exclusion of the tangential stresses, and it is much
to give an elementary exposition of the former than of the latter.
If, however, we go on to calculate the forces acting on any portion of the
gat in virtue of the stresses on its surface, we find that when the flow
of heat is steady, these forces are in equilibrium. Mr Crookes tells us that
there is no molar current or wind in his radiometer vessels. It is not easy
to prove this by experiment, but it is satisfactory to find that the system
of stresses here described as arising from inequalities of temperature will not,
when the flow of heat is steady, generate currents.

11. Consider, then, the case in which there are no currents of gas but
a steady flow of heat, the condition of which is

(In the absence of external forces such as gravity, and if the gas in contact
with solid bodies does not slide over them, this is always a solution of the
equations, and it is the only permanent solution.) In this case the equations
of motion shew that every particle of the gas is in equilibrium under the stresses
acting on it. Hence, any finite portion of the gas is also in equilibrium ;
also, since the stresses are linear functions of the temperature, if we superpose
one system of temperatures on another, we also superpose the corresponding
systems of forces.

Now the system of temperatures due to a solid sphere of uniform tem-
perature immersed in the gas, cannot of itself give rise to any force tending
to move the sphere in one direction rather than in another. Let the sphere

ARISING FROM INEQUALITIES OF TEMPERATURE. 685

be placed within the finite portion of gas which, as we have said, is already
in equilibrium. The equilibrium will not be disturbed. We may introduce
any number of spheres at different temperatures into the portion of gas, so as
to form a body of any shape, heated in any manner, and when the flow of
heat has become steady the whole system will be in equilibrium.

12. How, then, are we to account for the observed fact that forces act
between solid bodies immersed in rarified gases, and this, apparently, as long
as inequalities of temperature are maintained ?

I think we must look for an explanation in the phenomenon discovered in
the case of liquids by Helmholtz and Piotrowski *, and for gases by Kundt and
Warburg t, that the fluid in contact with the surface of a solid must slide
over it with a finite velocity in order to produce a finite tangential stress.

The theoretical treatment of the boundary conditions between a gas and
a solid is difficult, and it becomes more difficult if we consider that the gas
close to the surface is probably in an unknown state of condensation. We
shall therefore accept the results obtained by Kundt and Warburg on their
experimental evidence.

They have found that the velocity of sliding of the gas over the surface
due to a given tangential stress varies inversely as the pressure.

Q

The coefficient of sliding for air on glass was found to be G = -

centimetres, where p is the pressure in millionths of an atmosphere. Hence
at ordinary pressures G is insensible, but in the vessels exhausted by Mr Crookes
it may be considerable.

Hence, if close to the surface of a solid there is a tangential stress S,
acting on a surface parallel to that of the body in a direction h parallel
to that surface, there will also be a sliding of the gas in contact with the

solid over its surface in the direction h with a finite velocity =--.

r*

13. I have not attempted to enter on the calculation of the effect of
this sliding motion, but it is easy to see that if we begin with the case in
which there is no sliding, the instantaneous effect of permission being given

* Wiener Sitzb., xl. 1860, p. 607.
t Pogg. Ann., civ. 1875, p. 337.

RARIFIED OA8BS

U> the g*« to slide must be to diminish the action of all tangential stresses
oa the surface, without affecting the normal stresses, and in course of time
to wt up current* MMpi* «wr the surfaces of solid bodies, thus completely
<WtrorinjT the simplicity of oar first solution of the problem.

14. Wbja «ternal forces, such as gravity, act on the gas, and when the
thermal phenomena produce differences of density in different parts of the vessel,
flp» the well-known convection currents are set up. These also interfere with
the simplicity of the problem and introduce very complicated effects. All that
we know is that the rarer the gas and the smaller the vessel the less is
the effect of the convection currents, so that in Mr Crookes' experiments
they play a very small part

We now proceed to the calculations : —

(1) Encounter between Two Molecules.

The motion of the two molecules after an encounter depends on their
motion before the encounter, and is capable of being determined by purely
dynamical methods. If the encounter of the molecules does not cause rotation

• >r vibration in the individual molecules, then the kinetic energy of the centres
of "H"P« of the two molecules must be the same after the encounter as it was
before.

This will be true on the average, even if the molecules are complex
systems capable of rotation and internal vibration, provided the temperature is
constant. If, however, the temperature is rising, the internal energy of the
molecules is, on the whole, increasing, and therefore the energy of translation

• it" their centres of mass must be, on an average, diminishing at every encounter.
The reverse will be the case if the temperature is falling.

But however important this consideration may be in the theory of specific
heat and that of the conduction of heat, it has only a secondary bearing on
the question of the stresses in the medium ; and as it would introduce great
complexity and much guesswork into our calculations, I shall suppose that the
jzas here considered is one the molecules of which do not take up any sensible
amount of energy in the form of internal motion. Kundt and Warburg*
have shewn that this is the case with mercury gas.

* Pogg. Ann., clvii. 1876, p. 353.

ARISING FEOM INEQUALITIES OF TEMPERATURE. 687

Let the masses of the molecule be Ml and M3, and their velocity-
components £, fy, £u and £,, -r),, 4 respectively. Let V be the velocity of
Mt relative to Mr

Before the encounter let a straight line be drawn through M1 parallel to
V, and let a perpendicular 6 be drawn from J/a to this line. The magnitude
and direction of b and V will be constant as long as the motion is undisturbed.

During the encounter the two molecules act on each other. If the force
acts in the line joining their centres of mass, the product bV will remain
constant, and if the force is a function of the distance, V and therefore b
will be of the same magnitude after the encounter as before it, but their
directions will be turned in the plane of V and b through an angle 20, this
angle being a function of b and V, which vanishes for values of b greater
than the limit of molecular action. Let the plane through V and b make an
angle <f> with the plane through V parallel to x, then all values of (j> are
equally probable.

If £,' be the value of £, after the encounter,

M
When the two molecules are of the same kind, ,, V> = \, and in the

present investigation of a single gas we shall assume this to be the case.

If we use the symbol 8 to indicate the increment of any quantity due
to an encounter, and if we remember that all values- of <j> are equally probable,
so that the average value of cos <f> and of cos3 $ is zero, and that of cos5 (f>
is £, we find

§(£+£) =o ................. . ......................... .................. (2)

(3)
............. (4).

From these by transformation of coordinates we find

l)sm'e<x>B>0- ......................... (5)

- 3 (£tf + £tf)
7^ +F*)] sin' 0 cos' 0 ............... (6)

3 (M + &7,t) = - i [9 (M + &?,Q - 3(frfc& + M + f fl£ + &7,&

£.)] an1 6 cos2 0 ...... ...(7).

IS RAIUFIED OASES

[Application °f Spherical Harmonics to the Theory of Gases.

If we suppose the direction of the velocity of M, relative to Mt to be
d by the position of a point P on a sphere, which we may call the
of reference, then the direction of the relative velocity after the encounter
will be indicated by a point /x, the angular distance PP being 20, so that
the point I" lie* in a small circle, every position in which is equally probable.

We hare to calculate the effect of an encounter upon certain functions of
the six velocity-components of the two molecules. These six quantities may
be expressed in terms of the three velocity-components of the centre of mass
of the two molecules (say v, v, w), the relative velocity of Jf, with respect
to J/, which we call V, and the two angular coordinates which indicate the
direction of V. During the encounter, the quantities u, v, w and V remain the
Mine, but the angular coordinates are altered from those of P to those of P on
the sphere of reference.

Whatever be the form of the function of £„ 17, , £,, &, rj,, £,, we may
consider it expressed in the form of a series of spherical harmonics of the
angular coordinates, their coefficients being functions of u, v, w, V, and we
have only to determine the effect of the encounter upon the value of the
spherical harmonics, for their coefficients are not changed.

Let }'<-) be the value at P of the surface harmonic of order n in the
series considered.

After the encounter, the corresponding term becomes what I^"' becomes
at the point P, and since all positions of P in a circle whose centre is P
are equally probable, the mean value of the function after the encounter must
depend on the mean value of the spherical harmonic in this circle.

Now the mean value of a spherical harmonic of order n in a circle, the
cosine of whose radius is ft, is equal to the value of the harmonic at the pole of
the circle multiplied by P"> (p.), the zonal harmonic of order n, and amplitude p..

Hence, after the encounter, 7<" becomes P»>P<"> (^), and if Fn is the
corresponding part of the function to be considered, and SFn the increment of
Fm arising from the encounter, oFu = Fn(PM (/i)-l).

This is the mean increment of Fn arising from an encounter in which
cos 2$ = /t. The rate of increment is to be found from this by multiplying it

ARISING FROM INEQUALITIES OF TEMPERATURE. 689

by the number of encounters of each molecule per second in which p. lies
between p, and p. + dp., and integrating for all values of p. from — 1 to + 1 .

This operation requires, in general, a knowledge of the law of force be-
tween the molecules, and also a knowledge of the distribution of velocity
among the molecules.

When, as in the present investigation, we suppose both the molecules to
be of the same kind, and take both molecules into account in the final
summation, the spherical harmonics of odd orders will disappear, so that if we
restrict our calculations to functions of not more than three dimensions, the
effect of the encounters will depend on harmonics of the second order only, in
which case I*®(p.)- I =f (/t2- 1) = -f sin220.— Note added May, 1879.]

(2) Number of Encounters in Unit of Time.

We now abandon the dynamical method and adopt the statistical method.
Instead of tracing the path of a single molecule and determining the effects
of each encounter on its velocity-components and their combinations, we fix
our attention on a particular element of volume, and trace the changes in
the average values of such combinations of components for all the molecules
which at a given instant happen to be within it. The problem which now
presents itself may be stated thus : to determine the distribution of velocities
among the molecules of any element of the medium, the current-velocity and
the temperature of the medium being given in terms of the coordinates and
the time. The only case in which this problem has been actually solved is
that in which the medium has attained to its ultimate state, in which the
temperature is uniform and there are no currents.

Denoting by

dN =/, (£ r), £ x, y, z, t) d£di)dt,dxdydz

the number of molecules of the kind Ml which at a given instant are within
the element of volume dxdydz, and whose velocity-components lie between the
limits g±$d£, "n±:kd'n> £±^£> Boltzmann has shewn that the function / must
satisfy the equation

0 ............ (8)

VOL. II. 87

IN RARIFIED OASES

f f, K denote what / becomes when in place of the velocity-com-

gf jf before the encounter we put those of M, before the encounter,

nd those of Jf, and M, after the encounter, respectively, and the integration
M fflrtflmfrH to all values of 4> and b and of £, 17,, £,, the velocity-com-
ponents of the second molecule J/,.

It M impossible, in general, to perform this integration without a know-
ledge, not only of the law of force between the molecules, but of the form
of the functions /, /„ /T, /.'. which have themselves to be found by means of
the equation.

It is only for particular cases, therefore, that the equation has hitherto
been solved.

If the medium is surrounded by a surface through which no communica-
tion of energy can take place, then one solution of the equation is given by
the conditions

y dz

which (rive

ftmAfr****»*«* (9)

where ^, is the potential of the force whose components are Xlt Ylt Z1} and
At is a constant which may be different for each kind of molecules in the
medium,, but h is the same for all kinds of molecules.

This is the complete solution of this problem, and is independent of any
hypothesis as to the manner in which the molecules act on each other during
an encounter. The quantity h which occurs in this expression may be deter-
mined by finding the mean value of f , which is _, . Now in the kinetic

theory of gases,

Pe=p = RP8 (10)

where p is the pressure, /> the density, 6 the absolute temperature, and R a
constant for a given gas. Hence

<»)•

ARISING FROM INEQUALITIES OF TEMPERATURE. 691

We shall suppose, however, with Boltzmann, that in a medium in which there
are inequalities of temperature and of velocity

where F is a rational function of £ 77, £, which we shall suppose not to
contain terms of more than three dimensions, and /„ is the same function as
in equation (9).

Now consider two groups of molecules, each defined by the velocity-
components, and let the two groups be distinguished by the suffixes (,) and
(,). We have to estimate the number of encounters of a given kind between
these two groups in a unit of volume in the time St, those encounters only
being considered for which the limits of b and <£ are b±^db and (j>±^d<j).

Let us first suppose that both groups consist of mere geometrical points
which do not interfere with each other's motion. The group d^ is moving
through the group dNt with the relative velocity V, and we have to find
how many molecules of the first group approach a molecule of the second group
in a manner which would, if the molecules acted on each other, produce an
encounter of the given kind. This will be the case for every molecule of the
first group which passes through the area bdbd<j> in the time St. The number
of such molecules is d^Vbdbd^St for every molecule of the second group,
BO that the whole number of pairs which pass each other within the given
limits is

and if we take the time St small enough, this will be the number of encounters
of the real molecules in the time St.

(3) Effect of the Encounters.

We have next to estimate the effect of these encounters on the average
values of different functions of the velocity-components. The effect of an indi-
vidual encounter on these functions for the pair of molecules concerned is
given in equations (3), (4), (5), (6), (7), each of which is of the form

SP = Q sin*0 cos5 6 (13)

where P and Q are functions of the velocity-components of the two molecules,

87—2

IN RARIFIED OASES

and if we writ* /* for the average value of P for the N molecules in unit
of volume, then Uking the sum of the eflecte of the encounters—

(14).

Wa thn* find

-(is).

Now, «noe 6 i« a function of 6 and V, the definite integral

B ........................ (16)

will be a function of V only.

If the molecules are " rigid-elastic " spheres of diameter s,

(17).

If they repel each other with a force inversely as the fifth power of the
distance, so that at a distance r the force is *cr'§, then

I. (18)

where At is the numerical quantity T3682. In this case B is independent of V.

The experiments of O. E. Meyer*, Kundt and Warburg t, PulujJ, Von
Obermayer§, Eilhard Wiedemann|[, and HolmanI, shew that the viscosity of
air varies according to a lower power of the absolute temperature than the
first, probably the 077 power. If the viscosity had varied as the first power
of the 'absolute temperature, B would have been independent of V. Though
this is not the case, we shall assume, for the sake of being able to effect the
integrations, that B is independent of F.

We shall find it convenient to write for B,

where p is the hydrostatic pressure, N the number of molecules in unit of

• Pogg. Ann., 1873, Bd. 148, p. 222. t Ibid. 1876, Bd. 159, p. 403.

: Witner Sift., 1874 and 1876. § Ibid. 1875.

| Ank. det Sei Phyt. at Nat., 1876, t. 56, p. 273.

^ American Academy of Arto and Science*, June 14, 1876. Phil. May., a. 5, vol. 3, No. K>,
F«h., 1877.

ARISING FROM INEQUALITIES OF TEMPERATURE. 693

volume, and p a new coefficient which we shall afterwards find to be the
coefficient of viscosity.

Equation (15) may now be written

(20)

where the integrations are all between the limits — oo and + <x> , and ft and /,
are of the form

...................... (21)

(£> V' £) being small compared with unity.
We may write F in the form

F= (2A)» (o

(22)

where each combination of the symbols afty is to be taken as a single in-
dependent symbol, and not as a product of the component symbols.

(4) Mean Values of Combinations of £ 77, £.

To find the mean value of any function of f, 77, £ for all the molecules
in the element, we must multiply this function by f, and integrate with respect
to f, r), and £.

If the non-exponential factor of any term contains an odd power of any
of the variables, the corresponding part of the integral will vanish, but if
'it contains only even powers, each even power, such as In, will introduce a

into the corresponding part of the integral.
First, let the function be 1, then

1 = Ifl&ndt (23)

IN UAKIFIED GASES

.(24)

which giro the condition

a«4.0« + y» = 0 (25).

Let m n«t find the mean value of ( in the same way, denoting the result
by the symbol £

Since in what follows we shall denote the velocity-components of each
molecule by v + f, v+q, to+{, where u, v, w are the velocity-components of
the centre of mass of all the molecules within the element, it follows that the
values of £ 17, C are each of them zero. We thus obtain the equations

.(27).

Remembering these conditions, we find that the mean values of combina-
tions of two, three, and four dimensions are of the forms

= 7?0(l + a')l

.(28)

.(29)

2a')

.(30).

(5) Rates of Decay of these Mean Values.

If any term of Q in equation (20) contains symbols belonging to one
group alone of the molecules, the corresponding term of the integral may be
f«-und from the above table, but if it contains symbols belonging to both

ARISING FROM INEQUALITIES OF TEMPERATURE. 695

groups we must consider the sextuple integral (20). But we shall not find

it necessary to do this for terms of not more than three dimensions, for in

these, if both groups of symbols occur, the index of one of them must be
odd, and the integral vanishes.

We thus find from equations (3), (4), (5), (6), and (7)

St

St

S Ip ,

5-. « =•= - ( - 2a" + a/?- + ay") (33)

ot 2 a

St 6 it

8

St^r- ~2^r (35>-

[Any rational homogeneous function of £ 77 £ is either a solid harmonic, or
a solid harmonic multiplied by a positive integral power of (f + ^ + C2), or may
be expressed as the sum of a number of terms of these forms.

If we express any one of these terms as a function of u, v, iv, V and
the angular coordinates of V, we can determine the rate of change of each of
the spherical harmonics of the angular coordinates.

If we then transform the expression back to its original form as a function
of f» ~n\, £i> £> *?2> 4. and if we add the corresponding functions for both
molecules, we shall obtain an expression for the rate of change of the original
function.

Thus among the terms of two dimensions we have the five conjugate solid
harmonics

IS RARIFIED GASES

The rmte of inowwe of each of these arising from the encounters of the
i. found by multiplying it by -£. We may therefore call £ the
"moduli* of the time of relaxation" of this class of functions.

The function f +V + C1 is not changed by the encounters.

Homogeneous functions of three dimensions are either solid harmonics of
the thiid outer or solid harmonics of the first order multiplied by f + V + £',
or combinations of these.

o .

The time modulus for solid harmonics of the third order is ^-. — Note
•dded May, 1879.]

That of f 17, or (, multiplied by f + if + C is f*.

(6) Effect of External Forces.
The only effect of external forces is expressed by equations of the form

The average values of £, 17, £ and their combinations are not affected by
external forces.

(7) Variation of Mean Values within an Element of Volume.

We have employed the symbol 8 to denote the variation of any quantity
within an element, arising either from encounters between molecules or from
the action of external forces.

There is a third way, however, in which a variation may occur, namely,
by molecules entering the element or leaving it, carrying their properties
with them.

We shall use the symbol 3 to denote the actual variation within a specified
element.

It' \IQ is the average value of any quantity for each molecule within the
element, then the quantity in unit of volume is pQ. We have to trace the
variation of pQ.

ARISING FROM INEQUALITIES OF TEMPERATURE. 697

We begin with an element of volume moving with the velocity-com-

ponents U, V, Wt then by the ordinary investigation of the "equation of
continuity "

If after performing the differentiations we make U=u, V=v, W—w, the
equation becomes for an element moving with the velocity (u, v, w)

u dv dw\ d , ~. ,., d , ~ . d , ,.,.,. S ._

(8) Equation of Density.

Let us first make Q=l, then, since the mass of a molecule is invariable,
the equation becomes

d£ + p(d^ + ^ + <^\ = Q . ,..(39)

ri/ V ft or Citi dz /

which is the ordinary "equation of continuity."

Eliminating by means of this equation the second term of the general
equation (38) we obtain the more convenient form —

>P% («>).

(9) Equations of Motion.
Putting Q = u + g, this equation becomes

where any combination of the symbols £, 17, £ is to be taken as the average
value of that combination.

Substituting their values as given in (28)

which is one of the three ordinary equations of motion of a medium in which
stresses exist.

VOL. u. 88

RAUIFIED OASES

(10) Ttrmt of Two Dimensions.

Put Qm(u + f)\ Since the resulting equation is true whatever be the values
of H. r, v, we may, after differentiation, put each of these quantities equal to
•Nk We ahall thus obtain the same result which we might have obtained
by elimination between this and the former equations. We find

or by •ubatituting the mean values of these quantities from (29)

I a

with two other equations of similar form.
Similarly we obtain by putting Q = (i

with two other equations of like form for /?y and

(11) Terms of Three Dimensions.

Putting @ =(« + £)* and in the final equation making u = v = iv = Q and
eliminating - by (41) we find

ARISING FROM INEQUALITIES OF TEMPERATURE.

699

which gives

Since the combinations of a/Jy represent small numerical quantities, we
may at this stage of the calculation, when we are dealing with terms of the
third order, neglect terms involving them, except when they are multiplied
by the large coefficient pfp-. The equation may then be written approxi-
mately : —

fl .................. (48).

Similarly, by putting Q = (u + £) (v + rff, we obtain the approximate equation

r] ft R 1 IY)

r>2 /) lx/l/ / T>/)\S jr / 3 o«A32 t «-.J\ />1Q\

/v pf -j— = p (Jilv) — — \a. — octp ~r ay ) ^4i/ j}

djX D fJi

and in the same way we find

j n i ~*

(50).

(12) Approximate Values of Terms of Three Dimensions.
From equations (48), (49), and (50), we find

™ e* \ f\ I T^ ) **/-* "•/

2p\v / dx
From which by substitution we obtain

9u./R\*dd ,s
dy'

2p\0/ dx

.(51).

The value of a/3y is of a smaller order of magnitude, and we do not
require it in this investigation.

88—2

, STRESSES IK RAKUTKD OASES

(13) Equation of Temperature.

Adding the three equations of the form (44), and omitting terms con-
uining Bff*" quantities of two dimensions, and also products of differential

such as /* , we find
ax cij"

, .

The first term of the second member represents the rate of increase of
temperature due to conduction of heat, as in Fourier's Theory, and the second
term represents the increase of temperature due to increase of density. We
must remember that the gas here considered is one for which the ratio of
the specific beats is 1'6.

(14) Stresses in the Gas.

Subtracting one-third of the sum of the three equations from (44), we
obtain

2 fdu dv dw\

+ ^) W

This equation gives the excess of the normal pressure in x above the
mean hydrostatic pressure p. The first two terms of the second member
represent the effect of viscosity in a moving fluid, and are identical with those

u by Professor Stokes (Cambridge Transactions, Vol. vin., 1845, p. 297).
The last two terms represent the part of the stress which arises from inequality
of temperature, which is the special subject of this paper.

There are two other equations of similar form for the normal stresses in
y and :.

The tangential stress in the plane xy is given by the equation

fdu dv\ . „ u5 d* 0

Fhero are two other equations of similar form for the tangential stresses
in the planes of yz and zx.

ARISING FROM INEQUALITIES OF TEMPERATURE. 701

(15) Final Equations of Motion.

We are now prepared to complete the equations of motion by inserting in
(42) the values of the quantities a\ a/3, ay, and we find for the equation in x

du dj^ fd?u d'u d*u\ 1 d Idu dv dw\
9 u? d fd'd d-0 d

If we write

1 Idu dv dw 9 0*6 d*9 d'

Updt"

or, if the pressure p is constant, so that pd0 + 0dp =

10 u. W

then the equation (55) may be written

du dp' (d*u d'u d*u

-

If there are no external forces such as gravity, then one solution of the
equations is

u = v = w = Q, p' = constant,

and if the boundary conditions are such that this solution is consistent with
them, it will become the actual solution as soon as the initial motions, if any
exist, have subsided. This will be the case if no slipping is possible between
the gas and solid bodies in contact with it.

But if such slipping is possible, then wherever in the above solution there
is a tangential stress in the gas at the surface of a solid or liquid, there
cannot be equilibrium, but the gas will begin to slide over the surface till
the velocity of sliding has produced a frictional resistance equal and opposite
to the tangential stress. When this is the case the motion may become steady.

70S

IN RARIFIED GASES

not, however, attempted to enter into the calculation of the state of
•Ceady motion.

[I have recently applied the method of spherical harmonics, as described
in the note to auctions (l) and (5), to carrying the approximations two
onkt* higher. I expected that this would have involved the calculation of
two new quantities, namely, the rates of decay of spherical harmonics of the
fourth and sixth orders, but I found that, to the order of approximation
required, all harmonics of the fourth and sixth orders may be neglected, so
that the rate of decay of harmonics of the second order, the time-modulus
of which is /* + /', determines the rate of decay of all functions of less than
6 dimensions.

The equations of motion, as here given (equation 55) contain the second
tlwivatives of v, v, w, with respect to the coordinates, with the coefficient p..
I find tliat in the more approximate expression there is a term containing
the fourth derivatives of u, v, w, with the coefficient p-' + pp.

The equations of motion also contain the third derivatives of 0 with the
coefficient p? + p9, Besides these terms, there is another set consisting of the
fifth derivatives of 6 with the coefficient p.* -=-

It appears from the investigation that the condition of the successful use
of this method of approximation is that Z-j, should be small, where -^ denotes

differentiation with respect to a line drawn in any direction. In other words,
the properties of the medium must not be sensibly different at points within
a distance of each other, comparable with the "mean free path" of a molecule.
—Note added June, 1879.]

ARISING FROM INEQUALITIES OF TEMPERATURE. 703

APPENDIX.

(Added May, 1879.)

In the paper as sent in to the Royal Society, I made no attempt to
express the conditions which must be satisfied by a gas in contact with a
solid body, for I thought it very unlikely that any equations I could write
down would be a satisfactory representation of the actual conditions, especially
as it is almost certain that the stratum of gas nearest to a solid body is in
a very different condition from the rest of the gas.

One of the referees, however, pointed out that it was desirable to make
the attempt, and indicated several hypothetical forms of surfaces which might
be tried. I have therefore added the following calculations, which are carried
to the same degree of approximation as those for the interior of the gas.

