+
dA' dB' dk’
dA dB dK”
is to be determined. Itis geometrically evident that dAdBdK
must be equal to dA'dB'dK'; for both sets of differentials
may be supposed to be obtained by supposing that, for fixed
position and magnitude of v, v', V, V', the axis of abscissze
describes the whole interior of a cone of infinitely small aper- °
ture ; and the system of coordinates revolves about the axis
of abscissee at a very small angle. This follows analytically
in the following way. Wesee from fig. 2 that B’'=B+ {vXv'.
{vXv' is simply a function of A, K, and the now constant
angles. If, therefore, we now introduce A’, K’, B’ instead of the
variables A, K, B, we have dB’=dB. Therefore
5, DAY OB’ OK! _y, A! OK!
~ ON AOR ORAS ih? Ore
for the Theoretical Proof of Avogadro’s Law. 327
In the latter functional determinant, besides the angles
already put constant, B’ is to be regarded as constant,
Further
a’ =al+anrcosh,
sinj: sinh=a : @’;
whence
asin h asin h
V1i—a?—asinh ad—alcosh
We see, further, from the figure that
180°—K=h— gv'eV,
when the latter angle depends simply on the form of the im-
pact, and is therefore to be regarded at present as constant.
So also
j+180=K’+ ¢€V'v'r.
The latter angle again is constant ; whence it follows, since
nothing here depends upon the sign, that
Bee ee 15 Oe. OF.
tan7=
me OK aA... oe
Since in the equations for a’ and tan j also the angle L,
which equally depends only on the form of the impact, plays
the part of a constant, the determinant can be calculated
without difficulty, and we obtain for it the value “ We
might also have obtained this result without any calculation
by imagining the points v, v', V, and V’ as fixed. Since A
and h are spherical polar coordinates of the point X of the
spherical surface, so also A’, 7; the element of area ad Adh
expressed by the former polar coordinates must be equal to
the element of area a'd A'dj expressed by the latter. We
have then
a dA'dB' dK'=adA dB dK.
For a fixed position of the points v, v', V, and V’, A, K
and then A’, K’ may be regarded as spherical coordinates of
the point X, which would give at once
adA' dK =e' dA' dK’.
Since, further, from the definition of A (equation 13),
dv' dV' dT’ dO'=Adv dV aT dO,
it follows from equations (28), (29), (30), (81) that
dé dy’ de’ d&y' dn, d&'dO' vw? V"r'A
eal
dé dnd€dé,dn,djdO ~~ = vV*r
328 Prof. L. Boltzmann on the Assumptions necessary
Hquation (15) is therefore proved by proving the equation
dé’ dy! dt dk! dn! dt! dO'=d dn dEdE, dy, dt, dO, . (32)
and vice versa. :
O is here the angle between the planes ROR’ and ROvw of
fig. 1. If on the right-hand side of the equation (32) we
introduce, instead of O, the angle yy, which the former plane
makes with the plane ROX (compare fig. 3), & 7, & &1, m1, &,
and therefore also the angle between the planes ROX and
RQv remain constant ; and since this is equal to the difference
between ¢O and Ww, it follows that dyy=dO. If in the same
way we introduce upon the left-hand side of the equation (32)
vy’ instead of O', it follows that
| d0'=dy’,
and equation (32) becomes
dé dn} dd dé! dy! dey’ dyy! = dé dy dE dé, dm doy dp,
which is exactly the form which H. Stankewitsch gives to the
equation.
We will, however, further multiply each side by od8, by
which at the same time we indicate that S is to be chosen as
the eighth independent variable. The equation thus assumes
_ the form
dé dy! dt’ dé,’ dn, dt,’ dy’ odS= dé dn dg dé, dn, at, dip ods. (33) .
We now again draw all the
lines from the centre © of a
sphere of unit radius, and de-
note in fig. 3 the points of
intersection of the two relative
velocities before and after im-
pact with the surface of the
sphere by Rk and R’; the ends
of the two relative velocities
by R, and R,'. Let H be the
middle point of the arc R R’ of
a great circle, X the point in
which the axis of abscissa in-
tersects the surface of the
sphere. We now for constants
E,n, & &1, m, $ introduce the
angles ¢(N=XH and H=ZXH instead of ¢S=RH and
w=XRR'. Since, again, for a fixed position of the points
X, Z, and R, both 8 and wW as well as N and B are spherical
for the Theoretical Proof of Avogadro’s Law. 329
polar coordinates of the point H of the sphere, we have
_ vdN dh=cd8 dy.
The left side of equation (33) is next transformed into
dé dn d¢d&,dn,df,vdNdE. . . . . (34)
If, now, we denote the projections of the relative velocity
QR, before impact on the axes of coordinates by w, y, z, and
also the projections of the relative velocity OR,’ after impact |
on the axes of coordinates by 2’, y’, 2’, and with constant &, 7, ¢
introduce the variables
a=&—§& y=m—n, 2=&-4
expression (34) becomes
d&dnd¢dxdydzvdNdH. . . . . (85)
Then we leave wz, y, z, N, Ei constant, and instead of &, n, €
introduce the variables &', y', ¢. If x,y, z be the projections
of the line R, R,’ of the relative velocities drawn from © on
the axes of iene we have
Mz,
or m+ MM
. Since, now, all the lines drawn in fig. 3 remain altogether
unaltered in magnitude and position, a, ¥,, and z, are also
constant, and we have
dé dy! dt! =dé dn dé.
Hence expression (35) becomes
dé dy dG dadydzvdN dW...) ~ (86)
The next step consists in introducing for constant &', 7’, ¢,
N, E the variables wz’, y', z' instead of xz, y, z; that is, the
coordinates of the point Ry,’ instead of the sootdtnates of the
point R,. It is at once seen from fig. 3 that the element of
volume described by the point R, on change of its coordinates
is exactly equal to that which the point R,' describes for the
position of the point H remains unchanged. it follows,
therefore, that
Wes My, s (pees
; U ie Came m+M’ v=c—
LEO EET OME Sa EPS Mactan, aaa
and expression (36) becomes
dé dy dc da dy dz sv..dN.dB.... °. (38)
Now, again, inversely
E/=E Gh, m =n +y', Cit +2!
330 Prof. L. Boltzmann on the Assumptions necessary 7
are introduced instead of w',y’,z', so that the expression (38)
becomes |
dé dy dé’ dé,' dn, dt! vdN di. oF las a0 (39) |
Lastly, we introduce, instead of the spherical polar coordi- |
nates N, E of the point H, its spherical polar coordinates 8, W’;
so that we obtain
vadN .dH=od8S. dw’.
Lastly, expression (380) becomes therefore
dé’ dy’ de’ d&y' dn,' d&' odS.. df’,
by which equation (33) is proved.
If we prefer to prove equation (37) analytically, fig. 3
would give
L=rm, y=rpsnd, z=rpcosd,
where 0. Ry=OR,'=r.
d:d=pic, 6: p=p'ioa, p'd'=pd,
s=mnt+prf, oA? =o'—p' :
=1—(mn-+ pif)’ — pw’? = (mv —pnf)’,
m=ns+vod=mMn+ bs f,
where
N= cos2N, v= sin 2N.
From :
s=m'n-+ plyf'=mn-+ py,
it follows that
pf! — MVo5 ne pf Ng.
If in this equation and in the equation w/¢’=d we put
f'=ecos+esin@’, g'=esin O'—ecos G,
it follows that
pL! cos 6! =mev,— pefn, — ped,
wi! sin 6’ = mev,— pen, + wed.
By multiplying by 7 and observing that
rm =z', rp sinf'=y', rp'cosf' =z’,
f=ecos@+esin@, h=ecos é—esin 8,
rpfaeytez, rup=—ey +ez,
we obtain
x! = Nyx + Voey + V9€Z,
y' =Voen — (e? + €’na) y + 2veez,
2’ =veex + 2vecey —(e? + e’ng)z,
for the Theoretical Proof of Avogadro's Law. 331
and we can then convince ourselves directly that
02 Oy 02 _
2+ Qa" Oy 8
Although I have already deduced a great variety of rela-
tions from fig. 1, yet it would probably furnish several
other equations which might be of use in particular circum-
stances, ¢. g. by denoting the magnitude and position of the
straight lines v, V, v', V’ symmetrically by the magnitude
and position of the straight lines ©, P and of the line joining
the point P with the middle point of the straight line W’.
Symmetrical relationships of this kind are particularly conve-
nient when we wish to obtain equations in which the magni-
tudes before and after impact play the same part. as the
equation we have used.
a
Ver. x(a, X, 2')= V2(a+X—2’').y(2', c+ X—w', 2).
Second Appendix.
After correcting the foregoing for the press, I became ac-
quainted, by the kindness of the author, with Prof. Tait’s
paper “On the Foundations of the Kinetic Theory of
Gases”’*. While reserving for a future occasion my remarks
on Prof. Tait’s observations on the mean path, and on the case
when external forces act, I will here mention only one point.
Ifin a gas on which no external forces act,and whose molecules
are elastic spheres, F(z, y, z) dx dy dz be the probability that
components of the velocity of a molecule parallel to the axes
of coordinates shall at the same time lie between the limits 2
and #w+dz, y and y+dy, z and ¢+dz, then Maxwell bases
the first proof which he givesf of his law of distribution of
velocities on the assumption that F(a, y, z) is a product of
these functions, of which the first contains only 2, the second
only y, the third only z. Thisis the same as the assumption
that, fora given component of velocity at right angles to the
axis of abscissze, the quotient of two probabilities, viz. the pro-
bability that the component of the velocity of a molecule in
the direction of the axis of abscissz lies between x and x+dz,
and the probability that the same quantity lies between certain
other limits € and &+dé, is altogether independent of the
given value of the component of the velocity of the same
molecule at right angles to the axis of abscisse. In a
* Trans. Roy. Soc. Edin. xxiii. p. 65 (1886).
+ Phil. Mag. [4] vol. xix. p. 19 (1860).
332 Theoretical Proof of Avogadro's Law.
later paper* Maxwell himself speaks of this assumption as
precarious; and therefore gives a proof resting on a quite
different foundation. In fact, we should expect that greater
velocities in the direction of the axis of abscissee in comparison
with the smaller ones would be so much the more improbable
the greater the component of velocity of the molecule at right
angles to the axis of abscissee. If, for example,
mr ty ye ACen 3 e—hlatt 202(y2+27)] ,
— i :
I'(z, y, 2) =ce
then the quotient just mentioned would be
F(x, y,2z)\dx _ dx
WUE, y, 2)d& dé
The larger wf: y’? +2", the more would small values in com-
parison with large ones gain in probability. Now, by means
of the law of distribution of velocities, which is to be proved,
we obtain the proof of the very remarkable theorem: that the
relative probability of the different values of x is altogether
independent of the value, supposed to be given,which 4/7? + 2?
has for the same molecule; that therefore the quotient F(z, y,z) :
Fé, y, 2) is independent of y and z; or, what is the same,
since the three axes of coordinates must play the same part,
that F(z, y, z) may be represented as a product of three
functions of which the first contains only #, the second only
y, the third only z.
It is therefore an altogether inadmissible circulus viliosusto
make use of this assumption to prove Maxwell’s law of distribu-
tion of velocities. This therefore also holds good of the proof
which Prof. Tait has given (pp. 68 & 69 of the paper quoted),
and which is only a reproduction of Maxwell’s first proof,
which he himself later rejected. or, from the circumstance
that the distribution of velocities must be independent of the
special system of coordinates chosen for its calculation, we can
never show that F(w, y, z) must have the form f(x) d(y)W(),
only when this has been already proved. One might make use
of the circumstance to show the similarity of form of the three ~
functions /, @,and yr. I do not even need to enter upon known
geometrical investigations if the value of a function of three
rectangular coordinates x, y, z is independent of the choice of
the system of coordinates. For Prof. Tait has already shown
of the function denoted above by F, that it can only bea —
function of Vz2’+y?+2; but the value of the expression
V2 +y" +2 is already quite independent of the special posi-
* Phil. Mag. [4] xxxv. p. 145 (1868).
oh(Et—at) ph 2—a2)(y2-+ 22),
Arc-Lamp suitable for use with the Duboscg Lantern. 333
tion of the system of ened therefore evidently any
fanction whatever of /2?+y?+2’ fulfils the same condition,
and by this condition no other further property of the func-
tion F' can be disclosed. As, for example, the value of the
above-used function e—*@?++2"" is also entirely independent of
the special choice of the system of the coordinates, although
it does not permit of being reduced to the form /(z), $(y),
v(2).
XXXVIT. On an Arc-Lamp suitable to be used with the
Duboscg Lantern. - By, Professor Sirvanus P. THompPson,
D.Se*
[Plate IIL.]
HE lamp devised by Foucault and Duboscq, and supplied
for so many years by the famous house of Duboscq,
fails to fulfil the electrical requirements of the modern physi-
cal laboratory, though it has rendered excellent service in the
past. Yet the lantern and optical adjuncts of the standard
pattern of Duboseq are so widely used that it seemed desirable
to find some other arc-lamp which, while fulfilling the elec-
trical requirements of the case, could be used with the
Duboseq lantern.
Before describing the lamp which I have for twelve months
employed for this purpose, I propose to state the conditions
to be fulfilled, and the reasons why the old Duboscq lamp
fails to fulfil them.
The modern physical laboratory is usually supplied with
electric energy under one of two alternative conditions, namely
either at constant potential or with constant current ; more
usually under the former condition. If supplied from a
dynamo the dynamo may be either series-wound, shunt-
wound, or compound-wound. If supplied from accumulators
the accumulators will work at constant potential, and will
have a very small internal resistance.
The arc-lamp for laboratory use must be capable of working
under the given conditions. No doubt the Duboseq lamp
worked fairly when supplied with current trom 50 Grove’s
cells. But in a laboratory where there is another and better
and less wasteful source of supply, 50 Grove’s cells are not
desirable. Though 40 accumulators have an electromotive
force almost exactly equal to that of 50 Grove’s cells, the
Duboseq lamp does not work well with them unless a resist-
ance of several ohms is intercalated in the circuit to represent
* Communicated by the Physical Society.
334 Prof. 8. P. Thompson on an Arc-Lamp
the internal resistance of the Grove-cells ; and even then the
Duboseq lamp does not, for certain reasons, work as satis-
factorily as the lamp to be described, and its cost is about
three times as great.
In every arc-lamp for optical purposes there must be
mechanism adapted to perform the four following actions:—
1. To bring the carbons together into initial contact.
2. To part the carbons suddenly, and with certainty, to a
short distance—about 3 millimetres—apart. This action is
technically called “ striking ” the arc.
3. To supply carbon as fast as it is consumed, by moving
one (or both) of the pencils forward into the are. This
action is called “ feeding ” the arc.
4. To so move the carbons, or their holders, that the lumi-
nous points retain the same position in space at the proper
focus of the optical system. This action is called “ focusing”
the arc.
It may be remarked, in passing, that the feeding mechanism
of many lamps also performs the action, set down as No. 1 of
the above, of bringing the carbons into initial contact pre-
paratory to striking the are.
In many arc-lamps the attempt is made to unite the striking
and feeding mechanisms in one; but in many lamps, and in
the one I have to describe, the striking and feeding mechanisms
are distinct. The striking mechanism in all the arc-lamps of
commerce consists of an electromagnet or solenoid arranged
in the main circuit of the lamp, the armature or plunger of
the same being mechanically connected with one or both of
the carbons, so that when, by the turning on of the current
through the touching carbons, there is a great rush of current,
the attraction of the electromagnet or solenoid shall instantly
part the carbons and strike the arc. In the majority of the
commercial arc-lamps it is the upper carbon only that is
raised to strike the arc ; in a few other lamps, and in the one
I am using, the lower carbon is depressed. In one of the
older patterns of the Duboscq lamp the lower carbon was also
thus directly acted upon, its holder being attached to the
armature of an electromagnet beneath it. The same is true
of the Serrin lamp. But in the Duboscq-Foncault lamp the
arc is struck in a different way. ‘The two carbon-holders are
connected by racks to a clockwork gearing which either
parts them or brings them together, the movement being
driven by a double train of wheels, either of which can be
released in turn. The weight of the upper carbon-holder
drives the train that moves the carbons together; a coiled
spring drives the train that parts the carbons. Whether
suitable to be used with the Duboscq Lantern. 339
either of the trains, or neither of them, shall be released is
determined by the position of a double-toothed detent which,
placed between the final spur-wheels of the two trains, locks
both of them when in its mean position, but releases one or
other when shifted to right or left. The position of this de-
tent is determined by the current through the lamp, it being
attached to one end of a three-arm lever, the two other ends
of which are respectively attached to the armature of the con-
trolling electromagnet and to an opposing spiral spring.
When the moment of pull of the electromagnet upon its
armature is greater than that of the opposing spring, the
detent is pulled over one way, releasing the approximating
train of wheels while retaining locked the parting train.
When the moment of the pull of the opposing spring exceeds
that of the electromagnet on its armature, the detent is pulled
over the other way, locking the approximating train and
releasing the parting train. When the pull of the electro-
magnet exactly balances that of the opposing spring, both
trains are locked. Now when the current is at first turned on,
there isa sudden pull upon the armature of the electromagnet;
but the carbons are not instantly parted, partly because of
the inertia of the train of wheels, and partly because of the
backlash of the mechanism. Two or three seconds may
elapse before the arc is struck. This delay is serious, either
when working with dynamo or with accumulators. If the
dynamo is shunt-wound, the shortcircuiting even for this short
period demagnetizes the field-magnets. If the dynamo is
series-wound or compound-wound, or if accumulators are
being used, there is overheating during the period of delay.
Supposing, however, the arc to be struck, then the inertia of
the train of wheels makes itself evident in another way ; for
it parts the carbons too far, producing a long arc of consider-
able resistance ; and as the current then drops below its normal
value, the armature goes over the other way, and the other
train of wheels is momentarily released. ‘This alternation
between the two trains, which often lasts for some time, pro-
duces a disagreeable instability.
The feeding mechanism of arc-lamps next comes in for con-
sideration. The object of the feeding mechanism is to supply
carbon as fast as it is consumed, and so keep the light constant.
But the light cannot be kept constant unless the consumption
of electric energy in the arc is constant. The electric energy
is the product of two factors—the current through the arc, and
the difference of potential between the electrodes. Calling
the current i and the potential difference e, it is the product
ei which is to be kept constant. Now, as remarked at the
336 Prof. 8. P. Thompson on an Arc-Lamp
outset, the very conditions of modern electric supply are that
either ¢ or 7 is maintained constant, the usual arrangement in
commercial lighting being 7 constant for arc-lamps in series,
and e constant for glow-lamps in parallel. One of the two
factors being a constant by the conditions of the supply, the
other factor must be kept constant by the feeding mechanism.
Or, in other words, the variations of the other factor should
be made to control the action of the feeding mechanism. The
mechanical part of the feed may consist of a train of wheels
driven by the weight of the carbon-holder or by a spring, or
it may consist of a friction-clutch holding the carbon from
sliding forward, or of a worm-gearing or any other; but it
must be controlled by an electric mechanism of one of the two
following kinds. For keeping i constant, the feeding mecha-
nism must be controlled by an electromagnet (or solenoid)
placed in the main circuit, working against an opposing spring
or weight. For keeping e constant, the feeding mechanism
must be controlled by an electromagnet (or solenoid) placed
asashunt to the arc, and working against an opposing spring
or weight. In the latter case,if for any reason the arc grows
too long, the potential at the terminals will rise, more current
will flow around the shunt, which will then overcome its op-
posing spring (or weight), and will release the feeding
machinery until balance is restored. The use of the shunt,
introduced first by Lontin, enables arc-lamps to be connected —
two or more in series in one circuit. A less perfect solution
is the differential principle introduced by Von Hefner Alteneck,
where the difference between the attractions of a series and a
shunt-solenoid maintains constant, not the product e, but the
difference e—i.
The only perfect solution of the problem is a feeding
mechanism which, by combining in itself a shunt-coil and a
series-coil, shall keep the product e: a constant, however either
factor may vary. All the commercial arc-lamps for lighting
in series have shunt-circuits to control the feeding mechanism;
though often the arrangement takes the form of a shunt-coil
wound (differentially) outside the series-coil of the striking
mechanism ; so that feeding is accomplished by the shunt-coil
demagnetizing the striking electromagnet and momentarily
un-striking the arc.
Returning to the Duboscq lamp, it may be observed that, as
it possesses no shunt-coil, it can only feed by a weakening of
the current in the main circuit. Hence it is obvious that a
Duboseg lamp cannot possibly work in a constant-current
circuit. Also two Duboscq lamps will not work in series with
one another, as their individual feeding is not independent of
suitable to be used with the Duboscg Lantern. 337
the other. Neither will two work in parallel with one another;
for the weakening of current in one throws more current
through the other, and the instability before alluded to—called
“hunting” by electric engineers—becomes yet more pro-
nounced. |
The lamp that Ihave adapted to the Duboscq lantern is one
known in commerce as the “ Belfast ” arc-lamp, its principles
of construction being due to Mr. I’. M. Newton; but I have
had the design altered to suit the special work. In this lamp,
as previously mentioned, the striking and feeding mechanisms
are separate. ‘The arc is struck by means of an electromagnet
Hi of the tubular pattern, having as its armature an iron disk
A, which, when no current is passing, is held up by a short
spiral spring at about 3 millim. from the end of the electro-
magnet. The lower carbon-holder is mounted upon this disk,
so that the are is struck by the downward movement of
the lower carbon. The feeding mechanism is both simple
and effective. The upper carbon-holder is along straight
tube of brass: it passes through a collar in the frame of the
lamp, and also through a metal box Babove. This metal box
contains a piece of curry-comb with the steel bristles of the
comb set to point obliquely inwards and downwards. They
grip the carbon-holder and allow it to be pushed downwards,
but not upwards. ‘The box itself is mounted upon a strong
brass lever, L, close to the point of the lever. One end of this
lever is drawn downwards by an adjustable spiral spring 8,
whilst the other carries an iron armature which stands imme-
diately above the poles of an electromagnet, which is wound
with fine wire and placed as a shunt to the lamp. Above the
lever there is a contact-screw, platinum-tipped, making con-
tact with the lever, exactly as in the ordinary trembling electric
bell, and the lever and contact-screw are included in the shunt-
circuit. The attraction of the shunt-magnet for its armature
is opposed by the pull of the spiral spring. Whenever, by
reason of the resistance of the arc, a sufficient current flows
through the shunt-circuit, the opposing spring is overcome,
and the lever is set into vibration like the lever of an electric
bell, but more rapidly. The vibratory motion is thus com-
municated to the box containing the steel wire comb, which
at once, by an action well known in mechanism, wriggles
the carbon-holder downwards by innumerable small successive
impulses. So soon as the motion of the carbon has reduced
the resistance of the arc, the shunt-current diminishes and
the feeding action ceases, to reeommence when required. It
is found best for lantern-purposes to send the current upwards
through the lamp, the lower carbon being the positive one.
Phil. Mag. 8. 5. Vol. 23. No. 143. April 1887. 2A
338 Mr. R. H. M. Bosanquet on Electromagnets.
A thick cored carbon of 13 to 15 millim. diameter is preferred,
as it gives a good luminous crater and burns slowly. A 10-
millim. copper-plated carbon is used for the upper electrode,
and it is adjusted so that its centre falls slightly in front of
the centre of the lower carbon, thereby causing the crater to
send its light forward.
The lamp as used in commerce has no focusing-arrange-
ment. In adapting it to the Duboscq lantern, the frame was
made narrow; so that when the inner chimney of the Duboseq
lantern was removed, the lamp could be dropped entire down
the outer chimney, a metal sleeve of the same diameter as
the inner chimney being added to the lamp as a guide. At
the bottom of the lamp a gun-metal tube was added, tapped
inside with a screw-thread, into which works a steel screw
having a small hand-wheel near its lower end and a pointed
‘pivot at the extremity. The lamp slides down the chimney
of the lantern until the pivot touches the base-board. When
the are burns down the lower carbon, so that the luminous
crater is no longer in the optical focus, a turn given by hand
to the wheel suffices to raise it to the proper position; but
the lamp will burn for ten minutes without requiring any
readjustment on this account. The lamp shown to the Physical
Society was constructed by Mr. E. Rousseau, Instructor in
the Physical Workshop of the Finsbury Technical College,
assisted by Mr. A. D. Raine, now Demonstrator in the City .
and Guilds Central Institution.
XXXVITI. Hlectromagnets—VII. The Law of the Elec-
tromagnet and the Law of the Dynamo. By R. H. M.
BosanquEtT, St. John’s College, Oxford.
To the Editors of the Philosophical Magazine and ee
GENTLEMEN,
| ae present communication will consist of two parts. First,
the application of the measures of the bars with pole-
pieces, contained in No. V. of this series (Phil. Mag. [5] xxii.
p. 298), to the establishment of the type of law which governs
electromagnets of this description, and the comparison of
this law with the various assumptions which have been made
on the subject, and in particular with Frdlich’s law and the
law of tangents.
Secondly, a few propositions will be stated which offer a
general method of discussing the action of dynamos, inde-
pendent of the assumption of any particular law of magneti-
zation, and based on a consideration of the dynamic action.