It will be seen that the equations I have arrived at express both the
fact that the gas may slide over the surface with a finite velocity, the
previous investigations of which have been already mentioned;""" and the fact
that this velocity and the corresponding tangential stress are affected by
inequalities of temperature at the surface of the solid, which give rise to a
force tending to' make the gas slide along the surface from colder to hotter
places.

This phenomenon, to which Professor Osborne Reynolds has given the
name of Thermal Transpiration, was discovered entirely by him. He was the
first to point out that a phenomenon of this kind was a necessary consequence
of the Kinetic Theory of Gases, and he also subjected certain actual phenomena,
of a somewhat different kind, indeed, to measurement, and reduced his measure-
ments by a method admirably adapted to throw light on the relations between
gases and solids.

It was not till after I had read Professor Reynolds' paper that I began
to reconsider the surface conditions of a gas, so that what I have done is
simply to extend to the surface phenomena the method which I think most
suitable for treating the interior of the gas. I think that this method is, in

* Sect. 12 of introductiou.

IN RAIUFIED OASES

•one •••Ml it i batter than that adopted by Professor Reynolds, while I admit
I hi* method fe efficient to establish the existence of the phenomena, though
not to afford an estimate of their amount.

Tbe method which I have adopted throughout is a purely statistical one.
It eonaider* the mean values of certain functions of the velocities within a
givw element of the medium, but it never attempts to trace the motion of a
molecule, not even so for as to estimate the length of its mean path. Hence
•11 the * equations are expressed in the forms of the differential calculus, in
which the phenomena at a given place are connected with the space variations
of certain quantities at that place, but in which no quantity appears which
explicitly involves the condition of things at a finite distance from that place.

The particular functions of the velocities which are here considered are
thoM of one, two, and three dimensions. These are sufficient to determine
approximately the principal phenomena in a gas which is not very highly
rarined, and in which the space-variations within distances comparable to X are

not very great

The same method, however, can be extended to functions of higher degrees,
and by a sufficient number of such functions any distribution of velocities,
however abnormal, may be expressed. The labour of such an approximation is
considerably diminished by the use of the method of spherical harmonics as
indicated in the note to Section I. of the paper.

On the Conditions to be Satisfied by a Gas at the Surface of a Solid Body.

As a first hypothesis, let us suppose the surface of the body to be a
perfectly elastic smooth fixed surface, having the apparent shape of the solid,
without any minute asperities.

In this case, every molecule which strikes the surface will have the normal
component of its velocity reversed, while the other components will not be
altered by impact.

The rebounding molecules will therefore move as if they had come from
an imaginary portion of gas occupying the space really filled by the solid, and
such that the motion of every molecule close to the surface is the optical
reflection in that surface of the motion of a molecule of the real gas.

In this case we may speak of the rebounding molecules close to the surface
as constituting the reflected gas. All directed properties of the incident gas

ARISING FROM INEQUALITIES OF TEMPERATURE. 705

are reflected, or, as Professor Listing might say, perverted in the reflected gas ;
that is to say, the properties of the incident and the reflected gas are sym-
metrical with respect to the tangent plane of the surface.

The incident and reflected gas together constitute the actual gas close to
the surface. The actual gas, therefore, cannot exert any stress on the surface,
except in the direction of the normal, for the oblique components of stress in
the incident and reflected gas will destroy one another.

Since gases can actually exert oblique stress against real surfaces, such
surfaces cannot be represented as perfectly reflecting surfaces.

If a molecule, whose velocity is given in direction and magnitude, but
whose line of motion is not given in position, strikes a fixed elastic sphere,
its velocity after rebound may with equal probability be in any direction.

Consider, therefore, a stratum in which fixed elastic spheres are placed so
far apart from one another that any one sphere is not to any sensible extent
protected by any other sphere from the impact of molecules, and let the
stratum be so deep that no molecule can pass through it without striking one
or more of the spheres, and let this stratum of fixed spheres be spread over
the surface of the solid we have been considering, then every molecule which
comes from the gas towards the surface must strike one or more of the spheres,
after which all directions of its velocity become equally probable.

When, at last, it leaves the stratum of spheres and returns into the gas,
its velocity must of course be from the surface, but the probability of any
particular magnitude and direction of the velocity will be the same as in a
gas at rest with respect to the surface.

The distribution of velocity among the molecules which are leaving the
surface will therefore be the same as if, instead of the solid, there were a
portion of gas at rest, having the temperature of the solid, and a density
such that the number of molecules which pass from it through the surface in
a given time is equal to the number of molecules of the real gas outside
which strike the surface.

To distinguish the molecules, which, after being entangled in the stratum
of spheres, afterwards return into the surrounding gas, we shall call them,
collectively, the absorbed and evaporated gas.

If the spheres are so near together that a considerable part of the surface
of each sphere of the outer layer is shielded from the direct impact of the
incident molecules by the spheres which lie next to it, then if we call that

VOL. II. 89

.ft- •TRUBB IN RARIFIED OASES

point of each sphere which lies furthest from the solid the pole of the sphere,

• greater proportion of molecules will strike any one of the outer layer of
spheres near its pole than near ita equator, and the greater the obliquity of

adence of the molecule, the greater will be the probability that it will strike

• sphere near its pole.

The direction of the rebounding molecule will no longer be with equal
probability in all directions, but there will be a greater probability of the
tangential part of its velocity being in the direction of the motion before
impact, and of its normal part being opposite to the normal part before

impact.

The condition of the molecules which leave the surface will therefore be
intermediate between that of evaporated gas and that of reflected gas, approach-
ing most nearly to evaporated gas at normal incidence and most nearly to
reflected gas at grazing incidence.

If the spheres, instead of being hard elastic bodies, are supposed to act
on the molecules at finite, though small distances, and if they are so close
together that their spheres of action intersect, then the gas which leaves the
surface will be still more like reflected gas, and less like evaporated gas.

We might also consider a surface on which there are a great number of
minute asperities of any given form, but since in this case there is consider-
able difficulty in calculating the effect when the direction of rebound from the
first impact is such as to lead to a second or third impact, I have preferred
to treat the surface as something intermediate between a perfectly reflecting
and a perfectly absorbing surface, and, in particular, to suppose that of every
unit of area a portion f absorbs all the incident molecules, and afterwards
allows them to evaporate with velocities corresponding to those in still gas at
the temperature of the solid, while a portion 1 — / perfectly reflects all the
molecules incident upon it.

We shall begin by supposing that the surface is the plane yz, and that
the gas is on that side of it for which x is positive.

The incident molecules are those which, close to the surface, have their
normal component of velocity negative. We shall distinguish these molecules
by the suffix (,). For these, and these only, £, is negative.

The rebounding molecules are those which have £ positive. We shall
distinguish them by the suffix (,). Those which are evaporated will be further
distinguished by an accent.

ARISING FROM INEQUALITIES OF TEMPERATURE. 707

Symbols without any mark refer to the whole gas, incident, reflected, and
evaporated, close to the surface.

The quantity of gas which is incident on unit of surface in unit of time,
is -p&

Of this quantity the fraction I-/ is reflected, so that the sign of f is
reversed, and the fraction / is evaporated, the mean value of £ in evaporated
gas being g, where the accent distinguishes symbols belonging to unpolarized
gas at rest relative to the surface, and having the temperature, 6', of
the solid.

Equating the quantity of gas which is incident on the absorbing part of the
surface to that which is evaporated from it, we have

/Pi£+//H&' = 0 ................................. (60).

Equating the whole quantity of gas which leaves the surface to the reflected
and evaporated portions

If we next consider the momentum of the molecules in the direction of
y, that of the incident molecules is p^^. A fraction (I—/) of this is reflected
and becomes (1 —f) p^^, and a fraction f of it is absorbed and then evaporated,
the mean value of 77 being now —v, namely, the velocity of the surface rela-
tively to the gas in contact with it.

The momentum of the evaporated portion in the direction of y is there-
fore t-fpf&v, and this, together with the reflected portion, makes up the whole
momentum which is leaving the surface, or

p^ = (f-l)Pl^rll-fp.;^v ........................ (62).

Eliminating fp^3' between equations (61) and (62)

(l-f)p£r)1 + p.£r).> + v[_(l-f)p£ + p£~} = 0 ............... (63).

The values of functions of £ 77 and £ for the incident molecules are to
be found by multiplying the expression in equation (22) by the given function,
and integrating with respect to £ between the limits — co and 0, and with
respect to 77 and £ between the limits + co .

The values of the same functions for the molecules which are leaving the
surface are to be found by integrating with respect to £ from 0 to oo .

We must remember, however, that since there is an essential discontinuity
in the conditions of the gas at the surface, the expression in equation (22) is

89—2

ftTRJBSES or RARIFIED GASES

a much )a« aoearate approximation to the actual distribution of velocities in
the gM dote to the surface than it is in the interior of the gas. We must,
therafatL consider the surface conditions at which we arrive in this way as
liable If important corrections when we shall have discovered more powerful

i<xis of attacking the problem.

For the present, however, we consider only terms of three dimensions or
we find

»)-»

Substituting these expressions in equation (63), and neglecting a1 in com
pmrison with unity, we find

(2-S)pR0afi+f(2ir)-*pR0a'P + 2f(2ir)-* (1 + $a«) W/>u = 0 ...(66).
If we write

and substitute for o^S and a'/8 their values as given in equations (54) and (51),
and divide by 2(/>p)i, equation (66) becomes

, , _^-£L~u = 0 ..(68).

\dx 2 p0 dxdyj 4 pd dy

If there is no inequality of temperature, this equation is reduced to

~dv

If, therefore, the gas at a finite distance from the surface is moving
parallel to the surface, the gas in contact with the surface will be sliding over
it with the finite velocity v, and the motion of the gas will be very nearly
the same as if the stratum of depth O had been removed from the solid and
filled with the gas, there being now no slipping between the new surface of
the solid and the gas in contact with it.

The coefficient G was introduced by Helmholtz and Piotrowski under the
of Gleitungs-coefficient, or coefficient of slipping. The dimensions of G are

ARISING FROM INEQUALITIES OP TEMPERATURE.

709

those of a line, and its ratio to I, the mean free path of a molecule, is given
by the equation

*-i(H< <7°>-

Kundt and Warburg found that for air in contact with glass, G = 2l,
whence we find f=\, or the surface acts as if it were half perfectly reflecting
and half perfectly absorbent. If it were wholly absorbent, G = f I.

It is easy to write down the surface conditions for a surface of any form.
Let the direction-cosines of the normal v be I, m, n, and let us write

d e , d
-T- lor i-j-
dv dx

We then find as the surface conditions

u-G~[(l-f)u-lmv -
"

d

^-
ay

d

-j-.

dz

p6 \dx

v—G ~r f(l— m1) v— mnw— mlu~\ + - ^( ^ — m-p ) (d + iG^- | = i
dv LV 4 pd \dy dv] \ dv)

w—G-r\(\—n*)w — nlu — nmv~\ + -—/t(-r-
dv1^ J 4 pO \dz

-r
dv

^r
dv

...(71).

In each of these equations the first term is one of the velocity-components
of the gas in contact with the surface, which is supposed fixed; the second
term depends on the slipping of the gas over the surface, and the third term
indicates the effect of inequalities of temperature of the gas close to the
surface, and shows that in general there will be a force urging the gas from
colder to hotter parts of the surface.

Let us take as an illustration the case of a capillary tube of circular
section, and for the sake of easy calculation we shall suppose that the motion
is so slow, and the temperature varies so gradually along the tube that we
may suppose the temperature uniform throughout any one section of the tube.

Taking the axis of the tube for that of z, we have for the condition of
steady motion parallel to the axis

dp _ /d'w d?w\

no vnunw IN BARiriED OASES

Since •iwything i« «ymraetrical about the axis, if we write r» for z' + y'
we find at the solution of this equation

mA4- —^r* ..(73).

If Q denotes the quantity of gas which passes through a section of the
tube in unit of time

At the inner surface of the tube we have r-a, and

1 dp ,
« = A + ^~dza

JL+-L4* ..(75)

vpa* * 8/1 dz

dw 1 dp
d^-^dz"

The last of equations (71) may therefore be written

- ................... (77).

8/1 v ' dz ±p0 dz

Equation (77) gives the relation between the quantity of gas which passes
through any section of the tube, the rate of variation of pressure, and the
rate of variation of temperature in passing along the axis of the tube.

If the pressure is uniform there will be a flow of gas from the colder to
the hotter end of the tube, and if there is no flow of gas the pressure will
from the colder to the hotter end of the tube.

These effects of the variation of temperature in a tube have been pointed
out by Professor Osborne Reynolds as a result of the Kinetic Theory of Gases,
and have received from him the name of Thermal Transpiration : a name in
strict analogy with the use of the word Transpiration by Graham.

But the phenomenon actually observed by Professor Reynolds in his ex-
periments was the passage of gas through a porous plate, not through a
capillary tube ; and the passage of gases through porous plates, as was shown

ARISING FROM INEQUALITIES OF TEMPERATURE. 711

by Graham, is of an entirely different kind from the passage of gases through
capillary tubes, and is more nearly analogous to the flow of a gas through a
small hole in a thin plate.

When the diameter of the hole and the thickness of the plate are both
small compared with the length of the free path of a molecule, then, as Sir
William Thomson has shown, any molecule which comes up to the hole on
either side will be in very little danger of encountering another molecule before
it has got fairly through to the other side.

Hence the flow of gas in either direction through the hole will take place
very nearly in the same manner as if there had been a vacuum on the other
side of the hole, and this whether the gas on the other side of the hole is
of the same or of a different kind.

If the gas on the two sides of the plate is of the same kind but at
different temperatures, a phenomenon will take place which we may call thermal
effusion.

The velocity of the molecules is proportional to the square root of the
absolute temperature, and the quantity which passes out through the hole is
proportional to this velocity and to the density. Hence, on whichever side the
product of the density into the square root of the temperature is greatest,
more molecules will pass from that side than from the other through the
hole, and this will go on till this product is equal on both sides of the hole.
Hence the condition of equilibrium is that the density must be inversely as
the square root of the temperature, and since the pressure is as the product
of the density into the temperature, the pressure will be directly proportional
to the square root of the absolute temperature.

The theory of thermal effusion through a small hole in a thin plate is
therefore a very simple one. It does not involve the theory of viscosity at all.

The finer the pores of a porous plate, and the rarer the gas which effuses
through it, the more nearly does the passage of gas through the plate corre-
spond to what we have called effusion, and the less does it depend on the
viscosity of the gas.

The coarser the pores of the plate and the denser the gas, the further
does the phenomenon depart from simple effusion, and the more nearly does it
approach to transpiration through a capillary tube, which depends altogether on
viscosity.

IS KAJUnXD OASES ABISING FROM INEQUALITIES OF TEMPERATURE.

To return to the case of transpiration through a capillary tube. When
the temperature is uniform

By experiment* on capillary tubes of glass, MM. Eundt and Warburg
found* for the value of G for air at different pressures and at from 17° C.
to 2TC,

Q

G = - centimetres (79)

P

where p is the pressure in dynes per square centimetre, which is nearly the
•une as in millionths of an atmosphere. For hydrogen on glass

15
G = — centimetres (80).

P

When there is no flow of gas in a tube in which the temperature varies
from end to end, the pressure is greater at the hot end than at the cold
end. Putting Q = 0 we have

d&=6p0

M*

The quantity 6 ~. is just double of that calculated in section (3) of the

introduction, and is therefore in C.G.S. measure 0'63 xp for dry air at 15° C.
Let us suppose a = 0'01 centimetre, and the pressure 40 millimetres of mercury,
then </' ='00016 centimetre.

If ono end of the tube is kept at O'C. and the other at 100°C., the
pressure at the hot end will exceed that at the cold end by about 1-2
millionths of an atmosphere.

The difference of pressure might be increased by using a tube of smaller
bore and air of smaller density, but the effect is so small that though the
theoretical proof of its existence seems satisfactory, an experimental verification
of it would be difficult.

* Pogg. Ann., July, 1876.

[From the Cambridge Philosophical Society's Transactions, Vol. xii.]

XCIV. On Boltzmann's Theorem on the average distribution of energy in a system

of material points.

LUDWIG BOLTZMANN, in his "Studien liber das Gleichgewiclit der
lebendigen Kraft zwisclien bewegten materiellen Punkten " [Sitzb. d. k. Akad.
Wien, Bd. LVIII., 8 Oct. 1868], has devoted his third section to the general
solution of the problem of the equilibrium of kinetic energy among a finite
number of material points. His method of treatment is ingenious, and, as far
as I can see, satisfactory, but I think that a problem of such primary importance
in molecular science ought to be scrutinized and examined on every side, so
that as many persons as possible may be enabled to follow the demonstration,
and to know on what assumptions it rests. This is more especially necessary
when the assumptions relate to the degree of irregularity to be expected in
the motion of a system whose motion is not completely known.

Mr H. W. Watson, in his Treatise on the Kinetic Theory of Gases*, has
developed with great clearness the steps of the investigation of the distribution
of energy among a set of particles which are supposed to act on each other
only at very small distances. The particles may be acted on by external forces
such as gravity, but it is expressly stipulated that the time during which a
particle is encountering other particles is very small compared with the tune
during which there is no sensible action between it and other particles ; and
also that the time during which a particle is simultaneously within the distance
of molecular action of more than one other particle may be neglected.

Now this method of treating the question, however necessary it may be
in the subsequent investigation of the processes of diffusion, &c. in gases, is
inapplicable to the theory of the equilibrium of temperature in liquids and

* Clarendon Press Series, 1876.

VOL. ii. 90

714 BOLTMAJO*'* THEOREM ON THE AVERAGE DISTRIBUTION

•olid*. for in these bodies the particles are never free from the action of
neighbouring particles. It is true that in following the steps of the investi-
gttrn. M giTen either by Boltzmann or by Watson, it is difficult, if not
impossible, to see where the stipulation about the shortness and the isolation
of the encounters is made use of. We may almost say that it is introduced
rather f«-r the sake of enabling the reader to form a more definite mental
image of the material system than as a condition of the demonstration. Be
tfrfo M it may, the presence of such a stipulation in the enunciation of the
problem cannot fail to leave in the mind of the reader the impression of a
oomeponding limitation in the generality of the solution.

In the theorem of Boltzmann which we have now to consider there is no
such limitation. The material points may act on each other at all distances,
and according to any law which is consistent with the conservation of energy,
and they may also be acted on by any forces external to the system provided
also are consistent with that law.

The only assumption which is necessary for the direct proof is that the
system, if left to itself in its actual state of motion, will, sooner or later, pass
through every phase which is consistent with the equation of energy.

v it is manifest that there are cases in which this does not take place.
The motion of a system not acted on by external forces satisfies six equations
besides the equation of energy, so that the system cannot pass through those
phanra, which, though they satisfy the equation of energy, do not also satisfy
these su equations.

Again, there may be particular laws of force, as for instance that according
to which the stress between two particles is proportional to the distance between
them, for which the whole motion repeats itself after a finite time. In such
caws a jKirtictilar value of one variable corresponds to a particular value of
each of the other variables, so that phases formed by sets of values of the
variables which do not correspond cannot occur, though they may satisfy the
general equations.

But if we suppose that the material particles, or some of them, occasionally
encounter a fixed obstacle such as the sides of a vessel containing the particles,
tlion, except for special forms of the surface of this obstacle, each encounter
will introduce a disturbance into the motion of the system, so that it will

OF ENERGY IN A SYSTEM OP MATERIAL POINTS. 715

pass from one undisturbed path into another. The two paths must both satisfy
the equation of energy, and they must intersect each other in the phase for
which the conditions of encounter with the fixed obstacle are satisfied, but they
are not subject to the equations of momentum. It is difficult in a case of such
extreme complexity to arrive at a thoroughly satisfactory conclusion, but we may
with considerable confidence assert that except for particular forms of the surface
of the fixed obstacle, the system will sooner or later, after a sufficient number
of encounters, pass through every phase consistent with the equation of energy.

I shall begin with the case in which the system is supposed to be contained
within a fixed vessel, and shall afterwards consider the case of a free system,
or of a system contained in a vessel rotating uniformly about an axis which
itself moves uniformly in a straight line.

I have found it convenient, instead of considering one system of material
particles, to consider a large number of systems similar to each other in all
respects except in the initial circumstances of the motion, which are supposed
to vary from system to system, the total energy being the same in all. In
the statistical investigation of the motion, we confine our attention to the number
of these systems which at a given time are in a phase such that the variables
which define it lie within given limits.

If the number of systems which are in a given phase (defined with respect
to configuration and velocity) does not vary with the time, the distribution of
the systems is said to be steady.

It is shewn that if the distribution is steady, a certain function of the
variables must be constant for all phases belonging to the same path. If the
path passes through all phases consistent with the equation of energy, this
function must be constant for all such phases. If however there are phases
consistent with the equation of energy, but which do not belong to the same
path, the value of the function may be different for such phases.

But whether we are able or not to prove that the constancy of this
function is a necessary condition of a steady distribution, it is manifest that
if the function is initially constant for all phases consistent with the equation
of energy, it will remain so during the motion. This therefore is one solution,
if not the only solution, of the problem of a steady distribution.

Now we know from the empirical laws of the diffusion of heat that the
problem of the equilibrium of temperature in an isolated material system has

90—2

716 BOLTXMAHN'S THEOREM ox THE AVERAGE DISTRIBUTION

ooe and only one solution. But we have found one solution of the problem
of equilibrium of energy in a system of material points in motion. If, therefore,
the real material system in which the equilibrium of temperature takes place
M ««p«hLt of being accurately represented by a system of material points (as
•hfifyj in pure dynamics) acting on each other according to determinate, though
unknown, laws, then the mathematical condition of the equilibrium of energy
nmt be the dynamical representative of the physical condition of the equality
of temperature.

It appears from the theorem that in the ultimate state of the system the

avenge kinetic energy of two given portions of the system must be in the

• of the number of degrees of freedom of those portions. This, therefore,

must be the condition of the equality of temperature of the two portions of the

. -••

H.nce at a given temperature the total kinetic energy of a material system
must be the product of the number of degrees of freedom of that system
into a constant which is the same for all substances at that temperature, being
in feet the temperature on the thermodynamic scale multiplied by an absolute
constant

If the temperature, therefore, is raised by unity, the kinetic energy is
increased by the product of the number of degrees of freedom into the absolute
constant.

The observed specific heat of the body, expressed in dynamical measure, is
the increment of the total energy when the temperature is increased by unity.
The observed specific heat cannot therefore be less than the product of the
number of degrees of freedom into the absolute constant, unless the potential
energy diminishes as the temperature rises.

Dynamical Specification of the motion.

\\ B shall begin by supposing the material system to be of the most general

having ite configuration determined by the n variables q,, qt...qn, and its

motion determined by the corresponding momenta pv pt...pn. The state of the

system at any instant is completely defined if we know the values of these

variables for that instant.

We shall suppose the forces acting between the parts of the system to
be of the most general kind consistent with the conservation of energy. This

OF ENERGY IN A SYSTEM OF MATERIAL POINTS. 717

may be expressed by defining V, the potential energy of the system, as a
function of <&...<£,, the variables which define the configuration.

The kinetic energy of the system is denoted by T. We shall suppose it
to be expressed in terms of the q's and ^'s as in Hamilton's method. The
total energy is denoted by

E=V+T .................................... (l),

and is a constant during the motion of the system.

Hamilton's equations of motion for this system are

dt dpr '

dr dE

where qr and pr are the co-ordinate and the momentum corresponding to each
other.

Let us now consider a finite motion of the system. Let the initial co-
ordinates and momenta be distinguished by accented letters, and the final co-
ordinates and momenta by the same letters unaccented.

To define completely such a motion requires 2n+l variables to be given.
These may be the n initial co-ordinates, the n initial momenta, and the time
occupied by the motion.

There is another method however in which the 2n + 1 variables are the
n initial co-ordinates, the n final co-ordinates, and the total energy. When
these quantities are given there are in general only a finite number of possible
motions.

Definition of the "Action" of the system during the motion.

Twice the time integral of the kinetic energy, taken from the beginning
to the end of the motion, and expressed in terms of the initial and final
co-ordinates and of the total energy, is called the "Action" of the system
during the motion. If we denote it by A,

.................................... (4),

and is expressed as a function of '?/... </„', <li---qn, and E.

711 BOLWMAXN'* THEOREM ox THE AVERAGE DISTRIBUTION

1 1 U abewn in treatises on dynamics* that

' (6).

j-j-— — ' ' —_'/'' ..(7).

f] if* sis* \ /

The indices r and * in this equation may be the same or different.

Also if t' and t are the values of the time at the beginning and at the end

of the motion,

•/.!_, . .

dp, dt' , . , dp,' dt

Ik'noe ,//•:= -<i<]r <9> and -d-E=-dq~; <10)-

In the course of our investigation we shall have to compare the product
..f the differentials of the co-ordinates and momenta at the beginning of the
motion with the corresponding product at the end of the motion. We shall
write for brevity ds = dql...dqn for the product of the differentials of the co-
ordinates, and da- = dpl...dpn for the product of the differentials of the momenta,
and we shall use the product ds'dsdE as a middle term in comparing ds do-' <li'
with tlsda-iJt.

Now ds da' dt' = ds' ds dE2 ±(dfi-. . . ^ —'} • .-(ID

\»2i dqn dE)

_, (<1>\ dp' dt'\

where 2 ± ^- --p. )

\dql dqn dLJ

denotes the functional determinant

dp; dpt'
"dqn' dE

'//>.' dpS (12).

dqt' " dqn' dE

dqn' dE

* Thomson and Tail's Natural Philosophy, § 330.

OF ENERGY IN A SYSTEM OF MATERIAL POINTS.

719

Substituting for the elements of this determinant their values as given by
equations (7), (9), and (10) it becomes

dp, dpn dt

3q?' ~ dq~," ~d£

dfr _dp^ _dt_

dqn" ~dq~n' ~dq?

dp, dpn dt

~ ~dE

Now the rows in this determinant are the same as the columns in the former
one ; the accented and unaccented letters being exchanged and the signs of
all the elements changed. We may therefore express the relation between the
two determinants in the abbreviated form

- /dp/ dp^_dtf\_(_ /dp, dp^ dt \

- (dq, •' • dqn d£/~( > 2'± \dq,'" - dqn' dE)~

Hence ds' da-' dt' = dsf ds dEZ +

- \dql dqn

- ( - )n+lds ds dE^ + (^ —

- (dq, dqn' dE

= ( - )n+lda- ds dt

= ds da1 dt (15).

If we suppose the time, t — t', to be given, dt = dt' and

ds da-' = ds da- (1 G),

or dq,' dqn'dp,' dpn' = dq, dqndp, dpn (17).

The initial state of the system is a function of 2n variables. We have
hitherto supposed these to be the n co-ordinates and the n momenta, but
since the total energy E is a function of these variables we may substitute
for one of the momenta, say _p/, its value in terms of the n co-ordinates,
the n-l remaining momenta, and E, and thus express every quantity we

BOLTZMANX 8 THEOREM ON THE AVERAGE DISTRIBUTION*

have to deal with in terms of the latter set of variables. Then since by

dE

'-J>....(19).

Similarly we find for the final state of the system

dqt rfg.rfp, dpn = dqt dqndp dpn dE — (20).

The left-hand members of these equations have been proved equal, and in
the right-hand members dE is the same at the beginning and end of the
motion. Dividing out dE we find

1 1'
dqt' dq.'dp,' dp* — = dqt dqndp,\ dpn'— (21).

This equation is applicable to the case in which the total energy is supposed
not to vary from one particular instant of the motion to another, and in which,
therefore, the 2n variables are no longer independent, but, being subject to the
equation of energy, are reduced to 2»— 1.

Statistical Specification.

We have hitherto, in speaking of a phase of the motion of the system,
supposed it to be defined by the values of the n co-ordinates and the n momenta.
We shall call the phase so defined the phase (pq). We shall now adopt a
wider definition by saying that the system is in the phase (a,6) whenever the
values of the co-ordinates are such that 5, is between 6, and &, + <$>,, qt between
l>, and bt+dbt, and so on; also pt between a, and at + dalt and so on. The
limits of the first component of momentum,/),, are not specified, because the value
• •t" /i, is not independent of the other variables, being given in terms of E
and the other 2n — 1 variables in virtue of the equation of energy.

The quantities a, b are of the same kind as p and q respectively, only
they are not supposed to vary on account of the motion of the system. In
the statistical method of investigation, we do not follow the system during
ito motion, but we fix our attention on a particular phase, and ascertain whether

OF ENERGY IN A SYSTEM OF MATERIAL POINTS. 721

the system is in that phase or not, and also when it enters the phase and
when it leaves it.

Boltzmann defines the probability of the system being in the phase (a^)
as the ratio of the aggregate time during which it is in that phase to the
whole time of the motion, the whole time being supposed to be very great.
I prefer to suppose that there are a great many systems the properties of
which are the same, and that each of these is set in motion with a different
set of values for the n co-ordinates and the n — 1 momenta, the value of the
total energy E being the same in all, and to consider the number of these
systems which, at a given instant, are in the phase (a^). The motion of each
system is of course independent of the other systems.

Let N be the whole number of systems, and let the number of these
which, at the time t, are in the phase (aj)) be denoted by N (aly b, t). The
aim of the statistical method is to express JV (alt b, t) as a function of N,
of the co-ordinates and momenta with their limits, and of t. It is manifest
that N can only enter the function as a factor, for the different systems do
not act on each other. Also any differential as da or db can only enter as
a factor, for the number of systems within any phase must vary in the ratio
of the interval between the limits of that phase. We may therefore write

N(a1bt) = Nf(a,, an, b, bn, t)da2 dand\ dbn (22),

where we have to determine the form of the function f.

We shall now follow the motion of these systems from the time t', when
we begin to watch the motion, to the time t when we cease to watch it.

Since the systems which at the tune t form the group N(alt b, t) are
individually the same systems which at the time t' formed the group N (a{, b', t')

we have

N(au b, t) = N(a;, b', t) (23),

or Nf(a, t)da, dbn = Nf(a3' t')da,' dbn' (24).

But by equation (21)

da, dbn(\Yl = da; A.'&T1 - (25).

Hence /(a, «)W« t'}b^=C (26),

where C is a constant for all phases of the same motion, and we may write

/(«. t) = C(b1)-> (27),

and N(au b, t) = NC(b^da, dbn (28).

VOL. II. 9!

BOLTXXAXN'S THBORKM ON THE AVERAGE DISTRIBUTION

If the distribution of the N systems in the different phases is such that
tit* number in a given phase does not vary with the time, the distribution
m mvl to be steady. The condition of this is that C must be constant for
all pbJMea belonging to the same path. It will require further investigation to
determine whether or not this path necessarily includes all phases consistent
with the equation of energy.

If, however, we assume that the original distribution of the systems according
to the different phases is such that C is constant for all phases consistent
with the equation of energy, and zero for all phases which that equation shows
to be impossible, then the law of distribution will not change with the time,
and C will be an absolute constant.

We have therefore found one solution of the problem of finding a steady
distribution. Whether there may be other solutions remains to be investigated.

Let N(l) denote the number of systems in which q1 is between 6, and
6, + </6,, 9, between 6t and 6, +db,, and so on, and qn between 6n and bn + dbn,
the momenta not being specified otherwise than by their being consistent with
the equation of energy, then

N(b)=t...tN(al, l}d^...dan (29),

the integration being extended to all values of the momenta consistent with
the equation of energy.

To simplify the integration let us suppose the variables transformed so that
the kinetic energy is expressed in terms of the squares of the component momenta,

T=l(p1a1' + nta;+...+nnan*) (30),

where a,... a. are the transformed momenta, and /*,.../*„ are functions of the
co-ordinates, which we may call moments of mobility, and which, in the case
of material points, are the reciprocals of the masses.

Now let us assume

.(32),
•(33),

.(34).

OF ENERGY IN A SYSTEM OF MATERIAL POINTS. 723

Then by the equation of energy

Of these quantities, An is a function of the co-ordinates only, because E is
given and V is a function of the co-ordinates, An-1 is a function of the co-
ordinates and an, An_, of the co-ordinates and of an and an_l} and so on.

Also by equation (2)

6 -=— =

To integrate the expression

we begin by integrating with respect to aa, thus

|(7(61)-1c?a3=|(7()a1^)-4(^i-a25)-^a2 .................. (37),

the limits of integration being +AZ. The result is

-u; ........................... (38).

For the next integration we have

3(A?-aWda3=T^A? ................ (39).

Hence after r integrations, r being any number less than n, the result is

Putting r = n— 1 and remembering that pnAn* = 2E— 2V, we find

<lvf? .................... (41).

This is the number of systems whose configuration is specified by the
variables 6,... 6., while the momenta may have any values consistent with the
equation of energy.

91—2

ANN'S THEOREM OK THE AVERAGE DISTRIBUTION

The quantity E- F. which occurs in this equation, is, by equation (1),
equal in magnitude to T, the kinetic energy of the system. The quantity T,
m defined explicitly in terms of the velocities or the momenta of the
whoffQM K-V does not involve these quantities explicitly, but is
M a function of the configuration.

We t***H find it convenient, however, especially in the study of more
complicated problems, to remember that the number of systems in a given
j^lyifatfff" IB a function of the kinetic energy corresponding to that configuration.
If the kinetic energy is not expressed as a sum of squares, but in the
move general form,

+ i[22]a,' + [23]a,a, + &c (42),

where the quantities denoted by [11] &c. are functions of the co-ordinates,
which we may call the moments and products of mobility of the system ;
thm fjfusM the discriminant

[n], [121 M

A- ^J2^;;;;:;;;.12!1 («>

[»ll [»2J [nn]

is an invariant, its value is the same when T is reduced to a sum of squares,
in which case all the elements except those in the principal diagonal of the
determinant vanish, and we have

A =/*,/»,.../*» (44),

and we may write the value of N (b),

*' ^ (45).

If the system consists of n' material particles, whose masses are ml...mn-,
then the number of degrees of freedom is n = 3n' and

/*I = /AI = /A» = WI~I, pt = fr = p-t = ini~l and so on (46).

•e in this case we may write

(r(

db,...dbn (47).

OF ENERGY IN A SYSTEM OF MATERIAL POINTS. 725

These expressions give the number of systems in a given configuration
only when E— V is positive for that configuration, for since the kinetic energy
is necessarily positive, the potential energy cannot exceed the total energy.
For configurations specified in such a way that if they existed V would be
greater than E, the value of N(b) is zero.

The value of N(b) is also zero for configurations which, though they make
V less than E, cannot be reached by a continuous path from the original
configuration without passing through configurations which make V greater than E.

We shall return to this expression for the number of systems in a com-
pletely specified configuration, but in the mean time it will be useful to
consider how many of these systems have one of their momenta, pn, between
given limits. In this way we shall be able to determine completely the
average distribution of momentum among the variables without making any
assumptions about the nature of the system which might limit the generality
of our results.

In order to find the number of systems in the configuration (b) for which
one of the momenta, say pn, lies between an and an + dan, we must stop
before the last integration. Putting r = n — 2 in equation (40)

2 /

The whole number of systems in configuration (b) is given by (45). Hence
the proportion of these systems for which an lies between an and an + dan is

(49).

If we write

then kn denotes the part of the kinetic energy arising from the momentum an.
The proportion of the systems in configuration (6) for which kn is between kn
and ka + dkn is

JLTXMAJnc'« THBORKM ON THB AVERAOB DISTRIBUTION

-

Knee Mr one of the variable* may be taken for qn, the law of distribution
of ndoe. of the kinetic energy is the same for all the variables. The mean
nkhM of the kinetic eneigy corresponding to any variable is

Kss\(E-V) = - T (52).

n *i

The maximum value is T=nK (53).

The mean value of If is

When n is very large, the expression (51) approximates to

].,-"-* dk ................................. (55).

Recapitulation.

The result of our investigation may therefore be stated as follows :
(a) We begin by considering a set of material systems which satisfy the
general equations of dynamics (2) and (3), and the equation of energy (1). If
in these systems the distribution of configurations satisfies equation (45), and
the distribution of motion satisfies equation (51), these equations will continue
to be satisfied during the subsequent motion of the system. One result of
equation (51), to which we shall have to refer, is that the average kinetic
energy corresponding to any one of the variables is the same for every one
of the variables of the system.

(ft) We now turn our attention to a system of real bodies enclosed in
a rigid vessel impervious to matter and to heat. We know by experiment that
in such a system the temperature cannot remain steady in every part unless
the temperature of every part of the system is the same, and that this condition
is necessary in whatever manner the configuration of the system may be varied
by altering the position and mean density of the portions of sensible size into
which we are able to divide it.

Now if the system of real bodies is a material system which satisfies the
equations of dynamics, and if equations (45) and (51) are also satisfied, the
condition of the system will, as we have shewn, (a), be steady in every respect,

OF ENEHGY IN A SYSTEM OF MATERIAL POINTS. 727

and therefore in respect of temperature. Hence by (/3) the temperature of every
part of the system must be the same.

Therefore if equations (45) and (51) are satisfied, the condition of equality
of temperature is also satisfied.

But the condition of equality of temperature does not depend on the
configuration of the system, for though we can alter the configuration by external
constraint we cannot prevent the temperature from becoming equalized. It does
not depend, therefore, on equation (45). We must therefore conclude, that if
equation (51) is satisfied, the condition of equality of temperature is also satisfied,
or, in other words, that equation (51) is the condition of equality of temperature.

Hence when two parts of a system have the same temperature, the average
kinetic energy corresponding to any one of the variables belonging to these
parts must be the same.

If the system is a gas or a mixture of gases not acted on by external
forces, the theorem that the average kinetic energy of a single molecule is the
same for molecules of different gases is not sufficient to establish the condition
of equilibrium of temperature between gases of different kinds such as oxygen
and nitrogen, because when the gases are mixed we have no means of ascer-
taining the temperature of the oxygen or of the nitrogen separately. We can
only ascertain the temperature of the mixture by putting a thermometer into it.

We cannot legitimately assert that the temperatures of the oxygen and
of the nitrogen must be equal because they are in contact with each other,
for the only way in which we can conceive the oxygen or the nitrogen as
existing in the mixture is by picturing the medium as a system of molecules,
and as soon as we begin to see the molecules distinctly, heat becomes resolved
into motion.

But since our investigation is equally applicable to a system of any kind,
provided only it satisfies the equations of dynamics, we may suppose it to
consist- of pure oxygen and pure nitrogen separated by a solid diaphragm,
the solid diaphragm consisting of molecules capable of motion, but acting on
each other with forces which are sufficient to prevent any molecule from getting
far apart from its neighbours except under the action of disturbing forces
greater than any which would occur in a system at the given temperature.
In this system, though the oxygen and the nitrogen cannot mix, each can make

BOtmUlW'i tmOHKM ON THE AVKEAOE DISTRIBUTION

with the surface molecules of the diaphragm,

MM «f 1*~BT can go on within the solid diaphragm itself without
wage of tm4r™'«* between distant parts of the diaphragm.
Hence in this system, the average kinetic energy of a molecule of oxygen
will become equal to that of a molecule of nitrogen in the final state of the
k— ^m» if to say, when the temperatures of all parts of the system have
me equal, and since in that final state we have pure oxygen on one side
and pure nitrogen on the other, we can verify the equality of temperature by
mqiflt* of a thermometer, and we can now assert that the temperatures, not
only of oxygen and nitrogen, but of all bodies, are equal when the average
kinetic energy of a single molecule of each of these substances is the same.

Approximate value of the probability when V is small compared with E.

To find the number of systems the configuration of which is specified as
regards the limits of certain of the variables while the other variables are left
undetermined, we should have to integrate the expressions in equations (41),
(45), or (47) with respect to each of the undetermined variables in succession,
the integrations being extended to all values of these variables which are
consistent with the equation of energy.

These integrations cannot be performed unless the potential energy of the
system is a known function of the variables which determine its configuration.
We cannot therefore in general continue the integration so as to determine
the number of systems in which the limits are specified for some, but not all,
of the variables.

But when the number of variables is very great, and when the potential
energy of the specified configuration is very small compared with the total

energy of the system, we may obtain a useful approximation to the value of

•-i
' K- FJ«~ in an exponential form, for if we write, as in equation (53), E — nK,

[J0-F]«
nearly, provided n is very great and V is small compared with E. The

«-» _r

'"•* (56),

OF ENERGY IN A SYSTEM OP MATERIAL POINTS. 729

expression is no longer approximate when V is nearly as great as E, and it
does not vanish, as it ought to do, when V=E,

Hence when the potential energy of the system in the given configuration
is very small compared with its kinetic energy, we may use the approximately
correct statement, that the number of systems in a given configuration is
inversely proportional to the exponential function, the index of which is half
the potential energy of the system in the given configuration divided by the
average kinetic energy corresponding to each variable of the system.

If we divide the system into any two parts, A and B, we may consider
V, the potential energy of the whole system, as made up of three parts,
VA and VB, the potential energy of A and B, each on itself, and W, that of
B with respect to A.

When, as in the case of a gas, the parts of a system are in a great
degree independent of each other, the average values of VA and Vs may be
treated as constants, and the variations of V will be the same as those of W,
so that the variable part of the exponential function will be reduced to

w
e™ (57).

If we suppose that A denotes a single molecule of a particular kind of
gas, and that B denotes all the other molecules, of whatever kind, in the
system, then, since there are many molecules similar to A, we may pass, from
the number of systems in which A is within a given element of volume, to
the average number of molecules similar to A which are within that element,
or, in other words, the average density of the gas A within that element.

We may therefore interpret the expression (57) as asserting that the density
of a particular kind of gas at a given point is inversely proportional to the
exponential function whose index is half the potential energy of a single
molecule of the gas at that point, divided by the average kinetic energy
corresponding to a variable of the system.

We must remember that since the centre of mass of a molecule is

determined by three variables, the mean kinetic energy of agitation of the

centre of mass of a molecule is three times the quantity K which denotes
the mean kinetic energy of a single variable.

VOL. H. 92

BOLTtMAmft THWRW ON THE AVERAGE DISTRIBUTION

PART II. A Free system.

In a material system not acted on by external forces the motion satisfies
«x equation* besides the equation of energy, so tout we must not include in
KIT integration ail the phases which satisfy the equation of energy, but only

of them which also satisfy these six equations.

In what follows, we shall suppose the system to consist of n particles,
m are «i,...ro., and whose co-ordinates x, y, z, and velocity-components
«, », v, are distinguished by the same suffix as the particle to which they belong.

Let us now consider a system consisting of s of these particles, and write

ml + mt + &c.+mt = Mt (58),

tn.x, + »W + Ac. + mjc. = MtXt, '

m1yl+my1 + &c. + my. = MtY,, (59),

«j,2, + wiA +&c. + «V, = MtZt,

then M, will be the mass of the minor system and X,, Y,, Z, the co-ordinates
of its centre of mass. If we also write

m,w, + &c. + mtu, = M, Ut, 1

T/W + &c. + m.v. = M, V,, I (60),

mjMJ, + &c. + mtwt = M, W, , J

+ m.(y.v>. ~ a»v.) = F.+M.( Y.W, - Z. V.), 1
+ m.(z.u.-xlw'l) = G,+ M.(Z,U,-X.Wt), I ...(61),
&c. + mt(x.v. - y,u.) = H,+ M,(Xt V, - Y. U,) , j

then Utt Vtt Wt will be the velocity-components of the centre of mass, and
/•'„ (J,, If, the components of angular momentum round this point.

We shall also write

!«,(«,' + »,* + «»!«) + &c.+i«i»(tt.i + u.l + w.t) = r. (62).

The seven conditions satisfied by the whole system are that the seven
quantities Unt Vn, Wn, Fn, Gn, Hn and E are 'constant during the motion.

Under these conditions the 3n momentum-components are not independent.
We shall therefore transform equation (17) into one in which the differentials

OF ENERGY IN A SYSTEM OF MATEBIAL POINTS.

731

of the first seven velocity-components are replaced by the differentials of the
seven constants.

The functional determinant is found by differentiating the seven quantities
Un, Vn, Wn, Fn, Gn> Hn and E with respect to the momenta m^, ra^,
.iv, ; m2u2, m^\, mawa ; and m3u3. We thus obtain

1, 0, 0, 0, zlt -ylt u,

0, 1, 0, -zlt 0, xlt v,

0, 0, 1, y» -a;,, 0, wl

1, 0, 0, 0, z,, -y,, «, =A (63),

0, 1, 0, -z,, 0, • a:,, r,

0, 0, 1, y,, —x3, 0, w,

1, 0, 0, 0, «,, -ys, -w3

which we may write A = a r12 f 12 (64),

where a = (?/, - y.) (z, - z3) - (y, - y,) (z, - z2) (65),

or twice the projection on the plane of yz of the triangle whose vertices are
mlt ma, and m3, and

or the rate of increase of the distance between m1 and m2 multiplied into that
distance.

In a system composed of material particles, each component of momentum
is equal to the corresponding velocity-component multiplied into the mass of
the particle. We may therefore write ^1 = m1«1 and so on, and since the masses
are invariable we may omit them from both members of equation (17), and write it

dx{. . .dzn' du^. . .dwn' = dx^ ..dzndu.i..,dwn (67).

But d Ud Vd WdFdGdHdE = m* m? m3 0VM r'12 du.'dv.'dw.'du.'dv^dw.'du,'

dx1'.^dzn'dva'...dwn' dxl...dzndv3...dwn „ .

Hence - , . y p- = m*m»m ar r = °' (69)'

ll< \ I''-: //' .j U- ' U ' U //t-I //(.I //f-J Ur/ i" ' 1"

and equation (29) becomes

rll)-1dv,...dwn (70).

92—2

. •:

BO 41 i i: IS* I>I-II:II:ITIUN

We- shall find it uaeful in what follows to define the energy of internal
M the exce§« of the whole kinetic energy of the system over that
which it would have if it were moving like a rigid body with the same con-
the «ame components of momentum and of angular momentum.

If we Mippoeo the internal motion of the system to be destroyed in a
abort tirw by internal forces, so that the configuration is not sensibly
ahcmi during the process, then the work done by the system against these
foMOt i« the measure of the energy of internal motion.

Writing 7* for the kinetic energy referred to the origin, K for that of
the maw moving with the velocity of the centre of mass, J for the kinetic
due to the rotation of the system as a rigid body, and 7 for the
of internal motion, we have

where

I=T-K-J

J= 2 ftro («•

W)

(71),
(72),
(73),
(74),

where p, q, r are the components of angular velocity with respect to the
axes of x, y, z and are related to F, G, H by the equations

aF-nG-mH=p,
-nF+bG- lH=q,

Ap-Nq-Mr =
-Xr + Bq-Lr =
-Mp-Lq+ Cr =

-mF-

clt=r,

(75).

-. ••:.-

ft -2i» «-

= -£m(y-Y)(z-Z)

=S»i (z-Z)(x-X) [...(76).

Writing for the sake of brevity

-1, -N, -M
-N, B, -L
-M, -L, C

d=

a, — n, — m

— n, b, — I

— m, — I, c

•(77),

the relations between the moments and products of mobility and those of
inertia will be given by equations of the forms

> = BC-D Ad=bc-f

Dd=l.

Ld= —mn — al,

.(78).

OF ENERGY IN A SYSTEM OF MATERIAL POINTS.

733

If we write

.(79),

T) = V — V + rx —pz
£=w— W+py — qx

then £ 77, £ will be the velocity-components of a particle with respect to
axes passing through the centre of mass of the system and rotating with the
angular velocity whose components are p, q, r. We may therefore call £ 17, £
the velocity-components of the internal motion. If the system were to become
rigid, the internal motion would become zero. The energy of internal motion
may be expressed in terms of £ 77, £, thus : —

h£2) (80).

We have now to express the energy of internal motion of a system of
s — 1 particles in terms of the quantities U, V, W, F, G, H and T belonging
to the system of s particles, together with the position and velocity of the
,$th particle.

To avoid the repetition of suffixes we shall distinguish quantities belonging
to the minor system of s — 1 particles by accented letters, and quantities
belonging to the complete system of s particles and the particle mt by unaccented
letters. We shall also write

We thus find

Mm

' = MX-mx

' = MU-mu,

' = L-p.(y-Y)(z-Z)

Since the choice of the axes of reference is arbitrary, we may simplify
the expressions by taking for origin the centre of mass of the system M, and
for the axis of z the line passing through the particle m. We may also turn

- ; BOLTXMASN'8 THEOREM OS THB AVERAGE DISTRIBUTION

the axes of * and y about tliat of z till A becomes a maximum, the condition
of which is

We ahall abo reckon velocities with reference to the centre of mass of
the tyatem JA

With these simplifications we find

L'-L

V

•_• '

m

m

1-a/iz

"

(82).

We are now able to calculate the energy of rotation, J', of the minor

a'F*+b'G't+c'H't-2l'G'H'-'2m'H'F'-2nfF'G' (83),

-, jVa/iZ5 - 2vpz (Fa - Hm) + /tz' (Fa - JHrn)1]

-iir)+pt*(Gi}-iny]

(84).

Combining these results and reducing we find for the energy of internal
motion of the system M'
/' - /- i/i (1 - fyu1)-' (u - Gb + in)'- - $n (1 - a^Y1 (v - Fa + HI)" - ^w1.

Hence

the integration being extended to all values of u, v, and w which make /'
positive.

D'

Now (1 — a/iO*) (1 — fyi2*) = -yj,

and this is an invariant.

OF ENERGY IN A SYSTEM OF MATERIAL POINTS. 735

Hence in general, whatever axes we choose,

(({ i «

J J J [^ A-J ' 1.,'du.dv.dw. = - V - I [ifflJ-^Jf/ AT*/;*- ..... (87).

For the system consisting of the two particles m, and m2 the energy of
rotation is

and the energy of internal motion is

JW-jf*- (89)'

Hence we may write equation (70)

f3""7 / M \~i A

\ W) 'JW /

\ I Ill-ill tint

We have first to express J2 in terms of quantities having the suffix 3.

If we make the plane of yz pass through the three particles ml, m2, m3,
so that the origin coincides with their centre of mass and has the same velocity,
and the axis of z passes through m3, then a is twice the area of the triangle
whose vertices are mlt m3 and m3,

,u3, H, = H3 (91),

0 (92),

1 , M3m3 z3
^

M.GA v 2 _ _4 2

"h V + '

M,

,

We have now to integrate

I \I~*dv3dw3,

extending the integration to all values of v, and wt which make 73 positive, and

BOLTMAXX'* THBOWai OK THB ATEBAOB DISTRIBUTION

remembering that equation (92) shews that «. is independent of v, and «v

The result it

* (94\

.«• for the three particles m,, TO,, TO,,

l ............... (95),

where r», r. and rn ore the distances between the particles, and a is the area
of the triangle »«,«'.'">•

Also rBXmi+ r,,'ro,»»i + n.X™»= famr* + M*™**f) ......... (96).

We may now write equation (90) in the form

^l^..*«'. ........... (97).