Mr. R. H. M. Bosanquet on Electromagnets. 339
These will be applied to the law first obtained. A discussion
of the actual behaviour of my Gramme dynamo will follow in
a future paper. |
The average values of the magnetic resistance of a number >
of bars with pole-pieces throughout the whole course of
magnetization were given in the investigation above cited.
The bars in question all had cores of the same shape, viz.
length : diameter :: 20:1. Thenumbers for other shapes will
no doubt be different, and the laws which deal with the shapes
will be the subject of future investigation; but it is not likely
that the general type of the law will differ materially from
that here discussed.
It has been a matter for some consideration what scheme
should be adopted for the representation and comparison of
the different laws. I have adopted a scheme in which the
magnetic inductions are measured horizontally, and the
permeabilities, or, as I prefer to call them, conductivities,
vertically.
The reason for adopting the magnetic induction as the
chief variable, is the fact that the magnetic properties of the
metal depend only on the induction. It is enough to glance
at the figures which follow, or, better, at the reciprocal figures
representing magnetic resistance at p. 303 of the paper above
cited, to see that, whether the metal be in the form of rings,
or of bars with or without pole-pieces, the resistance to
magnetization (or its reciprocal the conductivity) changes
always in much the same way at the same values of the
induction. Any representation which overlooks this, overlooks
the principal law so far known as to the variation of the
magnetic properties of iron. And I dissent from the position
of those who say that the magnetizing current, or magneti-
zing force, has the chief claim to be regarded as the
independent variable in such a representation.
I have in this case chosen conductivity, rather than the
magnetic resistance, to be combined with the magnetic
induction, mainly because this combination represents F'ré-
lich’s law, which is so generally accepted, as a straight line ;
and this facilitates the comparison of Frolich’s law with other
laws.
A few words as to the precise meaning of the expressions
permeability and conductivity.
The word permeability was originally used in connection
with the old theory, and was the ratio
% magnetic induction
= or = ;
a) magnetizing force ”
2
340 Mr. R. H. M. Bosanquet on Electromagnets.
but the magnetizing force existing within the metal of the
bar, for instance, was supposed not to be the same as the
external magnetizing force, but to be diminished by demag-
netizing forces, which resided on the ends of the bar. In
rings, however, where there were no ends, the theoretical
magnetizing force was the same as the external magnetizing
force, and the permeability of a magnetic metal could be
determined by measures of rings.
In dealing practically with bars, what we want to know is
the connection between the magnetism developed and the
external magnetizing force. The ratio of these two quantities,
which I call conductivity, is a quantity analogous to the
permeability of the theory, but not identical with it ; the
conductivity does not take into account the supposed demag-
netizing forces at the ends of the bar. The system that I
have adopted prefers to attribute the diminished magnetism in
the case of bars with ends to the increased resistance experi-
enced by the magnetism in traversing space external to the
magnetic metal; and I employ the word conductivity in
connection with an entire magnetic circuit, with its air
resistances ; leaving permeability with its original application
to circuits or parts of circuits lying wholly within the
magnetized substance.
The permeability of rings and the conductivity of bars
have, then, precisely the same meaning. In both cases the
meaning is,
magnetic induction
external magnetizing force’
If we suppose the magnetizing force uniformly extended
along the bar, as by a uniformly wound coil, we have
Hxternal magnetizing force x length = external potential.
Conductivity __ magnetic induction _ 1
length =~ ~‘magnetic potential ~ p’
where p is magnetic resistance, according to our definition of
magnetic resistance.
Whence permeability of rings or conductivity of bars
_ length
p
I invariably reserve * for the permeability, since it is thus
* In a paper presented to the Royal Society, Dec. 20, 1880 (Proce. R.
S. vol. xxxiv. p. 445), in which the system of broken magnetic circuits
with air resistances, now so generally used by practical men, was first
developed, I distinguished insufficiently between permeability and conduc-
tivity, and used » to represent both ideas.
Mr. R. H. M. Bosanquet on Electromagnets. 341
used by Maxwell and the best writers. For conductivity I
write either ; or cy.
Having the magnetic resistances of the magnets with pole-
pieces mentioned above, we can obtain their conductivities ;
.)
p
The formula known as Frdélich’s may be obtained by
assuming that the conductivity of the magnet is proportioned
to its defect of saturation. (See my letter to ‘The Electrician,’
vol. xvi. p. 247, February 1886; and Prof. 8. P. Thompson,
Phil. Mag. xxii. p. 290, Sept. 1886.) Ifwe measure the mag-
netism as % (magnetic induction), we may write this,
Conductivity = k (B, — B),
which represents a straight line on the scheme of conduc-
tivities and inductions ; see figs. 1 & 2.
For the sake of clearness I have drawn on fig. 1 the
permeabilities of Rowland’s table i., and of my ring H, and
also the average conductivities of plain bars and bars with
pole-pieces, deduced from the magnetic resistances given in my
paper first above cited ; also both in figs. 1 & 2 the applica-
tion of Frélich’s law to the bars with pole-pieces.
It will be seen that, if a real state of things be represented
by any curve, a tangent drawn to that curve at any point will
represent a Frélich’s law, which will be true only so far as the
curve and the tangent coincide. In the present case there
appears to be a point of inflexion on the curve just before
approaching the region of what I may call super-saturation
(tendency of % to increase without actual limit ; see paper
first cited). The tangent drawn through this point of inflexion
coincides with the curve for a considerable distance in the
neighbourhood of % = 15,000 ; and I shall show later that the
excitation in the cores of a dynamo with such magnets may
be confined in actual practice within very moderate limits on
either side of this value.
The use of Frolich’s law to deduce consequences where
wide variations of the magnetic intensity take place, as, for
instance, where the magnetism is supposed to be reduced to
half its maximum value, appears to be fallacious in such cases
as the present. :
The curve in fig. 2 is the same as the curve marked “ Bars
with PP ”’ in fig. 1, but drawn to a larger vertical scale.
342 Mr. R. H. M. Bosanquet on Electromagnets.
Conductivity-=
2,500
His, 1,
Rowland’s Table No.1.
2,000
bt ae
500
me with. ieee. mae
700
1,000
=<,
=
IS
Magnetic Conductioity g& Bars 1-20 with pole pieces
Conductivity 2
200
700
1,000
Plain- Bars ee
Sie |
5,000 10,000
Fig. 2.
o
AUN
EEEEPTP EEE
FEECEEEELE EEE
15,000
5,000 10,000
15,000
[Ss
-..| _ |3$
20,000
20,000
Mr. R. H. M. Bosanquet on Electromagnets. 343
The following Table contains the conductivities of the bars
with pole-pieces, as obtained from the experimental magnetic
resistances, and their comparison with some of the different
methods which may be or have been adopted for representing
the law of magnetism. These methods are enumerated in the
following statement, the table only containing such compa~
risons as seem necessary.
Numerical Comparison with Experiments of some of the
Calculations representing the Magnetic Conductivity of
Bars 1 : 20 with Pole-pieces.
| |
| |
Conduc- :
a : s III. Rule with
pny or | II. Fourier Series. approximate p. IV. Tangent law.
B. = Oe seh Sth Uae eit eal Py ebue ne case
E 1 i I
eee: | Das. oe eT A Dilts,
periment. p p p
0,000 169 187 +18 171 + 2
1,000; 271 245 —26 252 —19 614
2,000; 301 290 --11 289 —12 610
3,000; 319 316 — 3 310 — 9 603
4,000} 332 326 — 6 323 9 592
5,000| 337 332 — 5 330 — 7 578
6,000| 340 337 — 3 334 — 6 561
7,000; 340 340 0 338 — 2 541
8,000} 340 340 0 309 — 1 518
9,000; 339 339 0 338 — 1 490
10,000} 336 337 +1 334 — 2 459
11,000} 332 331 — 1 330 — 2 424
12,000| . 324 327 +3 323 — 1 385 +61
13,000} 310 312 + 2 310 0 341 +31
14,000} 286 283 — 3 289 + 3 292 + 6
15,000| 288 233 — 5 252 +14 238 0
16,000 171 175 +4 171 0 17k 0
17,000} 136 130 — 6 110 — 26
18,000 91 104 +13 36 —55
Statement of Methods for representing Conductivities by
Calculation.
I, By value of D where p is derived from my theory: see
Phil. Mag. vol. xxii. p. 8308 (September 1886). (The formule
at the head of the tables at pp. 307 & 308 have been unfor-
tunately wrongly copied. I give the correct headings com-
plete in the Hrrata at the end of this paper.)
This representation is quite close, but the computations are
rather too laborious for ordinary use. It is not necessary to
exhibit the comparison.
Il. Empirical representation of conductivities by Fourier’s
344 Mr. R. H. M. Bosanquet on Electromagnets.
series. This is fitted specially in the region from %% =7000
upwards, which includes the dynamo range.
= 210-+150 sin 6+26 sin 30+ 8sin 50+2 sin 70,
where 6=-0124%3 —400}°.
III. Representation by means of my rule that
p = shape-constant + °
with rough approximate value of pw. The shape-constant
for unit length 00252 in this case is derived from my in-
vestigation cited above. General values for this will be the
subject of further investigation.
Dee ae
Di isl
00252 + —
be
where p= 300 + 2000 sin @,
— 800 13 e
This fits fairly up to % =16,000, but fails above. It is a
useful formula for an approximation to the general outline.
IV. Representation by means of the tangent law.
G g°
Magnetizing force = conluelney =k tan 505"
Fitting to the 15,000 entry,
k=1-28051 3
then |
The differences are only entered when there is some approxi-
mation to the truth.
V. Frolich’s law.
Conductivity = =
p 7 magnetizing force
=k(B,—B).
This is shown by the straight line on the figures. It is
unnecessary to exhibit the calculation of the numbers. The
correspondence with the truth is about the same as in the
tangent law ; but the error increases more rapidly, and ulti-
mately becomes much larger, as the magnetism diminishes.
Mr. R. H. M. Bosanquet on Electromagnets. 345
Thus the tangent and Frolich’s laws, upon one or other of
which almost all treatment of the theory of dynamo machines
has been based, are shown to be far from representing the
true laws which govern electromagnets.
In a paper by the Messrs. Hopkinson, reprinted in ‘ The
Hlectrician,’ Nov. 19th, 1886, we have an example of another
way in which it has been attempted to fit Frdlich’s law to
represent the law of magnetization. The intersection of the
Frélich law with the true law in diagram A there given is
made to take place at about % = 5500. If the case be
represented by a scheme of conductivity and induction, the
straight line representing [rélich’s law crosses the curve of
the true law at a considerable angle, and by the end of the
representation in about % = 11,000 the two diverge widely.
Now it seems unlikely that Frolich’s law, so used, can have
any bearing upon the action of the dynamo machine. The
advantage of the law is that, being easily manipulated, it can
be made to coincide exactly with the true law in the part of
the dynamo range in actual use. Such a case is represented
by a tangent drawn to the curve in my scheme. It is very
unlikely, however, that any dynamic action, such as to be of
practical utility, could take place in the region of B = 5000.
I shall now proceed to a few propositions, suitable for
application to the true laws of electromagnets as embodied in
series of numbers rather than in formule, founded chiefly on
the dynamic action itself.
The outlines of the theory have been explained to some
extent in my paper on Self-regulating Dynamo machines,
Phil. Mag. [5] xv. p. 275. But the application to laws
expressed numerically, and the line of reasoning now adopted
are new.
For the present I confine myself to the series dynamo.
According to the mode of statement now usually adopted,
the H.M.F. developed in an armature at n revolutions is
Bye AAAs ite ee
where 9%, is the field-intensity within the coils of the armature.
This differs only in arrangement of units from the formula
adopted in my previous paper.
The first thing is to express %, in terms of the % developed
in the field-magnets. We have measures of the % across the
equatorial sections of the field-magnets, and can connect it to
some extent with the potential of the magnetizing current.
In the present approximate purpose we assume
et fe a
346 Mr. R. H. M. Bosanquet on Electromagnets.
where f may be called the coefficient of efficiency ; it will
depend on the build of the machine, and may probably range
from 4 to 3/5 or less. Nothing in the theory depends on it, so
far as our present purpose is concerned. We neglect varia-
tions of distribution, which would give rise to change of f.
We then put H=CR, from Ohm’s electrical law in the
circuit, and the equation stands
Cl 4n ASB. ee
The next step is to express 93 in terms of the magnetizing
current. If we make here the usual assumption,
%3 = conductivity x magnetizing force,
4
or G = cy x a . 8 Oe eee
and substitute in (3), the current disappears from the equation,
and we have
[R = I6amn Af Xx cys: 4) a
a relation between the coefficients for a given value of the
conductivity, which is not without use, but does not help us
in the general problem.
The assumption (4) isnot, however, called for by the nature
of things, for it is clear that % is not generally proportional
to the magnetizing current. And our present treatment will
be founded on the assumption that, so long as the conditions
of the machine vary but little, there must be some power of
the magnetizing current or of its magnetic potential to which
$3 may be regarded as proportional. Assume, then, a general
form of law which can be fitted to any part of the range of
magnetization,
B= KO ch eee (6)
Substitute this in (3), and gather up the constants into the
coefficient ; then,
Cry
1
eae 62k at.
which expresses the current as a power of the velocity of
rotation.
: : 1
Here we may conveniently put #= i and assume Cy and
nm, to be a pair of corresponding values differing little from C
and n, so that és
n HH
Cami, ) (8)
By means of this formula we can determine the value of «
Mr. R. H. M. Bosanquet on Electromagnets. 347
experimentally, for any condition of the machine. We vary
the speed slightly, and measure the two speeds and the two
currents. Then we have « from the equation
2 (log n—log m)=log C—log GQ. . . . (9)
A rough determination of the values of 2 for my Gramme
machine, by this method, is given at Phil. Mag. [5] xv.
p- 285, It is as follows * :—
Current Amperes.
BUS SRT, ge yak
10
f oe
aC Zoe a
bo oo &
cH tolHoxito “SS
x=1, y=0, correspond to the condition of saturation, accor-
ding to the theoretical assumption of a saturation limit, which
we know is not quite justified in practice. .
I have now to show how the assumption (6) can be fitted
to a law of magnetization when the law is given by a series
of numbers representing the magnetic resistances or conduc-
tivities of the magnets of the machine, for the different induc-
tions used,
Rearrange (6) as follows :—
95 —K’ (magnetizing force)’; . . . . (7)
(yk, SRST at ets
magn. force
then
or
%-Y (conductivity )¥== ys) |. sei ie (9)
or, if %, cy; %o, cyo are pairs of values differing but little,
BB ~1(cy)¥=Bo (eyo), ~ » - « + = (10)
and
ae es a
Ee =xr—1;
fy Aes. rep}
° B, a * >]
whence
log B— log B .
Smt hime log cy’ ween)
* These numbers are so far justified by my later determinations that
it is not worth while to amend them at present. The conditions and
limitations to which they are subject will be touched ae in the discus-
sion of my dynamo.
848 Mr. R. H. M. Bosanquet on Electromagnets. ©
| where « is that power of the velocity with which the current
: varies, in the given state of the machine.
We immediately infer some propositions of interest with
regard to 2.
x—1 can only be finite and positive so long as the conduc-.
tivity diminishes as 9% increases. |
It is infinite when the conductivity is constant. It is_
negative when % and the conductivity increase together. .
Following the course of the changes usual in electromag-
nets, which we may illustrate by fig. 2, we then have
z—l. Sg oe ae i : Conductivity.
- - »« « Oup to about 5000 © increasing.
ie ie, “9000 to about 7000 maximum.
+0
“= Scant above 7000 diminishing.
Limit 0 at saturation 18,000 to 20,000.
- Thus the range of possible dynamic equilibrium is from
about %=7000 upwards. In the lower part of this z is great,
| or the current changes violently for small changes in the
velocity. In the saturation region # approximates to 1, and
| the current is more nearly proportional to the velocity.
I have calculated the values of x given by the successive
pairs of numbers on which fig. 2 is based (bars with pole-
pieces). » These. are:— ,
BB. ie
8/500) uae eae 3) Qe t
F500 me mee Eh) ese .
10,500: edge Bo 3 whos
11,500 eee ia dene
12,500 Rae ea a
13,500 eee eae teal
14,500; tee oe ilar
15,500.10 ee a Gale
16,500. /.k7. 5 Weg cee ileal
Plotting these on a scale, I took out the values of % corre-
sponding to those of w obtained from the Gramme machine.
And from the experimental conductivities (fig. 2) I calculated
the magnetizing forces which would be required by magnets
such as ours to produce this condition. These are :— reid
Mr. R. H. M. Bosanquet on Electromagnets. 349
Power of velocity Inductions in dynamo | Magnetizing forces
to which current having electromagnets required by bars
is proportional, such as in fig. 2. with pole-pieces.
2. 8.
3 12,500 39:4
2 13,640 455
1:25 15,320 729
We cannot of course assume that the magnets of the Gramme
machine follow the same law as that of our bars; in fact they
do not do so at all approximately, probably in consequence of
the large amount of cast iron in the machine. But these
numbers are enough to illustrate the limited nature of the
variations of the induction which may be possible during the
working of a dynamo, while the current produced varies in a
ratio of more than 2:1. By carrying the magnetization
higher still, we get a further considerable range of current
with a small change of the induction. Looking at fig. 2, we
see that it would be possible to draw a secant through the
point of inflection, representing a Frélich’s law, and deviating
but little from the curve from about 12,500 up to18,000. In
this way the law of magnetization would be approximately
represented by a Frélich’s law over a very wide range.
A word as to the physical meaning of the quantity y, which
is connected with w by the relation ~ =u. This y may be
said to be what determines the dynamic action. The dynamic
action consists of the summation ofan infinite number of ele-
ments, whether of magnetism, current, or H.M.F., which
originate in one small change of velocity. It is only where
these elements are successively less and less, and so form a
convergent series, that their sum is finite, and gives rise to a
definite behaviour of the machine, or to what we may call a
state of dynamic equilibrium.
Let y be then the ratio of diminution of the successive
elements, due to the additional element of induced magnetism
being less than the element of inducing current. It is easy
to see that we may express the whole change of C, say, due
to a small change of velocity dn, thus :—
AS =(ltyte+...)@
nr
Lek) On
~ l-y 2
390 Mr. R. H. M. Bosanquet on Electromagnets.
(see Phil. Mag. [5] xv. pp. 285 & 286) ; whence
log.n= (1—y) log C+ const.,.
or
N= comeby si! >,
or
n*= const. O;
and we have deduced the form of equation before assumed,
from the principles of the dynamic action of the machine.
In practice these considerations and laws are much modified,
chiefly by the enormous magnetic retentiveness of the field-
magnets.
I shall have to deal with this subject in discussing the
performance of my Gramme machine.
HRRATA in recent papers.
Phil. Mag, vol. xxii. p. 307, for the heading of the Table substitute
7 Bars with P.P. without Magnetizing-Force term.
pcale, ="00252-4 * 1=1centim., log < = 235933,
p='415 (18,366 — 33) cos 6, log f= ‘16500,
vat __ Kk 60°—w
Se aang tamer
Phil, Mag. vol. xxii. p. 308, for the heading of the Table substitute
Bars with P.P. with Magnetizing-Force term.
pcale. =-00252+ /, 7=1centim, logn=1-08160,
pe
u='69 (= +16,421-33) cos 6, loge : =2°57346,
78, 38 ra log f = ‘16878.
Phil. Mag. vol. xxii. p, 536, line 3, for ‘526 centim. in diameter read ‘526
inch in diameter. (This clerical error appears in the description of the
instrument, but does not affect the calculations.)
fBaE J
XXXIX. Note on the Tenacity of Spun Glass.
By H. Gisson and R. A. GREGORY *.
1 is well known that the tenacity of metallic wires increases
as the diameter diminishes, so that very fine wires will
carry much larger loads than those obtained by calculation
based upon the assumption that the breaking weight varies
as the square of the diameter. As glass can be drawn into very
fine fibres, we have made some observations on the tenacity
of this material, comparing the strength of very thin threads
with that of rods made from the same glass, but of much
greater thickness.
_ The experiments were carried out in the course of our work
in the Physical Laboratory of the Normal School of Science
and Royal School of Mines.
In dealing with a substance so brittle as glass, it is evident
that special care must be taken to ensure that the observation
is not vitiated by rupture due to a shearing stress, at or near
the points of support. Precautions were taken to prevent
this in all cases, and no experiments are quoted in this paper
in which rupture took place near the points of support, or of
attachment of the weight.
Three different thicknesses of glass were subjected to
experiment: viz., fibres the diameters of which were about
0-002 and 0-004 centim. respectively, and rods with dia-
meters varying between 0:05 and 0:09 centim.
The fibres were attached at theends of two strips of paper by
means of shellac varnish ; this on setting was found sufficiently
strong to carry more than the breaking weight, without
allowing the fibre to slip. A small paper basket suspended
from the lower strip carried the load, consisting of fine shot
and silica, the latter being added when the fibre was near its
breakin g-point.
The diameter of the thread was measured at the place of
rupture by means of a Compound Microscope with micrometer
eye-piece. From data thus obtained the tenacity was calcu-
lated with the following results :—
Tenacity, in dynes
Diameter, in centims. B. weight in grs. per sq. centim.
0-00186 11°76 424 x 10°
0:00159 8°70 425 x 10°
0:00315 32°26 405 x 10°
0-003840 43°23 466 x 10’
~ * Communicated by the Physical Society: read February 12, 1887,
352 Note on the Tenacity of Spun Glass.
Some observations were next made on rods about 1 millim.
in diameter ; the method of support and the loading being
changed. Two pieces of angle brass, each about 8 inches
long, were substituted for the slips of paper. Through a hole
drilled near the end of the angles, a piece of }” wire was
passed, turned up and soldered to the back. The free
extremities of the wires were plaited into rings, which served
to support the load and suspend the whole from a hook above.
The ends of the rod were laid in the angles, leaving the
glass free for about 12 inches. Small pieces of red-ochre
cement (a compound consisting of resin, red ochre, and bees-
wax) were placed at intervals along the glass, and a Bunsen
flame applied. The cement speedily melted, and imbedded
the glass; on cooling, the whole was suspended vertically.
A bottle was hung on the wire attached to the lower angle-
piece, into which a fine stream of mercury flowed from a
reservoir above. ‘The apparatus was so arranged that when
the rod broke mercury would no longer fall into the bottle.
The mode of measuring the diameters of the rods differed
from that adopted in the case of the fibres. About half an
inch of rod was broken away at the place of rupture, and
mounted in wax on a piece of looking-glass, the broken section
being upwards. Its diameterwas then measured by means of
a microscope-cathetometer, and the tenacity found as in the
ease of fibres. The following are the results of four
experiments :—
Tenacity, in dynes
Diameter, in centims. Weight, in grs. per sq. centim.
0-090 3908 60 x 107
0°082 4443 83 x 10’
0-050 1948 OT x 10F
0-042 1731 126 x 10°
These observations show, in the first place, that the tenacity
of fine fibres is very considerably greater than that of thick
rods, and that the strength of rods increases as the diameter
diminishes. It may be interesting to point out that the
tenacity of glass fibres studied by us is nearly as great as that
assigned by Wertheim to many ofthe metals ; e. g., the tenacity
given by him for annealed steel wire 1 millim. in diameter i 1s
499 x 10’ cent.-dynes, and even in the case of drawn steel the
tenacity is not greater than twice that of a glass fibre, viz.
998 x 10° cent. -dynes.
With steel pianoforte-wire the tenacity is, however, con-
siderably greater ; according to Sir William Thomson (Art.
On an Improved Form of Seismograph. 353
‘ Hlasticity,’ Encyc. Brit., new edition) the breaking-stress is
Cent.-dynes.
Best pianoforte steel-wire . . . . . 2318 x 10!
The question as to what is the most probable cause of this
increase in strength as the diameter diminishes, presents some
difficulty.
Quincke (Comptes Rend. de Vl Acad. de Berlin, 1868,
p. 132) has suggested that the great increase observed in the
case of metals is due to a surface tension, analogous to that
observed in liquids. If this were the true explanation, the
breaking-weight could be expressed by the sum of two terms
which vary as the diameter and the square of the diameter
respectively. This suggestion does not receive much support
from our observations, as the results cannot be satisfactorily
expressed by means of such a formula. It is, perhaps, more
probable that the heating and rapid cooling undergone by the
glass when it is drawn out into a fine fibre produces an
increase in tenacity ; and it is at all events certain that no
comparisons can be made between the strengths of different
materials unless they have undergone similar treatment, and
unless the sizes of the rods or wires submitted to experiment
are the same.
XL. On an Improved Form of Seismograph.
By Tuomas Gray, B.Sc., F.RS.L*
[Plate IV. ]
| ee apparatus described in this paper is an improved
form of a seismograph which was made for Prof. Milne
in the beginning of 1883, to be used by him in his investi-
gations for the Committee appointed by the British Associa-
tion to “ Investigate the Harthquake Phenomena of Japan.”