Continuing the integration by equation (87) we find

*i-8 /r>l\w-« , . Sn -8

C /_ (mi...mn}-*Mn-* I?.-*/."!- .......... (98),

where /, is what we have defined as the energy of internal motion of the
system, or the work which the system would do, in virtue of its motion,
against the system of internal forces which would be called into play if the
distances between the parts of the material system were in an insensibly small
time to become invariable.

In order to determine the number of systems in a given configuration for
which the velocity-components of the particle mn lie between the limits
•*±|C/H, v±ldv, w±$dw, we must form the expression for N (b, VH, rn, «•„)
by stopping short before the last triple integration.

We thus find N(b, un, vn, wn)

r /^"1""7"-0 -'

r

OF ENERGY IN A SYSTEM OF MATERIAL POINTS. 737

If, as in equations (82) to (86), we suppose the origin of co-ordinates to
be the centre of mass of the whole system, the axis of z to pass through
the particle mn, and the axes of x and y to be in the directions of the
principal axes of the section of the momental ellipsoid normal to z, then writing

g=u-qz, t) = v+pz, t, = w ....................... (100),

so that £ 77, £ are the velocity- components of mn relative to axes moving as
the system would do if it were then to become rigid, with the angular velocity
whose components are p, q, r, we may write

The sum of the last three terms of this expression, with its sign taken positive,
represents the part of the internal motion of the system which is due to the
fact that the particle mn is moving with the relative velocity whose components
are £ 77, £.

"We may also define it as the work which would be done by the particle
mn against the internal forces of the system, if these forces were suddenly to
become such as to render the whole system rigid in an infinitely short time.

Comparing this result with that obtained in equation (48), we see that
the law of distribution of the velocities of the particle mn is the same as
what it would be in a fixed vessel containing n — 2 particles, provided that we
substitute for wa, if, V? the quantities (l — fynz2)"^2, (1 — a/AZ2)"1??2, £" respectively.

Hence the mean square of the velocity in the direction of the line joining
the particle with the centre of mass is the same at all points of the system,
but the mean square of the velocity in other directions is less than this in
the ratio of 1— a/zz2 to 1, where z is the perpendicular from the centre of mass
on the line of relative motion of the particle, and a is the moment of mobility
of the system about an axis through the centre of mass and normal to the
plane through that centre and the line of motion.

When the product of the mass of the particle into the square of its
distance from the centre is so small that it may be neglected in comparison
with the moments of inertia of the system, then quantities like a/*z2 and fytz2
may be neglected in respect of unity, and we may assert that the mean square
of the relative velocity, for a particle of given mass, is the same in all directions
and at all points of the system; but that for different particles it varies

VOL. II. 9S

; , BOLT«IIAin*'i TBHMM* ON THB AVERAGE DISTRIBUTION

mrefttlr M their masses; eo that the average energy of motion relative to
the noring axes i« the aame for particles of all kinds throughout the system.

We h»re already learned from equation (98) that in a free system of n

particle* the number of cases in which the system is in a given configuration,

( othtr wnrft. the probability of that configuration, is proportional to the

! power of the energy of internal motion corresponding to that configuration.

next to consider the manner in which this probability depends
oo the position of a particular particle, say of the last particle, mn.

Let /.** denote the energy of internal motion of the complete system when
M. is at the centre of mass of the system and is without any velocity relative
to that centre. It is manifest that in this case mn contributes nothing towards
the energy of internal motion.

Now let w. be carried from the centre of mass to the point (0, 0, z) and
left there without any velocity (that is, let u = v = w=Q).

Let W be the work which must be done against the forces of the system
to eflect this transference, then since the total energy of the system and the
three angular momenta must be maintained constant, we shall have after this
displacement, for the energy of internal motion of the remaining n — 1 particles,

/.., = /.«- W .............................. (102).

But by equation (85)

Substituting the value of /,,_, from equation (102), and remembering that
•i = r = ir = 0, we find for the energy of internal motion in the new configuration

/. = /."- TT+i/itl-ft/u^-'oy + liL(l-a,a?)j& ........... (104).

The probability, therefore, of a configuration in which, the positions of all

the other particles being given, that of mn is varied, is proportional to /„"*",
/. being given by equation (104).

When, as in the case of a gas, there are a great many particles similar
to «., we may speak of the density of the medium consisting of such particles
in the element dxdydz. In this case, however, for reasons already given, neglect

OF ENERGY IN A SYSTEM OF MATERIAL POINTS. 739

the quantities c^z2 and fyz2, and we may write m for /n. We may also choose
our axes in the manner which is most convenient. We shall therefore make
the axis of z that round which the system, if it were rendered rigid, would
rotate with velocity w, and we shall suppose this axis to be vertical, as
otherwise a steady motion under the action of gravity could not exist, and we
shall denote the horizontal distance from this axis by r.

We may now write for the density of the gas at the point (z, r)

Sn-S

p = p0[l + (2ln(<>})~* (ma>V — 2mgz)^ 2 (105),

where pg is the density at the origin.

When n is a very large number and when the second term of the binomial
is very small compared with unity, we may write for this the exponential expression

P = P«e ............................... (106).

If m0 is the mass of a molecule of hydrogen, /zm0 will be the mass of a
molecule of the kind of gas considered, where p. is the chemical equivalent of
the gas.

Also if T is the temperature on the centigrade scale, and a the coefficient
of dilatation of a perfect gas, then since the " velocity of mean square " of
agitation of the molecules of hydrogen at 0"C. is I'844xl05 centimetres per
second, the kinetic energy of agitation of a system contain ing n molecules of
any kind will be

and the difference between this and the energy of internal motion may be
neglected.

We thus find for the density at any point

»' (1-844) HO'" (1+aiO /1A'7\

P = P«? ............................ (107).

Let us now consider a tube of uniform section placed on a whirling table
so that one end, A, of the tube coincides with the axis while the other end,
B, revolves about the axis with the angular velocity w. The linear velocity
of B is car, and we shall suppose, for the sake of easy calculation, that this
velocity is one-tenth of the velocity of agitation of the molecules of hydrogen.

93—2

740 Botmuira'a THBOBKM ON THE AVEIUGB DISTRIBUTION

The velocity of the end /? would be 184'4 metres per second. If the tube
• hydrogen at 0"C, the ratio of the density of the gas at B to the
at A will be «•*«, or approximately 1 +*fo-
If it contains a gas whose chemical equivalent is p, the ratio will be

JL
200'

If the tube contains hydrogen and carbonic acid, and if a certain volume
of the tube at A contains 200 parts of hydrogen and 200 of carbonic acid,
then an equal volume of the tube at B will contain 201 parts of hydrogen
and 222 part* of carbonic acid.

The time during which the experiment would require to be continued in
older to obtain a given degree of approximation to the ultimate distribution
of the mixed gases varies as the square of the length of the tube.

Thus in Loschmidt's experiments on the diffusion of gases he used a tube

about a metre long, and continued his experiments from half an hour to an

hour in order to obtain the results from which he could best deduce the
coefficient of diffusion.

In these experiments the inequalities of distribution of hydrogen and
carbonic acid were reduced to less than a third part of their original value
in h«lf an hour, and if the experiment had gone on for two hours the
differences from the ultimate distribution would have been reduced to a hundredth
part of their original value.

We may therefore consider two hours as ample time for an experiment
on the ultimate distribution of these two gases in a tube one metre in length.

But if we make the whirling tube 20 centimetres long, the differences of
distribution from the ultimate distribution would be reduced to a hundredth
part of their original value in a twenty-fifth part of the time, that is to say
in 4 minutes 48 seconds.

If it were found more convenient to have bulbs on the ends of the tubes,

so as to be able to secure the gas at each end before it got mixed up by

the violent commotion arising from the stopping of the whirling tube, we
should have to allow a longer time for the whirling.

OF ENERGY IN A SYSTEM OP MATERIAL POINTS. 741

In order to obtain a similar distribution of the two gases in a vertical
tube by the action of gravity the tube would require to be 1720 metres high,
and in order to obtain the same degree of approximation to the ultimate
distribution we should have to let the experiment go on for 675 years,
carefully preserving the tube during that time from all inequalities of tempera-
ture, which, by causing convection-currents, would continually mix up the gases
and prevent their partial separation.

[From Nature, VoL XVIIL]

XCV. The Telephone (Rede Lecture).

WHEN, about two years ago, news came from the other side of the
Atlantic that a method had been invented of transmitting, by means of
electricitv, the articulate sounds of the human voice, so as to be heard
hundreds of miles away from the speaker, those of us who had reason to
believe that the report had some foundation in fact, began to exercise our
in picturing some triumph of constructive skill — something as far

surpassng Sir William Thomson's Siphon Recorder in delicacy and intricacy as
that is beyond a common bell-pull. When at last this little instrument appeared,
consisting, as it does, of parts, everyone of which is familiar to us, and capable
of being put together by an amateur, the disappointment arising from its
humble appearance was only partially relieved on finding that it was really
able to talk.

But perhaps the telephone, though simple in respect of its material and
construction, may involve some recondite physical principle, the study of which
might worthily occupy an hour's time of an academic audience : I can only
aay that I have not yet met anyone acquainted with the first elements
of electricity who has experienced the slightest difficulty in understanding the
physical process involved in the action of the telephone. I may even go
further, and say that I have never seen a printed article on the subject, even
in the columns of a newspaper, which shewed a sufficient amount of mis-
apprehension to make it worth preserving — a proof that among scientific subjects
the telephone possesses a very exceptional degree of lucidity.

However, if the telephone has something to say for itself, it would seem
hardly necessary for me to take up your time with any tedious introduction.
It is unfortunate, however, that up to the present time the telephone has kept
all his more perfect utterances to be whispered into the privileged ear of a
dingle listener. When he is older, he may get more accustomed to public

THE TELEPHONE. 743

speaking, but if we force him, in his present immature state, to exert his voice
beyond what is good for him, it may sound rather too like the pot quarrelling
with the kettle, and may call for the criticism with which Mr Tennyson's
Princess complimented the disguised Prince on his " Song of the Swallow : " —

»

"Not for thee, she said,
O Bulbul, any rose of Gulistan
Shall burst her veil: marsh divers rather, maid,
Shall croak thee sister, or the meadow crake
Grate her harsh kindred in the grass."

Is it for this, then, that we are to forsake the luncheons and lawn tennis
and all the engrossing studies of the May Term, and to assemble in this
solemn hall, where the very air seems thick with the accumulation of unsolved
problems, or else redolent of the graces of innumerable congregations ?

It is not by concentrating our minds on any problem, however important,
but rather by encouraging them to expand, that we shall best fulfil the intention
of Sir Robert Rede when he founded this lecture.

It would be as useless as it would be tedious to try to explain the
various parts of this small instrument to persons in every part of the Senate
House. I shall, therefore, consider the telephone as a material symbol of the
widely separated departments of human knowledge, the cultivation of which has
led, by as many converging paths, to the invention of this instrument by
Professor Graham Bell.

For whatever may be said about the importance of aiming at depth
rather than width in our studies, and however strong the demand of the present
age may be for specialists, there will always be work, not only for those who
build up particular sciences and write monographs on them, but for those who
open up such communications between the different groups of builders as will
facilitate a healthy interaction between them. And in a university we are
especially bound to recognise not only the unity of science itself, but the
communion of the workers in science. We are too apt to suppose that we are
congregated here merely to be within reach of certain appliances of study, such
as museums and laboratories, libraries and lecturers, so that each of us may
study what he prefers. I suppose that when the bees crowd round the flowers
it is for the sake of the honey that they do so, never thinking that it is
the dust which they are carrying from flower to flower which is to render

THE TELEPHONE.

possible a more splendid array of flowers, and a busier crowd of bees, in the
years to come. We cannot, therefore, do better than improve the shining hour
Helping forward the cross-fertilization of the sciences.

pofmD ^ -o further, I wish to express my obligation to Mr Garnett for
the able sssistsntr he has given me. He has not only collected the apparatus
before you, but constructed some of it himself. But for him, I might have
given you some second-hand information about telephones. He has made it
ptmiMft ^ you to hear something yourselves. I have also to thank Mr Gower,
who has brought his telephone harp, and Mr Middleton, who has contributed
several instruments of his own invention.

We shall begin with the telephone in its most obvious aspect, as an
instrument depending on certain physical principles.

The apparatus consists of two instruments, the transmitter and the receiver,
doubly connected by a circuit capable of conducting electricity. The speaker
»«lfc« to the transmitter at one end of the line, and at the other end of the
line the listener puts his ear to the receiver, and hears what the speaker says.

The process in its two extreme stages is so exactly similar to the old-
fashioned method of speaking and hearing that no preparatory practice is
required on the part of either operator.

We must not, however, fell into the error of confounding the principle of
the electric telephone with that of other contrivances for increasing the distance
at which a conversation may be carried on. In all these the principle is the
same as in the ordinary transmission of sound through the air. The different
|H>rtions. of matter which intervene between the speaker and the hearer take
part, in succession, in a certain mechanical process. Each receives a certain
motion from the portion behind it and communicates a precisely similar motion
to the portion in front of it, in doing which it gives out all the energy it
received, and is again reduced to rest.

The medium which takes part in this process may be the open air, or air
confined in a long tube, or some other medium such as a brick wall, as when
we hear what goes on in the next house, or a long wooden rod, or a metal
wire, or even a stretched string. In all these it is by the actual motion of
the successive portions of the medium that the message is transmitted.

In the electric telephone there is also a medium extending from the one
instrument to the other. It is a copper wire, or rather two wires forming a
dosed circuit. But it is not by any motion of the copper that the message

THE TELEPHONE. 745

is transmitted. The copper remains at rest, but a variable electric current
flows to and fro in the circuit.

It is this which distinguishes the electric telephone from the ordinary
speaking tube, and from the transmission of vibrations along wooden rods by
which Sir Charles Wheatstone used to cause musical instruments to sound in
a mysterious manner without any visible performer.

On the other hand, we have to distinguish the principle of the articulating
telephone from that of a great number of electrical contrivances which produce
visible or audible signals at a distance. Most of these depend on the alternate
transmission and interruption of an electric current. In some part of the
circuit a piece of apparatus is introduced corresponding to this instrument which
is called a key. Whenever two pieces of metal, called the contact pieces, touch
each other, the current flows from the one to the other, and so round the
circuit. Whenever the contact pieces are separated the current is interrupted,
and the effects of this alternation of current and no current may be made to
produce signals at any other part of the circuit.

In the Morse system of signalling, currents of longer and of shorter duration
are called dashes and dots respectively, and by combinations of these the
symbols of letters are formed. The rate at which these little currents succeed
one another depends on the rate at which the operator can work the key,
and may be increased by mechanical methods till the receiving clerk can no
longer distinguish the symbols.

But the capability of the telegraph wire for transmitting signals is by no
means exhausted ; as the rapidity of the succession is increased, the ear ceases
to distinguish them as separate signals, but begins to recognise the impression
it receives as that of a musical tone, the pitch of which depends on the
number of currents in a second.

Tuning forks driven by electricity were used by Helmholtz in his researches
on the vowel sounds, and the periodically intermittent current which they
furnish is recognised as a most valuable agent in physical and physiological
research. The tuning forks are of the most massive construction, and the
succession of currents goes on with the most inflexible regularity, so that
whenever we have occasion to follow the march of a process which takes place
in a short time, such as the vibration of a violin string or the twitch of a
living muscle, the tuning fork becomes our appropriate timepiece.

Apparatus of this kind, however, the merit of which is its regularity, is quite

VOL. II. 94

TUB TELEPHONE.

of adapting itself to the transmission of variable tones such as those

of a melody.

The first suiMiiMsfiil attempt to transmit variable tones by electricity was
MiJT ky pjjijip jfcj^ a teacher in a school at Friedrichsdorf, near Homburg.
Oft October SI, 1861, Reis showed his instrument, which he called a telephone,
to the Ptmwml Society of Frankfort on the Main. He succeeded in transmitting
melodies which were distinctly heard about the room.

The transmitter of Beis's telephone is essentially a make and break key
of so ddint* a construction that the sound-waves in the air are able to

work it

The sir vibrations set in motion a stretched membrane like a drumhead,
with a piece of platinum fastened to it. This piece of platinum, when vibrating,
strikes against another piece of platinum, and so completes the circuit every
time contact is made.

At every point of the circuit there is thus a series of currents corresponding
in number to the vibrations of the drumhead, and by causing these to pass
through the coil of an electromagnet, the armature of the electromagnet is
attracted every time the current passes, and if the armature is attached to a
resonator of any kind, the succession of tugs will set it in vibration, and cause
it to emit a sound, the pitch of which is the same as that of the note sung
into the transmitter at the other end of the line.

[Mr Cower here played the "March of the Men of Harlech" on the
telephone harp placed in the Geological Museum. The instrument consists of
a set of steel reeds worked by percussion, which make and break contact on
the battery circuit, of which the primary wire of an induction coil forms part.
The receivers are worked by the secondary current. There were four receivers,
one of them Prof. Bell's original one, placed in different parts of the Senate-house.]

If the pitch of a sound were the only quality which we are able to
distinguish, the problem of telephony would have received its complete solution
in the instrument of Reia. But the human ear is so constructed, and we
ourselves are so trained by continual practice, that we recognise distinctions
in sound of a far more subtle character than that of pitch ; and these finer
distinctions have become so much more important for the purposes of human
intercourse than the musical distinction of pitch, that many persons can detect
the slightest variation in the pronunciation of a word who are comparatively
indifferent to the variations of a melody.

THE TELEPHONE. 747

Now, the telephone of Prof. Graham Bell is an articulating telephone,
which can transmit not only melodies sung to it, but ordinary speech, and
that so faithfully that we can often recognise the speaker by his voice as
heard through the telephone. How is this effected ? It is manifest that if by
any means we can cause the tinned plate of the receiving instrument to vibrate
in precisely the same manner as that of the transmitter, the impression on the
ear will be exactly the same as if it had been placed at the back of the
plate of the transmitter, and the words will be heard as if spoken at the
other side of a tinned plate.

But this implies an exact correspondence, not only in the number of
vibrations, but in the type of each vibration.

Now, if the electrical part of the process consisted merely of alternations
between current and no current, the receiving instrument could never elicit
from it the semblance of articulate speech. If the alternations were sufficiently
regular, they would produce a sound of a recognisable pitch, which would be
very rough music if the pitch were low, but might be less unendurable if the
pitch were high ; still, at the best, it would be like playing a violin with a
saw instead of a bow.

What we want is not a sudden starting and stopping of the current, but
a continuous rise and fall of the current, corresponding in every gradation and
inflexion to the motion of the air agitated by the voice of the speaker.

Prof. Graham Bell has recounted the many unsuccessful attempts which he
made to produce undulatory currents instead of mere intermittent ones. He
had, of course, to give up altogether the method of making and breaking
contact. Every method involving impact of any kind, whether between electric
contact pieces or between the sounding parts of the instrument, introduces
discontinuity of motion, and therefore precludes a faithful reproduction of
speech.

In the ultimate form which the telephone in his hands assumed, the electric
current is not merely regulated but actually generated by the aerial vibrations
themselves.

The electric principle involved in Bell's telephone is that of the induction
of electric currents discovered by Faraday in 1831. Faraday's own statement
of this principle has been before the scientific world for nearly half a century,
but has never been improved upon.

Consider first a conducting circuit, that is to say, a wire which after any

94—2

TUB TELKTHONK.

of coaroluuon» returns into iteelf. Round such a circuit an electric
my JIM, and will flow if there is an electromotive force to drive it.

lest a li»« of magnetic force, such a line as you see here made

by •prinkling iron filing* on a sheet of paraffin paper. This line, as
r also tint showed, ia a line returning into itself, or, as the mathema-

woald my. it ia a closed curve.

v if there are two closed curves in space, they must either embrace
•Doctor so aa to be linked together, or they must not embrace each other.
If the line of force as well as the circuit were made of wire, and if it
he copper circuit, it would be impossible to unlink them without
cutting one or other of the wires. But the line of force is more like one of
Milton'* spirit*, which cannot

"In their liquid texture mortal wound
Receire, no more than can the fluid air."

Now, if the copper circuit or the lines of force move relatively to each
other, then in general some of the lines of force which originally embraced the
circuit will CIQMQ to embrace it, or else some of those which did not embrace
it will become linked with it,

For every line of force which ceases to embrace the circuit there is a
certain amount of positive electromotive force, which, if unopposed, will generate
a current in the positive direction, and for every new line which embraces
the circuit there is a negative electromotive force, causing a negative current.

In Bell's telephone the circuit forms a coil round a small core of soft iron
fcatened to the end of a steel magnet. Now lines of magnetic force pass more
freely through iron than through any other substance. They will go out of
their way in order to pass through iron instead of air. Hence a large proportion
of the lines of force belonging to the magnet pass through the iron core, and,
therefore, through the coil, even though there is no iron beyond the core, so
that they have to complete their circuit through air.

Hut if another piece of soft iron is placed near the end of the core it
will afford greater facilities for lines which have passed through the core to
complete their circuit, and so the lines belonging to the magnet will crowd
•till closer together to take advantage of an easy passage through the core
and the iron beyond it. If then the iron is moved nearer to the core, there
will be an increase in the number of such lines, and, therefore, a negative
current in the circuit If it is moved away there will be a diminution in the

THE TELEPHONE. 749

number of lines, and a positive current in the circuit. This principle was
employed by Page in the construction of one of the earliest magneto-electric
machines, but it was reserved for Prof. Bell to discover that the vibrations of
a tinned iron plate, set in motion by the voice, would produce such currents
in the circuit as to set in motion a similar tinned plate at the other end of
the line.

It will help us to appreciate the fertility of that germ of science which
Faraday first detected and developed if we recollect that year after year he
had employed the powerful batteries and magnets and delicate galvanometers of
the Royal Institution to obtain evidence of what he all along hoped to discover
— the production of a current in one circuit by a current in another, but all
without success, till at last he detected the induced current as a transient
phenomenon, to be observed only at the instant of making or breaking the
primary circuit.

In less, than half a century, and by the aid of no second Faraday, but
in the course of the ordinary growth of scientific principles, this germ, so
barely caught by Faraday, has developed on the one hand into the powerful
currents which maintain the illumination of the lighthouses on our coasts ; and
on the other, into these currents of the telephone which produce an audible
effect, though the engine that drives them is itself driven by the tremors of a
child's voice.

Prof. Tait has recently measured the absolute strength of these telephone
currents. He produced them by means of a tuning fork vibrating in front of
the coil of the transmitter. Before the transmitted note ceased to be audible
at the other end of the line he measured by means of a microscope the
amplitude of the vibrations of the fork.

He then placed a very delicate galvanometer in the circuit and found what
deflection was produced by a measured motion of the fork.

Finally he measured the deflection of the galvanometer produced by a small
electromotive force of known magnitude. He thus found that the telephone
currents produced an audible effect when reversed 500 times a second, though
their strength was no greater than what a Grove's cell would send through a
million megohms, about a thousand million times less than the currents used
in ordinary telegraphic work.

One great beauty of Prof. Bell's invention is that the instruments at the
two ends of the line are precisely alike. When the tin plate of the transmitter

THE TELEPHONE.

appraaflhee the oore at ita bobbin it produces a current in the circuit, which
ha* alrr to circulate round the bobbin of the receiver, and thus the core of the
M .TrJuntf more or leas magnetic, and attracts its tin plate with greater
^ fam. Thus the tin plate of the receiver reproduces on a smaller
but with perfect fidelity, every motion of the tin plate of the transmitter.
Thia perfect symmetry of the whole apparatus — the wire in the middle,
the two telephone* at the end of the wire, and the two gossips at the ends
of the telephone* — may be very fascinating to a mere mathematician, but it
would not satisfy an evolutionist of the Spencerian type, who would consider
anything with both ends alike to be an organism of a very low type, which
BM4 hare ita functions differentiated before any satisfactory integration can
Uke place.

Accordingly, many attempts have been made, by differentiating the function
of the transmitter from that of the receiver, to overcome the principal limitation
to the power of the telephone. As long as the human voice is the sole motive
power of the apparatus it is manifest that what is heard at one end must
be fainter than what is spoken at the other. But if the vibration set up by
the voice is used no longer as the source of energy, but merely as a means
of modulating the strength of a current produced by a voltaic battery, then
there will be no necessary limitation of the intensity of the resulting sound,
ao that what is whispered to the transmitter may be proclaimed ore rotunda
by the receiver.

A result of this kind has already been obtained by Mr Edison by means
of * transmitter in which the sound vibrations produce a varying pressure on
a piece of carbon, which forms part of the electric circuit. The greater the
prwHure, the smaller is the resistance due to the insertion of the carbon, and
therefore the greater is the current in the circuit.

I have not yet seen Mr Edison's transmitter, but the microphone of Prof.

Hughes IB an application of carbon and other substances to the construction

of a transmitter, which modulates the intensity of a battery current in more

or leas complete accordance with the sound-vibrations it receives. The energy

the aound produced is no longer limited by that of the original sound. All

the original sound does is to draw supplies of energy from the battery,

very feeble sound may give rise to a considerable effect. Thus, when

a fly walks over the table of the microphone the sound of his tramp may be

heard miles off

THE TELEPHONE. 751

Indeed, the microphone seems to open up several new lines of research.
We shall have London physicians performing stethoscopic auscultations on patients
in all parts of the kingdom. The Entomological Society have been much interested
by Mr Wood-Mason's discovery of a stridulating apparatus in scorpions. Perhaps
ere long a microphone, placed in a nest of tropical scorpions, may be connected
up to a receiver in the apartments of the society, so as to give the members
and their musical friends an opportunity of deciding whether the musical taste
of the scorpion resembles that of the nightingale or that of the cat.

I have said that the telephone is an instance of the benefit to be derived
from the cross-fertilization of the sciences. Now this is an operation which
cannot be performed by merely collecting treatises on the different sciences, and
binding them up into an encyclopaedia. Science exists only in the mind, and
the union of the sciences can take place only in a living person.

Now, Prof. Graham Bell, the inventor of the telephone, is not an electrician
who has found out how to make a tin plate speak, but a speaker, who, to
gain his private ends, has become an electrician. He is the son of a very
remarkable man, Alexander Melville Bell, author of a book called "Visible
Speech," and of other works relating to pronunciation. In fact, his whole life
has been employed in teaching people to speak. He brought the art to such
perfection that, though a Scotchman, he taught himself in six months to speak
English, and I regret extremely that when I had the opportunity in Edinburgh
I did not take lessons from him. Mr Melville Bell has made a complete
analysis and classification of all the sounds capable of being uttered by the
human voice, from the Zulu clicks to coughing and sneezing; and he has
embodied his results in a system of symbols, the elements of which are not
taken from any existing alphabet, but are founded on the different configurations
of the organs of speech.

The capacities of this new mode of representing speech have been put to
the test by Mr Alexander J. Ellis, author of "The Essentials of Phonetics,"
a gentleman who has studied the whole theory of speech acoustically, philologically,
and historically. He describes the result in a letter to The Reader: —

" The mode of procedure was as follows : — Mr Bell sent his two sons, who
were to read the writing, out of the room — it is interesting to know that the
elder, who read all the words in this case, had only had five weeks' instruction
in the use of the alphabet — and I dictated slowly and distinctly the sounds which
I wished to be written. They consisted of a few words in Latin, pronounced

75S

THE TELEPHONE.

at Eton, then as in Italy, and then according to some theoretical
of how the Latins might have uttered them. Then came some English
and affected pronunciations, the words 'how odd' being given in
distinct ways. Suddenly German provincialisms were introduced ; then
of sounds often confiised. Some Arabic, some Cockney English,
with an intiwduoed Arabic guttural, some mispronounced Spanish, and a variety

of rowels and diphthongs.

"The remit was perfectly satisfactory — that is, Mr Bell wrote down my
and purposely exaggerated pronunciations and mispronunciations, and

: :. ••••!• tittd li> lOOa, licit liiivinn; hc;ml tlu-ni,

ao attend them as to surprise me by the extremely correct echo of my own

voice Accent, tone, drawl, brevity, indistinctness were all reproduced with

surprising accuracy. Being on the watch, I could, as it were, trace the
alphabet in the lips of the readers. I think, then, that Mr Bell is justified
in the somewhat bold title which he has assumed for his mode of writing—
'Visible speech.' I only hope that for the advantage of linguists, such an
alphabet may soon be made accessible, and that, for the intercourse of nations,
it may be adopted generally, at least for extra-European nations, as for the
Chinese dialect and the several extremely diverse Indian languages, where such
an alphabet would rapidly become a great social and political engine."

The inventor of the telephone was thus prepared, by early training in the
practical analysis of the elements of speech, to associate whatever scientific
knowledge he might afterwards acquire with those elementary sensations and
actions, which each of us must learn from himself, because they lie too deep
within us to be described to others. This training was put to a very severe
test, when, at the request of the Boston Board of Education, Prof. Graham
Bell conducted a series of experiments with his father's system in the Boston
School for the Deaf and Dumb. I cannot conceive a nobler application of the
scientific analysis of speech, than that by which it enables those to whom all
sound is

"expunged and rased
And wisdom at one entrance quite shut out,"

not only to speak themselves, but to read by sight what other people are saving.

lie successful result of the experiments at Boston is not only the most

valuable testimonial to the father's system of visible speech, but an honour

THE TELEPHONE. 753

which the inventor of the telephone may well consider as the highest he has
attained.

An independent method of research into the process of speech was employed
by Wheatstone, Willis, and Kempelen, the aim of which was to imitate the
sounds of the human voice by means of artificial apparatus. This apparatus
was in some cases modelled so as to represent as nearly as possible the form
as well as the functions of the organs of speech, but it was found that an
equally good imitation of the vocal sounds could be obtained from apparatus
the form of which had no resemblance to the natural organs.

Several isolated facts of considerable importance were established by this
method, but the whole theory of speaking and hearing has been so profoundly
modified by Helmholtz and Bonders, that much of what was advanced before
their time has come to possess only an historical interest.

Among all the recent steps in the progress of science, I know none of
which the truly scientific or science-producing consequences are likely to be so
influential as the rise of a school of physiologists, who investigate the conditions
of our sensations by producing on the external senses impressions, the physical
conditions of which can be measured with precision, and then recording the
verdict of consciousness as to the similarity or difference of the resulting sensa-
tions.

Prof. Helmholtz, in his recent address as Rector of the University of Berlin,
lays great stress on that personal interaction between living minds, which I
have already spoken of as essential to the life of a University. " I appreciate,"
he says, "at its full value this last advantage, when, looking back, I recall
my student days, and the impression made upon us by a man like Johannes
Muller, the physiologist. When one finds himself in contact with a man of
the first order, the entire scale of one's intellectual conceptions is modified for
life; contact with such a man is perhaps the most interesting thing life may
have to offer."

Now, the form in which Johannes Muller stated what we may regard as
the germ which fertilized the physiology of the senses is this, that the difference
in the sensations due to different senses does not depend upon the actions
which excite them, but upon the various nervous arrangements which receive
them.

To accept this statement out of a book, as a matter of dead faith, may not
be difficult to an easy-going student; but when caught like a contagion, as

95

VOL. II.

Tint TELEPHONE.

it, from the lip« of the living teacher, it has become the

of a life of research.

No 'taut 'ha* done man than llelmholtz to open up paths of communication
i^^ departments of human knowledge; and one of these, lying in
attract!™ region than that of elementary psychology, might be explored

udlv favourable conditions, by some of the fresh minds now
»c« j »« •"/

coming op to Cambridge.

Helmholu, by a aeries of daring strides, has effected a passage for himself
orer that untrodden wild between acoustics and music — that Serbonian bog
whole armies of scientific musicians and musical men of science have
without filling it up.
We mar not be able even yet to plant our feet in his tracts and follow
him right across. That would require the seven league boots of the German
ooloMUs; but to help us in Cambridge we have the Board of Musical Studies,
vindicating for music its ancient place in a liberal education. On the physical
we have Lord Bayleigh laying the foundation deep and strong in his
of Sound, On the aesthetic side we have the University Musical
Society doing the practical work, and in the space between, those conferences
of Mr Sedley Taylor, where the wail of the siren draws musician and mathematician
together down into the depths of their sensational being, and where the gorgeous
hoot of the phoneidoecope are seen to seethe and twine and coil like the

"Dragon boughs and elvish emblemings"

on the gates of that city where

"an ye heard a music, like enow
They are building still, seeing the city is built
To music, therefore never built at all,
And therefore built for ever."

The special educational value of this combined study of music and acoustics
w that more than almost any other study it involves a continual appeal to
what we must observe for ourselves.

The facts are things which must be felt; they cannot be learned from any
description of them.

All this has been said more than two hundred years ago by one of our
own prophets — Wdliam Harvey, of Gonville and Caius College. "For whosoever
they be that read authors, and do not by the aid of their own senses, abstract

THE TELEPHONE. 755

true representations of the things themselves (comprehended in the author's
expressions) they do not represent true ideas, but deceitful idols and phantasms,
by which they frame to themselves certain shadows and chimseras, and all their
theory and contemplation (which they call science) represents nothing but waking
men's dreams and sick men's phrensies."

Prof. Maxwell was assisted in his practical demonstrations by Mr Garnett,
of St John's College.

95—2

[Prom tfatttrt, VoL

XCVI. — Paradoxical Philosophy*.

On otumipg this book, the general appearance of the pages, and some of
UM phram on which we happened to light made us somewhat doubtful whether
\y within our jurisdiction, as it is not the practice of Nature to review
either norels or theological works.

In the dedication, however, the book is described as an account of the
of a learned society, a species of literature which we are under a
to rescue from oblivion, even when, as in this case, the proceed-
those of one of those jubilee meetings, in which learned men seem
to aim nther at being lively than scientific.

On the title-page itself there is no name to indicate whether the author
M one of those who by previous conviction have rendered themselves liable to
our surveillance, but on the opposite page we find The Unseen Universe; or,
Pkynaal Speculations on a Future State, to which this book is a " Sequel,"
Moribod to the well-known names of Balfour Stewart and P. G. Tait.

Mr Browning has expressed his regret that the one volume in which
lUfaelle wrote his sonnets, and the one angel which Dante was drawing when
he was interrupted by "people of importance," are lost to the world. We
ahall therefore make the most of our opportunity when two eminent men of
atteooe, "driven," as they tell us, "by the exigencies of the subject," have
laid down all the instruments of their art, shaken the very chalk from their
hands, and, locking up their laboratories, have betaken themselves to those
blissful country seats where Philonous long ago convinced Hylas that there
can be no heat in the fire and no matter in the world; and where in more
recent times, Peacock and Mallock have brought together in larger groups the
more picturesque of contemporary opinions.

In this book we do not indeed catch those echoes of well-known voices
in which the citizens of the "New Republic" tell us how they prefer to

ital PkilotojAy. A Sequel to The Untem Univerte (London : Macmillan <fe Co., 1878).

PARADOXICAL PHILOSOPHY. 757

regard themselves as thinking, taking care, all the while, that no actual thought
shall disturb their enjoyment of the luxury of extravagant opinion. The
members of the Paradoxical Society, with their guest, Dr Hermann Stoffkraft,
are far too earnest to adopt this pose of mind, but they exhibit that sympathy
in fundamentals overlaid with variety in opinions which is one of the main
conditions of good-fellowship. Dr Stoffkraft, in spite of his name and of his
office as the single-handed opponent of the thesis of the book, makes it his
chief care so to brandish his materialistic weapons as not to hurt the feelings
of his friends ; and when, near the end of the book, he gets a little out of
temper, it is about matters with which a materialist, as such, has no concern.

As the book is not a novel there is no literary reason for not telling
" what became of the Doctor," as narrated in the last chapter. He goes to
Strathkelpie Castle to take part in an investigation of spiritualistic phenomena.
He begins by detecting the mode in which one young lady performs her spirit-
rapping, but forthwith falls into an " electro-biological " courtship of another,
and, this proving successful, he is persuaded by his wife and her priest to
renounce the black arts in the lump as works of the foul fiend ; and then
we are told that, having quieted his spirit by a few evolutions in four
dimensions, he has now settled down to compose his Exposition of the Relations
between Religion and Science, which he intends to be a thoroughly matured
production.

The Doctor — and, indeed, most of the other characters — are no mere
materialised spirits, or opinions labelled with names of the Euphranor and
Alciphron type. They do not reduce their subject to a caput mortuum by an
exhaustive treatment, but take care, like well-bred people, to drop it and pass
on to another before we have time to suspect that the last word has been
said.

We cannot accuse the authors of leading us through the mazy paths of
science only to entrap us into some peculiar form of theological belief. On the
contrary, they avail themselves of the general interest in theological dogmas
to imbue their readers at unawares with the newest doctrines of science. There
must be many who would never have heard of Carnot's reversible engine, if
they had not been led through its cycle of operations while endeavouring to
explore the Unseen Universe. No book containing so much thoroughly scientific
matter would have passed through seven editions in so short a time without
the allurement of some more human interest.

rxBAPOXICAL PHILOSOPHY.

Nor need •» ftn>r to draw down on Nature the admonition which fell on
the inner ear of the poet—

-Tboo prabvt hew whore thou art leatt ;
Tfci* fcilli hath many » purer priest,
many an abler voice than thou.

For era* thof* words and phrases which seemed at first sight to remove the
book from the field of our criticism, are found on a nearer view to have
acquired a new, and indeed a paradoxical sense, for which no right of sanctuary

be claimed.

The words on the title-page : " In te, Domine, speravi, non confundar in
may recall to an ordinary reader the aspiration of the Hebrew
the closing prayer of the "Te Deum," or the dying words of Francis
Xarier; and men of science, as such, are not to be supposed incapable either
of the nobler hopes or of the nobler fears to which their fellow-men have
attained. Here, however, we find these venerable words employed to express
a conviction of the perpetual validity of the "Principle of Continuity," enforced
by the tremendous sanction, that if at any place or at any time a single
exception to that principle were to occur, a general collapse of every intellect
in the universe would be the inevitable result.

There are other well-known words in which St Paul contrasts things seen
with things unseen. These also are put in a prominent place by the authors
of the Unxcn Universe. What, then, is the Unseen to which they raise their
thoughts! .

In the first place the luminiferous aether, the tremors of which are the
dynamical equivalent of all the energy which has been lost by radiation from
the various systems of grosser matter which it surrounds. In the second place
a still more subtle medium, imagined by Sir William Thomson as possibly
capable of furnishing an explanation of the properties of sensible bodies; on
the hypothesis that they are built up of ring vortices set in motion by some
supernatural power in a frictionless liquid: beyond which we are to suppose an
indefinite succession of media, not hitherto imagined by any one, each mani-
foldly more subtle than any of those preceding it. To exercise the mind in
•peculations on such media may be a most delightful employment for those who
are intellectually fitted to indulge in it, though we cannot see why they should
on that account appropriate the words of St Paul

PARADOXICAL PHILOSOPHY. 759

Nature is a journal of science, and one of the severest tests of a scientific
mind is to discern the limits of the legitimate application of scientific methods.
We shall therefore endeavour to keep within the bounds of science in speaking
of the subject-matter of this book, remembering that there are many things
in heaven and earth which, by the selection required for the application of
our scientific methods, have been excluded from our philosophy.

No new discoveries can make the argument against the personal existence
of man after death any stronger than it has appeared to be ever since men
began to die, and no language can express it more forcibly than the words
of the Psalmist : —

" His breath goeth forth, he returneth to his earth ; in that very day his
thoughts perish."

Physiology may supply a continually increasing number of illustrations of
the dependence of our actions, mental as well as bodily, on the condition of
our material organs, but none of these can render any more certain those facts
about death which our earliest ancestors knew as well as our latest posterity
can ever learn them.

Science has, indeed, made some progress in clearing away the haze of
materialism which clung so long to men's notions about the soul, in spite of
their dogmatic statements about its immateriality. No anatomist now looks
forward to being able to demonstrate my soul by dissecting it out of my pineal
gland, or to determine the quantity of it by the process of double weighing.
The notion that the soul exerts force lingered longer. We find it even in the
late Isaac Taylor's Physical Theory of a Future State. It was admitted that
one body might set another in motion; but it was asserted that in every case,
if we only trace the chain of phenomena far enough back, we must come to
a body set in motion by the direct action of a soul.

It would be rash to assert that any experiments on living beings have as
yet been conducted with such precision as to account for every foot-pound of
work done by an animal in terms of the diminution of the intrinsic energy of
the body and its contents; but the principle of the conservation of energy has
acquired so much scientific weight during the last twenty years that no physio-
logist would feel any confidence in an experiment which shewed a considerable
difference between the work done by an animal and the balance of the account
of energy received and spent.

Science has thus compelled us to admit that that which distinguishes a

PARADOXICAL PHILOSOPHY.

lir hodr from a doid one is neither a material thing, nor that more refined

Mfcr» of energy." There are methods, however, by which the applica-

ioa of energy may be directed without interfering with its amount. Is the

sool like the engine-driver, who does not draw the train himself, but, by means

of certain ralves, direct* the course of the steam so as to drive the engine

or backward, or to stop it?

The dynamical theory of a conservative material system shews us, however,
I* general the present configuration and motion determine the whole course
of the system, exceptions to this rule occurring only at the instants when the
les through certain isolated and singular phases, at which a strictly
force may determine the course of the system to any one of a finite
of equally possible paths, as the pointsman at a railway junction directs
the train to one set of rails or another. Prof. B. Stewart has expounded a
theory of this kind in his book on The Conservation of Energy, and MM. de
8t Venant and Boussineeq have examined the corresponding phase of some
purely mathematical problems.

The science which rejoices in the name of " Psychophysik " has made con-
siderable progress in the study of the phenomena which accompany our sensations
and voluntary motions. We are taught that many of the processes which we
nippoit entirely under the control of our own will are subject to the strictest
laws of succession, with which we have no power of interfering; and we are
shewn bow to verify the conclusions of the science by deducing from it methods
of physical and mental training for ourselves and others.

Thus 'science strips off, one after the other, the more or less gross
materialisations by which we endeavour to form an objective image of the soul,
till men of science, speculating, in their non-scientific intervals, like other men
on what science may possibly lead to, have prophesied that we shall soon have
to confess that the soul is nothing else than a function of certain complex
. • • • . • • -

Men of science, however, are but men, and therefore occasionally contemplate
their souls from within. Those who, like Du Bois-Reymond, cannot admit that
sensation or consciousness can be a function of a material system, are led to
the conception of a double mind.

On tfce one aide the acting, inventing, unconscious material mind, which puts the muscles
into motion, and determine* the world's history; this is nothing else but the mechanics of atoms,
and a nbpct to the caoail law, and on the other side the inactive, contemplative, remembering,

PARADOXICAL PHILOSOPHY. 761

fancying, conscious, immaterial mind, which feels pleasure and pain, love, and hate; this one lies
outside of the mechanics of matter, and cares nothing for cause and effect."

We might ask Prof. Du Bois-Reymond which of these it is that does right
or wrong, and knows that it is his act, and that he is responsible for it, but
we must go on to the other view of the case, which Dr Stoffkraft alludes to
at p. 78, although by some law of the Paradoxical, he is not allowed to pursue
a subject which might have afforded excellent sport to the Society.

"I feel myself compelled to believe," says the learned Doctor, "that all
kinds of matter have their motions accompanied with certain simple sensations.
In a word, all matter is, in some occult sense, alive."

This is what we may call the "levelling up" policy, and it has been
expounded with great clearness by Prof, von Niigeli in a lecture, of which a
translation was given in Nature, Vol. xvi. p. 531.

He can draw no line across the chain of being, and say that sensation
and consciousness do not extend below that line. He cannot doubt that every
molecule possesses something related, though distantly, to sensation, " since each
one feels the presence, the particular condition, the peculiar forces of the other,
and, accordingly, has the inclination to move, and under circumstances really
begins to move — becomes alive as it were;". . . "If, therefore, the molecules feel
something which is related to sensation, then this must be pleasure if they can
respond to attraction and repulsion, i.e. follow their inclination or disinclination ;
it must be displeasure if they are forced to execute some opposite movement,
and it must be neither pleasure nor displeasure if they remain at rest."

Prof, von Niigeli must have forgotten his dynamics, or he would have
remembered that the molecules, like the planets, move along like blessed gods.
They cannot be disturbed from the path of their choice by the action of any
forces, for they have a constant and perpetual will to render to every force
precisely that amount of deflexion which is due to it. Their condition must,
therefore, be one of unmixed and unbroken pleasure.

But even if a man were built up of thinking atoms would the thoughts
of the man have any relation to the thoughts of the atoms? Those who try
to account for mental processes by the combined action of atoms do so, not by
the thoughts of the atoms, but by their motions.

Dr Stoffkraft explains the origin of consciousness at p. 77 and at p. 107.
We recommend to his attention Mr Herbert Spencer's statement in his Principles
of Psychology, § 179, where he shews in a most triumphant manner how, under

VOL. II. 96

PARADOXICAL PHILOSOPHY.

esrtain arcomstances, "there must arise a consciousness." Such statements, care-
fyiy itndkrt, mar contribute to the further progress of science hi the path
which we have been describing, by shewing more clearly that consciousness
be the result of a plexus of nervous communications any more than of

of plastidule souls.

Jity is often spoken of as if it were another name for the continuity

of oooscioasness as reproduced in memory, but it is impossible to deal with
personality at if it were something objective that we could reason about. My
knowledge that I am is quite independent of my recollection that I was, and
also of my belief that, for a certain number of years, I have never ceased to
be. But as soon aa we plunge into the abysmal depths of personality we get
beyond the limits of science, for all science, and, indeed, every form of human
tpeech. is about objects capable of being known by the speaker and the hearer.
Whenever we pretend to talk about the Subject we are really dealing with an
Object under a false name, for the first proposition about the Subject, namely,
•' I am," cannot be used in the same sense by any two of us, and therefore
can never become part of science at all.

The progress of science, therefore, so far as we have been able to follow
it. baa added nothing of importance to what has always been known about the
physical consequences of death, but has rather tended to deepen the distinction
between the visible part, which perishes before our eyes, and that which we
are ourselves, and to shew that this personality, with respect to its nature as
well as to its destiny, lies quite beyond the range of science.

[From Encyclopaedia Britannica.]

XCVII. Ether.

ETHER, or ^ETHER (aWijp, probably from aW<a, I burn, though Plato in his
Cratilus (410, b) derives the name from its perpetual motion — on del del irepl
rov depa peiav, aeiQerjp SIKCUWS av /caXoiro), a material substance of a more
subtle kind than visible bodies, supposed to exist in those parts of space which
are apparently empty.

The hypothesis of an sether has been maintained by different speculators
for very different reasons. To those who maintained the existence of a plenum
as a philosophical principle, nature's abhorrence of a vacuum was a sufficient
reason for imagining an all-surrounding aether, even though every other
argument should be against it. To Descartes, who made extension the sole
essential property of matter, and matter a necessary condition of extension,
the bare existence of bodies apparently at a distance was a proof of the
existence of a continuous medium between them.

But besides these high metaphysical necessities for a medium, there were
more mundane uses to be fulfilled by aethers. ^Ethers were invented for
the planets to swim in, to constitute electric atmospheres and magnetic effluvia,
to convey sensations from one part of our bodies to another, and so on, till
all space had been filled three or four times over with aethers. It is only
when we remember the extensive and mischievous influence on science which
hypotheses about aBthers used formerly to exercise, that we can appreciate the
horror of aethers which sober-minded men had during the 18th century, and
which, probably as a sort of hereditary prejudice, descended even to the late
Mr John Stuart Mill.

The disciples of Newton maintained that in the fact of the mutual gravi-
tation of the heavenly bodies, according to Newton's law, they had a complete
quantitative account of their motions; and they endeavoured to follow out the
path which Newton had opened up by investigating and measuring the attrac-

96—2

- .

ETHER.

and rwwlsions of electrified and magnetic bodies, and the cohesive forces
i the interior of bodies, without attempting to account for these forces.

Newton himself, boweTcr, endeavoured to account for gravitation by dif-
fojao,, ,/ puHHiro in an wther (see Art. ATTRACTION*, VoL m. p. 64); but
k* did not publish his theory, "because he was not able from experiment and
ubtm fit inn to give a satisfactory account of this medium, and the manner of
to opffmtfr" in producing the chief phenomena of nature."

On the other hand, those who imagined aethers in order to explain phe-
nosMBa could not specify the nature of the motion of these media, and could
not prow that the media, as imagined by them, would produce the effects
UMT were meant to explain. The only {ether which has survived is that
which was invented by Huygens to explain the propagation of light. The
for the existence of the luminiferous aether has accumulated as addi-
fitnnm**** of light and other radiations have been discovered; and the
of this medium, as deduced from the phenomena of light, have been
to be precisely those required to explain electromagnetic phenomena.

Function o/ the other in the propagation of radiation. — The evidence for
the undulatory theory of light will be given in full, under the Article on
LIGHT, but we may here give a brief summary of it so far as it bears on
the existence of the nether.

That light is not itself a substance may be proved from the phenomenon
of interference. A beam of light from a single source is divided by certain
optical methods into two parts, and these, after travelling by different paths,
are made to reunite and fall upon a screen. If either half of the beam is
stopped, the other falls on the screen and illuminates it, but if both are allowed
to pass, the screen in certain places becomes dark, and thus shews that the
two portions of light have destroyed each other.

Now, we cannot suppose that two bodies when put together can annihilate
each other; therefore light cannot be a substance. What we have proved is
that one portion of light can be the exact opposite of another portion, just
•f +o is the exact opposite of — a, whatever a may be. Among physical
quantities we find some which are capable of having their signs reversed, and
others which are not. Thus a displacement in one direction is the exact
opposite of an equal displacement in the opposite direction. Such quantities

* [p. 485 of the present vol.]

ETHER. 765

are the measures, not of substances, but always of processes taking place in a.
substance. We therefore conclude that light is not a substance but a process
going on in a substance, the process going on in the first portion of light
being always the exact opposite of the. process going on in the other at the
same instant, so that when the two portions are combined no process goes
on at all. To determine the nature of the process in which the radiation of
light consists, we alter the length of the path of one or both of the two
portions of the beam, and we find that the light is extinguished when the
difference of the length of the paths is an odd multiple of a certain small
distance called a half wave-length. In all other cases there is more or less
light ; and when the paths are equal, or when their difference is a multiple
of a whole wave-length, the screen appears four tunes as bright as when one
portion of the beam falls on it. In the ordinary form of the experiment these
different cases are exhibited simultaneously at different points of the screen,
so that we see on the screen a set of fringes consisting of dark lines at
equal intervals, with bright bands of graduated intensity between them.

If we consider what is going on at different points in the axis of a beam
of light at the same instant, we shall find that if the distance between the
points is a multiple of a wave-length the same process is going on at the
two points at the same instant, but if the distance is an odd multiple of
half a wave-length the process going on at one point is the exact opposite
of the process going on at the other. -

Now, light is known to be propagated with a certain velocity (3<004xl010
centimetres per second in vacuum, according to Cornu). If, therefore, we sup-
pose a movable point to travel along the ray with this velocity, we shall
find the same process going on at every point of the ray as the moving point
reaches it. If, lastly, we consider a fixed point in the axis of the beam, we
shall observe a rapid alternation of these opposite processes, the interval of time
between similar processes being the time light takes to travel a wave-length.