That apparatus was exhibited to the Geological Society of
London, and a description of it by the present writer was
published in the Quarterly Journal of that Society in May of
the same year. It consisted of a combination of instruments
which had been devised by Prof. Milne and the writer, and
descriptions of which had appeared from time to time in the
‘ Transactions of the Seismological Society of Japan,’ and in
the ‘ Philosophical Magazine.? The object of the apparatus
was to determine the time of occurrence, the amount, the
period, and the direction of the different motions in an earth-
quake shock. Arrangements were made for recording three
components of the motion, one vertical and two horizontal, at
* Communicated by the Author.
Phil, Mag. 8. 5. Vol. 23. No. 143. April 1887. 2B
Se o--
354 Mr, T. Gray on an Improved
right angles to each other, on a band of smoked paper which
covered the surface of a cylinder. The cylinder was intended
to be kept continuously in motion round its axis by clock-
work ; and the recording points were, on the supposition of
no motion of the earth, expected to trace continuoasly the
same line on the smoked paper in a sim‘lar manner to that
introduced by Prof. J. A. Hwing, and used by him in his
experiments in Japan*. Prof. Hwing used smoked glass for
his record-receiving surfaces, and that is a very good arrange-
ment when it can be conveniently adopted. It had been pre-
viously used by Prof. Milne in apparatus in which the
record-receiver was either stationary or automatically started
into motion by the earthquake ; and it has since been much
used by him and the writer in earthquake investigations.
Smoked paper was adopted in the apparatus here referred to,
and, when smoked surfaces are used, it is still recommended
for the present form, because it is desirable to obtain straight
records, written side by side and to the same scale, of all the
three components. ‘This, combined with continuous motion,
could only be got on a cylindrical surface ; and, considering
the risk of breakage, cylinders of glass sufficiently true and
inexpensive could not be readily obtained.
The apparatus used for recording the motions was in prin-
ciple the same as that described in this paper, but differed
considerably in detail. A separate clock was provided for
the purpose of recording the time of occurrence, the record
being made on the dial of the clock, which was, at the time
of an earthquake, automatically pushed forward into contact
with ink-pads fixed to the ends of the hands, a mark being at
the same time made on the record-receiver to show at what
part of the earthquake the time was recorded. In subsequent
instruments this method of recording time was abandoned
because, with the improved form of record-receiving appa-
ratus, it became unnecessaryt. This will be more particularly
referred to when the method of recording time now adopted
is being described.
The instrument above referred to was set up in the Meteo-
rological Observatory in Tokio, where it is still in use. Hx-
perience with it, however,soon suggested many improvements,
* See “A new Form of Pendulum Seismograph,” Trans. Seis. Soc.
Japan, vol. i. part 1, p. 38; and “On a New Seismograph,” Proc. R. S.
no. 210 (1881).
+ This refers only to the instruments here described, which are made
in this country by White, of Glasgow. In a less complete form of the
apparatus made in Japan, and a considerable number of which are in use
in different parts of that country, the clock with movable dial is still used.
Form of Seismograph. 355
which have been introduced into later instruments. It was
found that when the “ conical pendulums”’ (see below, p. 362)
used for actuating the recording indices were, as in that in-
strument, made to turn with very little ee and were
adjusted to have a long period of free oscillation, that is to
say to have very little positive stability, the lines traced by
the recording points gradually broadened to a very incon-
venient extent. This rendered good records of small motions
impossible after the record-receiver had been in motion for a
short time, and introduced a risk that such records might be
obliterated after they had been obtained. Such considerations
as these led Prof. Milne to abandon the continuous motion
element, and adapt the instrument to the comparatively old
method of automatic starting at the time of the earthquake.
There are, besides the difficulty experienced due to the
broadening of the lines by the recording points, several other
important objections to the use of a band of paper of such
limited length as that provided by a single turn round a
cylinder of moderate dimensions. The record may, for ex-
ample, extend more than once round the cylinder ; that is,
the earthquake may last longer than the time taken by the
cylinder to make a complete turn. This produces great con-
fusion in the record, rendering it difficult to interpret.
Again, two earthquakes may occur before the record-sheet
has been changed ; and in such a case both records are practi-
cally lost. Considerations such as these have led us to adopt
one or other of the forms of apparatus described in this paper.
The new form of apparatus has for its object the determi-
nation of the same elements as have been already enumerated
with reference to the old instrument. Provision is, however,
now made for the whole of the record being obtained on fresh
surtace, and for any number of earthquakes which may occur
within a limited period, say a week, being recorded on the
same sheet. The record-receiver is kept continuously in
motion at a very slow rate, and time is marked on it at
regular intervals by means of a good clock; the object being
to secure with perfect certainty that most important element
in earthquake investigation—the time of occurrence of the dis-
turbance. In the most complete form of the apparatus the
record-receiving surface is a long ribbon of thin paper, which
is gradually unwound from a supply drum on to another,
which may be called the hauling-off drum, by means of a
weight or spring and a train of wheelwork. The speed is
rendered uniform by taking the paper in its passage from the
one drum to the other round a third drum, which is kept
continuously in uniform motion by a train of clockwork and
2B2
396 Mr. T. Gray on an Improved
a suitable governor. A somewhat simpler arrangement is
obtained by using a single drum covered with paper, or a
smoked glass or metal cylinder, and giving to this cylinder a
slow motion of translation in the direction of its axis, so that
the record takes the form of a spiral line round it. As, how-
ever, the rate of motion must be such as to give the time of
occurrence with fair accuracy within a second of time, it is
difficult to obtain a good record on a cylinder of moderate
size, which will extend over more than twelve hours with
this arrangement. It is of course easy to adapt the apparatus
to be used either way, if that were desirable; but the con-
tinuous ribbon of paper is so much the better form of re-
ceiving-surface that the description given in this paper, in so
far as it refers to earthquales, only includes that form. The
spiral record has some advantages in apparatus adapted to
record slow changes of level of the earth’s surface ; and it will
be again referred to in that connection. For such purposes
the rate of motion may be made excessively slow; end hence
the records for a considerable length of time may be written
on one sheet.
At the time of occurrerce of an earthquake, the rate of
motion of the paper is automatically greatly increased, and a
chronographic reed is simultaneously set into vibration, and
made to mark equal intervals of time on the ribbon, thus -
showing accurately the rate of motion at any instant. The
actual rate of motion of the paper on the slow speed may be
varied from about a quarter of an inch to an inch per minute,
and on the fast speed from about 25 to 50 inches per minute,
with the present form of instrument. This change of speed
is generally obtained by including in the driving clockwork
two governors, one of which can be automatically thrown out
of gear, either electromagnetically or mechanically. The
latter method has been found the best and the simplest in
practice. The arrangement commonly used is described
below, page 361, and need not be more particularly referred
to here than in a general statemert of the operations it is in-
tended to perform. At the time of an earthquake three
operations take place simultaneously. One is the introduction
in train with the clockwork of an adjusting mechanism which
is intended to readjust the starting apparatus, wuatever that
may be, so thati t may be in readiness for another earthquake
should that occur. Another is to throw out of gear the slow-
speed governor, or, if that method is adopted, to work a change-
wheel lever, so as to shorten the train between the driving
power and the governor. A third is to close the circuit of
the chronographic reed, so as to cause it to mark time on the
Form of Seismograph. 357
record sheet. It will thus be seen that the instrument is
intended to be absolutely self-acting, so long as its supply of
paper lasts and the driving mechanism continues to go. The
supply-drum can take as much paper as Is required in a week
on the slow speed.
The record is made in ink by means of fine glass siphons,
in very much the same manner as that which was introduced
by Sir William Thomson in his siphon-recorder for submarine
telegraph-cable work. This is extremely well adapted for the
continuous ribbon method of working, and, besides, gives an
excellent clear record which requires no further preparation
before it is filed for reference ; and, what is of great im-
portance, the record is obtained with exceedingly little dis-
turbance from friction at the marking-point.
The siphons which write the horizontal components of the
motion are controlled by two pendulums, the suspending wires
of which are held out of the vertical by horizontal struts ter-
minating in knife-edges which rest against the bottoms of
flat V-grooves fixed to a cast-iron pillar rigidly attached to
the sole plate of the instrument. These pendulums, when set
in vibration, describe cones, and hence they have been called
“conical pendulums.”’ The degree of deflection from the
vertical can be varied from about one and a half inches toa
foot, by sliding the pendulum-bob along the strut. The strut
is made in two pieces, so that a part of it can be removed
when high sensibility is required, and in consequence the
mass is used near the knife-edge. The bob of the pendulum
is suspended by a fine platinum or steel wire from an arrange-
ment which permits the suspending wire to be lengthened and
shortened, and also allows the po‘nts of suspension to be put
in such positions above the knife-edges as causes the struts to
place themselves in positions at right angles to each other,
and at the same time provides the means of adjusting their
periods of free vibration to any desired length *.
It is of great importance in apparatus of this kind that the
mass which, through its inertia, enables the record of the
motion of the earth to be written, should be as far as possible
from the knife-edge or poins fixed to the earth; a long
period of free vibration can thus be obtained combined with
considerable stability of position, while the greatest motion to
which the knife-edge is likely to be subjected does not turn
the strut through a large angle. If this latter condition be
* This pendulum isa modification of one designed by the Author in
the beginning of 1880, in which the weight was supported by a thin wire
in line with a rigid vertical axis fixed to the end of the strut and resting
against bearings so as to keep the strut horizontal.
358 Mr. T. Gray on an Improved
not provided for, the interpretation of the record becomes
exceedingly difficult ; and this difficulty is likely to be greatly
increased by the mass acquiring oscillations in its own free
period of such large angular amplitude that the direction of the
component whichis being recorded becomes avariable quantity.
The siphon which writes the vertical component of the
motion is controlled by a compensated horizontal lever instru-
ment, on the same principle as that introduced by the present
writer and exhibited to the Seismological Society of Japan,
and described in the Transactions of that Society, vol. i.
part 1, p. 48, and vol. ii. p. 140, and also in the Philo-
sophical Magazine for September 1881. This instrument
consists of a horizontal lever carrying near one end a heavy
mass, and provided at the other end with knife-edges in a
line at right angles to the length of the lever. The lever
is supported by two flat springs, acting, through a link, on
a knife-edge attached to it at a point between the mass and
the knife-edges before mentioned, which are by this means
held up against the apex of inverted V-grooves rigidly
fixed to the framework. In the form of this instrument
previously described in the Philosophical Magazine, the
supporting springs were of the ordinary spiral type; but in
subsequent instruments two flat springs have been adopted ,
because for the same period of oscillation of the lever with-
out compensation they give a more compact arrangement.
These springs are now made of such variable breadth between
the fixed and the free ends that, when they are supporting the
lever, each part is equally bent. They may either be initially
straight, and bent into a circular form when in use, or they
may be initially set to a circular form and straight when in
use. When the lever is supported in this way it has a fairly
long period of free vibration ; and this may be increased to —
any desired extent by means of a second pair of springs,
which pull downwards on a light bar fixed vertically above
the axis of motion of the lever. This second pair of springs,
besides providing the necessary compensation for the positive
stability of the lever and supporting-spring system, gives a
ready means of obtaining a fine adjustment for bringing the
lever to the horizontal position. This is accomplished either
by giving to the points of attachment of the compensating
springs a screw-adjustment so that they can be moved a short
distance backward or forward, or by making the point of
attachment of one spring a little in front of, and of the other
a little behind, the vertical plane through the knife-edge.
The lever can then be raised or lowered by increasing the
pull on one spring and diminishing that on the other. Sir
Form of Seismograph. 399
William Thomson has recently suggested to the writer that a
flat spring, which in its normal state is bent to such a curva-
ture that it is brought straight by supporting a weight on its
end, might be found a good arrangement for a vertical motion
seismometer. This would certainly have considerable advan-
tage in the way of simplicity, and with proper compensation
applied, say to the index-lever, so as to lengthen the period,
may be found very suitable. The only doubtful point seems
to be whether the want of rigidity in the spring may not lead
to false indications in the record due to the horizontal motions.
The application of a rigid horizontal lever, pivoted on knife-
edges and supported by springs as a vertical-motion seismo-
meter, was first described in the earlier of the two papers to
the Seismological Society of Japan, quoted above. The ad-
vantage of this arrangement, as rendering it possible to obtain
a long period of free vibration by placing the intermediate
point of support below the line joining the other two, was also
pointed out. The advantage obtained by the lever itself,
without compensation, over an ordinary stretched string was
more specifically pointed out in the other papers referred to ;
and a method of obtaining very perfect compensation, either
for a lever or an ordinary spring arrangement, by means of a
liquid, was then given. The idea of increasing the period of
a vibrating system by the addition, as it were, of negative
stability, which was first brought forward in these papers, has
been worked out in various ways ; but the method described
in this paper is the most perfect yet adopted. Its application
to the ordinary pendulum was also brought forward and dis-
cussed at a subsequent meeting of the Seismological Society
of Japan*.
The apparatus above referred to for recording the horizontal
components of the motion during an earthquake may, when
properly adjusted, be used for registering minute tremors and
slow changes of level of the earth’s surface. It is, however,
absolutely necessary for such a purpose that friction of the
different parts should be reduced to a minimum ; and hence
the siphons, or the marking-points when a smoked surface is
used, are only brought for a few seconds at a time into contact
with the paper, thus recording a series of dots close enough
together to form practically acontinuous line. Anothermethod,
which gives excellent results and is simple, has been much
used by Prof. Milne in Japan. It consists in passing from the
point of the index, through the paper, to the drum a series of
sparks from an electric induction-coil. The sparks can be
* “On a Method of Compensating a Pendulum so as to make it
Astatic,” by Thomas Gray, Trans, Seis. Soc. Japan, vol. iii. p. 145.
360 Mr, T. Gray on an Improved
made to pass at regular intervals by a clockwork circuit-
closing arrangement; and, by the perforations they leave, a
record both of their position and the corresponding time is
obtained*. This method is absolutely frictionless so far as the
recording-point is concerned, and has the advantage that the
sheet can afterwards be used as a stencil-plate for printing
copies of the record. An ordinary simple pendulum, furnished
with a very light vertical index of thin aluminum tube giving
a multiplication of 200, has been for some time in use. The
record of the position of the end of the index is taken on two
strips of paper which are being slowly pulled along, in direc-
tions at right angles to each other, under it. The sparks per-
forate both sheets simultaneously, thus automatically breaking
up the motion into two rectangular components. The details of
some forms of apparatus for this purpose will form the subject
of a separate communication.
Mechanical Details.
The record-receiver consists of a long ribbon of thin paper,
about five inches broad, which is slowly wound from the
drum A, situated behind the drum C (Plate IV. fig. 1), on
to the drum, B, by means of a train of clockwork driven
by a spring or a weight of sufficient power to keep the
ribbon taut. The rate at which the paper is fed forward
is governed by a second train of clockwork, driven by a
separate weight and governed by means of two Thomson
spring-governors. In gear with this train of wheelwork
there is a third drum, C, round which the paper is taken as
it passes from the drum A to the drum B. This drum is
kept moving at a uniform rate, and serves to regulate the
motion of the paper. The object of the double set of clock-
work mechanism is to render the rate at which the paper is
fed forward independent of the size of the coil on the drums
A and B. The surface of the drum C is covered with several
thicknesses of blotting-paper for the purpose of giving a soft
surface for the siphons to write upon, and of preventing the
ribbon blotting or adhering to the drum in consequence of ink
passing through the paper. This blotting-pad is of some
importance, because a cheap kind of thin paper is found to
answer perfectly for the siphons to write upon. They move
with less friction on a moderately rough surface and on paper
which rapidly absoros the ink. Under ordinary circumstances
the paper is fed forward from a quarter of an inch to an inch
* This method of recording the motions of an index was used by
Sir William Thomson in his “Spark Recorder.” ‘Mathematical and
Physical Papers,’ vol. ii. p. 168,
Form of Seismograph. 361
per minute, this being kept up continuously for the purpose of
allowing the magnitude and the time of occurrence of any dis-
turbance, which is of sufficient amplitude to leave a record, to be
accurately obtained. This obviates the unavoidable uncertainty
which exists as to the action of any automatic contrivance de-
signed to come into action at the time of the disturbance. The
time of occurrence is obtained by causing the siphon, D (figs. 1
and 3), to mark equal intervals of time on the paper ribbon.
The siphon is fixed to a light index-lever which is pivoted on
the end of the lever, H, and the link, F. The lever E turns
round an axis at G, and rests with its end in contact with the
wheel, H, which is fixed to the end of the hour-spindle of the
clock, K (fig. 1). As each tooth of the wheel H passes the
end of the lever H a mark is made on the paper, and the end
of the hour is distinguished by putting a larger or a double
tooth at that part of the wheel. ‘The time at which an earth-
quake has occurred can thus be found by measuring the dis-
tance of the record of the disturbance from the last time-mark,
then counting the number of intervals from the last hour-
mark, and then the number of hours to a known point. It is
convenient to mark the hour once or twice a day on the paper,
so as to save trouble in the reckoning should an earthquake
occur.
The ordinary rate of motion is much too slow for the record
to show the motions of the earth in detail; and, as has been
already stated, this is obtained by automatically increasing
the speed at the commencement of the shock. The arrange-
ment for doing this is shown at O (fig. 1), and is also illustrated
diagrammatically in fig. 2. Referring to the diagram, a and
b represent two levers, which are pivoted at ¢ and d respec-
tively. On the right-hand end of the lever 6 a ball ¢ is fixed,
and the weight of this is counterpoised by another ball /,
which rests on a rocking platform g, pivoted on the other end
of the lever. Opposite the end of the rocking platform g, and
fixed to the end of the lever a, there is another platform, h,
which receives the ball / when it rolls off the platform g. The
ball is prevented from rolling sideways by light springs, 7 i,
fixed to the sides of the platforms. On the end of the lever a,
or on another lever connected with it, the end of the spindle
of the wheel) is supported. This wheel is in gear with the
pinion &, which is on the shaft of the most distant of the two
governors from the driving-power. The ball / is so adjusted
over the pivot of the rocking platform g that an exceedingly
slight disturbance causes it to roll forward on to A, tilting g
over, and at the same time pushing down the end of a and
raising the wheel 7 out of gear with the pinion 4, thus allowing
362 Mr. T. Gray on an Improved
the clockwork to run on without the governor which regulates
the slow speed. The rate of motion then rapidly increases
until the second governor acquires sufficient velocity to con-
trol the speed, after which the paper moves forward at a rapid
but uniform rate. In order to again reduce the speed after a
sufficient interval has elapsed, the rolling forward of the ball
f allows the unbalanced weight of e to bring a wheel /, on the
spindle of which a “ snail,’ m, is fixed, into gear with the
pinion, », which forms part of the clockwork mechanism.
The spindle of / rests on a spring, 0, which is adjusted so as
to push the lower part of the “snail” just into contact with a
pin, p, fixed in the lever b. The weight of e acting through
the pin p on the “ snail” deflects the spring o and brings the
wheel / into gear with the pinion. The “snail” is then gra-
dually moved round and raises the ball e and the end of the
lever b, at the same time lowering the rocking platform g.
After this has proceeded so far as to cause the platform g to
come below the lever of A the ball rolls back to its original
position; and, as the “snail”? moves round, the platforms
are gradually raised to their original positions, the wheel 7
again comes into gear with the pinion &, and the speed is re-
duced. The wheel / remains in gear with the pinion n for a
short time after the speed is reduced, so as to allow the final
adjustment in position of the platform g and the ball f to be
made gently. After this is accomplished a hollow in m allows
the spring o to push the wheel / out of gear, and everything
is left in readiness for the next disturbance.
In order to obtain the rate at which the paper is moving at
any instant during the transition period between the slow and
the quick speed, the lever a is made to close an electric circuit
at g, which causes an electromagnetic vibrator, indicated at J
(fig. 3), to come into action and write equal short intervals of
time on the record-sheet. ‘The short intervals are sometimes
given by a vibrating reed, which is the most convenient
arrangement if the intervals are to be fractions of a second ;
but, for marking seconds, a break-circuit arrangement worked
by the clock, &, is preferable. The way in which the siphon,
D, is made to record both the long and the short time-intervals
is sufficiently explained by the diagram, fig. 3.
One of the “conical pendulums”’ used for actuating the
siphons which record the two horizontal components of the
motion is illustrated in plan in fig. 4, and in elevation in
fig. 5. It consists of a thin brass cylinder 7, filled with lead,
and held deflected by a light tubular strut, s, furnished with
a knife-edge at ¢, which rests against the bottom of a vertical
V-groove fixed to the support wu. The weight of the pendu-
Form of Seismograph. 363
lum-bob and strut is supported by a thin wire, v, attached at
the lower end to a stirrup, w, pivoted at wa little below and
in front of the centre of gravity of 7, and taken at the upper
end over a small wheel, y, to a drum, z, round which the wire
may be wound, so as to adjust the level of the strut, s. The
position of the pivot, w, is so arranged that the knife-edge at
t has little or no tendency to rise or fall, no matter at what
part of the strut the cylinder 7 may be clamped. The wheel
y is provided with adjusting screws, a, and 6,, by means of
which the top of the wire can be placed vertically above the
knife-edge, or as much in front of or behind that point as may
be necessary to make the period of free vibration of the pen-
dulum have any desired length. A light aluminium lever
is hinged to the strut s at d,, and is provided at its outer end
with a small hollow steel cone e,, which may be placed over
one or other of a series of sharp points /;, fixed to the vertical
arm of the cranked lever g,. The lever g, turns round a
horizontal axis at h; in bearings fixed to the ink-well 7,, and
the vertical arm is hinged at j,, so as to be free to turn in a
direction at right angles to the plane of the crank. A siphon,
ky, is fixed to the horizontal arm of the lever g,, and, drawing
ink from the well 7, writes a continuous line on the paper
ribbon. The horizontal arm of the lever g, is made very
flexible in a horizontal direction, and besides can be turned
round a vertical axis to such an extent as allows the pressure
of the point of the siphon on the paper to be adjusted until it
is only sufficient to give a record.
The horizontal-lever pendulum used for actuating the
siphon which writes the vertical motion is illustrated dia-
grammatically in fig. 6. It consists of a horizontal lever, /,,
carrying at one end a cylindrical weight m,, and free to turn
round knife-edges m,, fixed to the other end of the lever.
The lever is supported in a horizontal position by two flat
springs, clearly shown in fig. 1, and indicated at 0, fig. 6.
A light aluminium index, p,, pivoted at q,, and connected by
a thin wire or thread to the end of the lever /,, carries a fine
siphon, 7, which rests with one end in the ink-well, s,, and
the other end touching the surface of the paper. The end of
the index is weighted sufficiently to cause it to follow the
motions of the lever. This arrangement gives a period of
free vibration of about two seconds in the actual instrument ;
and in order to increase this period a second set of springs,
indicated at t, are made to act on knife-edges, w, fixed ver-
tically above 7,,so as to add negative stability to the arrange-
ment. When the lever is deflected downwards the pull on
the supporting spring is increased, but at the same time the
= Sa es
SSS eee
= = 2S = ——
SS
2 a ee
364 Mr. F. Y. Edgeworth on
knife-edge u, comés in front of the vertical plane through 1;
and, since the lower point of attachment of the compensating
spring ¢, is far below 7, a couple is introduced which com-
pensates for the greater upward force. The same is the case
in the reverse order, when the lever is deflected upwards.
Hence if the pull exerted by ¢, and the other conditions
mentioned below be properly adjusted, the horizontal lever
may be made to have any desired period of free oscillation.
In actual practice some positive stability must be given to
the lever in order that its position of equilibrium may be
definite ; but its period may be made so great that, even if
oscillations of considerable amplitude in its own period are
set up, they will be so slow compared with those of the earth-
quake, that the undulating line so drawn will still be practi-
cally straight, so far as the earthquake record is concerned.
In order to insure good compensation, the condition must be
fulfilled that the rate of variation of the compensating couple
is always the same as that of the supporting couple. If
this be not the case, the pendulum must either be left with
excessive positive stability for small deflections, or it will be
continually liable to become unstable by the compensating
couple becoming too great when the deflection exceeds a cer-
tain limit. In the present instance, let the modulus of the
supporting spring be M, the arm at which it acts a; let the
modulus of the compensating spring be M,, and the distance
between 1, and wu, be a;. Then for a deflection of the lever
equal to @ we have, on the supposition that the length of the
supporting spring and link is great compared with a,, for the
return couple Ma’ cos 0 sin 0@— Mya,’ cos 6 sin 0—M,8 sin 8,
where 8 +a, is the total elongation of the spring for the hori-
zontal position of the lever. Now our condition necessitates
B being either zero or negative; and in order to keep within
this condition the length of the unstretched spring and link
are made to reach a little above m, and the height of w, is
made adjustable, so that M,a,” can be adjusted to be as nea
Ma? as may be desired.
XLI. On Discordant Observations. By F. Y. Hpgnworta,
M.A., Lecturer at King’s College, London*.
ANT observations may be defined as those which
present the appearance of differing in respect of their
law of frequency from other observations with which they are
combined. In the treatment of such observations there is
great diversity between authorities ; but this discordance of
* Communicated by the Author.
Discordant Observations. 365
methods may be reduced by the following reflection. Different
methods are adapted to different hypotheses about the cause of
a discordant observation ; and different hypotheses are true, or
appropriate, according as the subject-matter, or the degree of
accuracy required, is different.
To fix the ideas, I shall specify three hypotheses : not pre-
tending to be exhaustive, and leaving it to the practical reader
to estimate the & priori probability of each hypothesis.
(a) According to the first hypothesis there are only two
species of erroneous observations—errors of observation proper,
and mistakes. The frequency of the former is approximately
represented by the curve y= a e—?; where the constant h
J 7
is the same for all the observations. But the mathematical
law* only holds for a certain range of error. Beyond certain
limits we may be certain that an error of the first category
does not occur. On the other hand, errors of the second
category do not occur within those limits. The smallest
mistake is greater than the largest error of observation proper.
The following example is a type of this hypothesis. Suppose
we have a group of numbers, formed each by the addition of
ten digits taken at random from Mathematical Tables. And
suppose that the only possible mistake is the addition or sub-
traction of 100 from any one of these sums. Here the errors
proper approximately conform to a probability curve (whosef
modulus is 4/165), and the mistakes{ are quite distinct from
the errors proper.
Here are seven such numbers: each of the first six was
formed by the addition of ten random digits, and the seventh
by prefixing a one to a number similarly formed—
Wein 9B 431, 50. (49. 45, | 136)
* This follows from the supposition that an error of observation is the
joint result of a considerable, but finzte, number of small sources of error.
The law of facility is in such a case what Mr. Galton calls a Binomial, or
rather a Multinomial. (See his paper in Phil. Mag. Jan. 1875, and the
remarks of the present writer in Camb. Phil. Trans. 1886, p. 145, and
Phil. Mag. April 1886.)
+ I may remind the reader that I follow Laplace in taking as the
constant or parameter of probability-curves the reciprocal of the coefficient
of x: that is 2 according to the notation used above. It is 2 times
the “Mean Error” in the sense in which that term is used by the
Germans, beginning with Gauss, and many recent English writers
(e.g. Chauvenet) ; and it is 7 times the Mean Error in the (surely more
natural) sense in which Airy, after Laplace, employs the term Mean Error
(Chauvenet’s Mean of the Errors).
{ In physical observations the limit of errors proper must, I suppose,
be more empirical than in this artificial example.
366 Mr. F. Y. Edgeworth on
The hypothesis entitles us to assert that 23 is an error-proper
—an accidental deviation from 45; though the odds against
such an event before its occurrence are considerable, about 100
to 1. On the other hand, we may know for certain that 136
is a mistake. |
(8) According to the second hypothesis, the type of error
is still the probability-curve with unvarying constant. But
the range of its applicability is not so accurately known before-
hand. We cannot at sight distinguish errors proper from mis-
takes. We only know that mistakes may be very large, and
that the large mistakes are so infrequent as not to be likely to
compensate each other in a not unusually numerous group of
observations. This hypothesis may thus be exemplified :—
As before, we have a series of numbers, each purporting to be
the sum of ten random digits. But occasionally, by mistake,
the sum (or difference) of two such numbers is recorded. The
mistake might be large, but it would not always exceed the
limits of accidental deviation (100 and 0); which need not be
supposed known beforehand. Here is a sequence of seven
such numbers, which was actually obtained by me (in the
course of 280 decades) —
d0, 54, 41, 78, 46, 38, 49.
The hypothesis leaves it doubtful whether 73 may not be a
mistake ; the odds against it being an ordinary accidental
deviation being, before the event, about 250 to 1.
(y) According to the third hypothesis all errors are of the
type y = eine But the A is not the same for different
observations. Mistakes may be regarded as emanating from
a source of error whose / is very small. This hypothesis may
be thus illustrated. Take at random any number n between
certain limits, say 1 and 100. ‘Then take at random (from
Mathematical Tables) digits, add them together and form
their Mean (the sum + 7), and multiply this Mean by ten.
The series of Means so formed may be regarded as measure-
ments of varying precision ; the real value of the object mea-
sured being 45. The weight, the h’, being proportionate to n,
one weight is a priori as likely as another. In order that the
different degrees of precision, the equicrescent values of h,
should be & priori equiprobable, it would be proper, having
formed our 7 as above, to take the mean of (and then mul-
tiply by 10), not n, but n? digits. Here is a series formed
in this latter fashion :—
lOp oe So oseeanenon” Dalida 6 Lo Oe as 1
PAu teeue sana 25 49 36 1 100 64 1
n
10x Mean ofm" (3) 45 43 100 48 47:5 100
random digits ;
Discordant Observations. 367
In this table the first row is obtained by taking at random
ten digits from a page of Statistics, 0 counting for ten. The
second row consists of the squares of these numbers. The
third row was thus formed from the second :—I took 25
random digits, and divided their sum by 25; then multiplied
this mean by 10. I similarly proceeded with 49 (fresh) digits,
and so on. It will be noticed how the defective precision of
the fourth and seventh observations makes itself felt. It was,
however, a chance that they both erred as far as they could,
and in the same direction.
In the light of these distinctions I propose now to examine
the different methods of treating discordant observations. For
this purpose the methods may be arranged in the following
groups :—
I. The first sort of method is based upon the principle that
the calculus of probabilities supplies no criterion for the cor-
rection of discordance. All that we can do is to reject certain
huge errors by common sense or simple induction as distin-
guished from the calculation of a posteriori probability.
II. Or, secondly, we may reject observations upon the
ground that they are proved by the Calculus of Probability
to belong to a much worse category than the observations
retained.
IIT. Or, thirdly, we may retain all the observations, affecting
them respectively with weights which are determined by
a postertort probability.
IV. In a separate category may be placed a method which,
as compared with* the simple Arithmetical Mean, reduces the
effect (upon the Mean) of discordant observations—the method
which consists in taking the Medianf or ‘“ Centralwerth’’ t of
the observations.
I propose now to test these methods by applying them in
turn to all the hypotheses above specified.
I. (a) The first method—which is none other than Airy’s,
as I understand his contribution$ to this controversy—is
adapted to the first hypothesis. Upon the second hypothesis
(@) the first method is liable to error, which, as will be shown
under the next heading, is avoidable. (y) Upon the third
hypothesis the first method is not theoretically the most
precise ; but it may be practically very good.
II. Under the second class 1 am acquainted with three
* This is pointed out by Mr. Wilson in the Monthly Notices of the
sa enmicel Society, vol. xxxviii., and by Mr. Galton, Fechner, and
others. i
+ Cournot, Galton, &c.
{ Fechner, in Abhandl. Sax. Ges. vol. [xvi.].
§ Gould’s Astronomical Journal, vol. iv. pp. 145-147.
368 Mr. F. Y. Pilsen orth on
species : the criteria of Prof. Stone*, Prof. Chanvoneli and
Prof. Peircet.
IT. (1) Prof. Stone’s method is to reject an observation
when it is more likely to have been a mistake than an error
of observation of the same type as the others. In deter-
mining this probability he takes account of the a prior
probability of a mistake. He puts for that probability =
admitting that m cannot be determined precisely. The use of
undetermined constants like this is, I think, quite legitimate§,
and, indeed, indispensable in the calculation of probabilities.
This being recognized, Prof. Stone’s method may be justified
upon almost any hypothesis. Hypothesis (a) presents two
cases: where the discordant observation exceeds that limit of
errors proper which is known beforehand, and where that
limit is not exceeded. For example, in the instance|| given
above—where 45 is the Mean, and the Modulus is about 13—
the discordant observation might be either above 100 (e.g. 110)
or below it (e.g. 84). Now let us suppose that the a priori
probability of a mistake is not infinitesimal, but say of the
order yj55: Since the deviation of 110 from the Mean is
about five times the Modulas, the probability of this deviation
occurring under the typical law of error is nearly a millionth.
This observation is therefore rejected by Method II. (1), which
so far agrees with Method I. Again, the probability of 84
being an accidental deviation is less than a forty-thousandth;
84—45 beingabout three times the Modulus. Therefore 84 also
is rejected by the criterion. And we thus lose an observation
which is by hypothesis (a) a good one. But this loss occurs
very rarely. And the observation thrown away is, to say the
least, not** a particularly good one, though doubtless it may
happen that it is particularly wanted—as in the case of Gen.
Colby, adduced tf by Sir G. Airy.
II. (1) (@) The second hypothesis is that to which Prof.
Stone’s criterion is specially adapted. Upon this hypothesis,
84 may be a mistake. In rejecting such discordant observa-
tions, we may indeed lose some good observations, especially if
* Month. Not. Astronom. Soc. Lond. vol. xxviii, pp. 165-168.
+ ‘Astronomy,’ Appendix, Art. 60. t Ibid. Art. 57,
§ See my paper on @ priort Probabilities, in Phil. Mag. Sept. 1884; also
‘Philosophy of Chance,” Mind, 1884, and Camb. Phil, Trans. 1885,
pp: 148 ef seg. | Page 365,
q 165, exactly. As determined empirically by me from the mean-
square- -of-error of 280 observations (ze. sums of 10 digits), the Modulus
was 4/160.
** See the remark made under II. (2) ({).
+t Gould’s Astronom. Journ. vol. iv. p. 138.
Discordant Observations. 369
we have exaggerated the a priori probability of a mistake. But
it may be worth while paying this price for the sake of getting
rid of serious mistakes. specially is this position tenable
according to the definition of the quesitum in the Theory
_ of Errors*, which Laplace countenances. According to this
view, the destderatum in a method of reduction is not so
much that it should be most frequently right, as that it should
be most advantageous ; account being taken, not only of the
Frequency, but also of the seriousness, of the errors which it
incurs. Prof. Stone’s method might diminish our chance of
being right (in the sense of being within a certain very small
distance from the true markt); and yet it might be better than
Method I., if it considerably reduced the frequency of large
and detrimental mistakes.
II. (1) (¥) Prof. Stone’s method is less applicable to the
third hypothesis. ‘Though even in this case, if the smaller
weights are @ priori comparatively rare, it may be safe enough
to regard (m—1) of the m observations as of one and the
same type ; and to reject the mth if violently discordant with
that supposed type.
The only misgiving which I should venture to express
about this method relates, not to its essence and philosophy,
but to a technical detail. Prof. Stone says:—“ If we find that
value which makes J-| eVdy = wy [ where p is the devia-
Te) P. nr
tion of a discordant observation, and a is the modulus of the
probability-curve under which the other observations range,
and = is the & priori probability of a mistake], all larger
values of p are with greater probability to be attributed to
mistakes.” But ought we not rather to equate to > not the
left-hand member of the equation just written, which may be
called (2), but 6” = where m is the number of observa-
tions. Jam aware that the point is delicate, and that high
authority could be cited on the other side. There is some-
thing paradoxical in Cournot’s{ proposition that a certain
* See my paper on the “ Method of Least Squares,” Phil. Mag. 1883,
vol. xvi. p. 363; also that on “ Observations and Statistics,” Camb. Phil.
Tr. 1885; and a little work called ‘ Metretike’ (London: Temple Co., 1887).
+ The sense defined by Mr. Glaisher, ‘ Memoirs of the Astronomical
Society,’ vol. xl. p. 101.
_{ Exposition de la théorie des Chances, Arts. 102, 114, “Nous ne nous
dissimulons pas ce qu'il y a de délicat dans toute cette discussion,” I
may say with Cournot.
Phil. Mag. 8. 5. Vol. 23. No. 143. April 1887, 2C
370 Mr. F. Y. Edgeworth on
_ deviation from the Mean in the case of Departmental returns
of the proportion between male and female births is signifi-
cant and indicative of a difference in kind, provided that we
select at random a single French Department; but that the
same deviation may be accidental if it is the maximum of the.
respective returns for several Departments. There is some-
thing plausible in De Morgan’s* implied assertion that the
deficiency of seven in the first 608 digits of the constant 7 is
theoretically not accidental; because the deviation from the
Mean 61 amounts to twicet the Modulus of that probability-
curve which represents the frequency of deviation for any
assigned digit. I submit, however, that Cournot is right, and
that De Morgan, if he is serious in the passage referred to, has
committed a slight inadvertence. When we select out of the ten
digits the one whose deviation from the Mean is greatest, we
ought to estimate the improbability of this deviation occurring
by accident, not with De Morgan as 1—@(1°63), corresponding
to odds of about 45 to 1 against the observed event having
occurred by accident ; but as 1—6"(1°63), corresponding to
odds of about 5 to 1 against an accidental origination.
II. (2) Prof. Chauvenet’s criterion differs from Prof.
Stone’s in that he makes the & priori probability of a mistake
—instead of being small and undetermined—definite and con-
siderable. In effect he assumes that a mistake is as likely as not
to occur in the course of m observations, where m is the number
of the set which is under treatment. Itis not within the scope
of this paper to consider whether this assumption is justified
in the case of astronomical or of any other observations. It
suffices here to remark that this assumption coupled with
hypothesis («) commits us to the supposition that huge mis-
takes occur on an average once in the course of 2m observa-
tions. Upon this supposition no doubt Method II. (2), is a
good one. Hypothesis (8) expressly t excludes this suppo-
sition ; the mistakes which, according to II. (2), are as likely
as not, must, according to this second hypothesis, be of
moderate extent. Thus, in the case above put of sums of ten
digits, suppose that the number of such sums under observa-
tion is ten. According to Prof. Chauvenet’s criterion we
must reject any sum which lies outside 45+, where
k 2n—1 19
3) Ton) 0 ee
* Budget of Paradoxes,’ p. 291.
t If we take many batches of random digits, each batch numberin
608, the number of sevens per batch ought to oscillate about the Mean 61,
according to a probability-curve whose Modulus is a a 608 = 10-4,
t Above, p. 366. 10
Discordant Observations. 371
This gives for the required limit about 15. According, then,
to II. (1) (@), any observation greater than 60, or less than 30,
is more likely than not to be a mistake in the sense of not
belonging to the same law of frequency as the observations
within those limits. But why on that ground. should the
discordant observation be rejected ? Suppose there were not
merely a bare preponderance of probability, but an actual
certainty, that the suspected observation belonged to a different
category in respect of precision from its neighbours, the best
course certainly would be if possible (as Mr. Glaisher in his
paper ‘On the Rejection of Discordant Observations” sug-
gests) to retain the observation affected with an inferior weight.
But if we have only the alternative of rejecting or retaining
whole, it is a very delicate question whether retention or re-
jection would be in the long run better. There is not here
the presumption against retention which arises when, as in
IT. (1), the discordant observation is large and rare ; so that,
if it is a mistake, it is likely to be a serious and an uncom-
pensated one. However, Prof. Chauvenet’s method may
quite possibly be better than the No-method of Sir G. Airy.
Much would turn upon the purpose of the caleulator—whether
he aimed at being most frequently right* or least seriously
wrong. ‘The same may be said with reference to hypo-
thesis (7).
There is a further difficulty attaching particularly to this
species of Method II. In its precise determination of a limit,
it takes for granted that the probability-curve to which we
refer the discordant observation is accurately determined.
But, when the number of observations is small, this is far
from being the case. Neither of the parameters of the curve,
neither the Mean, nor the Modulus, can be safely regarded as
C
accurate. The “probable error” of the Mean is *477 a.
e
where ¢ is the Modulus. The probable error of the Modulus
is conjectured to be not inconsiderable from the fact that, if
we took m observations at random, squared each of them and
formed the Mean-square-of-error, the “‘ probable error ” of that
2
Mean-square-of-error would be ‘477 — }. This, however, is
vn ,
not the most accurate expression for the probable error of the
Modulus-squared as inferred { from any given n observations.
* See the remarks above, p. 369.
+ Todhunter, art. 1006 (where there is no necessity to take the origin
at one of the extremities of the curve).
} LI allude here to delicate distinctions between genuine Inverse Pro-
bability and other processes, which I have elsewhere endeavoured to
draw, Camb. Phil. Trans, 1885,
2C2
372 Mr. F. Y. Edgeworth on
To appreciate the order of error which may arise from these
inaccuracies, we may proceed as in my paper of last Octo-
ber*. First, let us confine our attention to the Mean, sup-
posing for a moment the Modulus accurate. Let k have
been determined according to Prof. Chauvenet’s method, so
; EO.
To determine more accurately the probability of an observa-
tion not exceeding a we must put for a, a+z, where z is the
error of the Mean subject to the law of frequency
mz2
i J m ia
WV 16
The proper course is therefore to evaluate the expression
{ a(“7*) = vie me ae
Expanding 0, and neglecting the shee powers of zt, we
have for the correction of a(t ) the subtrahend See he
where £ is put for = e Call this modification of 8, 00. To see
how the primd facie limit 8 is affected by this modification,
let us put
[0+00](8+46) =},
whence eae tale i
Beal eae Ret?
Whence A vem a i:
an extension of the limit which may be sensible when 7 is
small.
In the example given by Prof. Chauvenet the uncorrected
limit as found by him is 1:22. This divided by the Modulus
[which= V 2e=°8] is 1°5. This result, our 6, divided by 15
the number of observations, gives ‘1 as the correction of 8 ;
08 as the correction of the limit a. The limit must be ad-
vanced to 1:30. This does not come up to the discordant
observation 1:40. But we have still to take into account
that we have been employing only the apparent Modulus (and
Mean Error), not the real one. In virtue of this consideration
I find—by an analysis analogous to that given in the paper
* Phil. Mag. 1886, vol. xxii. p. 371. + See the paper referred to.
Discordant Observations. - aes
just referred to—that the limit must be pushed forward as
much again; so that the suspected observation falls within
the corrected limit. I have similarly treated the example
given by Prof. Merriman in his Textbook on The Method of
Least Squares (131). The limit found by him is 4°30, and
he therefore rejects the observation 4°61. But I find that
this observation is well within the corrected limit *.
II. (3) Prof. Peirce’s criterion is open to the same objec-
tions as that of Prof. Chauvenet. Indeed it presents additional
difficulties. If by y the author designates that quantity which
Prof. Stone calls 2 and which I have termed the “a priori”’
probability of a mistake, I am unable to follow the reasoning
by which he obtains a definite value for this y. But I am
aware how easy it is on such subjects to misunderstand an
original writer.
III. We come now to the third class of method, of which
Tam acquainted with three species. (1) There is the procedure
indicated by De Morgan and developed t by Mr. Glaisher ;
which consists in approximating to the weights whic are to be
assigned to the observations respectively, after the analogy of
the Reversion of Series and similar processes. (2) Another
method, due to Prof. Stone {, is to put
es iieas ke are Pe eo OX dig Dhgea
as the a posteriori probability of the given observations having
resulted from a particular system of weights h,’ h,” &e., and a
particular Mean 2 ; and to determine that system so that P
should be a maximum. (3) Another variety is due to Prof.
Newcomb §.
Ill. (1) & (2) Neither of the first two Methods are well
adapted to the first two hypotheses. Both indeed may success-
fully treat mistakes by weighting them so lightly as virtually to
reject them. But both, I venture to think, are liable to err in
underweighting observations, which, upon the first two hypo-
theses, have the same law of frequency as the others. Both, in
fact, are avowedly adapted to the case where the observations
* These corrections may be compensated by another correction to which
the method is open. In determining whether the suspected observation
belongs to the same type as the others, would it not be more correct to
deduce the characters of that type from those others, exclusive of the
suspected observation? The eftect both on the Mean and the Modulus
would be such as to contract the limit.
+ Memoirs of the Astronomical Society.
t Monthly Notices of the Astronomical Society, 1874. This Method
was proposed by the present writer in this Journal, 1883 (vol. xvi.
p. 360), in ignorance of Prof. Stone’s priority. :
§ American Journal of Mathematics, vol. viii. No. 4.
374 On Discordant Observations.
are not presumed beforehand to emanate from the same source
of error. The particular supposition concerning the a priort
distribution of sources which is contemplated by the De-
Morgan-Glaisher Method, has not perhaps been stated by
its distinguished advocates. The particular assumption made
by the other Method is that one value of each h is as likely as
another over a certain range of values—not necessarily between
infinite limits. I have elsewhere* discussed the validity of
this assumption. I have also attempted to reduce the in-
tolerable labour involved by this method. Forming the equa-
tion in x of (n—1) degrees,
nx”-!— (n—1)Sa, a-? + (n—2)Sax, 4, x" —&e.=0, —
I assume that the penultimate (or antepenultimate) limiting
function or derived equation will give a better value than the
last-derived equation |nv—jn—1S2,, which gives the simple
Arithmetic Mean. Take the observations above instanced
under hypothesis (+), |
31, 45, 48, 100, 438, 47:5, 100.
For convenience take as origin the Arithmetical Mean of
these observations 58°5, say 58. Then we have the new
series
429, 18) 216, 492 15
Here S2,2,= —2494. And the penultimate limiting equa-
tion is
7x6xX5xX4x8a74+5xX4x%38xX2x1x —2494=0,
Whence. #?=119. And #=+11 nearly. To determine
which of these corrections we ought to adopt, the rule is to
take the one which makes P greatest; which ist the one
which makes («—.)(@—2) (w—a3) . . . (@—2;) smallest ;
each of the differences being taken positively.
The positive value, +11, gives the differences
38, 24,26," 21). 36; 22a
For the negative value, —11, the differences are
16,0 25) 4,..03,): 4 HOR ae
(where 0 of course stands for a fraction). The continued
product of the second series is the smaller. Hence —11 is
* Camb. Phil. Trans. 1885, p. 151.
Tt See Phil. Mag. 1888, vol. xvi. p. 371.
Action of Heat on Potassic Chlorate and Perchlorate. 375
the correction to be adopted. Deducting it from 58, or rather
58°5, we have 47:5, which is a very respectable approximation
‘to the real value, as it may be called, viz. 45.
III. (8) Prof. Newcomb* soars high above the others, in
that he alone ascends to the philosophical, the utilitarian,
principles on which depends the whole art of reducing obser-
vations. Here are whole pages devoted to estimating and
minimizing the Hvil incident to malobservation. With Gauss,
Prof. Newcomb assumest “that the evil of an error is pro-
portional to the square of its magnitude.”” He would doubt-
Jess admit, with Gauss, that there is something arbitrary in
this assumption. Another somewhat hypothetical datum is
what het describes as the “distribution of precisions.” In view
of this looseness in the data, it becomes a nice question
whether it is worth while expending much labour upon the
calculation. The answer to this question depends upon an
estimate of probability and utility, concerning which no one
is competent to express an opinion who has not, on the one
hand, a philosophical conception of the Theory of Errors, and,
on the other hand, a practical acquaintance with the art of
Astronomy. The double qualification is probably possessed
by none in a higher degree than by the distinguished astro-
nomer to whom we owe this method.
IV. It remains to consider the fourth Method. But the
length and importance of this discussion will require another
paper.
XLII. On the Action of Heat on Potassic Chlorate and Per-
chlorate. By Kyuunp J. Miuts, D.Sc., AS.
|? has been pointed out by Teed||, and subsequently by
P. Frankland and Dingwall{, that potassic chlorate and
perchlorate may be decomposed by heat in such a manner as
to lead in each case to various relations among the products
of decomposition.
It has occurred to me that both of these chemical changes
are instances of Cumulative Resolution**, from which point
of view they admit of very simple, and at the same time
perfectly adequate, representation.
* American Journal of Mathematics, vol. viii. No. 4.
Tt § 3, p. 348. t § 9, p. 359.
§ Communicated by the Author.
| Proc. Chem. Soc. xii. p. 105; xvi. p. 141; xxxiil. pp. 24 & 25.
q Ibid. xvi. p.141; xxx. p. 14; and Trans, Chem. Soe. 1887, p, 274.
** Phil. Mag. [5] ili. p. 492 (1877).
376 Dr. E. J. Mills on the Action of Heat
Action of Heat on Potassic Chlorate.
The products of this action are potassic chloride, oxygen,
and perchlorate. All known relations among these products
may be expressed by the cumulative equation
2n KC1O3— (n—2) O.= (n+ 1) KC1O,+ (n—1) KCI.
In order to compare theory with experiment, I have selected
the quotient of the percentage of chloride produced by that
of the oxygen formed as the specific measure of the change ;
the percentage being calculated on the weight of chlorate
taken for trial. If this quantity be called 7, the equation
alleges that
OFT Der!
Y RCAC) 20.
or
AGTH oe
n—2
It will be seen from the following Table that this is the case ;
a rational value ‘of m always corresponding to the specitic
measure r. No attention has been paid to instances in which
perchlorate is known to have been decomposed. Whenn= @,
the equation becomes
2K'C1O;—O,=KCI10,+ KCl.
Taste I.
|
No. of| Oxygen, |Chloride, ss i Authority.
exp. | per cent.) per cent.
166 | 526 | 31687 | 47910 Teed.
3°49 | 10°86 J 1117 | 49949 Ps
6:00 | 18:25 3°0417 | 52906 i
2°66 |x 9°4916| 3°5684 | 3°8879 | Frankland and Dingwall.
5:19 |x18°566 | 3°5774 | 38744 i
647 |*21°609 | 3:3399 | 4:3164 55
6°89 /*21°533 | 3:1253 | 4:9438 ‘S
6°78 |x20°147 | 2°9715 | 5°6523 i
36! | 11:58 3°2167 | 4°6392 Teed.