These phenomena may be summed up in the mathematical expression

u = A coa(nt-px + a)

which gives u, the phase of the process, at a point whose distance measured
from a fixed point in the beam is x, and at a time t.

We have determined nothing as to the nature of the process. It may
be a displacement, or a rotation, or an electrical disturbance, or indeed any

ETOER.

which it capable of assuming negative as well as positive
be the nature of the process, if it is capable of being
by an equation of this form, the process going on at a fixed point

• ribratiom; the constant A is called the amplitude; the time - is
the period; and nt-px+a is the phase.
The configuration at a given instant is called a wave, and the distance —

the MMt-fayft. The velocity of propagation is -. When we contem-
plate the different parts of the medium as going through the same process in
•oeceaaion, we nee the word undulatory to denote this character of the process
without in any way restricting its physical nature.

A further insight into the physical nature of the process is obtained from
the fact that if the two rays are polarized, and if the plane of polarization
of one of them be made to turn round the axis of the ray, then when the
two planes of polarization are parallel the phenomena of interference appear as
above doeffribfld. As the plane turns round, the dark and light bands become
less distinct, and when the planes of polarization are at right angles, the
illumination of the screen becomes uniform, and no trace of interference can be
discovered.

Hence the physical process involved in the propagation of light must not
only be a directed quantity or vector capable of having its direction reversed,
but this vector must be at right angles to the ray, and either in the plane
of polarization or perpendicular to it. Fresnel supposed it to be a displace-
ment of the medium perpendicular to the plane of polarization. Maccullagh and
Neumann supposed it to be a displacement in the plane of polarization. The
comparison of these two theories must be deferred till we come to the phenomena
of dense media.

The process may, however, be an electromagnetic one, and as in this case
the electric displacement and the magnetic disturbance are perpendicular to each
other, either of these may be supposed to be in the plane of polarization.

All that has been said with respect to the radiations which affect our
eyes, and which we call light, applies also to those radiations which do not
produce a luminous impression on our eyes, for the phenomena of interference

ETHER. 767

have been observed, and the wave-lengths measured, in the case of radiations,
which can be detected only by their heating or by their chemical effects.

Elasticity, tenacity, and density of the ceiher. — Having so far determined the
geometrical character of the process, we must now turn our attention to the
medium in which it takes place. We may use the term sether to denote this
medium, whatever it may be.

In the first place, it is capable of transmitting energy. The radiations which
it transmits are able not only to act on our senses, which of itself is evidence
of work done, but to heat bodies which absorb them ; and by measuring the
heat communicated to such bodies, the energy of the radiation may be calculated.

In the next place this energy is not transmitted instantaneously from the
radiating body to the absorbing body, but exists for a certain time in the
medium.

If we adopt either Fresnel's or Maccullagh's form of the undulatory theory,
half of this energy is in the form of potential energy, due to the distortion of
elementary portions of the medium, and half in the form of kinetic energy,
due to the motion of the medium. We must therefore regard the aether as
possessing elasticity similar to that of a solid body, and also as having a finite
density. If we take Pouillet's estimate of 17633 as the number of gramme-
centigrade units of heat produced by direct sunlight falling on a square centi-
metre in a minute, this is equivalent to 1'234 x 10" ergs per second. Dividing
this by 3'004 x 1010, the velocity of light in centimetres per second, we get for
the energy in a cubic centimetre 4'1 x 10~" ergs. Near the sun the energy in
a cubic centimetre would be about 46,000 times this, or T886 ergs. If we

further assume, with Sir W. Thomson, that the amplitude is not more than one

n i

hundredth of the wave-length, we have Ap= . Q-, or about — ; so that we have-
Energy per cubic centimetre =|/>F2^y = 1'886 ergs.*

Greatest tangential stress per square centimetre = pVAp =30'176 dynes.

Coefficient of rigidity of ether = />F2 = 842'8.

Density of sether = p =9'36xlO-I§.

The coefficient of rigidity of steel is about 8 x 10", and that of glass
2-4 x 10".

* [The numbers in this column are incorrectly deduced from the data. They should be 1-886,
60-352, 965-632 and 1-07 x HT18.]

ETHER.

If the temperature of the atmosphere were everywhere 0* C., and if it were
in equilibrium about the earth supposed at rest, its density at an infinite distance
from the earth would be 3x10"** which is about T8 x 10W times less than
I <rttirf^ density of the aether. In the regions of interplanetary space the
dsositr of the sHher is therefore very great compared with that of the at-
taaoaled atmosphere of interplanetary space, but the whole mass of aether
a sphere whose radius is that of the most distant planet is very small
with that of the planets themselves *.

Tkt <s*Arr distinct from gross matter. — When light travels through the
it is manifest that the medium through which the light is propagated
not the air itself, for in the first place the air cannot transmit transverse
vibrations, and the normal vibrations which the air does transmit travel about
• million times slower than light. Solid transparent bodies, such as glass
and crystals, are no doubt capable of transmitting transverse vibrations, but
the velocity of transmission is still hundreds of thousand times less than that
with which light is transmitted through these bodies. We are therefore obliged
to suppose that the medium through which light is propagated is something
distinct from the transparent medium known to us, though it interpenetrates
all transparent bodies and probably opaque bodies too.

The velocity of light, however, is different in different transparent media,
and we must therefore suppose that these media take some part in the process,
and that their particles are vibrating as well as those of the aether, but the
energy of the vibrations of the gross particles must be very much smaller
than that of the aether, for otherwise a much larger proportion of the incident
light would be reflected when a ray passes from vacuum to glass or from
to vacuum than we find to be the case.

Rrfatirt motion of the tether.— We must therefore consider the jether within
bodies as somewhat loosely connected with the dense bodies, and we
hare next to inquire whether, when these dense bodies are in motion through
the great ocean of aether, they carry along with them the {ether they con-
tain, or whether the aether passes through them as the water of the sea
passes through the meshes of a net when it is towed along by a boat. If
were possible to determine the velocity of light by observing the time it
takes to travel between one station and another on the earth's surface, we
• See Sir W. Thomson, Trans. R. S. Edin. Vol. xxi. p. 60.

ETHER. 769

might, by comparing the observed velocities in opposite directions, determine
the velocity of the aether with respect to these terrestrial stations. All
methods, however, by which ,it is practicable to determine the velocity of light
from terrestrial experiments depend on the measurement of the time required
for the double journey from one station to the other and back again, and the
increase of this time on account of a relative velocity of the aether equal to
that of the earth in its orbit would be only about one hundred millionth
part of the whole time of transmission, and would therefore be quite insensible.

The theory of the motion of the aether is hardly sufficiently developed to
enable us to form a strict mathematical theory of the aberration of light,
taking into account the motion of the aether. Professor Stokes, however, has
shewn that, on a very probable hypothesis with respect to the motion of the
aether, the amount of aberration would not be sensibly affected by that motion.

The only practicable method of determining directly the relative velocity of
the aether with respect to the solar system is to compare the values of the
velocity of light deduced from the observation of the eclipses of Jupiter's satellites
when Jupiter is seen from the earth at nearly opposite points of the ecliptic.

Arago proposed to compare the deviation produced in the light of a star
after passing through an achromatic prism when the direction of the ray within
the prism formed different angles with the direction of motion of the earth
in its orbit. If the aether were moving swiftly through the prism, the deviation
might be expected to be different when the direction of the light was the
same as that of the aether, and when these directions were opposite.

The present writer* arranged the experiment in a more practicable manner
by using an ordinary spectroscope, in which a plane mirror was substituted
for the slit of the collimator. The cross wires of the observing telescope were
illuminated. The light from any point of the wire passed through the object-
glass and then through the prisms as a parallel pencil till it fell on the
object-glass of the collimator, and came to a focus at the mirror, where it was
reflected, and after passing again through the object-glass it formed a pencil
passing through each of the prisms parallel to its original direction, so that
the object-glass of the observing telescope brought it to a focus coinciding
with the point of the cross wires from which it originally proceeded. Since

* Phil. Trans. CLVIII. (1868), p. 532. [Communicated by Prof. Maxwell to Dr Huggins and
included by him in his paper on the spectra of some of the stars and nebulae.]

VOL. II. 9?

ETHER.

coincided with the object, it could not be observed directly, but by
the penal by partial reflection at a plane surface of glass, it was
nd that U»e image of the finest spider line could be distinctly seen, though
light which Conned the image had passed twice through three prisms of
Tne appaimtua was fiwt turned so that the direction of the light in first
^rf through the second prism was that of the earth's motion in its orbit.
He apparatus WM afterwards placed so that the direction of the light was
opposite to that of the earth's motion. If the deviation of the ray by the
WM increased or diminished for this reason in the first journey, it would

be diminished or increased in the return journey, and the image would appear
i one side of the object When the apparatus was turned round it would
appear on the other side. The experiment was tried at different times of the
rear, hut only negative results were obtained. We cannot, however, conclude
absolutely from this experiment that the sether near the surface of the earth
is earned along with the earth in its orbit, for it has been shown by Professor
Stokes* that according to Fresnel's hypothesis the relative velocity of the aether
within tlif prism would be to that of the aether outside inversely as the
aquare of the index of refraction, and that hi this case the deviation would
not be sensibly altered on account of the motion of the prism through the aether.

Fizeaut, however, by observing the change of the plane of polarization of
light transmitted obliquely through a series of glass plates, obtained what he
supposed to be evidence of a difference in the result when the direction of
the ray in space was different, and Angstrom obtained analogous results by
diffraction. 'The writer is not aware that either of these very difficult experi-
ments has been verified by repetition.

In another experiment of M. Fizeau, which seems entitled to greater con-
fidence, he has observed that the propagation of light in a stream of water takes
place with greater velocity in the direction in which the water moves than
in the opposite direction, but that the change of velocity is less than that
which would be due to the actual velocity of the water, and that the
phenomenon does not occur when air is substituted for water. This experiment
seems rather to verify Fresnel's theory of the aether; but the whole question
• •f the state of the luminiferous medium near the earth, and of its connexion
with gross matter, is very far as yet from being settled by experiment.

• I'kiL Hag. 1846, p. 53. t Ann. de Chimie et de Physique, Feb. 1860.

ETHER.

Function of the tether in electromagnetic phenomena.— Faraday conjectured
that the same medium which is concerned in the propagation of light might
also be the agent in electromagnetic phenomena. "For my own part," he says,
"considering the relation of a vacuum to the magnetic force, and the general
character of magnetic phenomena external to the magnet, I am much more
inclined to the notion that in the transmission of the force there is such an
action, external to the magnet, than that the effects are merely attraction and
repulsion at a distance. Such an action may be a function of the aether;
for it is not unlikely that, if there be an aether, it should have other uses
than simply the conveyance of radiation*." This conjecture has only been
strengthened by subsequent investigations.

Electrical energy is of two kinds, electrostatic and electrokinetic. We
have reason to believe that the former depends on a property of the medium
in virtue of which an electric displacement elicits an electromotive force in the
opposite direction, the electromotive force for unit displacement being inversely
as the specific inductive capacity of the medium.

The electrokinetic energy, on the other hand, is simply the energy of the
motion set up in the medium by electric currents and magnets, this motion not
being confined to the wires which carry the currents, or to the magnet, but
existing in every place where magnetic force can be found.

Electromagnetic Theory of Light. — The properties of the electromagnetic
medium are therefore as far as we have gone similar to those of the lumi-
niferous medium, but the best way to compare them is to determine the
velocity with which an electromagnetic disturbance would be propagated through
the medium. If this should be equal to the velocity of light, we would have
strong reason to believe that the two media, occupying as they do the same
space, are really identical. The data for making the calculation are furnished
by the experiments made in order to compare the electromagnetic with the
electrostatic system of units. The velocity of propagation of an electromagnetic
disturbance in air, as calculated from different sets of data, does not differ
more from the velocity of light in air, as determined by different observers,
than the several calculated values of these quantities differ among each other.

If the velocity of propagation of an electromagnetic disturbance is equal
to that of light in other transparent media, then in non-magnetic media the

* Experimental Researches, 3075.

97—2

ETHER.

capacity should be equal to the square of the index of

Bohjonann* has found that this is very accurately true for the gases which
he bM examined. Liquids and solids exhibit a greater divergence from this
, but we can liardly expect even an approximate verification when we
bar* to compare the result* of our sluggish electrical experiments with the
*it*ntatiom of light, which take place billions of times in a second

The undulatory theory, in the form which treats the phenomena of light
M UM motion of an elastic solid, is still encumbered with several difficulties!.

The first and most important of these is that the theory indicates the possi-
bility of undulations consisting of vibrations normal to the surface of the wave.
The only way of accounting for the fact that the optical phenomena which
would arise from these waves do not take place is to assume that the aether
is incompressible.

The next is that, whereas the phenomena of reflection are best explained
on the hypothesis that the vibrations are perpendicular to the plane of polari-
mfa* those of double refraction require us to assume that the vibrations are
in that plane.

The third is that, in order to account for the fact that in a doubly re-
fracting crystal the velocity of rays in any principal plane and polarized in
that plane is the same, we must assume certain highly artificial relations among
the coefficients of elasticity.

The electromagnetic theory of light satisfies all these requirements by the
•ingle hypothesis | that the electric displacement is perpendicular to the plane of
polarization. No normal displacement can exist, and in doubly refracting crystals
the specific dielectric capacity for each principal axis is assumed to be equal
to the square of the index of refraction of a ray perpendicular to that axis,
and polarized in a plane perpendicular to that axis. Boltzmann§ has found
•that these relations are approximately true in the case of crystallized sulphur,

SUsk., 53 April, 1874.

t 8w Prot Stoke* " Report on Double Refraction," British Ass. Report, 1862, p. 253.
Our 4* Aeon* der tcruykaaUing en brekimj van ftet licfU,— Academisch Proefschrift door H. A.

Ambrm, K. T«n dcr Zande, 1875.

Uetor di» Venchiedenheit dor Dielektricitatsconstante des krystallisirten Schwcfels nach verscliie-
" by Ludwig Boltzmann, Wiener Sitzb., 8th Oct, 1874.

ETHER. 773

a body having three unequal axes. The specific dielectric capacity for these
axes are respectively

4773 3-970 3-811

and the squares of the indices of refraction

4-576 3-886 3'591

Physical constitution of the (ether.— What is the ultimate constitution of the
aether ? is it molecular or continuous ?

We know that the aether transmits transverse vibrations to very great
distances without sensible loss of energy by dissipation. A molecular medium,
moving under such conditions that a group of molecules once near together
remain near each other during the whole motion, may be capable of trans-
mitting vibrations without much dissipation of energy, but if the motion is
such that the groups of molecules are not merely slightly altered in configura-
tion but entirely broken up, so that their component molecules pass into new
types of grouping, then in the passage from one type of grouping to another
the energy of regular vibrations will be frittered away into that of the irregular
agitation which we call heat.

We cannot therefore suppose the constitution of the aether to be like that
of a gas, in which the molecules are always in a state of irregular agitation,
for in such a medium a transverse undulation is reduced to less than one five-
hundredth of its amplitude in a single wave-length. If the aether is molecular,
the grouping of the molecules must remain of the same type, the configuration
of the groups being only slightly altered during the motion.

Mr S. Tolver Preston* has supposed that the aether is like a gas whose
molecules very rarely interfere with each other, so that their mean path is far
greater than any planetary distances. He has not investigated the properties
of such a medium with any degree of completeness, but it is easy to see that
we might form a theory in which the molecules never interfere with each
other's motion of translation, but travel in all directions with the velocity of
lio-ht ; and if we further suppose that vibrating bodies have the power of im-
pressing on these molecules some vector property (such as rotation about an
axis) which does not interfere with their motion of translation, and which is
then carried along by the molecules, and if the alternation of the average

* Phil. Mag., Sept. aud Nov. 1877.

774

valoe of ihis teeter for all the molecules within an element of volume be the

oh w* mil light, then the equations which express this average

be of the same form as that which expresses the displacement in the

theory.

It is often uscrtH that the mere fact that a medium is elastic or com-
M • proof that the medium is not continuous, but is composed of
part* having void spaoea between them. But there is nothing incon-
with experience in supposing elasticity or compressibility to be properties
of ewty portion, however small, into which the medium can be conceived to
be di\ul«l. in which case the medium would be strictly continuous. A medium,
|a<|^¥irt though homogeneous and continuous as regards its density, may be
rendered heterogeneous by its motion, as in Sir W. Thomson's hypothesis of
vorlex-moleculea in a perfect liquid (see Art ATOM)'.

The wilier, if it is the medium of electromagnetic phenomena, is probably
molecular, at least in this sense.

Sir W. Thomsont has shewn that the magnetic influence on light discovered
by Faraday depends on the direction of motion of moving particles, and that it
indicate* a rotational motion in the medium when magnetized. See also
Maxwell's Electricity and Mmjin'tism, Art. 806, &c.

Now, it is manifest that this rotation cannot be that of the medium as a
whole about an axis, for the magnetic field may be of any breadth, and there is
no evidence of any motion the velocity of which increases with the distance from
• single fixed line in the field. If there is any motion of rotation, it must be a
rotation of very small portions of the medium each about its own axis, so
that the medium must be broken up into a number of molecular vortices.

We have as yet no data from which to determine the size or the number
of these molecular vortices. We know, however, that the magnetic force in the
region in the neighbourhood of a magnet is maintained as long as the steel
retains its magnetization, and as we have no reason to believe that a steel
magnet would lose all its magnetization by the mere lapse of time, we
conclude that the molecular vortices do not require a continual expenditure of
work in order to maintain their motion, and that therefore this motion does
not necessarily involve dissipation of energy.

' (p. 445 of the present volume.] t Proceedings of llie Royal Society, June, 1856.

ETHER. 775

No theory of the constitution of the aether has yet been invented which
will account for such a system of molecular vortices being maintained for an
indefinite time without their energy being gradually dissipated into that irregular-
agitation of the medium which, in ordinary media, is called heat.

Whatever difficulties we may have in forming a consistent idea of the con-
stitution of the aether, there can be no doubt that the interplanetary and
interstellar spaces are not empty, but are occupied by a material substance or
body, which is certainly the largest, and probably the most uniform body of
which we have any knowledge.

Whether this vast homogeneous expanse of isotropic matter is fitted not
only to be a medium of physical interaction between distant bodies, and to
fulfil other physical functions of which, perhaps, we have as yet no conception,
but also, as the authors of the Unseen Universe seem to suggest, to con-
stitute the material organism of beings exercising functions of life and mind
as high or higher than ours are at present, is a qiiestion far transcending the
limits of physical speculation.

[From Nature, Vol. «.]

XCVIII. Thomson and Tait's Natural Philosophy.

TH« yew 1867 will long be remembered by natural philosophers as that
of the publication of the first volume of "Thomson and Tait." They had long
Irrn waiting for the book, and in the preface the delay was accounted for
by the necessity of anticipating the wants of the other three volumes, in which
the remaining divisions of Natural Philosophy were to be treated. The reader
wa* also reminded, that if in any passage he failed to appreciate the aim of
the authors, the reason might be that what he was studying was in reality
a prospective contrivance, the true aim of which would not become manifest
until oA«>r the perusal of that part of the work for which it was designed to
prepare the way.

Wliat we have had before us now for twelve years was. the authors
reminded us, strictly preliminary matter. The plan of the whole treatise
could only be guessed at from the scale on which its foundations were con-
structed.

In these days, when so much of the science of our best men is dribbled
••ut of them in the fragmentary and imperfectly elaborated form of the memoirs
which they contribute to learned societies, and when the work of making
Un»ka is relegated to professional bookmakers, who understand about as much
of one subject as of another, it was something to find that even one man of
known power had not shrunk from so great a work ; it was more when it
appeared tliat two men of mark were joined together in the undertaking ; and
when at hist the plan of the work was described in the preface, and the
scale on which its foundations were being laid was exhibited in the vast sub-
structure of Preliminary Matter, the feeling with which we began to contemplate
the mighty whole was one in which delight was almost overpowered by awe.

THOMSON AND TAIT's NATURAL PHILOSOPHY. 777

This feeling has been growing upon us during the twelve years we have
been exploring the visible part of the work, marking its bulwarks and telling
the rising generation what manner of a palace that must be, of which these
are but the outworks and first line of defences, so that now, when we have
before us the second edition of the first part of the first volume, we are
impelled to risk the danger of criticising an unfinished work, and to say some-
thing about the plan of what is already before us.

The first thing which we observe in the arrangement of the work is the
prominence given to kinematics, or the theory of pure motion, and the large
space devoted under this heading to what has been hitherto considered part
of pure geometry. The theory of the curvature of lines and surfaces, for
example, has long been recognised as an important branch of geometry, but
in treatises on motion it was regarded as lying as much outside of the subject
as the four rules of arithmetic or the binomial theorem.

The guiding idea, however, which, though it has long exerted its influence
on the best geometers, is now for the first time boldly and explicitly put
forward, is that geometry itself is part of the science of motion, and that it
treats, not of the relations between figures already existing in space, but of
the process by which these figures are generated by the motion of a point
or a line.

We no longer, for example, consider the line AB simply as a white
stroke on a black board, and call it indifferently AB or BA, but we conceive
it as the trace of the motion of a point from A to B, and we distinguish
A as the beginning and B as the end of this trace.

This method of regarding geometrical figures seems to imply that the
idea of motion underlies the idea of form, and is in accordance with the
psychological doctrine which asserts that at any given instant the attention is
confined to a single and indivisible percept, but that as time flows on the
attention passes along a continuous series of such percepts, so that the path
of investigation along which the mind proceeds may be described as a con-
tinuous line without breadth. Our knowledge, therefore, of whatever kind, may
be compared to that which a blind man acquires of the form of solid bodies
by stroking them with the point of his stick, and then filling up in his
imagination the unexplored parts of the surface according to his own notions
about continuity and probability. The rapidity, however, with which we make
our exploration is such that we come to think that by a single glance we

VOL. n. 98

THOMSON AKO TAIT'S NATURAL PHILOSOPHY.

«M thonxMrhlT «ee the whole of that surface of a body which ia turned
if indeed, we are not prepared to assert that we have seen the'
•da too. when after all, if our attention were to leave a trace behind

a. 0* point of the blind man's stick might do, this trace would appear
M » men line meandering over the surface in various directions, but leaving
between iU convolutions unexplored areas, the sura of which is still equal to
whole surface. We are at liberty no doubt to course over the surface and
> subdivide the meshes of the network of lines in which we envelope it, and
to conclude that there cannot be a hole in it of more than a certain diameter,
bat no amount of investigation will warrant the conclusion, which nevertheless
we dimw at once and without a scruple, that the surface is absolutely con-
tinuous and has no hole in it at all. Even when, in a dark night, a flash
of lightning discloses instantaneously a whole landscape with trees and buildings,
we discover these things not at once, but by perusing at our leisure the
picture which the sudden flash has photographed on our retina.

The reason why the phenomena of motion have been so long refused a
place among the most universal and elementary subjects of instruction seems
to be, that we have been relying too much on symbols and diagrams, to the
neglect of the vital processes of sensation and thought.

It is no doubt much easier to represent in a diagram or a picture the
instantaneous relations of things coexisting in space than to illustrate in a full
and complete manner the simplest case of motion. When we have drawn our
diagram it remains on the paper, and the student may run his mind over the
lines in any order which pleases him. But when we are either perceiving
real motions, or thinking about them without the aid or the encumbrance of
a diagram, the mind is carried along the actual course of the motion, in a
manner far more easy and natural than when it is rushing indiscriminately
hither ami thither along the lines of a diagram.

Having pursued kinematics from its elementary principles till its intricacies
.begin to be appalling, we resume the study of the elements of science in the
opening of the chapter on " Dynamical Laws and Principles." It is here that
we first have to deal with something which claims the title of Matter, and
our authors, one of whom never misses an opportunity of denouncing meta-
physical reasoning, except when he has occasion to expound the peculiarities
of the Unconditioned, make the following somewhat pusillanimous statement : —

cannot, of course, give a definition of Matter which will satisfy the

THOMSON AND TAIT S NATURAL PHILOSOPHY. 779

metaphysician, but the naturalist may be content to know matter as that which
can be perceived by the senses, or as that which can be acted upon by, or can
exert, force."

The authors proceed to throw out a hint about Force being a direct
object of sense, and after telling us that the question What is matter? will
be discussed in a future volume, in which also the Subjectivity of Force will
be considered, they retire to watch the effect of the definition they have thrown
into the camp of the naturalists.

Now all this seems to us very much out of place in a treatise on
Dynamics. We have nothing of the kind in treatises on Geometry. We have
no disquisitions as to whether it is by touch or by sight that we come to
know in what way a triangle differs from a square. We have not even a
caution that the diagrams of these figures in the book do not exactly corre-
spond with their definitions. Even in kinematics, when our authors speak of
the motion of points, lines, surfaces, and solids, though they introduce several
modern phrases, the kind of motion they speak of is none other than that
which Euclid recognises, when he treats of the generation of figures.

Why, then, should we have any change of method when we pass on from
kinematics to abstract dynamics? Why should we find it more difficult to
endow moving figures with mass than to endow stationary figures with motion ?
The bodies we deal with in abstract dynamics are just as completely known
to us as the figures in Euclid. They have no properties whatever except those
which we explicitly assign to them.