10. 4:27 4-73 3°7244 | 36764 3
Ie 161 6:00 3°7267 | 3°6736 0
£9 OC ST Se St BO DO
12. 1:60 6:14 3°8375 | 3°5502 .
13. 1:47 4°84 3°2925 | 4:4307 ”
14, 0°80 2°18 2°7250 | 7:9488 5
It is remarkable that the value of n should, amongst so
many experiments, prove to be so very restricted in its range.
There seems to be some tendency for r to be preferably about
equal tom. The exact fulfilment of this condition requires
* Recalculations.
on Potassie Chlorate and Perchlorate. BTL
r=n=3'7028 or *6300,—values which indicate the reduction
of the chemical change to a mere action of mass. __
Action of Heat on Potassie Perchlorate.
The equation of Cumulative Resolution is
(n+1)KCIO,—(2n—1)0,=2KCI1O; + (n—1) KCI;
the products of the reaction being chlorate, chloride, and
oxygen. Its starting-point is a point in the chlorate equa-
tion, viz., (n+1)KCIiO,. In this case the percentage of
chlorate cannot exceed a certain amount, viz., that indicated
by the relation given by n=1, or
. 2KC1O,—O0,=2KCI1Os;,
= 88°46 per cent.
A comparison of theory with experiment can be made on
a basis similar to that previously taken, viz. :—
Oz ~ nol
eb, eee
or n—1
42867 r = apa i
When n = «, the equation reduces to
KC10,—20, = KCL.
TaB_e ILI.
Number of | Oxygen, | Chloride, :
Experiment. | per cent.| per cent. e oo
310 | 297 | -95806| 32992 Teed.
447 4-41 ‘98658 | 3°7430 Ss
7°30 7°82 10712 | 66275 “i
35°21 40°33 1:1454 |28:278 Bs
6°34 * 67148 | 10591 | 59348 | Frankland & Dingwall.
780 | * 82600 | 1:0590 | 5-9300 ‘5
24:05 | *27:145 | 11287 |15-970
STS Ure Chore
In this case r cannot be equal to n. As regards the pro-
portion of chlorate formed, it has been stated by all three
investigators that this diminishes as the reaction proceeds.
Frankland and Dingwall have made actual determinations of
itsamount. In order to compare this part of their work with
theory, I have taken their experimental ratio p-of chloride
to chlorate, and calculated it from the estimations, made in
* Recalculations.
378 Action of Heat on Potassie Chlorate and Perchlorate.
Table II., of the corresponding values of n. The relation
required for this purpose is
Oe Pa Da
or 4:2860 p = n.
TaBLE III.
= eae a (Chloride p cale. N.
eR | =ehilonasbes)
5 1:5630 1:3847 59348
6. 15781 1:3836 59300
7 68442 37261 15-970 ;
There is a fair agreement in comparisons 5 and 6. The
discrepancy in 7 arises in great part from the fact that the
form of the function renders it difficult to deduce accurately
such high values of n as 15:970 from experiments of not
exceptional accuracy. If, for example, n=30, r=1:1275,
which differs very little Hdesd from r=l1' 1287, when n=
15:970r. It is probable also that the chlorate (never actually
exceeding more than about 4 per cent. of the perchlorate)
was decidedly underestimated. Additional experiments on
this subject are much to be desired.
Equal Weight Relations.
It is usual in chemical change for a critical relation to be
established when certain of the reagents are present in equal
weights. Thus, in the chlorate reaction, if the ratio of
chloride to oxygen be that of equality in weight, r=1; and
the equation
42867 r ==>
n—2
then gives n='24970.
Similarly, in the case of the perchlorate, where
asa
2n—1’
if r=1, n=4:0048—i. e. the reciprocal of the previous value
of appears then that, subject to the condition indi-
cated, the reaction whereby perchlorate is decomposed is the
exact inverse of the chlorate reaction.
42867 r=
Biiaes
XLII. Reply to Prof. Wilhelm Ostwald’s criticism on my
paper * On the Chemical Combination of Gases.”
To the Editors of the Philosophical Magazine and Journal.
(GENTLEMEN,
ROFESSOR WILHELM OSTWALD, in a work en-
titled Lehrbuch der Allgemeinen Chemie (Bd. i.
p- 745), has criticised my paper on the Chemical Combina-
tion of Gases published in the Philosophical Magazine, Octo- —
ber 1884, in which I applied the Williamson-Clausins theory
of dissociation to the solution of several problems in the theory
of the combination of gases. I wish in this letter to answer
this criticism, and, in order to make my meaning clear, I
must recapitulate one part of the paper. According to the
Williamson-Clausius hypothesis, the molecules of a gas are
continually splitting up into atoms, so that the atoms are
continually changing partners. I defined the “ paired”’ time
of an atom to be the average time an atom remained in part-
nership with another atom, and the ‘free time ” the average
time which elapses between the termination of one partner-
ship and the beginning of the next. Now the free time will
evidently depend upon the number of free atoms in the unit
volume, for before an atom can be paired again, it must come
into collision with another atom ; and though it need not get
paired at the first collision, yet it is evident that the time it
remains “‘ free ” will be proportional to the time between two
collisions, and, therefore, inversely proportional to the number
of free atoms in unit volume. But after the atom has got
paired with another, there is no reason why the time they
remain together should depend upon the number of molecules,
unless we assume that the atoms are knocked apart by colli-
sion with other molecules.
As one of my reasons for undertaking the investigation
was, that an eminent spectroscopist had mentioned to me that
there was spectroscopic evidence to show that the molecules
got split up independently of the collisions, and as I wished
to see if I could get any evidence of this from the phenomena
of dissociation, it would have been absurd on my part to beg
the question by assuming that the paired time was inversely
proportional to the number of atoms. I therefore made no
supposition as to the dependence of the paired time on the
number of atoms, except when the dissociation was produced
by an external agency, such as the electric discharge, but left it
to be determined from the experiments.
The above reasoning seems to me to be clear enough, but
as it is substantially the same as that in my paper, and Prof.
Ostwald says it is difficult to conceive how it is that I have
380 On the Chemical Combination of Gases.
not noticed that the paired time is inversely proportional to
the number of atoms, I must endeavour to find some way of
explaining myself which shall not entail the necessity of form-
ing any abstract conceptions. Let us then illustrate the
pairing of a molecule by the act of getting into a cab, and a
gas by a number of men and cabs, the men riding about in the
cabs, getting out,and after walking about for a time getting into
acab again. To fix our ideas, let us suppose that after leaving
a cab, each man gets into the sixth cab he meets. Then it is
evident that the time he spends on foot (his “ free time’’)
will depend upon the number of cabs, the more cabs the
shorter the time ; and if the cabs are evenly distributed, his |
“‘ free time ” will be inversely proportional to the number of
cabs. But after getting into a cab, unless he is upset by a
collision with another cab, there is no reason why the time
he stays in his cab should depend upon the number of cabs.
Prof. Ostwald’s remark, when applied to this case, is—it
is difficult to conceive how it is that I have not noticed that
the only way of getting out of a cab is to wait until one is
shot out by the collision of one’s own cab with another. But
difficult as the conception is, Prof. Ostwald is equal to it, for
in a footnote he suggests that the reason is that I knew
what the result ought to be, and so “ cooked’’ my equations
accordingly. Now I should not have thought it worth while
to reply to criticism of this order had it not been that the
subject of the application of mathematics to chemistry is only
dealt with in a few text-books, so that it is important to point
out any misrepresentations and misstatements in those which
profess to explain this subject. The amusing part of Prof.
Ostwald’s criticism is that when, after his tirade, he attempts
to obtain one of my equations, he implicity assumes that the
molecules are not split up by the collisions, for he assumes
that the number of molecules split up in a given time is pro-
portional to the average number of molecules. Now, if we
refer to the illustration of the cabs, it will be evident at once
that this is equivalent to assuming that the collisions have
nothing to do with the breaking-up of the molecules, for if
the men were shot out of their cabs by collisions with cabs
with men inside, the number leaving their cabs in any time
would be increased fourfold if the number of men in cabs
were doubled, for the number of men in cabs would be
doubled, and the average time they spend in the cabs would
be halved.
It may illustrate the care with which the book has been
written, and the reliance to be placed on its contents, if I
mention that within about half a page Prof. Ostwald makes
three misstatements. He says that an equation he obtains by
Intelligence and Miscellaneous Articles. 381
a process of his own is the same as one of mine, though it is
not ; he says that I sometimes suppose the free time to be
constant, and sometimes to depend on the number of atoms,
when I do not; and, lastly, that I have not stated what
meaning I attach to r, when on page 238, line 44, I have
defined it to be the free time multiplied by the number of atoms.
Iam, Gentlemen,
Your obedient servant,
Trinity College, Cambridge, J.J. THOMSON.
Feb. 14, 1887.
XLIV. Intelligence and Miscellaneous Articles.
ON CERTAIN MODIFICATIONS OF A FORM OF SPHERICAL
INTEGRATOR.
To the Editors of the Philosophical Magazine and Journal.
GENTLEMEN,
if HAD occasion recently to read in the Philosophical Magazine
(August 1886) the very interesting description of a ‘‘ Spherical
Integrator,” designed by Mr. Frederick John Smith, and which
appears to be a modification of that of Prof. Hele Shaw. But the
first conception of these apparatus, and it is to this that I wish to
call your attention, belongs without doubt to me, as in No. 630 of
‘Nature’ (Nov. 24, 1881) I gave a description of an ‘‘ Anemometer
Integrator” founded on the same principle, and which was after-
wards mentioned in the Quarterly Journal ot the Royal Meteoro-
logical Society, No. 48(1882), by Mr Laughton (‘‘ Historical Sketch
of Anemometry and Anemometers ”).
The modification designed by Mr. F. J. Smith tending to do
away with or lessen, as much as possible, the moment of inertia of
the sphere, appears to me excellent, especially if it is to transmit
velocities of small magnitude. But when it is simply required to
register that of the wind upon a moderate scale, I believe that the
primitive form suffices; and after several trials which I have made,
an ivory ball rolling on bronze cylinders is that which gives the best
results.
I beg, Gentlemen, that you will allow this claim of priority to
appear in your valuable Journal, and also that you will accept my
most sincere thanks and the assurance of my marked regard.
Madrid Observatory, March 12, 1887. V. VENTOSA.
ON THE STRENGTH OF THE TERRESTRIAL MAGNETIC FIELD
IN BUILDINGS. BY M. AIME WITZ.
In consequence of the removal of my laboratory to a new
building in which the joists and framework are of iron, I have
been led to determine exactly the values of the horizontal com-
ponent in the various rooms used for Physics, with a view to
certain researches which I have undertaken. I have observed
382 Intelligence and Miscellaneous Articles.
astonishing discrepancies ; and I think it useful to draw the attention
of physicists to this subject, which has been but little studied.
A simple method of measuring the horizontal intensity consists
in passing a constant current through a circuit containing a weight
voltameter, and a tangent-galvanometer. By determining the
absolute strength of the current on the one hand by the results of
electrolysis, and on the other by the deflection of a compass-
needle, and equating these two values, we can get the value of T at
the spot where the galvanometer was placed. This method was
of sufficient exactitude for the work of comparison in which I was
engaged.
A Poggendorff’s battery may be used; this is a very constant
source when the chromic liquid is strongly acid, and the external
resistance is great. As an electrolyte I took a 10 per cent,
solution of pure copper sulphate; the copper electrodes at a
distance of about 30 mm. had 12 square centimetres immersed ;
from this resulted a favourable density of current, and therefore a
beautiful deposit of metal which was continuous and _ perfectly
adherent. The loss of the soluble electrode was always equal to
within 5 mer. to the gain of the negative electrode. The intensity
of the current, which was about + of an ampere, was determined to
within 5,55 of an ampere; it was assumed that 1190 mgr. was
deposited per ampere-hour. Two good tangent-galvanometers
were used simultaneously ; their constants are as follows :—
Length Mean Number
Galva- -——_—*-—oa~ radius. of :
nometer. of needle of wire R. windings. 20
mm. mm. cm.
PA VGN 15 1258 16°68 12 0-221
Lae ae 20 1114 16°12 11 0-233
The needles are suspended to a cocoon-thread ; the long pointers
of aluminum enable us to read ;4, of a degree. The relative
dimensions of the needles and of the frames are in these two
instruments in such a ratio, that we may dispense with the use of
the term of correction, which I have considered proportional to the
tangent of the deflection 6.
The manipulation was very simple; the element having been
shortcircuited for a few minutes, the current was passed for an
hour through the voltameter and the galvanometer. Two double
readings were made after five and twenty-five minutes; the current
being reversed in the galvanometer after thirty-five and fifty-five
minutes. The mean of these eight readings gives the value of the
mean deflection of the needle in the course of the operation. It
remained to weigh the electrodes, and to take the mean p of the
loss and gain of the plates in miligrammes. The formula
Pate ha
10 1190 ~ 2xn
leads to the value of T in C.G.S. units; and the same operations
repeated in various places enable us to discover considerable
T=tan 6
Intelligence and Miscellaneous Articles. 383
variations of the horizontal component in a building where much
iron has been used in the construction.
The following are the details of an experiment; they enable us
to judge of the value of the method, and the agreement of the
observations. This experiment was made on the 13th April 18386,
at La Solitude in the suburbs of Lille, in the centre of an open
space of several acres, and at a distance from any buildings and
from water- or gas-pipes.
Deflections of the Galvanometer.
Right. Left.
Time. a we FT!
m ° ) fo) ro)
BPE Osea a, 0 Wee ie ae
PN ss as 3o°/0 34:15
Bi .. 43 OO ad 33°50 is LE
24 55 Weel i ay 33°85 33°50
4 55 30°40 33°00
General Mean 33°- Bae 33° 33,
Observations of the Voltameter.
mer.
Loss of the Soluble Electrode .......... 360
Gain of the Negative Electrode ........ 355
Veer, Ate. ORAS ee ged 307'°O
WEE Gs
10 1190 = 0°233 T tan Soo
T = 0:187.
This value of T will serve as a point of comparison; it is less
than the value observed at Paris on the 1st January 1886, as was
to be expected from the position of Lille. We consider it exact to
within 3 or 4 thousandths ; in fact an observation made after the
first gave 0:185, and a great many experiments made in the
laboratory aes that the result of an experiment never differs
by more than zoo0 from the mean of a month of investigations.
The table given below gives the value of T obtained in various
parts of our septs sc
Date. aps
La patos mieaw: Dille: wi taeulailly. April 13 0°186
esquinimearullle i G4 ay oo aed brit 0-191
Outer court of the Faculty ...... us 1 0°183
Inner court of the Faculty ...... May 21 0-190
Protessor's Room 1.7. oh wue elo I 0:152
nyisical:Cabimeti jetties os Soe a oes Mar. 23 0-134
PARA VAY sae is SY HOS YU Wiel 3 29) 0-133
Peles eR Sey A. cihaite Sand OR dhol a oh 330 0-114
Vaulted Hall ....... Y FA, July 21 0-194
It follows from these researches that T may be reduced by 40 or
50 per cent. in a building made of iron ; hence the same current
will give in the same galvanometer a ‘deflection of 33° to 45°
+
384 Intelligence and Miscellaneous Articles.
according to its position. It will thus be seen that the calibration
of instruments of this kind must not be forgotten when they are
moved from one place to another.—Journal de Physique, Jan. 1887.
ON METALLIC LAYERS WHICH RESULT FROM THE VOLATILIZA-
TION OF A KATHODE. BY BERNHARD DESSAU.
The results of the present investigation may be summed up as
follows :—
By appropriate electrical discharge in highly rarefied spaces, the
metal which acts as kathode is volatilized and settles on a glass plate
as a reflecting layer or mirror. If the oxygen has not been most
carefully removed, all metals seem to undergo oxidation under these
circumstances. There is perhaps in all cases a combination with
the traces of residual gas (hydrogen or nitrogen), yet the mirrors
obtained in hydrogen are not materially different from those of pure
metals. With suitable arrangement of the electrodes the layer of
metal is obtained as a flat cone ; and when viewed in reflected light,
under as acute an angle as possible, coloured interference-rings are
obtained, which prove the presence of a dispersion in the metals. 1t
may be concluded with some certainty that this dispersion is normal
in platinum, iron, nickel, and silver, and abnormal with gold and
copper. The layer directly produced by the discharge, whether it
be metal or oxide, is always double refracting, probably in conse-
quence of an electrical repulsion between the particles expelled, and
the regular stratification thereby produced; in the metals the ray
which vibrates tangentially is accelerated in respect of the others.
In the metals the cross of double refraction was also observed in
reflected light, and in reflection from the metal side the action
was the reverse, and from the glass side the same as in transmitted
light. Double refraction disappears on oxidation of the double-
refracting metals, as well as by reduction of the layers of oxide,
while heating without any chemical change has no effect.— Wiede-
mann’s Annalen, No. 11, 1886.
ON THE PASSAGE OF THE ELECTRIC CURRENT THROUGH AIR
UNDER ORDINARY CIRCUMSTANCES. BY J. BORGMANN.
One end of the coil of a Wiedemann’s galvanometer is connected
with the earth, and the other with a platinum wire, which is placed
in the flame of an insulated spirit-lamp. Ata distance of 14 metre
from this lamp is an ordinary Bunsen burner, which is connected
with a conductor of the Holtz machine ; the other conductor is put
to earth.
When the lamp is lighted the galvanometer indicates no current ;
but when the disk is rotated a distinct current at once appears in
the galvanometer, and the deflection of the needle does not alter so —
long as the machine works at a uniform rate. If the Bunsen
burner is connected with the other conductor of the machine, a
current in the opposite direction is at once set up.—Bezblatter der
Physik, January 1887.
THE
LONDON, EDINBURGH, ann DUBLIN
PHILOSOPHICAL MAGAZINE
AND
JOURNAL OF SCIENCE.
[FIFTH SERIES.]
MAY 1887.
XLV. On the Expansion of Salt-Solutions. By W.W. J.
NicoL, M.A., D.Sc., F.R.S.E., Lecturer on Chemistry,
Mason College, Birmingham*.
[Plates V. & VI.]
NHIS is a subject which has at no time attracted much
attention. With the exception of the experiments of
Bischoff, Muncke, Despretz+, and Rosetti t, which deal
with special cases, such as the expansion of sea-water, we are
indebted to Gerlach§ and Kremers|| for the whole of our
knowledge of the subject ; and this may be summed up as
follows :—
1. The rate of expansion of a salt-solution is the more
uniform the more concentrated the solution. Thus, while the
line representing the volume of pure water at various tempe-
ratures is very pronounced in curvature, the lines correspond-
ing to the volumes of various solutions of a salt approximate
more and more to a straight line the stronger the solution
(Gerlach, loc. cit.).
2. As a consequence of the above it follows that salt-solu-
tions expand faster than water at low temperatures ; but that
at high temperatures, on the other hand, the rate of expansion
is less than that of water.
* Communicated by the Author.
Tt Pogg. Ann. xli. p. 58.
t Ann. de Chim. et Phys. (4) xvii. p. 370 (1869).
§ Spec. Gew. der Salzlosungen. Freiberg, 1859.
|| Pogg. Ann. vols. c.—cxx. (1857-62).
Phil. Mag. 8. 5. Vol. 23. No. 144. May 1887. 2D
386 Dr. W. W. J. Nicol on the
3. So markedly is this the case that at or below 100° C.
the difference between the volumes of water and of a salt-
solution of the same volume at 0° C. actually changes its sign
in many cases.
4. It is therefore possible in the case of every salt-solution
to find a temperature at which its coefficient of expansion is
the same as that of water at that temperature. According to
de Heen* this temperature is dependent only on the nature
of the salt, and is the same whatever be the strength of the
solution. Kremerst, on the contrary, holds that this last has
some slight influence.
5. No connexion can be traced between the expansion of
‘a salt in the solid state and that of its solution (Gerlach, loc.
cit.).
In the course of experiments on the nature of solution it
appeared to me probable that there exists a connexion be-
tween the increase of solubility with rise of temperature and
the rate of expansion of solutions of the salts. With the
object of ascertaining whether or not such a connexion exists
I made a series of experiments on the subject; for I found
that the results obtained by previous experimenters either
were not numerous enough or required confirmation. While
in all cases they were not suited for my purpose, owing to
the solutions experimented with being of percentage com-
position, and thus requiring recalculation into terms of mole-
cules of salt per 100 H,O. Even when this was done the
differences in the strengths of the various solutions were so
irregular that any conclusions derivable from the results were
extremely unsatisfactory.
The present paper contains the results of experiments on
solutions of the four salts NaCl, KCl, NaNO, and KNOs, at
temperatures between 20° C. and 80° C. The solutions were
as nearly as possible molecular, and differed from one another
in the case of each salt by two molecules of salt; for the
reasons given below, it will be seen that it is almost impossible
to use solutions of the precise composition aimed at. Still
the error thus introduced is practically eliminated by the
method of calculation employed.
In the determination of the expansion of a liquid two
general methods are available, each presenting certain points
of advantage over the other. I finally decided to employ the
dilatometric in preference to the pycnometric method ; and I
had the less hesitation in doing so, as I found it possible to
* Physique Comparée. Bruxelles, 1883, p. 76.
+ Pogg. Ann. evi. p. 882 (1858).
Expansion of Salt-Solutions. 387
construct a constant-temperature bath, which removed all the
difficulties and inaccuracies attending the use of long tubes.
As I wished, so far as possible, to experiment with solu-
tions of definite molecular composition, and at the same time
to avoid the multiplication of calibration and other corrections,
it was not possible to employ dilatometers with tubes suf-
ficiently large in internal diameter to permit of the introduc-
tion of the solution through the graduated tube. I therefore
modified the form of dilatometer devised by Kremers (Joc. cit.),
so that it presented the appearance shown in Pl. V.fig.1. The
bulb A is furnished with a tube at either end. One of these,
B, is short and bent round parallel with the side of the
bulb; it is about 3 millim. external and 1 millim. internal dia-
meter, but at the free end is thickened and narrowed to about
0:2 millim. A shoulder is formed about 20 millim. from the
free end, by which the closing apparatus is attached. The
measuring-tube, C, is about 700 millim. long, and is divided
into millimetres from —10 millim. to 600 millim. The gra-
duated tubes were obtained from Geissler, of Bonn, and after
calibration were sealed on to bulbs of suitable capacity.
The dilatometer is filled through the short tube, the end of
which is flat, and is closed by an indiarubber pad screwed
down by the clamp shown in fig. 2.
The calibration of the tubes was performed as follows:—
A short thread of mercury was passed through the tube
and measured at every 20 millim. It was found that the
bore was extremely uniform in all the tubes, no abrupt change
being perceptible. As the tubes were so long and so uniform, it
was considered unnecessary to do more than calibrate them
for more than every 100 millim. Thus, in the case of one
of the dilatometers Dv,a thread of mercury had the following
lengths at various parts of the tube :-—
At Omillim. length was 110 millim.
HODES 0 ah, nhl WALA a,
200 ” oD 111 ”
POU iy imctinas volt beets
A00 aK % Tray
500 ” ” 110 ”
Mean value 110°7.
The mean value in grammes of mercury of each millimetre
was obtained by weighing the mercury required to fill nearly
the whole of the graduated part of the tube. In the case
above 594 millim. contained 2°723 grm. mercury, or 1 millim.
contained 0:00459 grm.
2D 2
388 Dr. W. W. J. Nicol on the
From the data thus obtained the bulbs were proportioned
to each tube, so that the value of each millim. in terms of the
total capacity of the bulb should lie between 0-00004 and
0:00006. When the bulbs had been sealed on, the dilato-
meters were filled with mercury at 20° C. up to the zero
mark on the stem, and the weight of the mercury was deter-
mined. In the case above this was 79°93 grm. Thus the
mean value of 1 millim. of the stem was 5:74, that of the bulb
and stem up to the zero being 100,000. This was then cor-
rected according to the calibration results for every 100 millim.,
giving :—
0-100=5°778 300-400 =5°713
100-200 =5°713 ~ 400-500 = 5-701
200-300 =5°718 900-600=5°778.
The coefficient of expansion of the glass was determined in
each case by both mercury and water. With the above dila-
tometer the apparent expansion of mercury between 20° C.
and 78°°8 C. was found to be 100,914°6. Calculated from
Landolt’s tables the true volume is 101,069, difference 154°4.
The volume of water was 102,483°8, calculated 102,637°8,
difference 154:0, giving coefficient for glass=0-00002°62.
Of the various dilatometers thus made only three were used
in the following experiments. In these the mean value of
1 millim. of the stem was Dr=4°81, Div=5'51, Dv=5:74.
The thermometers employed were two by Geissler and two
by Negretti and Zambra. These last were verified at Kew.
Those by Geissler were from 20°-60° C., and from 40°-100° C.,
and were divided into 10ths; one of the others was from
—10°-40° C., also in =1,ths, and the fourth from —10°-110° C.,
divided into half degrees (a very open scale). These were
carefully compared together and corrected at 20°, 45°-46°,
50°-51°, 56°-57°, 61°-62°, 67°-68°, 72°-73°, and 78°-80°—
the temperatures at which determinations were to be made.
The comparisons were made in two constant-temperature
baths, one at 20° C., the other being the one employed for
heating the dilatometers.
The constancy of temperature was in this last case obtained
by means of the vapour of a liquid boiling under a constant -
pressure variable at will. The liquid in this case was a mixture
of alcohol and water boiling at about 82° at 760 millim. The
apparatus consists of two parts, the dilatometer-bath with boiler
and condenser, and the pressure-regulator.
The bath is shown in fig. 3. It consists of two glass tubes,
one within the other, diameters 65 and 45 millim., and re-
spective lengths 900 ‘and 700 millim: The longer and wider
of the tubes is drawn out at one end to about 15 millim.
EKzpansion of Salt-Solutions. 389
diameter ; the shorter tube is closed and rounded at the lower
end. A brass cap, firmly cemented on to the wide end of the
outer tube, carries the side tube bent at right angles, by which
communication is made with the Y-shaped condenser. The
inner tube is secured air-tight in the cap by an indiarubber
cork, the lower end being kept in the centre of the outer tube
by a ring with three projecting arms. ‘The free upper end
of the Y-condenser communicates with the pressure-regulator,
a Woulff’s bottle being interposed to retain any liquid boil-
ing over. The lower end of the Y passes to the bottom of the
boiler, which is a stout copper cylinder 150 millim. high and
120 millim. in diameter, and stands on a solid flame-burner.
When the boiler has been one third filled with alcohol, the
whole apparatus is made as nearly air-tight as possible, and
connected with the pressure-regulator. The inner tube is
filled with water, the gas is lighted, and the pressure is re-
duced the desired amount. The vapour of the boiling alcohol
passes up between the two tubes, entirely surrounding the
inner for its whole length. At first the condensed alcohol
flows back into the boiler; but as the temperature of the water
in the inner tube rises, the alcohol vapour passes into the
condenser and thus back to the boiler, complete condensation
being insured by the second limb of the Y-tube.
The pressure-regulator is shown at fig. 4: it is based partly
on that proposed by Meyer, and partly on the modification
introduced by Stadel and Schummann”*. It consists of a
firm wooden stand some 900 millim. high by 200 millim.
wide. =6°35 centim.
& by half turn of 1000 coil, in 8, throughout.
Galvano-|Make & break
Arrange- |
Date. | Set. ment. a Bb. 0. meter. or reversal. M.
49P A 1 Olen oe a j
Feb. 23.] 1] osgg |2022 | 142 | 3156 2 m. b. 10° x 5-9390 | ||
2) 2207 | e82| 173! |iaeoo |) IB rev. x61075 | |
Hobe 24 sla) oo. 86:5 | 175 | 43.55°5 a ne x 60030 | |
MTR fae 85:2 | 175 | 435 nie 3 x6 0893 | ||
3 | oayg |1902 | 1447] 2948 2 m. b. x 59862.
Calculation by Maxwell’s original formula . 10° x 5-275 } |
Comparison of arrangements :—
49 'P 250 P
2508 428 ;
Feb. 23. 1 10°x5-9390 Feb. 23. 2 10°» 6:1075 i il
Bade, he x 59862 sip), ee peal x 6:0030
| ag LOE ” 99 2 x 6:0893
Mean x5:9626 | ae
Mean x 6:0666 —
Mean of both arrangements 10° x 6:0146.
Determinations of Coefficient of Induction of Dynamo Field-
magnets and Armature. (A Gramme.) :
Feb. 24. P field-magnets, with two bichromates and 10
B.A. units. }
S armature, ballistic galvanometer, and 1000 coil.
By reversal.
Galvano- ei!
ae B. 0. meter. Amperes. Me} q
Meamvalue li i. | L464 (0990 27° 11! 18 181 107 x 1:85
Highest value . 176 Ee ecco 4 ie pais x 2°23)
Lowest _,, Se) Levee as nie0's 23 aul x1 9
of Coefficients of Mutual Induction. 419
Feb. 25. P field-magnets, with two bichromates.
S armature, ballistic galvanometer, and 1000 coil,
with and without 1000 B.A. units, so that
R=1045, Ro=40.
By reversal.
Galvano-
meter. Amperes. M.
a. B. 6.
ies = 179 39° 24 2 9°54 10'x 3-685
By formula M=~ 100. 1074-95
ADDENDUM.
March 5. Supplementary Notes.
1. Calculation of coefficients by the elliptic integral table
in the new edition of Maxwell, combined with the formula
for approximation to the effect of the sections of the coils.
I have now calculated the experiments of Feb. 22 and
Feb. 23-24 by this method, which is the most complete that
exists, short of calculating all the single combinations of
circles in the two coils. The accordance with experiment is
somewhat better, but still far from close.
Observed. Calculated.
Hen 22.1108 x 1-259 10° x 1:176
yy 28-24. 10°x 6-015 10° x 5-585
2. Simple formula for approximate calculation of the
coefficient.
Assume that the field in a due to A is everywhere the same
as at the centre of a. Then the total lines of force for unit
current are Dar A?
A alan
(A? == b?)#
where b is the distance between the planes of the circles, or,
if estan 6, this becomes
27a” sin °0
o% 7
which is simpler to calculate than Maxwell’s approximate
formula.
__ The following is the comparison of this formula with
observation.
Observed. Calculated.
Web.) 21. 40%x 7690 10’ x 8-991
«22. 108 x 1-259 108 x 1:133
», 28-24. 10°x 6015 10° x 5:036
22
420 Mr. W. Brown on the Effects of Percussion and
3. Course of values of the coefficient of field-magnets and
armature of a dynamo.
The numbers stated in the paper may possibly be mislead-
ing, as it is not sufficiently explained that the number deter-
mined from the motion of the machine, ( =) is not of the
same nature as the two results of determinations made at rest, |
which precede. 3
The number determined from (=) is necessarily infinite
when the current is evanescent, if there is any retention of
magnetism in the machine, and diminishes continually as the
current increases. The following corresponding values of
this coefficient and current are given by the data referred to
in the paper.
Coefficient. Current in amperes.
10° x 7-054 3°8
x 6°250 D°D
x 5°600 76
x 4°732 10°5
XLIX. The Hfects of Percussion and Annealing on the
Magnetic Moments of Steel Magnets. By Witu1AmM Brown,
Thomson Haperimental Scholar, Physical Laboratory, Uni-
versity of Glasgow”. ,
Parr II.
A Part I. of this paper, which appeared in the March
number, certain preliminary results were given, showing
the effects of percussion on the magnetic moments of steel
magnets. In the present communication these effects are
considered in greater detail, with tables giving the results of
an extended series of experiments, and the question of an-
nealing is treated with respect to exact measurements of the
annealing temperature.
The steel experimented on in this case was furnished to Sir
William Thomson for experimental purposes by two different
steel-makers,
The following Table gives approximately the relative per-
centage proportions of all the substances found in the steel,
the quantities in specimen I. being taken as unity. They are
taken, not from analyses of the particular pieces experimented
on, but from a general analysis of the sample in each case.
* Communicated by Sir William Thomson.
Annealing on the Magnetic Moments of Steel Magnets. 421
The proportions are on this account probably only roughly
approximate, and until special analyses are obtained it seems
unnecessary to give the actual quantities.
TABLE I,
Comparative Composition of the Specimens.
Number of specimen.
Substance.
iL, II. III.
SILC T1T a aR ee 1:00 0:08 0-17
Manganese ............ 1:00 1:28 3°25
Phosphorus .j)....-0:5- 1:00 eral 1:55
SUN OLA0 eens ame 1-00 0:00 0-00
Gambon: cesses cen sg 1-00 0:25 0-25
MOM ac deen iL” 1:00 0-994. 0:987
All the specimens contain, as a matter of course, nearly the
same amount of iron, but the other constituents differ con-
siderably. The magnets were prepared in the same manner
as those referred to in Part I. of this paper. They were all
made glass-hard to begin with ; and this was done by bringing
them to a bright red heat, and then dropping them, with their
lengths vertical, into a vessel 60 centim. deep, which was filled
with water at a temperature of 7° C.
A greater number of magnets than were actually required
were prepared, but only those which were found to be straight
and of uniform glass-hardness throughout, chosen for the ex-
periments. The hardness was tested by means of a file run
longitudinally along and around the magnet ; in this way any
marked divergence from uniformity in hardness was detected.
Also, to make sure that all the pieces of the same sample
should be as nearly as possible alike, they were one by one
let fall on to a block of hard wood, and those which gave the
same kind of metallic ring were taken for the experiments.
They were then thoroughly cleaned and polished, and their
lengths, diameters, and weights accurately determined ; these
measurements being, for ease of reference, given below in
Table IT.
There were fifteen magnets in all, 7. e. five samples of each
specimen, and each one was made exactly 10 centimetres in
length.
422 Mr. W. Brown on the Effects of Percussion and
TasiE I1.—Dimensions of Magnets.
Length of | Diameter of ;
Number of | magnet, in | magnet, in | Dimension, Weight of
specimen. | centimetres, | centimetres, | ratio d/d. be aa
7 in grms.
Tee ae 10 0:300 i Oo 5D
TT: fos 10 0-265 | 38 4:3
ALT dees 10 0-270 OF uk 4:5
The five pieces of each sample were then magnetized by
placing them between the poles of a powerful Ruhmkorif
electromagnet, which was excited by a current from twenty-
four Thomson tray-cells joined in series. The magnetizing
current was approximately 5:3 amperes, producing a field of
900 C.G.S. units intensity. The field was measured by rota-
ting a coil of known dimensions between the poles of the
magnet and observing the deflection produced on a ballistic
galvanometer ; and this was reduced to absolute measure by
comparing with the deflection (on the same galvanometer)
obtained by rotating another coil of known dimensions in a
field the strength of which was known.
When the field due to the electromagnet was being measured,
there was nothing between the poles except the meagsuring-
coil. In the process of magnetizing, the magnets were reversed
three times between the poles of the electromagnet and then
finally magnetized. This was also done in every case when the
magnets were remagnetized between two sets of experiments.
After being magnetized the magnets were laid aside for a
period of eighteen hours, and then the deflections were taken
for the purpose of calculating their magnetic moments. They
were put through the same series of operations as the magnets
used in the former experiments, described in Part I. ; that is
to say, the deflection produced by each magnet on a magneto-
meter-needle was observed ; each was then allowed to fall
once perpendicularly through a height of 150 centimetres,
with the true north end downwards, on to a thick glass plate;
and the deflection on the magnetometer again taken with each
magnet in exactly the same position. Hach was then allowed
to fall three times in succession through the same height, and
the deflection again taken.
The following Table gives the results obtained for the
magnets when they were all glass-hard ; and also after they
had been magnetized and left undisturbed for a period of
eighteen hours. |
Annealing on the Magnetic Moments of Steel Magnets. 423
Taste III.—Glass-hard.
Specimen I,
Percentage loss due to
Magnetic :
ever ot moment, falling Total loss.
ore per gram. : ;
one time. three times.
i es ae 0-79 0-40 1-19
i.e 60°42 0:90 0:20 1:10
2 ae 60°18 111 0°20 1°31
Bae 3fh), 60°96 . 0:49 0:30 0°79
es. oo oe 59:03 1:64 0°83 2°46
Mean ...... 60°33 _ 0:99 0:39 1:37
Specimen II.
Rees 72°10 1-72 0:87 257
hee 72:70 213 1:30 3°53
= peas 71°50 1:30 0°88 2°16
AI ais css 72°70 2°13 0:87 2:98
2) one 71:80 2°15 0:88 3:02
Mean ...... 72°16 1°88 0:96 2°85
Specimen ITI.
eh 68°89 4-72 2°25 6°89
7, 69°78 1°27 1-72 2°96
eee 68-10 4:78 1°83 6°52
73 Ee: 70°80 2°49 2-13 4:56
5 re 72:40 4:08 1:28 531
Mean ...... teas OO Ne ty BA | 1:84 5:25
The above table, as far as it goes, seems to show that the
percentage loss in the magnetic moment varies in the order
of the quantity of manganese which the specimen contains.
Thus specimen III. has a mean total loss of 5:25 per cent.,
and it has about three times as much manganese as either of
the other two ; and specimen II. has about 20 per cent. more
manganese than I., and its loss is 3 per cent. nearly, whilst
that of I. is approximately 1-4 per cent.
Specimen I., however, differs very much from the other
specimens in the quantity of silicon it contains, and it alone
contains sulphur.
These same fifteen magnets were now all fastened to a
424 Mr. W. Brown on the Effects of Percussion and
piece of wood by means of soft copper wire, and annealed for
one hour in a bath of linseed oil at a temperature of 100° C.
They were then taken out and allowed to lie at the ordinary
temperature of the room (8° C.) for a period of 6 hours,
after which they were magnetized with the same battery-
power, and every precaution taken, as formerly. Then, after
lying aside undisturbed for a period of 20 hours, they were
put through a similar series of observations for the purpose
of finding the effects of percussion in changing their magnetic
moments. The results are given in the following Table:—
Taste TV. (Annealed for one hour at 100° C.)
Specimen I.
Magnetic Percentage loss due to
Number of moment, fallin Total loss.
magnet. per gram.
one time. three times.
ilo 63-4 0-76 1-92 2:67
AREA 62°4 1:94 0°39 2°32
2 ee 617 274 | 0:60 3:33
AINA So a 62°6 1°54 0°78 2°70
Le eens 61:4 1:97 1:20 319
Mean ...... 62°3 1-79 0:98 2:84.
Specimen II.
Eee teh W1-2, 1°74 1°33 2:67
ae trscccs 72-1 2°57 1°32 3°86
Be fiers 7271 3°43 0:89 4:99
Aen, 72°4. 171 0°87 2:56
Wap es 72:4 2:99 0°88 3°85
Mean ...... 72:04 2°49 1:06 38°45
Specimen III.
LEAS aapelle 66°8 2°54 0°92 4:42
Fr 69°8 2°54 1°74 4:24
See 65:1 4:09 237 6°36
(5 TAN AR 67:4 2°63 1:80 4:38
SOAR ter 68°3 2°16 1:33 3°46
Rica i 67:5 2:79 1-63 457
From the above Table we see that annealing for one hour
in an oil-bath at temperature 100° C. has slightly raised the
Annealing on the Magnetic Moments of Steel Magnets. 425
magnetic moment of specimen I. and lowered it in III,
whilst that of II. remains unaltered. Also that the total
percentage loss in I. and II. is increased, whilst in ILI. it is
slightly diminished ; indeed, we find it is doubled in speci-
men I., and in II. it is increased 17 per cent., whilst it is
diminished 12 per cent. in specimen III.
We must remember, however, that specimen I. alone con-
tains sulphur, and has the least quantity of manganese, and by
far the most silicon of the three, while II. contains the least
amount of silicon.
The same fifteen magnets were again annealed for a period
of two hours in the same oil-bath at a temperature of 100° C.
They were allowed to cool and lie for six hours, as formerly,
at the ordinary temperature of the room. They were also
magnetized and treated similarly in every way as in previous
experiments. ‘Then, after lying aside undisturbed for a period
of twenty hours, they were put through the same series
of observations for determining the loss in their magnetic
moments. ‘The results are given in the following Table:—
Tasie. V.
(Annealed for two hours at 100° C.)
Specimen I.
; Percentage loss due to
Rneiher of meee falling Ro?
ones moment, otal loss.
per gram.
One time, three times.
: 62:09 156 1-19 2-73
De 0 ae 60°72 1:99 1:02 2-99
5. pease 60°72 1:99 061 2°59
Le 62:40 1:94 0°39 252,
5 aaa 61:20 2°37 Lot 2°56
fe oe 61:42 1-99 ROBeHt ben) cet
Specimen IT.
1) eae ai ay 2-13 1-70 3°83
rt 72°72 2:97 0°88 3°83
2 SA ak 72-10 Dey) lire 3°86
0) Nee 73°00 2°96 0:87 3°81
i 72°46 3°42 0°88 4°27
mee. 72:60 2:8] 113 3-92
ee I
ee. Ee EF
426 Mr. W. Brown on the Effects of Percussion and
Table V. (continued).
Specimen IIT.
Magnetic Percentage loss due to
jie moment, falling. Total loss.
peretan. one time. three times.
RE See 63°86 1°85 0:94 2°78
ee 71°55 5:00 1:09 5-99
St aS 68°89 4:72 5:40 9°87
ZA ae he 71:55 5:00 1°74 6°61
Sys ek ae W25 3'O2 2°14 5°39
Mean ...... 69°42 3°98 2°26 6°13
From this Table we see that the second annealing for two
hours has had no effect on the magnetic moment per gramme
in the case of specimens I. and fI., and has only slightly in-
creased that of specimen III. We also see that the total
percentage loss is unaltered in I., and but slightly increased
in II., but in specimen III. there is an increase of about
33 per cent.
All the magnets were now annealed for a period of thirty
minutes in an oil-bath at a temperature of 236° C.; they were
then taken out and allowed to cool, as usual, to the ordinary
temperature of the room (8° C.). Then, after lying aside
for six hours, they were magnetized in the same manner and
with the same battery-power as in the previous operations.
The temperature of the oil was at first determined approxi-
mately by means of a mercury in glass thermometer ; it was,
however, accurately determined by an air-thermometer con-
structed on a method introduced by Mr. J. T. Bottomley, and
communicated by him to the Birmingham Meeting of the
British Association in 1886. ‘This method will be explained
further on.
After being magnetized, the magnets were laid aside for a
period of twenty hours and then put through another series
of observations, the results of which are given in the following
Table :—
Annealing on the Magnetic Moments of Steel Magnets. 427
TABLE VI.
(Annealed for half an hour at 236° C.)
Specimen I.
Magnetic Percentage loss due to
Ae moment, coals Total loss.
magueb per gram.
one time. three times.
BS saa 63°48 514 321 8°19
7, ee 61-90 5°27 3°08 8:20
Sharepee 61-60 7°45 2:96 10:20
AN isibe 62°90 6°73 2°94 8:84
Dara oeeee 61°66 5:49 4°56 9°80
Mean ...... 62°32 6-01 3°35 9°04
Specimen IT.
eee 69:13 671 BVP lr LOSI
2 ae 69°13 7°80 7:28 | 14°54
Bie tiaras 68-05 6°36 7°76 13°63
2 Scere 69°60 L377 2°83 16°22
Be aaa 68°36 12°88 4-93 17:20
Mean ...... 68°85 9:50 5:37 (14-42
Specimen ITI.
i Ae 65°8 17-97 4-11 21°34
2) eee 67°9 10°43 5°34 15:22
Ser | 64:7 17:80 5°55 20°10
A Telateiwes 65:0 13°63 4:2] 17-27
DS aeeern 64:0 13°85 617 19°16
Mean ...... 65:5 473 | 507 | 1861
Here we get a very interesting result: we find that, by
annealing for half an hour at approximately three times the
temperature, we get three times more percentage loss. It is
also interesting to note that in every case the total percentage
loss is almost exactly tripled ; but the three specimens still
preserve the same relative behaviour throughout.
The same magnets were again immersed for half an hour in
428 Mr. W. Brown on the Hffects of Percussion and :
an oil-bath at temperature 236° C. and then allowed to cool
in the air as formerly. But this time they were allowed to
lie for three weeks, and then magnetized in a manner every
way similar to that formerly employed. After being mag-
netized they lay undisturbed for a further period of twenty
hours, and were then put through the same series of observa-
tions as on the previous occasions. The results are contained
in the following Table :—
Taste VII.
(Annealed for half an hour at 236° C.)
Specimen I.
Magnetic | Percentage loss due to
Number of moment, falling Total loss.
ME =
; one time. three times.
Eerie 57-0 76 105 14:6
PAE ene 60°9 Ia 5:4 16°8
SS ep sigan 61:1 10°8 4:4 14:8
4 Oh aan 60:4 86 53 13:6
Ey ane sete 60°9 15°4 54 19°8
Mien, G00 | aoe 56 15-9
Specimen II.
1 Pees eae 60°3 22:0 70 29°5
ey es 60°9 16°4 14-1 27:8
Se aa 58:9 10°6 16:1 25:0
2 et 60:4. 17:0 16°9 aH es |
Ryn cecsan 58°5 25:0 10°8 32'°8
Mean’... ... 59°8 18:2 12:9 29-2
Specimen III.
LS ae 5671 16°6 9°5 24-6
CA aaa 56°9 10°6 14°6 22°71
2 th ie ee 56'1 17°4 9:9 25°9
Di ea 582 23°0 6:2 27°6
Posh hate 58:2 20:0 14:2 31:0
Meant ee 571 17-5 10°8 62
We here find that the second annealing at a high tempera-
Annealing on the Magnetic Moments of Steel Magnets. 429
ture has diminished the magnetic moment per gramme by
13 per cent. in specimen II., and by about 12 per cent. in
III., whilst in specimen I. itis decreased by nearly 4 per cent.
We also find that the total percentage loss due to falling four
times through a height of 1:5 metre has increased above the
results of the last experiment as much as 100 per cent. in
specimen IJ., and 70 per cent. in I., and 40 per cent. in the
case of specimen III.
The magnets were not again magnetized, but were allowed
to lie undisturbed in the varying temperature of the room for
a period of nine months, that is from May 15, 1886, till
February 12, 1887. This was done merely to see what would
be the effect of time upon them in their annealed condition.
They were put through a similar series of observations, with
the exception that they were not remagnetized. The follow-
ing Table contains the results :-—
TasLe VIII.
Magnets not remagnetized and left undisturbed for 9 months.
Specimen [.
: . Percentage loss due to
Number of HEGEL
moment, ee Total loss.
magnet.
per gram : .
one time. three times.
i so av7g | «1-02 1-03 2:04
5) ae ~ 49-48 0:50 0:50 0:98
aes 48:51 1:50 0:00 1:50
AG! 4% 50:70 0-96 1:45 2-40
ht a 48:01 1:01 0:00 1:01
Mean ...... 48:89 1-00 0:59 1:58
Specimen IT.
egy a's 41:47 | 2-25 4-23 5-64
Fh) ee 43:02 | 2:89 1:50 4-34
SO veal 39-28 1:58 0:80 2:38
Jay Sea 41:00 1:14 0:00 ue Pace!
ee 38°66 0:80 2:52 3-22
4380 Mr. W. Brown on the Effects of Percussion and
Table VIII. (continued).
Specimen III.
. Percentage loss due to
Magnetic Lee
Number of moment, falling Total loss.
magnet. a ;
acess one time. three times.
‘peyote 8 42°31 211 2°16 4:22
De, Bee 43°45 1:37 1:38 2:82
oe park a 4111 0:00 0:00 0:00
2a eens 41-11 1:45 1:47 2°90
Lap eats 39°36 IE 0-77 2:27
Mean ...... 41-46 1-34 1-15 2:44
From the above Table we find that the relative losses of
magnetism in the different specimens due to lying undisturbed,
as indicated by the diminished magnetic moments, is in the
reverse order to what has taken place throughout the whole
series when the magnets were subjected to percussion.
The total percentage loss all through these experiments due
to percussion has been in the order of the number of the spe-
cimen. ‘Thus specimen I. has always decreased the least and
specimen III. the most; but, in the case of lying undisturbed
for nine months, the decrease in the magnetic moment of spe-
cimen I. is 3°5 per cent., and of II. 3:7 per cent., while ILI.
has diminished only 1:6 per cent. Specimen III., however,
contains about three times as much manganese as either I.
or II.
In Joule’s Scientific Papers, vol. i. page 591, some results
are given on the effects of time and temperature on hard mag-
nets. ‘The magnets used by him were either one inch long or
half an inch, and were made up of a number of thin bars
placed side by side so as to form compound magnets of ya-
riously shaped sections but with plane ends; the magnetic
moments of these magnets diminished about 33 per cent. on
lying aside for a period of eighteen years.
The rate of diminution of magnetism in different kinds of
steel, with annealing, time, and temperature, is at present
under investigation in this laboratory. In connection with
this investigation, further experiments are being made on the
same kind of steel as is referred to in this paper, and it is
hoped that further results will be ready for publication at an
early date. I will now give a tabular view of the results
obtained up to this point.
431
Annealing on the Magnetic Moments of Steel Magnets.
TaBLE IX.—Showing the changes in the magnetic moment per gramme, and the total percentage loss due to the
whole four falls through a height of 150 centims.; also showing the effects of annealing on the different specimens.
Not remagne-
Length Annealed Annealed other | Annealed half Annealed Ficeel acl et
and Dinaiaien Weight Glass-hard. one hour two hours at an hour at another half adi eee ee
Speci- | diameter |" "atin | OF at 100° C. 100° ©. 236° 0. _| hour at 236° ©, | UNdisturbed for
peci- of ratio magnet 9 months.
men. U/q 2 z
tape a Per Per Per Per Per 12
In brane. Mag. Mag. Mag. ; Mag. Mag Mag. iss
cents mom. | | mom, | gE | mom SEM | mom. | $2" | mom [PE | mom | Se
Te ep LOX O'S 33 5D 60°33 | 1:37 62°3 2:84 61:42 | 2°84 62°32 | 9:04} 60:00 | 15:9 | 48:9 16
ncecsul sO >< O:26D 38 4'3 72:16 | 2°85 | 72°04 | 3-45 72°60 | 3:92 68°85 | 1442] 59:80 | 292 | 407 | 3°34
IIT.......| 10 0:27 37 4:5 70:00 | 5:25 675 | 457 | 69°42 | 611 65°50 | 18°61) 57:10 | 26:2 | 41°5 | 2:44
TABLE X.—Showing the effects of the separate falls.