Again, at p. 222, the capacity of the student is called upon to accept
the following statement : —

" Matter has an innate power of resisting external influences, so that
every body, as far as it can, remains at rest or moves uniformly in a,
straight line."

Is it a fact that " matter" has any power, either innate or acquired, of
resisting external influences ? Does not every force which acts on a body
always produce exactly that change in the motion of the body by which its
value, as a force, is reckoned? Is a cup of tea to be accused of having an
innate power of resisting the sweetening influence of sugar, because it persist-
ently refuses to turn sweet unless the sugar is actually put into it?

But suppose we have got rid of this Manichsean doctrine of the innate
depravity of matter, whereby it is disabled from yielding to the influence of

98—2

TflOMMM AXP TATrt NATURAL PHILOSOPHY.

fr- tan* anfeM that tone actually spends itself upon it, what sort of

»laft «• to be the subject-matter of abstract dynamics ?
W« aw MppOMd to have mastered so much of kinematics as to be able
> describe all poMble motions of points, lines, and figures. In so far as real
to|Bt< 1^ figare. and motions, we may apply kinematics to them.

*t^ new ym appropriate to dynamics is that the motions of bodies are
not independent of each other, but that, under certain conditions, dynamical
UmiMBctioM take place between two bodies, whereby the motions of both bodies

mn aftoted.

Brery body and every portion of a body in dynamics is credited with a
attain quantitative value, called its mass. The first part of our study must
ikmforo be the distribution of mass in bodies. In every dynamical system
there M a certain point, the position of which is determined by the distribution
of nan. This point was called by Boscovich the centre of mass — a better
HfH^ we think, than centre of inertia, though either of these is free from
the enor involved in the term centre of gravity.

In every dynamical transaction between two bodies there must be some-
thing which determines the relation between the alteration of the motions of
the two bodies. In other words, there must be some function of the motions
of the two bodies which remains constant during the transaction. According
to the doctrine of abstract dynamics it is the motion of the centre of mass
of the two bodies which is not altered on account of any dynamical trans-
action between the bodies. This doctrine, if true of real bodies, gives us the
means of ascertaining the ratio of the mass of any body to that of the body
adopted as the standard of mass, provided we can observe the changes in the
motions of the two bodies arising from an encounter between them.

We then confine our attention to one of the bodies, and estimate the
magnitude of the transaction between the bodies by its effect in changing the
momentum of that body, momentum being merely a term for a quantity mathe-
matically defined in terms of mass and motion. The rate at which this change
of momentum takes place is the numerical measure of the force acting on the
body, and, for all the purposes of abstract dynamics, it is the force acting on
the body.

We have thus vindicated for figures with mass, and, therefore, for force
and stress, impulse and momentum, work and energy, their places in abstract
science beside form and motion.

THOMSON AND TAIT's NATURAL PHILOSOPHY. 781

The phenomena of real bodies are found to correspond so exactly with the
necessary laws of dynamical systems, that we cannot help applying the language
of dynamics to real bodies, and speaking of the masses in dynamics as if they
were real bodies or portions of matter.

We must be careful, however, to remember that what we sometimes, even
in abstract dynamics, call matter, is not that unknown substratum of real
bodies, against which Berkeley directed his arguments, but something as perfectly
intelligible as a straight line or a sphere.

Real bodies may or may not have such a substratum, just as they may
or may not have sensations, or be capable of happiness or misery, knowledge
or ignorance, and the dynamical transactions between them may or may not
be accompanied with the conscious effort which the word force suggests to us
when we imagine one of the bodies to be our own, but so long as their
motions are related to each other according to the conditions laid down in
dynamics, we call them, in a perfectly intelligible sense, dynamical or material
systems.

In this, the second edition, we notice a large amount of new matter, the
importance of which is such that any opinion which we could form within the
time at our disposal would be utterly inadequate. But there is one point of
vital importance in which we observe a marked improvement, namely, in the
treatment of the generalised equations of motion.

Whatever may be our opinion about the relation of mass, as defined in
dynamics, to the matter which constitutes real bodies, the practical interest of
the science arises from the fact that real bodies do behave in a manner
strikingly analogous to that in which we have proved that the mass-systems
of abstract dynamics must behave.

In cases like that of the planets, when the motions we have to account
for can be actually observed, the equations of Maclaurin, which are simply a
translation of Newton's laws into the Cartesian system of co-ordinates, are
amply sufficient for our purpose. But when we have reason to believe that
the phenomena which fall under our observation form but a very small part
of what is really going on in the system, the question is not — what phenomena
will result from the hypothesis that the system is of a certain specified kind ?
but — what is the most general specification of a material system consistent with
the condition that the motions of those parts of the system which we can
observe are what we find them to be ?

THOMSON AND TACT'S NATURAL PHILOSOPHY.

It » to Ugrange. in the first place, that we owe the method which
mil— m fa answer this question without asserting either more or less than
•II that ean be legitimately deduced from the observed facts. But though this
nethod ha* been in the hands of mathematicians since 1788, when the Mtcanique
Amtlytww was published, and though a few great mathematicians, such as Sir
W. R Hamilton, Jacobi, Ac., have made important contributions to the general
theory of dynamics, it is remarkable how slow natural philosophers at large
hare been to make Vie of these methods.

Now, however, we have only to open any memoir on a physical subject
in order to eee that these dynamical theorems have been dragged out of the
•utctuary of profound mathematics in which they lay so long enshrined, and
have been set to do all kinds of work, easy as well as difficult, throughout
the whole range of physical science.

Hie credit of breaking up the monopoly of the great masters, of the spell,
and making all their charms familiar in our ears as household words, belongs
in great measure to Thomson and Tait. The two northern wizards were the
first who, without compunction or dread, uttered in their mother tongue the
true and proper names of those dynamical concepts which the magicians of old
were wont to invoke only by the aid of muttered symbols and inarticulate
equations. And now the feeblest among us can repeat the words of power
and take part in dynamical discussions which but a few years ago we should
hare left for our betters.

In the present edition we have for the first time an exposition of the
general theory of a very potent form of incantation, called by our authors the
Ignoration of Co-ordinates. We must remember that the co-ordinates of Thomson
and Tait are not the mere scaffolding erected over space by Descartes, but
the variables which determine the whole motion. We may picture them as so
many independent driving-wheels of a machine which has as many degrees of
freedom. In the cases to which the method of ignoration is applied there are
certain variables of the system such that neither the kinetic nor the potential
energy of the system depends on the value of these variables, though of course
the kinetic energy depends on their momenta and velocities. The motion of
the rest of the system cannot in any way depend on the particular values of
these variables, and therefore the particular values of these variables cannot
be ascertained by means of any observation of the motion of the rest of the
system. We have therefore no right, from such observations, to assign to them

THOMSON AND TAITS NATURAL PHILOSOPHY. 783

any particular values, and the only scientific way of dealing with them is to
ignore them.

But this is not all. Since these variables do not appear in the expression
for the potential energy, there can be no force acting on them, and therefore
their momenta are, each of them, constant, and their velocities are functions
of the variables, but, since their own variables do not enter into the expressions,
we may consider them as functions of the other variables, or, as they are here
called, the retained co-ordinates, and of the constant momenta of the ignored
co-ordinates.

From the velocities as thus expressed, together with the constant momenta,
we obtain the contribution of the ignored co-ordinates to the kinetic energy
of the system in terms of the retained co-ordinates and of the constant momenta
of the ignored co-ordinates. This part of the kinetic energy, being independent
of the velocities of the retained co-ordinates, is, as regards the retained co-
ordinates, strictly positional'", and may be considered for all experimental purposes
as if it were a term of the potential energy. The other part of the kinetic
energy is a homogeneous quadratic function of the velocities of the retained
co-ordinates. In the final equations of motion neither the ignored co-ordinates
nor their velocities appear, but everything is expressed in terms of the retained
co-ordinates and their velocities, the coefficients, however, being, in general,
functions of the constant momenta of the ignored co-ordinates.

We may regard this investigation as a mathematical illustration of the
scientific principle that in the study of any complex object, we must fix our
attention on those elements of it which we are able to observe and to cause
to vary, and ignore those which we can neither observe nor cause to vary.

In an ordinary belfry, each bell has one rope which comes down through
a hole in the floor to the bellringers' room. But suppose that each rope, in-
stead of acting on one bell, contributes to the motion of many pieces of
machinery, and that the motion of each piece is determined not by the motion
of one rope alone, but by that of several, and suppose, further, that all this
machinery is silent and utterly unknown to the men at the ropes, who can
only see as far as the holes in the floor above them.

Supposing all this, what is the scientific duty of the men below? They
have full command of the ropes, but of nothing else. They can give each
rope any position and any velocity, and they can estimate its momentum by
* The division of forces into motional and positional is introduced at p. 370.

TUOMBOX AKD TArrt KATUBAL PHILOSOPHY.

Mopping all the ropes at onoe, and feeling what sort of tug each rope gives.

If the? Uke the trouble to ascertain how much work they have to do in

to dime the rope* down to a given set of positions, and to express

i in tenM of tbej* positions, they have found the potential energy of the

,1,1MB Jn term* of the known co-ordinates. If they then find the tug on any

one rope arising from a velocity equal to unity communicated to itself or to

any other rope, they can express the kinetic energy in terms of the co-ordinates

These data am sufficient to determine the motion of every one of the
when it and all the others are acted on by any given forces. This is
all that the men at the ropes can ever know. If the machinery above has
degrees of freedom than there are ropes, the co-ordinates which express
degrees of freedom must be ignored. There is no help for it.

Of course, if there are co-ordinates for which there are no ropes, but which
into the expression for the energy, then, if the motion of these co-
ordinates is periodic, there will be "adynamic vibrations" communicated to the
ropes, and by these the men below will know that there is something peculiar
going on above them. But if they pull the ropes in proper time, they can
cither quiet these adynamic vibrations or strengthen them, so that in this case
these co-ordinates cannot be ignored.

There are other cases, however, in which the conditions for the ignoration
of co-ordinates strictly apply. For instance, if an opaque and apparently rigid
body contains in a cavity within it an accurately balanced body, mounted on
frictionless pivots, and previously set in rapid rotation, the co-ordinate which
expresses the angular position of this body is one which we are compelled to
ignore, because we have no means of ascertaining it. An unscientific person on
receiving this body into his hands would immediately conclude that it was
bewitched. A disciple of the northern wizards would prefer to say that the
body was subject to gyrostatic domination.

Of the sections on cycloidal motions of systems, we can only here say
that the investigation of the constitution of molecules by means of their
vibrations, as indicated by spectroscopic observations, will be greatly assisted by
.1 thorough study of this part of the volume.

We have not space to say anything of what to many readers must be
one of the most interesting parts of the book — that on continuous calculating
machines, in which pure rolling friction is taken from the class of unavoidable

THOMSON AND TAIT's NATURAL PHILOSOPHY. 785

evils, and raised to the rank of one of the most powerful aids to science.
Rolling and sliding have been more than once combined in the hope of obtaining
accurate measurements, but the combination is fatal to accuracy, and these
new machines, one at least of which has been actually constructed and used,
are the first in which pure rolling friction has had fair play given it as a
method of mechanically accurate integration.

A method is also given of combining a number of disk, globe, and cylinder
integrators, so as to form a machine the motions of two pieces of which are
related to each other by a differential equation of any given form. These
machines all work in a purely statical manner, that is, in such a way that
the kinetic energy of the system is not an essential element in the practical
theory of the machine (as in the case of pendulums, &c.), but has to be taken
into account only in order to estimate the magnitude of the tangential forces
at the points of contact which might, if great enough, produce slipping between
the surfaces. Thus, by means of a machine, which will go as slowly as may
be necessary to keep pace with our powers of thought, motions may be
calculated, the phases of which in nature pass before us too rapidly to be
followed by us.

In the original preface some indications were given of what we were to
expect in the remaining three volumes of the work. We hope that the reason
why this part of the preface is omitted in the new edition is that the work
will now go on so steadily that it will be unnecessary to preface performance
by promise.

VOL. II.

[From Encyclopaedia Britannica.]

XCIX.— Faraday.

FARADAY, MICIIABL* chemist, electrician, and philosopher, was bora at New-
Surrey, 22nd September, 1791, and died at Hampton Court, 25th August,
1867. Hit parent* had migrated from Yorkshire to London, where his father
worked as a blacksmith. Faraday himself became apprenticed to Mr Riebau,
a bookbinder. The letters written to his friend Benjamin Abbott at this time
give a lucid account of hia aims in life, and of his methods of self-culture,
when his mind was beginning to turn to the experimental study of nature.
In 1812 Mr Dance, a customer of his master, took him to hear four lectures
by Sir Humphry Davy. Faraday took notes of these lectures, and afterwards
wrote them out in a fuller form. Under the encouragement of Mr Dance, he
wrote to Sir H. Davy, enclosing these notes. "The reply was immediate, kind,
and favourable." He continued to work as a journeyman bookbinder till 1st
March, 1818, when, at the recommendation of Sir H. Davy, he was appointed
it in the laboratory of the Royal Institution of Great Britain. He

appointed director of the laboratory 7th February, 1825; and in 1833
he was appointed Fullerton Professor of Chemistry in the Institution for life,
without the obligation to deliver lectures. He thus remained in the Institu-
tion 54 years. He accompanied Sir H. Davy on a tour through France, Italy,
Switzerland. Tyrol, Geneva, etc. from October 13th, 1813, to April 23, 1815.

Faraday's earliest chemical work was in the paths opened by Davy, to whom
he acted as assistant. He made a special study of chlorine, and discovered two
new chlorides of carbon. He also made the first rough experiments on the
diffusion of gases, a phenomenon first pointed out by Dalton, the physical
importance of which has been more fully brought to light by Graham and
Loscbmidt He succeeded in liquifying several gases ; he investigated the alloys
of steel, and produced several new kinds of glass intended for optical purposes.

FAEADAY. 787

A specimen of one of these heavy glasses afterwards became historically im-
portant as the substance in which Faraday detected the rotation of the plane
of polarization of light when the glass was placed in the magnetic field, and
also as the substance which was first repelled by the poles of the magnet.
He also endeavoured with some success to make the general methods of
chemistry, as distinguished from its results, the subject of special study and of
popular exposition. See his work on Chemical Manipulation.

But Faraday's chemical work, however important in itself, was soon com-
pletely overshadowed by his electrical discoveries. The first experiment which
he has recorded was the construction of a voltaic pile with seven halfpence,
seven disks of sheet zinc, and six pieces of paper moistened with salt water.
With this pile he decomposed sulphate of magnesia (first letter to Abbott,
July 12, 1812). Henceforward, whatever other subjects might from time to
time claim his attention, it was from among electrical phenomena that he
selected those problems to which he applied the full force of his mind, and
which he kept persistently in view, even when year after year his attempts
to solve them had been baffled.

His first notable discovery was the production of the continuous rotation
of magnets and of wires conducting the electric current round each other.
The consequences deducible from the great discovery of Orsted (21st July,
1820) were still in 1821 apprehended in a somewhat confused manner even by
the foremost men of science. Dr Wollaston indeed had formed the expectation
that he could make the conducting wire rotate on its own axis, and in April,
1821, he came with Sir H. Davy to the laboratory of the Royal Institution
to make an experiment. Faraday was not there at the time, but coming in
afterwards he heard the conversation on the expected rotation of the wire.

In July, August, and September of that year Faraday, at the request of
Mr Phillips, the editor of the Annals of Philosophy, wrote for that journal
an historical sketch of electro-magnetism, and he repeated almost all the ex-
periments he described. This led him in the beginning of September to dis-
cover the method of producing the continuous rotation of the wire round the
magnet, and of the magnet round the wire. He did not succeed in making
the wire or the magnet revolve on its own axis. This first success of Faraday
in electromagnetic research became the occasion of the most painful, though
unfounded, imputations against his honour. Into these we shall not enter, re-
ferring the reader to the Life of Faraday, by Dr Bence Jones.

99—2

TARAPAY.

We m*j remark, bowerer, that although the fact of the tangential force
, electric current and a magnetic pole was clearly stated by Orated,
Uttriy •ppwbended by Ampere, Wollaston, and others, the realization of
U» owtinooai^tatto of the wire and the magnet round each other was a
requiring no mean ingenuity for its original solution. For on

» one band tbe electric current always forms a closed circuit, and on the
^^ ^ two poles of the magnet have equal but opposite properties, and
•re iMeparmbly connected, BO that whatever tendency there is for one pole to
drenkte round the current in one direction is opposed by the equal tendency
of the other pole to go round the other way, and thus the one pole can
dreg the other round the wire nor yet leave it behind. The thing
be done unless we adopt in some form Faraday's ingenious solution, by
the current, in some part of its course, to divide into two channels,
one on <»ch side of the magnet, in such a way that during the revolution
of the magnet the current is transferred from the channel in front of the
to the channel behind it, so that the middle of the magnet can pass
the current without stopping it, just as Cyrus caused his army to pass
dryibod over the Gyndes by diverting the river into a channel cut for it in
hi* rear.

We must now go on to the crowning discovery of the induction of electric

.

In Dec. 1824 he had attempted to obtain an electric current by means
of a magnet, and on three occasions he had made elaborate but unsuccessful
attempts to . produce a current in one wire by means of a current in another
wire or by a magnet He still persevered, and on the 29th August, 1831, he
obtained the first evidence that an electric current can induce another in a
different circuit On September 23 he writes to his friend R. Phillips — " I am busy
just now again on electromagnetism, and think I have got hold of a good
thing, but can't say. It may be a weed instead of a fish that, after all
my lab«nr, I may at last pull up." This was his first successful experiment.
In nine more days of experimenting he had arrived at the results described
in hifl first series of "Experimental Researches" read to the Royal Society,
November 24, 1841.

By the intense application of his mind he had brought the new idea, in
lew than three months from its first development, to a state of perfect maturity.
The magnitude and originality of Faraday's achievement may be estimated by

FARADAY. 789

tracing the subsequent history of his discovery. As might be expected, it was
at once made the subject of investigation by the whole scientific world, but
some of the most experienced physicists were unable to avoid mistakes in
stating, in what they conceived to be more scientific language than Faraday's,
the phenomena before them. Up to the present time the mathematicians who
have rejected Faraday's method of stating his law as unworthy of the preci-
sion of their science have never succeeded in devising any essentially different
formula which shall fully express the phenomena without introducing hypotheses
about the mutual action of things which have no physical existence, such as
elements of currents which flow out of nothing, then along a wire, and finally
sink into nothing again.

After nearly half a century of labour of this kind, we may say that,
though the practical applications of Faraday's discovery have increased and are
increasing in number and value every year, no exception to the statement of
these laws as given by Faraday has been discovered, no new law has been
added to them, and Faraday's original statement remains to this day the only
one which asserts no more than can be verified by experiment, and the only
one by which the theory of the phenomena can be expressed in a manner
which is exactly and numerically accurate, and at the same time within the
range of elementary methods of exposition.

During his first period of discovery, besides the induction of electric cur-
rents, Faraday established the identity of the electrification produced in different
ways ; the law of the definite electrolytic action of the current ; and the fact, upon
which he laid great stress, that every unit of positive electrification is related in
a definite manner to a unit of negative electrification, so that it is impossible
to produce what Faraday called "an absolute charge of electricity" of one
kind not related to an equal charge of the opposite kind.

He also discovered the difference of the capacities of different substances
for taking part in electric induction, a fact which has only in recent years
been admitted by continental electricians. It appears, however, from hitherto
unpublished papers that Henry Cavendish had before 1773 not only disco-
vered that glass, wax, rosin and shellac have higher specific inductive capacities
than air, but had actually determined the numerical ratios of these capacities.
This, of course, was unknown both to Faraday and to all other electricians of

his time.

The first period of Faraday's electrical discoveries lasted 10 years. In 1841 he

FARADAV.

that be required net, and it was not till 1845 that he entered on his
iwt period of research, in which he discovered the effect of magnetism
on poluJMid light, and the phenomena of diamagnetism.

Faraday had for a long time kept in view the possibility of using a ray
of polarised light as a means of investigating the condition of transparent
bediss when acted on by electric and magnetic forces. Dr Bence Jones (Life
tf Parmiay, voL L p. 362) gives the following note from his laboratory book,
I0th September, 1822:—

ray of lamp-light by reflexion, and endeavoured to ascertain whether any depolarizing
(WM) curted on it by water placed between the poles of a voltaic battery and in a glass
. MW WoUMton'* trough tiled; the fluids decomposed were pure water, weak solution of
at cafe, and strong mlphuric acid; none of them had any effect on the polarized light,
out of or to the voltaic circuit, so that IK> particular arrangement of particles could be
in Uii* way."

yean afterwards we find another entry in his note-book, on 2nd
May, 1833 (Lift by Dr Bence Jones, vol. ii. p. 29). He then tried, not only
the effect of a steady current, but the effect on making and breaking contact.

" I do not think, therefore, that decomposing solutions or substances will be found to have (as
• MMtyMM of decomposition or arrangement for the time) any effect on the polarized ray. Should
now tnr noa-dceofnponng bodies, as solid nitre, nitrate of silver, borax, glass, etc. whilst solid, to see
if aay iatanud cut* induced, which by decomposition is destroyed, Le. whether, when they cannot
any *Ute of electrical tension is present My borate of glass good, and common electricity
Uun voltaic."

On May 6 he makes further experiments, and concludes — " Hence I see
no reason to expect that any kind of structure or tension can be rendered
evident, either in decomposing or non-decomposing bodies, in insulating or con-
ducting states."

:«riments similar to the last-mentioned have recently been made by Dr
Kerr, of Glasgow, who considers that he has obtained distinct evidence of
action on a ray of polarized light when the electric force is perpendicular to
the ray and inclined 45* to the plane of polarization. Many physicists, how-
ever, have found themselves unable to obtain Dr Kerr's result.

At last, in 1845, Faraday attacked the old problem, but this time with

wnplete success. Before we describe this result we may mention that in

he made the relation between magnetism and light the subject of his

5ry last experimental work. He endeavoured, but in vain, to detect any

FAEADAY. 791

change in the lines of the spectrum of a flame when the flame was acted on
by a powerful magnet.

This long series of researches is an instance of his persistence. His energy
is shewn in the way in which he followed up his discovery in the single
instance in which he was successful. The first evidence which he obtained of
the rotation of the plane of polarization of light under the action of mag-
netism was on the 13th September, 1845, the transparent substance being his
own heavy glass.

He began to work on August 30, 1845, on polarized light passing through
electrolytes. After three days he worked with common electricity, trying glass,
heavy optical glass, quartz, Iceland spar, all without effect, as on former trials.
On September 13 he worked with lines of magnetic force. Air, flint, glass,
rock-crystal, calcareous spar, were examined but without effect.

" Heavy glass was experimented with. It gave no effects when the same magnetic poles or the
contrary poles were on opposite sides (as respects the course of the polarized ray), nor when the
same poles were on the same side either with the constant or intermitting current. But when con-
trary magnetic poles were on the same side there was an effect produced on the polarized ray,
and thus magnetic force and light were proved to have relations to each other. This fact will most
likely prove exceedingly fertile, and of great value in the investigation of the conditions of natural
force."

He immediately goes on to examine other substances, but with "no effect,"
and he ends by saying, "Have got enough for to-day." On September 18 he
"does an excellent day's work." During September he had four days of work,
and in October six, and on 6th November he sent in to the Royal Society the
19th series of his "Experimental Researches," in which the whole conditions
of the phenomena are fully specified. The negative rotation in ferromagnetic
media is the only fact of importance which remains to be discovered afterwards
(by Verdet in 1856).

But his work for the year was not yet over. On November 3, a, new
horseshoe magnet came home, and Faraday immediately began to experiment
on the action in the polarized ray through gases, but with no effect. The
following day he repeated an experiment which had given no result on October 6.
A bar of heavy glass was suspended by silk between the poles of the new
magnet. "When it was arranged, and had come to rest, I found I could
affect it by the magnetic forces and give it position." By the 6th December
he had sent in to the Royal Society the 20th, and on 24th December the
21st series of his " Researches," in which the properties of diamagnetic bodies

m FARADAY.

are felly described. Thus these two great discoveries were elaborated, like his
isr one, b about three months.

The dkeoverj of the magnetic rotation of the plane of polarized light,
it did not lead to such important practical applications as some of
earlier discoveries, has been of the highest value to science, as fur-
eomplrt* dynamical evidence that wherever magnetic force exists there
fp^ll portions of which are rotating about axes parallel to the

direction of that force.

We hare given a few examples of the concentration of his efforts in seeking
to identify the apparently different forces of nature, of his far-sightedness in
selecting subjects for investigation, of his persistence in- the pursuit of what he
set before him, of his energy in working out the results of his discoveries, and
of the accuracy and completeness with which he made his final statement of
the laws of the phenomenon.

The characteristics of his scientific spirit lie on the surface of his work,
and are manifest to all who read his writings. But there was another side of
his fJHHTcH*"'. to the cultivation of which he paid at least as much attention,
and which was reserved for his friends, his family, and his church. His letters
and his conversation were always full of whatever could awaken a healthy interest,
and free from anything that might rouse ill-feeling. When, on rare occasions, he
was forced out of the region of science into that of controversy, he stated the
facts, and let them make their own way. He was entirely free from pride and
undue self assertion. During the growth of his powers he always thankfully
a correction, and made use of every expedient, however humble, which

would make his work more effective in every detail. When at length he found
his memory foiling and his mental powers declining, he gave up, without ostenta-
tion or complaint, whatever parts of his work he could no longer carry on
according to his own standard of efficiency. When he was no longer able to
apply his mind to science, he remained content and happy in the exercise of
those kindly feelings and warm affections which he had cultivated no less care-
fully than his scientific powers.