Gisacand Annealed one hour | Annealed two hours Annealed half an Annealed another After being left un-
: at 100° C. at 100° C. < hour at 286° C. half hour at 286° C. |disturbed for 9 months.
. - No. of - : No. of : : Nowot Sis : No. of f : No. of : ; No. of j
Speci-| 8 falls. | 2 | & falls, |-2 | & falls. | 2 | 8 falls. g g falls. | @ | 9 falls. |
men. q as S| Bo S| ee | = 5 = q a
Pel se lk a. 8 See ee ee a) ee eal a eel eles
Slee oie ee SLES erste ee eee Sep ake E
T....| 60°33 | 99 | -39 /1:37 | 62:3 {1°79 | 98 |2:84 | 61:42 |1-99 |1-08 |2:84 | 62:32 | 6:01 |3:35| 9-04! 60:00 |10°9| 5°6|15°9| 48-9 |1:00 0-59 |1°6
II....} 72°16 [1-88 | -96 |2:85 | 72-04 |2-49 /1-06 |3-45 | '72°6 /2°81 /1-13 |3-92| 68-85 | 9:50 [5-37 |14-42 | 59-80 |18-2 112-9 29-2] 40:7 1-73 |1-81 [3-34
TIT....| 70-00 [3:47 |1-84 |5:25 | 67-5 |2°'79 |1-63 |4:57 | 69-42 [3:83 |2-26 |6-11 | 65:50 |14-78 |5-07 |18°61 | 57-10 {17-5 |10°8 |26:°9 | 41:5 [1:34 )1-15 [2°44
432 Effects of Percussion and Annealing on Steel Magnets.
Mr. Bottomley’s modification of the air-thermometer, re-
ferred to above, which was used for measuring the high tem-
peratures, is constructed and employed as follows :—
Suppose a glass tube, 4 inch or ? inch internal diameter,
is made to the shape shown in fig. 1, which Fie. 1
is a quarter of the full size of the tubes used a ;
in these experiments. |
The parts AB and DC are drawn out to
fine capillary tubes, very small in volume in :
comparison with the bulb BD of the ther-
mometer. When ready for use it is com-
pletely filled with pure dry air and closed at
C, but open at A.
The parts CDB and the greater portion of
AB are now inserted into the liquid, the
temperature of which we wish to measure ; B
and when it has been in long enough to be :
at the same temperature as the liquid, it is
sealed at A with a blowpipe flame, thus en-
closing a sample of the air at the required
temperature. The height of the barometer
at the time of closing is also noted.
It is then taken out and allowed to cool, and also thoroughly
cleaned, with alcohol if the bath has been of oil, as it was in
the case under consideration.
It is now carefully weighed in a chemical balance ; then
the end C is opened under water at a known temperature; the
height of the barometer being again noted.
By this operation the water is allowed to rush into the bulb
BD and to compress the contained air to the volume consistent
with the barometric height and temperature at the given instant.
The thermometer with the contained air and water is again
carefully weighed, at the same time taking care to add the
small piece of tube which was broken off in the act of open-
ing the end C. The remaining part of the tube AB is now
filled with water by breaking off the end A, and the whole
again carefully weighed.
In the following calculation the weight of the air displaced
during this last operation is assumed to be so very smail that
for our present purpose we may neglect it.
Let now :
g= Weight of the glass, in grammes.
g+w,= Weight of the glass and the contained air, in
rammes.
g+w.= Weight of the glass and water, in grammes.
t= Temperature of the water employed.
D
Assumptions required for the Proof of Avogadro's Law. 433
T=The absolute temperature of the oil-bath.
H=Observed barometric height at the time of
sealing.
H’=The barometric pressure at the time of opening,
corrected for pressure of vapour of water, at the
. temperature of the water used in filling the tube.
Then we have
H! Wg—- Wy, _2id+t
He pe OS AR Gh
p— Uwe (273 +t)
H! (w2—wy) ©
In these experiments the observed values were, after making
all corrections,
H=752:4 millim.
i 78:
W2== 9°310 grammes.
W4= 2:278 ”
R= de ©.
oe 752-4 x 5°31 x 288
See S02 Ce
And the temperature of the oil was therefore 509—273=
236° C,
L. The Assumptions required for the Proof of Avogadro’s
Law. By Professor Tart*.
1 ee months ago (in consequence of a chance hint in
‘Nature’) I managed to procure a copy of Prof. Boltz-
mann’s paper (anté, p. 305), and inserted a reply to it in the
(forthcoming) Part II. of my investigations ; but, as there
may be some delay in the publication, I send a short abstract
to the Philosophical Magazine.
Prof. Boltzmann says that I do not expressly state that my
work applies only to hard spheres. This is an absolutely
unwarrantable charge, as I have taken most especial care
throughout to make this very point clear.
Prof. Boltzmann, while objecting to my remark about
“playing with symbols,’ has unwittingly furnished a very
striking illustration of its aptness. His paper bristles through-
out with formule, not one of which has the slightest direct
bearing on the special question he has raised !
He asserts that, in seeking a proof of Clerk-Maxwell’s
Theorem, I have made more assumptions than are necessary.
To establish this, he proceeds to show that the Theorem can
* Communicated by the Author.
Phil, Mag. 8. 5. Vol. 23. No. 144. May 1887. 2G
OT ae a
434 Assumptions required for the Proof of Avogadro’s Law.
be proved by the help of a different and much more compre-
hensive set of assumptions! “ “Hrép@ ye tud@, Avoyeves”! He
allows that my proof is correct ; and I am willing (without
reading it) to allow asmuch for his. The point at issue, then,
is :— Which of us has made the fewer, or the less sweeping,
assumptions? Another question may even be :— Whose as-
sumptions are justifiable ?
My assumptions are (formally) three, but the first two are
expressly regarded as consequences of the third, which is thus
my only one, viz. :—
There is free access for collision between each pair of par-
ticles, whether of the same or of different systems; and the
number of particles of one kind is not overwhelmingly greater
that that of the other.
From this I conclude (by general reasoning as to the be-
haviour of communities) that the particles will ultimately
become thoroughly mixed, and that each system (in conse-
quence of its internal collisions) will assume the ‘‘special state.”
Prof. Boltzmann denies the necessity for internal collisions
in either system, and assumes that (merely by coliisions of
particles of different kinds) uniform mixing, and distribution
of velocities symmetrically about every point, will follow!
Surely this requires proof, if proof of it can be given. So
sweeping is the assumption that it makes no proviso as to
the relative numbers of the particles in the two systems! The
character of this absolutely tremendous assumption is so totally
different from that of mine that 1¢ is impossible to compare
the two. My assumption has, to say the least, some justifica-
tion ; but I fail to see even plausible grounds for admitting
that of Prof. Boltzmann. There is noneed to inquire as to its
truth, at present; for I am not now discussing his extension
of Maxwell’s Theorem which, of course, is implied in it. The
question is :—Is Prof. Boltzmann’s assumption, even if cor-
rect, sufficiently elementary and obvious to be admitted as an
axiom? It is so wide-reaching as, in effect, to beg the whole
question ; and I venture to assert that, on grounds like these,
it cannot possibly be shown that any of my assumptions are
unnecessary.
The objection raised in Prof Boltzmann’s “Second Ap-
pendix ”’ (which is not in my German copy) was made long
ago to me by Prof. Newcomb and by Messrs. Watson and
Burbury.* I have replied to this also in my Part II., and
I will not discuss it now. I need only say that Prof. Boltz-
mann, while causelessly attributing to me a silly mathema-
tical mistake, has evidently overlooked the special importance
which I attach to the assumed steadiness of the “ average
behaviour of the various groups of a community.”
a BOBS
LI. On Evaporation and Dissociation—Part V1.* On the
Continuous Transition from the Liquid to the Gaseous State
of Matter at all Temperatures. By WitttaAM Ramsay,
Ph.L)., and SypNEY Youne, D.Sc.F
[Plates VIL, VIIT., IX., & X.]
ie was proved by Boyle, in 1662, that the volume of a gas,
provided temperature be kept constant, varies inversely
as the pressure to which it is subjected ; this relation may be
expressed by the equation p= - , or pv = constant, where p
and v respectively stand for pressure and volume. But sub-
sequent experiments by Van Marum, Oersted, Despretz, and
others showed that certain gases do not obey this law; and it
is now well known that Boyle’s statement is only approximate;
for it has been proved by experiment by Regnault, Natterer,
and more recently by Amagat, that no gas, under high pres-
sures, is diminished in volume in inverse ratio to the rise of
pressure. Indeed Boyle’s law could hold only on the assump-
tion that the actual molecules of matter possess no extension
in space and exert no attraction on each other. A gas, such
as hydrogen, at low pressures, and consequently at large
volumes, fills a space very great when compared with the
space occupied by the actual molecules ; and these molecules
are comparatively so distant from one another, that the attrac-
tion which they mutually exercise is inappreciable. But, on
compression, the actual space occupied by the molecules bears
an increased ratio to the space which they inhabit; and, by
their approach, the attraction which they exert is also increased.
The gas, then, deviates appreciably from Boyle’s law.
Gay-Lussac, in 1808, enunciated the law that the volumes
of all gases increase by a constant fraction of their volume at
0° for each rise of 1° in temperature. It was subsequently
ascertained by Magnus, and confirmed by Regnault, that cer-
tain gases deviate from this law, expanding more rapidly than
others. Such gases, as a rule, are at temperatures not far re-
moved from those at which they condense to liquids ; that is,
their volumes are comparatively small, and the actual size of
the molecules and their mutual cohesion begin to manifest
themselves within the range of experimental observation.
Again, it is evident that no gas can perfectly follow Gay-
Lussac’s law; but the larger the volume it occupies the
smaller is the influence of the disturbing factors. The usual
* Parts I. and IL., Philosophical Transactions, parti. 1886, pp. 71 and
123; Part III., ibid. part ii., 1886, p. 1; Part IV., Trans. Chem. Soe.
1886, p. 790; Part V., in the hands of the Royal Society.
+t Communicated by the Physical Society: read February 26, 1887.
2 G2 ;
re
a
if
|
436 Drs. Ramsay and Young on
expression for Gay-Lussac’s law is v= =c(1+at), or, if the
absolute temperature-scale be employed, v=cT.
As a deduction from these laws, it follows that if the volume
‘of unit mass of a gas, supposed to follow them rigorously, be
kept constant, the pressure varies dinecer as the absolute
temperature ; or p=clT.
Now, so long as the volume of unit mass of a gas is kept
constant, the average distance of its molecules from one an-
other will remain constant; and it is a fair assumption that
the attraction of the molecules for each other will not vary.
It may, of course, be the case that the effect of a rise of tem-
perature on any individual molecule is to alter its actual
volume ; but of this we know nothing; and, in default of
knowledge, it has been assumed by us that no such alteration
takes place. If these assumptions are correct, it follows
that the temperature and pressure of gases—and indeed the
same assumptions may be extended to liquids—should then
bear a simple relation to each other. We have obtained ex-
perimental proof of a convincing nature that this is the case ;
and in a preliminary note to the Royal Society, read on
January 6, we promised such a proof. ‘This proof is the sub-
ject of the present paper; and we must ask for indulgence
in quoting a large array of figures, some of which have
already been published, on the ground that such an important
generalization requires as much experimental evidence as can
be brought to bear on it.
The relation between the pressures and temperatures afie a
liquid or-a gas at constant volume is expressed by the equation
p=bT—a;
where pis the pressure in millimetres, T the absolute tempera-
ture, and b and a constants. The values of these constants
depend on the nature of the substance and on the volume. It
follows from this, that if a diagram be constructed to express
the relations of pressure, temperature, and volume of liquids
and gases, where pressure and temperature form the ordinates
and abscissee, the lines of equal volume are straight”.
We have proved this to be the case for ethyl oxide (ether)
between the temperatures 100° and 280°, and for volumes
varying from 1°85 cubic centim. per gram to 300 cubic centim.
per gram. This proof we now proceed to give.
The data for the calculations are at present in the press, and
will shortly appear in the Philosophical Transactions for 1886,
p. 10. A diagram (which will accompany that memoir) was
constructed with the greatest care, showing isothermal lines,
' * Amagat (Comptes Rendus, xciv. p. 847) has stated a similar relation
for gases; his data are, however, imperfect, and he expressly states that
the law does not apply to liquids.
Evaporation and Dissociation. 437
the ordinates and abscisse being respectively pressures and
volumes. It was possible to read pressure accurately to
within 20 millim.; and volume, up to a volume of 3:1, to
within 0°001 cubic centim. per gram ; and, at volumes greater
than 3:1 cubic centim. per gram, to 0:01 cubic centim. per
gram. Pressures corresponding to each isothermal were then
read off on the equal volume-lines, from curves constructed to
fit the experimental points as accurately as could be drawn
with the help of engineers’ curves. These pressures and
temperatures were then mapped as ordinates and abscisse ;
and it was found that points corresponding to each volume
lay in a straight line. Again, two points were chosen on
these equal volume-lines, as far apart as the scale of the dia-
gram would permit, and the values of the change of pressure
: dp :
per unit change of temperature, a were ascertained for each
separate volume chosen. To eliminate irregularities, these
values were smoothed graphically ; but it was difficult to find
any very satisfactory method. The method employed for
ether, which we found to give the best results, was to map as
ordinates the ratios between these values, and similar values
calculated on the supposition that the gas or liquid followed
the usual gaseous laws, against the reciprocals of the volumes
as abscisse. A curve was then drawn, taking a mean course
d
among the actual points, and the values of - were calculated
from readings at definite volumes. This expression, ~ is the
6 of our formula. Having thus obtained the most probable
value of b for each volume, the value of a at each volume was
ascertained by calculation from each individual point read
from the original curves, and at each volume the mean of all
was chosen.
Isothermals were then calculated by means of the equation
p=bl—a, T being kept constant ; and those values of a and
6 corresponding to the volumes required being selected. These
calculated isothermals are shown on Plate VII. ; and the lines
of equal volume, or isochors*, on Plate VIII. It is evident,
from inspection of the former, that the calculated lines corre-
spond as closely as possible with the actual observations.
Tt is necessary now to give the data on which these deduc-
tions are based. The following Table gives those points
corresponding to lines of equal volume read from the diagram
constructed from experimental observations.
* From icos, equal, and yawpeiv, to contain. Another suitable word
would be “:soplethe,” but we have Professor Jowett’s preference for the
one selected. Hither of these terms seems preferable to that ( zsometrics )
already proposed.
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Evaporation and Dissociation. 44]-
Lines of equal volume, or isochors, were then mapped from
these results, pressures and temperatures being ordinates and
abscissee. The values of = or 6, were then read for each
separate volume and smoothed, as before described. The fol-
lowing Table shows the read values, and the values after
smoothing. ‘The values of a are also given, calculated by
the equation a=bi—vp, from the results shown on the previous
table ; the means of all the individual results for each volume
are stated. TABLE II.
| | }
b | 6dcaleu- |
Volume. found. | lated | log b. ae
eg oi 3. 2034 30826 826860
1:90 1746 1861 326986 767670
ch he oe ae 1716 323452 715860
210 Sch ee 1597 320335 672820
2-05 1561 1492 317372 633070
21 1500 1405 314777 600110
2-15 1342 1320 3-12061 566170
2-2 1211 1243 309462 535100
2-25 1155 1175 307022 507170
- 23 1117 1115 304744 482160
2-4 1020 1010 300423 437240 —
25 920°8 919-7 269363 397970
2-75 732-0 732-0 2-26451 313605
3-0 623-5 621-7 2:79357 262917
3:3 585-7 5328 | 2°72659 221630
37 453°1 454°6 | 265762 185109
4:0 413-7 413-7 | 261669 165996
5:0 321-5 319-1 | 2-50892 121895
6-0 257-15 254-2 2-40511 91906
7 2136 208-7 2-31959 71464
8 178-0 176-1 | 9-94584 57203
9 150-95 1514 | 918092 | 46742
10 130-8 132°7 2°12280 39079
11 116-25 117°6 2-07027 33037
12 104-45 1055 | 2.02336 28401
13 94-29 95°54 | 1:98017 24659
14 85:00 87:09 1:93998 21567
15 78°82 80-06 1:90339 19125
16 72-29 73-95 1:86893 17049
17 67°58 68°76 1:83733 15313
18 62:53 64-24 1-80780 13854
19 59:23 60-11 1:77893 12533
20 55°70 56-43 175151 11386
25 43°75 43-26 163612 7529
30 35-60 34-99 1:54393 5412
40 26:23 2509 1:39945 3159
AO 20:38 19-46 1:28925 2077
75 12:82 12°50 1:09674 994
100 9-32 9-119 095997 571
150 5:87 5-923 077254 270
200 4°38 4-396 064306 160°5
250 3-46 3-483 0-54198 105
300 2-93 2-858 0°45605 59
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444 Drs. Ramsay and Young on
The lines of equal volume calculated from these values of
a and 6 are reproduced graphically in Plates VIII. and IX.
We propose to discuss the form of the diagram later.
We have calculated the values of p for various isothermals,
at the above volumes. These are shown in Plate VII.; the
actual experimental observations, of which a detailed account
is given in the ‘ Philosophical Transactions,’ 1886, p. 10 et
seq., are represented by circles. It is evident that the curves
through the calculated points represent the actual measure-
ments very closely, indeed as nearly as unavoidable error of
experiment allows. It is to be noticed that the greatest
divergence is at the temperatures 250° and 280°, but the de-
viations are in opposite directions, and must therefore be
ascribed to experimental error.
Table III. (pp. 442, 448) gives the data from which the cal-
culated isothermals were constructed.
As volumes above 30 cubic centim. per gram are not given
in the diagram, we have thought it advisable to show the cor-
respondence of calculated and observed results by a table ;
the calculated numbers were read from curves specially con-
structed from the formula p=6t—a, and the observed results
are those actually furnished by our experiments. It will be
seen that the correspondence is very close.
Table III. (continued).
Temperature 100°.
Waolures Pressure Pres. cal- Wee Pressure | Pres. cal-
: found. culated. ; found. culated.
Cc. ¢. millim. millim, c. ¢. millim. wmillim.
54:06 4817 4875 74:42 3668 3685
55°50 4720 4760 77:34 3546 3550
56°95 4627 4670 83°14 3334 3320
59°84 4434 4470 86:02 3243 3215
62°73 4265 4290 88 89 3150 3120
65°63 4102 4100 94-63 2978 2970
68°56 3946 3970 O77bL 2893 2900
71:48 3818 3815 |
Temperature 150°.
31:09 9066 9110 68°64 4651 4675
33°94 8497 8520 74-51 4304 4310
39°66 7484 7505 80°35 4027 4010
45°43 6674 6745 86:13 3783 3760
51-23 6024 6100 91:87 3564 3530
57-02 5480 5560 97:63 3375 3045
62°81 50384 5090
Evaporation and Dihociation. 445
Table III. (continued).
Temperature 175°.
Pressure | Pres. cal- i Pressure | Pres. cal-
7465 5232 5225 97°81 4044 4020
Volume. found. culated. | Volume. found. culated.
c. ¢. millim. millim. ib - €..G millim. millim.
31-11 9906 9975 | 68°69 4987 5000
33°96 9248 24) ee 74:56 4626 4630
39°68 8065 8130 80-40 4307 4310
45°46 7208 7205 86:18 4035 4030
51°26 6485 6510 91:93 3803 3800
57:06 5903 5940 97-69 3589 3600
62°85 5396 5450
Temp. 185°. Temp. 190°.
31°12 10284 10300 || 381-12 10455 10460
45°47 7434 7460 45°47 7549 7545
59:97 5811 5865 | 59°97 5902 5940
74:58 4755 4770 74:59 4828 4810
86°21 4159 4140 86:21 4210 4190
91-95 3915 3900
97°72 38692 3690
Temp. 192°. | Temp. 193°°8.
Fe 10544 10520 | 44 al 10587 10590
45°48 7590 7600s 45°48 7627 7630
59°98 5930 5970 59°98 5951 6600
74:60 4847 4850 74:60 4867 4875
86°42 4230 4200 | 86°42 4252 4220
Temp. 195°. | Temp. 197°.
31°12 10631 10620 33°17 10108 | 10055
45°48 7651 MOTO «| 38:07 8972 8965
59:98 5969 6020 47°84 7312 7320
74:60 4884 4895 | 57:59 6166 6130
86°42 4260 4230 67°33 5356 5340
91:97 4007 4000 77:05 4718 4735
97°74 3797 38775 | 86°75 4219 4220
96-44 3820 3820
| Temperature 223°°25.
| 31-15 11567 11550 || 8050 4870 4880
45°51 8260 8240 86:29 4556 4550
| 60:02 6412 6420 92°04 4289 4270
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446 Drs. Ramsay and Young on
These complete the data for ether. It appeared necessary
to examine the relations between volume, temperature, and
pressure for carbon dioxide, because it is chemically entirely
different from ether, and also because the data are furnished
by Dr. Andrews, whose experimental skill was very great; and
we shall prove that his results entirely corroborate our views.
It was first necessary to correct the pressures given by him
in atmospheres by means of Amagat’s results, so far as that is
possible. As Amagat’s experiments on the compressibility of
dry air do not extend beyond 65,000 millim., no correction was
possible above that pressure ; and extrapolation is inadmissible,
inasmuch as the minimum value of pv for air is at60,000 millim.
Data for obtaining the actual volume of carbon dioxide used
are given by Andrews. The weight was calculated in the
following manner :—Andrews gives the volume of carbon di-
oxide filling his tubes at 0°and076 millim.,and from Regnault’s
data the weight was calculated. This refers to Andrews’s first
paper (Phil. Trans. 1869, i. p. 575). In his second paper
(Phil. Trans. 1876, p. 421) no direct data are given from
which the weight can be determined ; but we succeeded, by
combining the results given in his various tables, in arriving
at the weight without any serious error.
His results are reproduced in an available form in the
following Tables :—
TasLE IV. (The first quantity weighed 0:000612 gram.)
emp. Vd": Temp. 21°°5. Temps" aly
Vol. of Vol. of Vol. of
= 1 gram i 1 gram. 1 gram.
millim. c. ¢. millim. Cc. €. millim. Cc. ¢.
68726 * 1-038 46600 1-232 63462 © 1:389
56333 1EOS9 46383 1:241 61416 1:425
40725 1:104 45490 1-484 59540 1:470
37631 1:124 45155 2:270 57847 1-495
37459 1°145 44962 3'124 56178 1812
37074 1377 44787 4-760 55010 2234
36942 1547 34907 8118 54588 2°338
36816 1-972 53089 3814
36719 2°758 51709 4-192
36668 3°733 50390 4-534
36610 5004 49118 4°855
36528 6:554 47910 5147
36483 6595 46725 5471
35497 6:964 ; 45622 5°750
44577 6009
43611 6°265
43182 6°515
41761 6°765
40895 7:003
* The pressures to which an asterisk is affixed are not corrected.
Evaporation and Dissociation. 447
Table IV. (continued).
Temp. 32°°5. Temp. 35°°5. | Temp. 48°°1.
|
_ Vol. of Vol. of || Vol. of
| - hit gram. = 1 gram. | = 1 gram.
millim | €..€. millim. c. ¢. | millim. c. ¢.
63246 1461 || 81775* 1:330 83144 * 1-999
59425 1612 || 75673* 1:392 72344 * 2:999
58501 1-318 | ‘70406* 1-476 62837 4-061
56813 2954 || 68035 * 1-532 56311 5-057
55151 3543 | 64516 1-624 51012 5991
54839 3°629 | 60552 2-511 46675 6°887
53292 4034 | 57057 2°549
42808 6°595 53977 4-233
51166 4-831 |
48625 5°396
| 46340 5935 |
| 44263 6-432
| 42383 6917
Table IV. (continued).
(The weight of the second quantity was 0°0018075 gram.)
Temp. 0°. Temp. 6°-6°°9. || Temp. 63°°6-64°. || Temp. 99°:5-100°°7.
Vol. of Vol. of Vol. of Vol. of
1 gram. e- 1 gram. o 1 gram. | = 1 gram
millim. | ¢. ¢. millim. | ¢. ¢. millim. | ¢. ¢. | millim. €, e.
25865 | 10:53 || 25870 | 11:25 ||169420*| 1-401 |} 169910*| 1-816
23325 | 12°36 23291 | 13:09 |/110610*| 1:907 | 110530*} 3-161
20800 | 14°53 20822 | 15°16 81229* | 3363 || 80323*| 5:057
18674 | 16°77 18681 | 17-11 60422 5-477 || 59985 7-192
15155 | 21°73 || 16777 | 19-90 || 48425 | 7-485 | 47910 | 9301
12277 | 27-83 || 15161 | 22°54 40554 9-462 | 40169 11°52
9081 | 39°09 || 12907 | 27-16 34806 | 11:45 | 34382 13°72
| 11093 | 32:18 30351 | 13-48 29910 15°96
| 9994 | 36°12 26212 -| 15:99 25930 18°74
9087 | 40:06 23568 | 18-00 23325 21-03
21106 | 20-41 || 20876 23°70
18866 | 23°08 | 18711 26°65
17000 | 25°86 || 16858 29°74
15356 | 28°88 15214 33°09
13288 | 33°74 13153 38°58
As with ether, these numbers were plotted graphically up
to a pressure of 75,000 millim. Above 65,000 millim. no true
correction for deviation of Andrews’s air-gauge was possible;
* The pressures to which an asterisk is affixed are not corrected.