The parents of Faraday belonged to the very small and isolated Christian
sect which is commonly called after Robert Sandeman. Faraday himself attended
the meetings from childhood; at the early age of 30 he made public profession
of his faith, and during two different periods he discharged the office of elder.
His opinion with respect to the relation between his science and his religion is

FARADAY. 793

expressed in a lecture on mental education delivered in 1854, and printed at the
end of his Researches in Chemistry and Physics.

"Before entering upon the subject, I must make one distinction which, however it may appear
to others, is to me of the utmost importance. High as man is placed above the creatures around him,
there is a higher and far more exalted position within his view; and the ways are infinite in
which he occupies his thoughts about the fears, or hopes, or expectations of a future life. I
believe that the truth of that future cannot be brought to his knowledge by any exertion of his
mental powers, however exalted they may be; that it is made known to him by other teaching
than his own, and is received through simple belief of the testimony given. Let no one suppose for
an instant that the self-education I am about to commend, in respect of the things of this life,
exteuds to any considerations of the hope set before us, as if man by reasoning could find out God.
It would be improper here to enter upon the subject farther than to claim an absolute distinction
between religious and ordinary, .belief. I shall be reproached with the weakness of refusing to apply
those mental operations which I think good in respect of high things to the very highest. I am
content to bear the reproach. Yet even in earthly matters I believe that 'the invisible things of
Him from the creation of the world are clearly seen, being understood by the things that are
made, even His eternal power and Godhead'; and I have never seen anything incompatible between
those things of man which can be known by the spirit of man which is within him, and those
higher things concerning his future which he cannot know, by that spirit."

Faraday gives the following note as to this lecture : —

"These observations were delivered as a lecture before His Royal Highness the Prince Consort
and the members of the Royal Institution on the 6th of May, 1854. They are so immediately con-
nected in their nature and origin with my own experimental life, considered either as cause or
consequence, that I have thought the close of this volume not an unfit place for their repro-
duction."

As Dr Bence Jones concludes —

"His standard of duty was supernatural. It was not founded on any intuitive ideas of right
and wrong, nor was it fashioned upon any outward experiences of time and place, but it was
formed entirely on what he held to be the revelation of the will of God in the written word,
and throughout all his life his faith led him to act up to the very letter of it."

Published Works. — Chemical Manipulation, being Instructions to Students in Chemistry, 1 vol.,
John Murray, 1st edition 1827, 2nd 1830, 3rd 1842; Experimental Researches in Electricity, vols.
i. and II., Richard and John Edward Taylor, vols. I. and n. 1844 and 1847; vol. in. 1844; vol. in.,
Richard Taylor and William Francis, 1855; Experimental Researches in Chemistry and Physics,
Taylor and Francis, 1859; Lectures on l/w Chemical History of a Candle (edited by W. Crookes),
Griffin, Bohn, and Co., 1861 ; On the various Forces in Nature (edited by W. Crookes), Chatto and
Windus (no date).

Biographies. — Faraday as a Discoverer, by John Tyndall, Longmans, 1st edition 1868, 2nd edition
1870; The Life and Letters of Faraday, by Dr Bence Jones, Secretary of the Royal Institution, in
2 vols., Longmans, 1870; Michael Faraday, by J. H. Gladstone, Ph.D., F.R.S., Macmillan, 1872.

VOL. H. 100

[From Britith Attociation Report.]

C. — Report* on Special Branches of Science.

KcroBTB on special branches of science may be of several different types,
to every stage of organisation, from the catalogue up to the

When a person is engaged in scientific research, it is desirable that he
be able to ascertain, with as little labour as possible, what has been
written on the subject and who are the best authorities. The ordinary method
is to get hold of the most recent German paper on the subject, to look up
the references there given, and by following up the trail of each to find out
who are the most influential authors on the subject. German papers have the
most complete references because the machinery for docketing and arranging
•dentine papers is more developed in Germany than elsewhere.

The " Fortachritte der Physik" gave an annual list of all papers, good
and bad, arranged in subjects, with abstracts of the more important ones.
Wiedemann's "Beiblatter" is a more select assortment, given more in full.

I think it doubtful whether a publication of this kind, if undertaken by
the British Association, would succeed. Lists of the titles of the proceedings
of Societies and of the contents of periodicals are given in Nature. These
are useful for strictly contemporary science, and I do not think that a more
elaborate system of collection could be kept up for long.

The intending publisher of a discovery has to examine the whole mass of
science to see whether he has been anticipated, but the student wishes to
read only what is worth reading. What he requires is the names of the best
authors. The selection or election of these is constantly done by skimming
individual authors, who indicate by the names they quote the men whose
opinions have had most influence. But a report on the history and present

REPORTS ON SPECIAL BRANCHES OF SCIENCE. 795

state of a science- has for its main aim to enumerate the various authors and
to point out their relative weight, and this has been very well done in several
British Association Eeports, some of which are nearly as old as the British
Association.

There are some branches of science whose position with respect to the
public, or else to the educational interest, is such that treatises or text-books
can be published on commercial principles, either as books to be read by the
free public, or to be got up by the school public.

There is little encouragement, however, for a scientific man to write a
treatise so long as he can, with much less trouble, produce an original memoir,
which will be much more readily received by a learned society than the treatise
would have been by a publisher.

The systematisation of science is therefore carried on under difficulties
when left to itself; and I think that the experience of the British Association
warrants the belief that its action in asking men of science to furnish reports
has conferred benefits on science which would not otherwise have accrued to it.

There are so many valuable reports in the published volumes that I shall
indicate only a few, the selection being founded on the direction of my own work
rather than on any less arbitrary principle.

First, when a branch of science contains abstruse calculations as well as
interesting experiments, it is desirable that those who cultivate the experimental
side should be conscious that certain things have been done by the mathema-
ticians. The matter to be reported on in this case is not voluminous, but it is
hard reading, and those who are not experts require a guide.

Thus, Professor Challis in 1834 gave a most useful report on the mathe-
matical investigations by Young, Laplace, Poisson, and Gauss on Capillary
Attraction, and Professor Stokes in 1862 reports on Theories of Double Re-
fraction. This report may, indeed, be accepted as an instalment of the treatises
which, if the desire of the scientific world were law, Professor Stokes would
long ago have written. It is meant, no doubt, as a guide to other men's
writings, but it is intelligible in itself without reference to those writings. Such
a report is a full justification of the existence of the British Association, if it had
done nothing else.

Another type of report is that of Professor Cay ley on Dynamics (1857
and 1862). This seems intended rather as a guide in reading the original
authors than as a self-interpreting document, though, of course, besides the

100—2

794 MPOBIV OV aPBCIAL BRANCHES OF SCIENCE.

the methodic*! arrangement, there is much original light thrown
oa Uw MM of memoir* diacumed in it. It will be many years before the
valoe of thk report will be superseded by treatises.

The Report of the Committee on Mathematical Tables deals with a subject
which, though not to abstruse, is larger and drier than any of the preceding.
It •. however, a most interesting as well as valuable report, and supplies in-
wfaieh would never have been printed unless the British Association
for the Report, and which never would have been obtained if the
of the report had not been available.
There are eevernl other reports which are not mere reports, but rather
original paper* preceded by a historical sketch of the subject. No special en-
M needed to get people to write reports of this kind.

[From the Encyclopaedia Britannica.}

CI. — Harmonic Analysis.

HARMONIC ANALYSIS is the name given by Sir William Thomson and
Professor Tait in their treatise on Natural Philosophy to a general method of
investigating physical questions, the earliest applications of which seem to have
been suggested by the study of the vibrations of strings and the analysis of
these vibrations into their fundamental tone and its harmonics or overtones.

The motion of a uniform stretched string fixed at both ends is a periodic
motion ; that is to say, after a certain interval of time, called the fundamental
period of the motion, the form of the string and the velocity of every part
of it are the same as before, provided that the energy of the motion has not
been sensibly dissipated during the period.

There are two distinct methods of investigating the motion of a uniform
stretched string. One of these may be called the wave method, and the other
the harmonic method. The wave method is founded on the theorem that in
a stretched string of infinite length a wave of any form may be propagated
in either direction with a certain velocity, V, which we may define as the
" velocity of propagation." If a wave of any form travelling in the positive
direction meets another travelling in the opposite direction, the form of which
is such that the lines joining corresponding points of the two waves are all
bisected in a fixed point in the line of the string, then the point of the
string corresponding to this point will remain fixed, while the two waves pass
it in opposite directions. If we now suppose that the form of the waves
travelling in the positive direction is periodic, that is to say, that after the
wave has travelled forward a distance I, the position of every particle of the
string is the same as it was at first, then Z is called the wave-length, and the
tune of travelling a wave-length is called the periodic time, which we shall
denote by T, so that

l=VT.

HARMONIC ANALYSIS.

If we now suppose a set of waves similar to these, but reversed in position,
lo b» travelling in the opposite direction, there will be a series of point*.
dMtent 4/ from each other, at which there will be no motion of the string;
I therefore make no difference to the motion of the string if we suppose
UM string frstmnd to fixed supports at any two of these points, and we may
ll^g mum+ii the part* of the string beyond these points to be removed, as it
taM1rt afleot the motion of the part which is between them. We have thus
arriTjd ^ the caee of a uniform string stretched between two fixed supports,
and w» conclude that the motion of the string may be completely represented
a« the reraltant of two sets of periodic waves travelling in opposite direc-
tion*, thffir wave-lengths being either twice the distance between the fixed
or a submultiple of this wave-length, and the form of these waves,
to this condition, being perfectly arbitrary.

To make the problem a definite one, we may suppose the initial displace-
and Telocity of every particle of the string given in terms of its distance
from one end of the string, and from these data it is easy to calculate the
farm which is common to all the travelling waves. The form of the string
at any subsequent time may then be deduced by calculating the positions of
the two sets of waves at that time, and compounding their displacements.

Thus in the wave-method the actual motion of the string is considered
as the resultant of two wave-motions, neither of which is of itself, and without
the other, consistent with the condition that the ends of the string are fixed.
Each of the 'wave-motions is periodic with a wave-length equal to twice the
distance between the fixed points, and the one set of waves is the reverse of
the other b respect of displacement and velocity and direction of propagation ;
hot, subject to these conditions, the form of the wave is perfectly arbitrary.
The motion of a particle of the string, being determined by the two waves which
pass orer it in opposite directions, is of an equally arbitrary type.

In the harmonic method, on the other hand, the motion of the string is
regarded as compounded of a series of vibratory motions which may be infinite
in number, but each of which is perfectly definite in type, and is in fact a
particular solution of the problem of the motion of a string with its ends
fixed.

\\ •..-..

A simple harmonic motion is thus defined by Thomson and Tait (§ 53) :

in a point Q moves uniformly in a circle, the perpendicular QP, drawn from

HARMONIC ANALYSIS. 799

its position at any instant , to a • fixed diameter A A' of the circle, intersects the
diameter in a point P whose position changes by a simple harmonic motion.

The amplitude of a simple harmonic motion is the range on one side or the
other of the middle point of the course.

The period of a simple harmonic motion is the time which elapses from
any instant until the moving-point again moves in the same direction through
the same position.

The phase of a simple harmonic motion at any instant is the fraction of
the whole period which has elapsed since the moving point last passed through
its middle position in the positive direction.

In the case of the stretched string, it is only in certain particular cases
that the motion of a particle of the string is a simple harmonic motion. In
these particular cases the form of the string at any instant is that of a curve
of sines having the line joining the fixed points for its axis, and passing
through these two points, and therefore having for its wave-length either twice
the length of the string or some submultiple of this wave-length. The ampli-
tude of the curve of sines is a simple harmonic function of the time, the
period being either the fundamental period or some submultiple of the funda-
mental period. Every one of these modes of vibration is dynamically possible
by itself, and any number of them may coexist independently of each other.

By a proper adjustment of the initial amplitude and phase of each of these
modes of vibration, so that their resultant shall represent the initial state of
the string, we obtain a new representation of the whole motion of the string,
in which it is seen to be the resultant of a series of simple harmonic vibra-
tions whose periods are the fundamental period and its submultiples. The
determination of the amplitudes and phases of the several simple harmonic
vibrations so as to satisfy the initial conditions is an example of harmonic
analysis.

We have thus two methods of solving the partial differential equation of
the motion of a string. The first, which we have called the wave-method, ex-
hibits the solution in the form containing an arbitrary function, the nature of
which must be determined from the initial conditions. The second, or harmonic
method, leads to a series of terms involving sines and cosines, the coefficients
of which have to be determined. The harmonic method may be defined in a

HARMONIC ANALYSIS.

j, a method by which the solution of any actual problem

mar be obtained M the sum or resultant of a number of terms, each of which
ie "a solution of a particular case of the problem. The nature of these par-
ticular cases is defined by the condition that any one of them must be con-
jugal* to any other.

The msthf^t^rfi1 test of conjugacy is that the energy of the system
•rising from two of the harmonics existing together is equal to the sum of
the energy arising from the two harmonics taken separately. In other words,
no put of the energy depends on the product of the amplitudes of two dif-
fcrant harmonic*. When two modes of motion of the same system are conju-
gate to fw4i other, the existence of one of them does not affect the other.

The simplest case of harmonic analysis, that of which the treatment of
the vibrating string is an example, is completely investigated in what is known
M Fourier's Theorem.

Fourier's theorem asserts that any periodic function of a single variable
period p, which does not become infinite at any phase, can be expanded in
the form of a series consisting of a constant term, together with a double
series of terms, one set involving cosines and the other sines of multiples of
the phase.

Thus if j(£) is a periodic function of the variable f having a period p,
then it may be expanded as follows :

The part of the theorem which is most frequently required, and which
also is the easiest to investigate, is the determination of the values of the
A, A, B,. These are

P o

t 2
'"

HARMONIC ANALYSIS. 801

This part of the theorem may be verified at once by multiplying both
sides of (1) by dg, by cos - — d£, or by sin - — dg , and in each case inte-
grating from 0 to p.

The aeries is evidently single-valued for any given value of £ It cannot
therefore represent a function of £ which has more than one value, or which
becomes imaginary for any value of £ It is convergent, approaching to the
true value of <£(£) for all values of f such that if £ varies infinitesimally
the function also varies infinitesimally.

Sir W. Thomson, availing himself of the disk, globe, and cylinder inte-
grating machine invented by his brother, Professor James Thomson, has
constructed a machine by which eight of the integrals required for the ex-
pression of Fourier's series can be obtained simultaneously from the recorded
trace of any periodically variable quantity, such as the height of the tide, the
temperature of the pressure of the atmosphere, or the intensity of the different
components of terrestrial magnetism. If it were not on account of the waste
of time, instead of having a curve drawn by the action of the tide, and the
curve afterwards acted on by the machine, the time axis of the machine
itself might be driven by a clock, and the tide itself might work the second
variable of the machine, but this would involve the constant presence of an
expensive machine at every tidal station.

It would not be devoid of interest, had we opportunity for it, to trace
the analogy between these mathematical and mechanical methods of harmonic
analysis and the dynamical processes which go on when a compound ray of
light is analysed into its simple vibrations by a prism, when a particular over-
tone is selected from a complex tone by a resonator, and when the enormously
complicated sound-wave of an orchestra, or even the discordant clamours of a
crowd, are interpreted into intelligible music or language by the attentive
listener, armed with the harp of three thousand strings, the resonance of
which, as it hangs in the gateway of his ear, discriminates the multifold
components of the waves of the aerial ocean.

VOL. II.

101

INDEX TO VOL. II.

Action at a distance, 311

Air, Viscosity of, 1

Airy's function of stress, 92, 102, 180, 192, 200—

205

Ampere, 317
Anaxagoras, 445, 449
Andrews, 371, 409
Apjohn, Dr, 637
Atom, 445

Atomic theory, forms of, 448
Attraction, 485 ; A Paradox in the Theory of, 599

Bell, A: M., 751

Bell, Professor Graham, 742—755

Benson, W., 232, 272

Bernoulli, D., 364

Bois-Raymond, 760

Boltzmann, 366, 431, 433, 622, 691 ; his Theorem

on the Kinetic Theory of Gases, 714
Boole, 229

Boscovich, 448, 471, 480
Boussinesq, 760

Bow's method of drawing diagrams, 492
British Association : Address to Physical Section,

215 ; Lecture before, on molecules, 361

Capacity, electrical, 672
Capillary Action, 542
Capillary Surface, Form of, 569
Cavendish, 301, 317, 322, 612, 789
Cayley, 233, 235
Challis, 338

Chemical Society, lecture to, 418
Chrystal, Professor, 537
Clairaut, 54l>

Clausius, 28, 55, 77, 222, 226, 344, 347, 365, 369,

409, 421, 427, 431, 451, 458, 664
Colour Blindness, 277
Colour Vision, 230, 267
Constitution of Bodies, 616
Continuity of Gaseous and Liquid State, 407
Cornu, 765
Cotes' Theorem, 390
Coulomb, 302, 317, 322
Cremona, 494
Crookes, 682, 685
Currents, Electric, maintenance of by mechanical

work, 79
Cyclide, 144; Confocal Cyclides, 155; Conjugate

Isothermal Functions of, 153 ; Construction of,

146, 148; Forms of, 148
Cylinder, Electrical capacity of, 672

Dalton, 277, 367

Descartes, Doctrine of continuity, 450

Diagrams, classified, 647 — -659 ; Reciprocal, 102

Diffusion, 625—646

Diffusion of Gases, 57, 61, 343, 501, 740

Disk, Electrical capacity of, 678

Dupre, 549, 551

Duprez, 585

Effusion, Thermal, 711

Electrical problems, Solution of, 256

Electrostatic and Electro-magnetic units compared
experimentally, 125

Ellis, A J., 751

Envelope, spherical, Equilibrium of, 86

Ether, 763; its function in Electro-magnetic pheno-
mena, 771; Physical constitution of, 773

101—2

INHEX TO Vol.. II.

«. 901 SI9-J1I, S49, 4*8, 7

KbetrioU work. >: Kxperi
«• Uftrt, 790; k» character. 793; hi.
wort. 7M ; RflMrfa on hi* character

M

tnuufbmatim, of, 298
rnuns of Pore*., 161
ropafation of Light, 770
•placement in a case of,

pencil, .132

of a gas: Mutual action

35 ; Mutual

two systems, 37 ; Reasons for adopting
UM> iaTcrM fifth power of distance, 32; External

of molecules, 49
107. 112.353

:•

Freedom and Constraint of, 171

r of Diffusion, 60; Conduction of
feat in, ' < 'ooling of, by expansion,

'*• I Definition of the Action of a system. 717:
Diihriuu of, 57; Dynamical Specification of
tke Motion of. 716 ; Dynamical theory of, 26 ;
Iqartinm of motion of, 66, 69 ; Interdiff usion
of, 61 ; Lav of Equivalent Volumes, 63 ;
Frame of, M; Rigidity of, 32, 71 ; Specific
hml of, 65, 66 ; Statistical Specification, 7i'U
Rarifled: Conditions at surface of a solid,

illations of motion, 697, 701
0M«, M8, !»«, 546
Oftjr-UMc, 455, 546
OMm, Profwor: Methods in Thermodynamics,

490, 625, 659
Gorman, 105
Urafcam. ll.M, 59, 61, 71, 73, 222, 343, 347, 367,

301. MS, 7 In
'•'•phioalSutica. 492
Onm, SS3.599

(irovp, !<ir W., Letter to, on Magneto-Electric In

dui-ticiii, 121; Review of his treatise, 400
Outhrie, Professor, 500

Hamilton's Characteristic Function for a narrow

Beam of Light, 381; applied to the theory of

optical instruments, 439
Hamilton. Sir W. R., 144, 259, 381
Harmonic Analysis, 797
Heat: Coefficient of conductivity, 76; Conduction

of in a gas, 54. 74
Ht-lmlioltz. .'L'3, 230, 255, 276, 306, 342, 402, 416,

466, 708, 74"), "f>3 ; Remarks on him as a man

of science,' 592
Herapath, 364

Herschel, Sir J., 277, 376, 483
Heterogeneous substances, equilibrium of, 498
Hills and Dales, 233
Hockin, 127, 133, 134
Holman, 692
Hopkinson, 6i''J
Huggins, 322, 465
Button, 637

Induction, Magneto-Electric, 121

Induction of currents in an infinite plane, 286

Institution, Royal : Lecture on Action at a distance,

311 ; Lecture on Colour Vision, 267
Integration, Multiple, 604

Jenkiii, 106, 107, 110, 164, 494
Jets, instability of, 582
Joule, 136, 305, 323, 364, 454, 501
Jurin, Dr, "> !_'

Kempelen, 7-V!

Kirchhoff, 96, 406

Kohlrausch, 1, 31, 126, 136, 641

Kopp, 349

Kronig, 365

Kundt, 685, 692, 7 1 2

Laplace, 545, 571

Leidenfrost, 543

Leonardo da Vinci, 541

Le Sage, explanation of gravitation, 474

INDEX TO VOL. II.

805

Leslie, 544, 637

Light, Electro-magnetic Theory of, 138, 771

Lightning, Protection of Buildings from, 538

Lines of Force, Experiment on, 319

Liquid : Drop between two plates, 571 ; Rise of,

between two plates, 568 ; Rise of, in a Tube,

567
Liquid films: Stability of, 578—583; Tension of,

553; films which are surfaces of revolution, 576
Lockyer, N., 465
Lorenz, 137, 228
Loschmidt, Experiments on Diffusion, 343, 369, 460,

740

Lucretius, 445, 471, 472
Ludtge, 549

Maccullagh, 766

Mayer, 402

Mean, the geometrical mean distance between two

figures, 280
Meyer, L., 349

Meyer, O. E., 12, 24, 25, 29, 71, 344, 459, 621, 692
Molecular aether, 437
Molecular Constitution of Bodies, 418
Molecular Science, 451
Molecular Vortices, 774
Molecules : Encounter between, 686 ; Final state

of, when subject to forces, 351 ; Lecture on,

361

Monge, 544
Motion of a connected system, Equations of, 308

Neumann, 228, 766

Newton, 230, 268, 314, 316, 327, 419, 452, 487,
542, 764

Obermayer, 692
Ohm's Law, 533
Optics, its relation to other parts of Mathematics,

391
Orsted, 317

Page, 749

Paradoxical Philosophy, 756

Peaucellier's Linkage, 495

Physical Forces, Correlation of, 400

Physical quantities, mathematically classified, 257

Physics, Introductory Lecture on, 241

Piotrowski, 685, 708

Plateau, 393, 548

Poisson, 301, 303, 317, 547, 560

Polyhedra, Relation between the numbers of edges,

summits and faces of, 170
Priestley, 367
Problem, in Discontinuity, 310

Quadratic functions, 329
Quaternions, 260 — 266
Quincke, 395, 541

Rankine, 163, 259, 494, 664

Rayleigh, Lord, 435, 465, 644, 754

Reading instruments, methods of, 514

Reciprocal Diagrams, 164, 182

Rede lecture, 742

Regnault, 454

Reis, Philip, 746

Retina, Yellow spot in, 278

Review : A work by Plateau, 393 ; A work by
Van der Waals, 407; Challis's Mathematical
Principles of Physics, 338 ; Grove's "Cor-
relation of Physical Forces," 400 ; Paradoxical
Philosophy, 756 ; Sir W. Thomson's papers,
301 ; Tait's Thermodynamics, 660 ; Thomson
and Tait's Nat. Phil., 324, 776; Whewell's
Writings and Correspondence, 528

Reynolds, Professor, 704

Riemann, 137, 170

Scientific Apparatus classified, 505 — 527

Segner, 542

Siemens's Governor, 108, 116

Slipping, Coefficient of, in a gas, 708, 712

Somerville, Mrs, 401

Spectrum, Best Arrangement for producing a pure

spectrum, 96
Spherical Harmonics, Application to Theory of

Gases, 688

Stefan, 344, 459, 503
Stereograms of Surfaces, 97

TO VOL. II.

f ;j4 Transpiration, Thermal, 710

u H m »*. 444, 749 '»"• - ' :>. 221

at i» tluw dfaMsrioas, 178; Van der Mensbrugghe, 548
, of. u. • fetid Body, 197 Van der Waal*, 407, 426, 560

, |B tUftfcd OMM. Ml Velocities of molecules of a gas, final distribution

hs» T«MM* Mse* «t oa velocity of waves, of, 43

»*T; Uw» of, 543; ?>•(••! arising from Venant, St, 760

at 67), IMaUoo to temperature, Virial, defined, 421

M, 690 Vucosity, Coefficient of, 7, 71 ; Table of, for several
r, SI4 gases, 347; of Air, 1—25, 692; of a mixture

of gases, 1-1

Is* MB, $44, M4. 749. 764, 776, 797 Viscous fluid, double refraction in, 379

1s*s TWnMlyHu^oi, 440-471 Voit, 626, 633 .

Taylor, Hwllr; Volumes, Law of Equivalent Volumes, 63, 430

T«jr«or. W. P^ 144, 494 Vortex Atoms, Theory of, 466

Itdb, 434 Warburg, 685, 692, 712

W, J, 16, 107, 112, 127, 136, 222, Weber, 31, 126, 136, 228, 245, 291, 621

S64, S68, J9i 301—307, 312, 321, Wheatetone, 84, 745, 753

« I. 344 371 460, 4K <70, 476, 488, Whewell's Writings and Correspondence, 528

01. 608, 649, 687, 431, 434, 711, 742, 767, Wiedemann, E., 692

T74, 798, 801 Willis, 753

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