448 Drs. Ramsay and Young on
but approximate corrections were introduced. It was possible
to read pressures to within 30 millim.on the scale employed, and
volume to within 0:02 cub. centim. per gram. Pressures corre-
sponding to even volumes were read off, as with ether ; and on
mapping the isochors with temperatures as abscissze and pres-
sures as ordinates, the gens points lay in straight lines. The
values of 0, i.e. of ——, were then read off, and smoothed,
by mapping them against the reciprocals of the volumes. After
smoothing, the values of a were calculated at each volume,
making use of the pressures previously read from the curve
representing isothermals. The diagram (1) on Plate X. was
then constructed from these smoothed values. The crosses
denote our readings of pressures at the temperatures chosen
by Andrews for his isothermals. These values of a and 6
were then made use of in recaiculating isothermals at the
above temperatures, and the diagram (2) on Plate X. repre-
sents the curves complete so far as Andrews’s data allow. The
circles represent Andrews’s actual measurements ; and it is
evident that no better concordance could be expected. The
tables which follow give the data afforded by Andrews’s
experiments.
TaBLe V.
Pressures read from Curves originally drawn from Andrews’s
experimental data, and represented by circles in the
diagram (2), Plate X.
Temperature.
Vol. |)
0°. Go. WTSEr VATS be Bel, (8205. 35°°5. | 489°1.| 64°. | 100°.
Cc. c. mm. mm. | mm, mm, mm. mm. mm, mm, mm. | mm.
30 11430) 41790) 2.2 yen oe te ie ... | 14820} 16710]
45) 13480 | 13830]... As Bis aS se .-- | 17500] 19860
20 16200| 16570)... pa gig Je he .-. | 21450 | 243800
15 20300 | 20970| ... a Me a pis 3M .-- | 27700} 31570 | :
12 23730 | 24800]... ae ok Ne ra ..- | 000900 | 88730 |
10 26715} 28000] ... tae a: as Ae ... | 38970 | 45330]: . -
8 id. one ... |853840 |87560?.37800?/38400?| 41930 | 46200 | 54630 |
ff hs ... | 85340 |38220?|40970 41370 |42060 | 46200 | 50850 | 61500 |
6 sens Le, ... |41100?/44700 45060 |46000 | 51000 | 56700 baa |
5 ne ne ... |44100?/48600 49200 |50370 | 56760
4:5 see Bs ae ... {50550 |51400 [52830 | 60000 |
4:0 Bee nan ie ... |02420 |538430 |55140 | 63500 . |
oD Re is ae ... |54250 155260 {57240 | 67000
30 55800 |56700 |59130 | 72400
Evaporation and Dissociation.
Tasxe VI.
Read and Smoothed Values of 4, and Values of a.
449
Vol. b, read. 6, smoothed. log b. a.
Cc. CG
30 52:3 52°5 1-72016 2877
25 63°3 64:0 1:80618 4024
20 81-9 82:0 1913881 6256 |
Peck 113°85 114°5 2:05881 10990
pith 150-0 149-9 2°17580 17103
188°5 2:27531 24718
8 255°0 25271 240175 39120
7 300°0 302°0 2°48001 50970
6 368°1 3730 2°57171 68877
5 472°3 475°5 2°67715 96008
45 5d3°7 548°5 2°73918 116230
4:0 6540 638°0 2°80482 141525
3°5 7500 759°5 2°88053 176860
| 30 933°6 936°5 2°97151 229420
TaBLE VII.
Calculated Pressures on Isothermal Curves, at definite
volumes.
Temperature.
Vol. . react WOON DPN BW oy SES
0° Grete lay 2h Ot heli 20D: | OD Oe aor dst, Gao.) LOOP.
c. Cc mm. mm mm mm mm. mm. mm. mm. mm. mm.
30 11456 | 11771 ... | 13088 13981 | 14816 | 16706
25 13448 | 13832 ... | 15439 16526 | 17544 | 19848
20 16130 | 16622 os 18680 20074 | 21378 | 24330
15 20269 | 20956 .. |23830 ... | 25776 | 27597 | 31719
12 23719 | 24719 | 25783 | 27043 | 28482 29140 | 31030 | 33413 | 38810
10 26742 | 27873 | 29212 | 30796 | 32605| ... | 33433 | 35809 | 38806 | 45592
8 29731 | 31244 | 33036 | 35155 | 37576 | 37928 | 38684 | 41862 | 45872 | 54952
7 31476 | 33288 | 35433 | 37971 | 40870 | 41291 | 42197 | 46003 | 50800 | 61680
6 32973 | 35190 | 37843 | 40973 | 44553 | 45073 | 46193 | 50893 | 56820 | 70250
5 33802 | 36652 | 40032 | 44032 | 48592 | 49262 | 50682 | 56672 | 64232
45 | 33510 | 36800 | 40700 | 45310 | 50570 | 51340 | 52980 59890 | 68620
4:0 | 32645 | 36475 | 41005 | 46365 | 52495 | 53385 | 55295 | 63335 | 73485
3°5 | 30480 | 35040 | 40440 | 46820 54110 | 55170 | 57440 | 67020
3°0 | 26240 | 31860 | 38520 | 46380 | 55370 | 56680 | 59620 | 71290) .
It will be seen that the highest calculated pressure is about
73,500 millim, Andrews gives measurements at much higher
pressures ; but these are few in number and uncertain, and
the correction for the compressibility of air is moreover
unknown. Hence it was impossible to make use of them in
determining the values of 0.
On reference to Andrews’s paper (Phil. Trans. 1876, p. 435)
Phil. Mag. 8. 5. Vol. 23. No. 144. May 1887.
2H
RTE SS ee Se ee ee OAS SS ae ee
Se re i Fe I ye ES TS ME a ee Bie oe Tn as. ee
450 Drs. Ramsay and Young on
it will be seen that he compared the relation of increase of
pressure to temperature-ditference at constant volumes, and
came to a conclusion opposed to ours. This is owing to his
having made very few observations, and having accidentally
chosen those which support his statement. If the coefficients of
increase of pressure for unit rise of temperature be calculated
by means of Table V., it may be noticed that, although irre-
gular, there is no tendency towards a rise or fall of the
coefficient.
Regnault has measured the rise of pressure of gaseous car-
bonic anhydride at constant volume. He gives the results of
four experiments, none of which are available for our purpose,
inasmuch as the volumes of a gram are too large.
Reverting to the behaviour of ether, as shown on Plate VII.,
it will be seen that the curves have been drawn in the region
where measurements are impossible. These curves have all
the same general form. After rise of pressure and decrease of
volume have proceeded for some distance, the curves bend
downward, presenting the abnormal feature of decrease of
volume with fall of pressure. The pressure continues thus to
fall, and at 160° the isothermal touches zero-pressure. At
lower temperatures, with small volumes, the pressure becomes
negative, and may even represent an enormous tension. At
0° the isothermal at vol. 1°85 cub. centim. per gram reaches
the almost incredibly great tension of —271,700 millim. ;
and it has at that volume (the smallest our results allow us
to calculate) by no means reached its limit. At still smaller
volumes the tension would doubtless still increase, until the
curve turned, and further decrease of volume would be repre-
sented, as it is at higher temperatures, by increase of pressure.
The existence of these unrealizable portions of such iso-
thermal curves was, we believe, first suggested by Prof. James
Thomson, in a paper in the ‘ Proceedings of the Belfast
Natural History and Philosophical Society,’ Nov. 29, 1871.
Since that time attempts have been made to express relations
between the pressure, temperature, and volume of gases and
liquids by Van der Waals and by Clausius ; and the formulze
which they propose, and which we hope to consider in a sub-
sequent paper, give isothermals of similar form. Portions of
these curves have, indeed, been experimentally verified. In
Professor Thomson’s paper, above referred to, he points out
that Donny, Dufour, and others have observed the phenomenon |
commonly alluded to as ‘boiling with bumping.” This is
usually the effect of a rise of temperature at constant pressure:
But it may equally well be produced, as we have frequently
had occasion to remark, at constant temperature by lowering
Evaporation and Dissociation. 451
pressure. If the diagram on plate ill. in our memoir on
‘alcohol (Phil. Trans. 1886, part i. p. 156) be referred to, it
will be seen that our actual measurement of such reduced
pressure was made on the isothermal 181°°4. Mr. John Aitken,
in an extended series of experiments on this subject (Trans.
Royal Scott. Soc. of Arts, vol. ix.), has shown that such
“ superheating”’ can take place only in absence of a free
surface, 7. e. the existence of gaseous nuclei in the liquid, into
which evaporation may take place. And Mr. Aitken has also
shown that a gas may be compressed to a volume smaller
than that at which liquefaction usually occurs, at any given
temperature, without formation of liquid. The space, again,
if no nuclei be present on which condensation may take place,
remains “ supersaturated with vapour.’ It is evidently, there-
fore, only the instability of such conditions which prevents
their complete realization*.
The formule: of Clausius and Van der Waals are based on
the assumption that two causes are in operation—those
referred to in the beginning of this paper—viz. the actual
size of the molecules, and their mutual attraction. It is
possible, by help of these assumptions, to realize the nature of
the continuous change from the gaseous to the liquid state of
matter. When a gas at a given temperature is reduced in
volume its molecules necessarily approach each other, and
their attraction for one another increases. This attraction
aids the increase of pressure in reducing volume. When a
certain volume is reached, the attraction has become so marked
that further reduction of volume is accompanied by fall of
pressure. If a certain volume be chosen on the descending
portion of an isothermal, a state of balance may be imagined
where pressure and cohesion unite in maintaining the volume
constant against the kinetic energy of the molecules, tending
to cause expansion.
The conception of negative pressure, or tension, is that at
low temperatures and small volumes the cohesion is such that,
‘In order to overcome ‘it and increase volume, it would be
necessary to apply tension to each molecule. But after the
lowest pressure or greatest tension has been attained, the
actual size of the molecules presents a bar to closer approach;
and to cause further decrease of volume pressure must again
* The reasoning of a recent paper by Wroblewski (Monatsheft der
Chemie, Wien, July 1885, p. 383) rests on the assumption that such con-
ditions are inconceivable. He supposes lines of equal density to be curves,
and on their close approach to the vapour-pressure curve to run parallel
with it. His conclusions are therefore not borne out by experimental
facts.
20a ge
452 Drs. Ramsay and Young on
be applied. It is not to be supposed that at any given volume
only one of these factors is operative ; the actual size of the
molecules exerts its influence even at large volumes, and the
cohesion does not disappear, but no doubt immensely increases,
as the volume is reduced, even when that reduction requires
rise of pressure Still, a mental picture of the process may,
we think, best be attained by directing attention to cohesion,
when volume is being decreased with fall of pressure, and to
the influence of the actual size of the molecules when volume
is small.
When a liquid is converted into gas, heat is absorbed, or
work is done on the liquid. We have previously (loc. cit.)
given tables showing the heats of vaporization of ether at
various temperatures. Our experiments have confirmed the
prediction that the heat of vaporization of stable liquids
decreases with rise of temperature, and in all probability
becomes zero at the critical temperature. Now the volume of
a fluid may be changed, either keeping the pressure constant
or allowing it to vary during the operation ; but if the initial
pressure and final pressure are the same, the variation of
pressure during the operation does not affect the total work
done. A liquid may be changed into saturated vapour at any
given temperature in the usual manner, when the intermediate
states are represented by non-homogeneous mixtures of liquid
and saturated vapour. ‘The area enclosed between the vapour-
pressure line and lines drawn vertically from its terminal points,
cut by the line of zero pressure (or pressure xchange of
volume), represents graphically the external work performed
in evaporating a liquid. If, however, the change of condition
be not abrupt, but continuous, the area enclosed by the iso-
thermal below the vapour-pressure line must be equal to that
above the vapour-pressure line (see Plate VII.). If this were
not the case, the amount of work required to effect the con-
tinuous change would differ from that required for the abrupt
change of state.
Now it is evident that a slight alteration in the position of
the vapour-pressure line would have great influence on the
relative areas enclosed by the isothermal above and below the
vapour-pressure line ; and it may also be seen that, when these
areas are rendered equal by a horizontal line, the position of
that horizontal line must represent the true vapour-pressure.
We have determined the position of the horizontal line in
the following manner :—
Knowing approximately the position of the vapour-pressure
line at a given temperature, three pressures were chosen—the
highest above and the lowest below the experimentally deter-
Evaporation and Dissociation. 453
mined vapour-pressure ; and by means of a planimeter (by
Stanley of Holborn) the areas enclosed between each hori-
zontal line and the curves respectively below and above it,
formed by the isothermal lines, were measured. To ascertain
what position of the horizontal line would render these areas
equal, the values of each set of three areas were mapped on
sectional paper as abscissz, the pressures corresponding to the
position of the horizontal lines being ordinates. The curves
passing through the resulting points cut each other at a point
which represents the true pressure and the true area. This
method is rendered clearer by inspection of the following
figure :—
20,400
Pf ime ST) ON Po ft
pf Nf euuged
HA NE ff
LN Bees ene Des a eae
TN ERE SERRE ARE RE SRSA
at aS J eee ef
See ERE
It is evident that these vapour-pressures ey Lae on
measurements represented in the diagram outside the area
bounded by the curve representing orthobasic volumes of gas
and liquid. It will be seen, on reference to the table on p. 444,
that the agreement between these calculated vapour-pressures
and those experimentally determined is a very close one, the
greatest difference being about 1 per cent. This agreement
between experimentally observed vapour-pressures and those
depending on the formula p=bt—a is very remarkable, and
it is difficult to believe that, if the isochoric lines were curves,
such an agreement could exist.
What are usually termed vapour-pressures, then, are those
pressures at which horizontal lines drawn through them render
the areas enclosed by the isothermal lines below the horizontal
lines equal to those above them. But there are other two
conditions of matter, each of which has its characteristic
pressures. One of these is represented by the highest pressure
attainable on any isothermal, or the summit of the curve above
the vapour-pressure line ; and the other the apex of the curve
below the vapour-pressure line. Hach temperature chosen has
454 Drs. Ramsay and Young on
its particular value for each of these conditions ; and it is
evident that the relations between the temperatures and
pressures corresponding to the inferior or reversed apices, as
well as those corresponding to the superior apices, would each
form a special curve.
The following Table gives the final results of the calculation
of vapour-pressures by the method of areas; and, for the
sake of comparison, the actually found vapour-pressures are
appended. ‘The pressures at the superior and inferior apices
of the isothermal curves, and also the enclosed areas, are
given”.
TABLE VIII.
Vapour- Vapour- P ep ese e ee Area
Tempe-| pressures, pressures, motes. sa are, | above or below
rature. deduced mean of superior | mrerior |yapour-pressure
from areas. observed. aie a line.
- millim. millim. | millim. millim. sq. in.
192 26350 26331 26490 26125 0-0425
190 25554 25513 25870 24960 0°1245
Old, 23623 r ; Bae
185 | 23708 {| Noy, d3yeg }| 24510 | 21660 04550
i a Old, 20189 ) ‘
ws | 20259 {| Nov Snort t| 22100 | 14060 1-6520
160 15900 15778 19090 |— 20 4-710
150 13405 13262 17380 | —10400 7551
The three pressure-curves—which we shall name the “ ordi-
nary’ vapour-pressure curve, the “ superior’’ vapour-pressure
curve, formed by the superior apices of the isothermal lines,
and the “ inferior’? vapour-pressure curve, produced by
the lower apices of the isothermal lines—must, it is evident,
meet at the critical point; and on mapping them, it was
found that this was the case. Points were chosen on these
curves at equal intervals of temperature, and the constants
for formule of the type logpy=a+ba‘ were calculated for
each. As the pressures on the inferior curve below a certain
temperature were negative, it was found convenient to add
30,000 millim. to each, which was subsequently subtracted
from the result. The constants for the curves are—
* The areas are in square inches; the scale was 2000 millim. and
2 cub. centim. per gram to the inch. It would be easy, if necessary, to
conyert these data into actual work.
ee
Evaporation and Dissociation. 455
Superior curve : .
a=3°59797 ; log b=1°8343195 ; log «a =0:00257762.
Ordinary vapour-pressure :
a=6°72909 ; log b=0°4027232 ; log «2 =1:99876897
(b is here negative).
Inferior curve :
a=4:867404 ; log b=1:5913793 ; log a=1-98382413
(b is again negative).
In each case t=%° Cent. —160°.
The results are given in the following Table :—
TABLE IX.
Ordinary Superior Inferior
Tempe- vapour-pressures. curye-pressures. curve-pressures.
rature. | Ss
Read. Calculated.| Read. Calculated. | Read. Calculated.
millim. millim. millim. millim. millim. millim.
150 13405 13084 17380 17437 | —10400 —10185
160 15900 15900 19090 19090 — 20 — 100
Lis 20259 20259 22100 22100 +14060 +14060
185 23703 23678 24510 24549 21660 21704
190 25554 25556 25870 25935 24960 24900
192 26350 26341 26490 26523 26125 26065
ia! ZOO, WEN Mace 2OSZH) Se 1) 8a 26624
13331 DAO Dikaale py d Se seSe PAROLE rata sks 27077
With exception of the lowest temperature, the agree-
ment between the read and the calculated pressures is close.
The extrapolation amounts to only 3°°83. The agreement
is close at 192°, and above that temperature the extrapolation
is only 1°83. It will be seen that at that temperature
(193°'83) the pressures coincide. The apparent critical point
was 193°8,
Isochoric Lines.
Plate IX. represents the whole of the isochors which we
have calculated between the volumes of 1°85 and ,300 cub.
centim. per gram.
If the gas followed Boyle’s and Gay-Lussac’s laws abso-
lutely, under all conditions, the isochoric lines would all radiate
from zero pressure, and would become more and more vertical
as the volumes decreased ; and the tangents of the angles
formed by these lines with the horizontal line of zero pressure
456 Drs. Ramsay and Young on
would be proportional to cin the equation p=ct, where c varies
inversely as the volume. But our equation, p=bt—a, intro-
duces another term, a, which is negative. These values of a
are represented on the diagram by the extremities of the
isochoric lines, where they cut the vertical line representing
absolute zero of temperature. The tangents of the angles
made by these lines with a horizontal line are proportional to
the values of b in our equation.
On referring to Plate IX. it will be noticed that, beginning
at the largest volume, two adjacent isochors cut each other at
a point, as regards pressure and temperature, not far above
zero. With decreasing volumes the points of intersection of
adjacent isochors occur at higher and rapidly increasing tem-
peratures and slowly increasing pressures ; and this proceeds
until the critical volume is reached. With still smaller volumes,
however, the points of intersection of adjacent isochors oceur
at lower and decreasing temperatures and pressures ; the ~
former decrease slowly, but the latter with great rapidity, and
soon extend into the region of negative pressures.
It is evident from the diagram that each isochor between
the largest and the critical volume is the tangent of a curve,
representing the relations of pressure to temperature ; while
the isochors below the critical volume are tangents to another
curve, also exhibiting the like relations. Neither of these
curves is identical with the vapour-pressure curve, which falls
in the area between them. | ais
Tt will be noticed that, in the area included between the
line of zero pressure and these two curves, each isochoric line
is cut by two others at every point along its whole length;
but outside this surface, and above the line of zero pressure,
no two lines cut each other, and below the line of zero pressure
each isochor is cut at each point by one other. The physical
meaning of the fact that within the first-mentioned region
three isochors intersect each other at one point is, that a gram
of the substance may occupy three different volumes at the
same temperature and pressure. Now, on referring to the
diagram on Plate VII., representing the experimentally un-
realizable portions of the isothermal curves, it is evident that
on each isothermal line, at pressures limited by the superior
or inferior apices of the isothermal, there are, corresponding
to each pressure, three volumes. At any pressure above or
below these pressures the isothermal line is cut only once, by a
horizontal line of equal pressure ; so that, for each pressure,
there is only one corresponding volume. At each apex a
horizontal line of equal pressure cuts the isothermal line
Evaporation and Dissociation. 457
at one point, and is also a tangent to the apex. There are,
therefore, two volumes corresponding to each of these pres-
sures. Since no gas can be submitted to a negative pressure,
those portions of an isothermal line representing the truly
gaseous condition of matter never extend below the horizontal
line of zero pressure ; only those portions of the isothermal
which proceed towards the inferior apex fall below this line.
An isothermal line below zero pressure is therefore cut only
twice by a line of equal pressure, and there are therefore two
volumes corresponding to each pressure. At each inferior
apex, however, the horizontal line is a tangent to the curve,
and there is therefore only one volume corresponding to a
given pressure.
On referring back to Plate IX., it will be seen that the
pressures corresponding to the superior apices of each iso-
thermal line, when mapped, produce the curve AC; and
those corresponding to the inferior apices, the curve BC.
The surface bounded by these curves and the line of zero
pressure corresponds to portions of the isothermal lines,
including pressures between the two apices, and each point in
the surface is the locus of intersection of three isochoric lines.
Below the line of zero pressure the isochoric lines cor-
responding to the gaseous state are absent ; and hence each
point is the locus of intersection of only two isochors. The
isothermal lines above and below the limits of pressure given
by the apices are cut only once by any line of equal pressure;
hence the isochors outside the area ACD, and above the line
of zero pressure, do not intersect. The apex C of the curvi-
lateral triangle ACD is the point of highest temperature and
pressure at which intersection can take place, and therefore
represents the critical point ; it is also the common point of
intersection of the three pressure-temperature curves.
Referring now to Plate VIII., in which the isochoric lines
in the neighbourhood of the critical point are shown on a
larger scale, it will be seen that the isochoric lines above
a volume not far removed from 4 cub. centim. per gram cut
the ordinary vapour-pressure curve CH on one side, while
those below the volume 3°75 evidently cut the vapour-pressure
line on its other side. There must therefore be an isochoric
line which does not cut the curve at all, but forms a tangent
to its end-point. That isochor gives the critical volume. It
may be determined by calculating the value of s at the
critical temperature. This value of 2 is identical with
458 On Evaporation and Dissociation.
the value of } in our equation p=bt—a at the critical volume.
Until a mathematical expression is discovered, representing b
as a function of volume, the only means at our disposal for
ascertaining the true volume corresponding to 0 is by inter-
polation of the original curve by which the values of 6 were
smoothed. The common point of intersection of the three
pressure-temperature curves has been shown on p. 450 to lie
at the temperature 193°°83. The value of a on the vapour-
pressure curve at this temperature, calculated by the formula
of which the constants have already been given, is 405 millim.,
which is also the value of } at that temperature. The volume
corresponding to this value is 4°06 cub. centim. per gram ;
and the specific gravity of ether at its critical point is there-
fore 0:2463. |
Unfortunately, Dr. Andrews’s measurements of the constants
of carbon dioxide are not sufficiently numerous to warrant an
attempt to obtain the critical temperature, pressure, and volume
by this method. The critical volume of carbon dioxide is
evidently less than 3 cub. centim. per gram ; but the values
of b below that quantity are unascertainable. It may be
noticed that the curves below volume 3 are inserted in broken
lines, showing a probable course ; but no reading from them
would be permissible.
The two liquids, ether and carbon dioxide, have no chemical
analogy with one another ; and we therefore feel justified in
concluding that the law which is the subject of this paper is
generally applicable to all stable substances. We have, how-
ever, other less complete data for methyl and ethyl alcohols,
which, so far as they go, are confirmatory of the results
described. We have also data available for the examination
of acetic acid—a substance which differs from those men-
tioned, inasmuch as it undergoes dissociation when heated ;
and we hope shortly to be able to communicate an account of
its behaviour. |
Professor Fitzgerald, to whom we gave a short account of
this law, has recently communicated to the Royal Society a
paper in which its thermodynamical bearings are considered.
Bristol, 12th February, 1887. |
[ 459 J
LII. On the Stability of Steady and of Periodic* Fluid Motion.
By Sir Witi1am THomson ft.
1. FAXHE fluid will be taken as incompressible; but the
results will generally be applicable to the motion of
natural liquids and of air or other gases when the velocity is
everywhere small in comparison with the velocity of sound in
the particular fluid considered. I shall first suppose the fluid
to be inviscid. The results obtained on this supposition will
help in an investigation of effects of viscosity which will follow.
2. I shall suppose the fluid completely enclosed in a con-
taining vessel, which may be either rigid, or plastic so that
we may at pleasure mould it to any shape, or of naturai solid
material and therefore viscously elastic (that is to say, return-
ing always to the same shape and size when time is allowed,
but resisting all deformations with a force depending on the
speed of the change, superimposed upon a force of quasi-
perfect elasticity). The whole mass of containing-vessel and
* By steady motion of a system (whether a set of material points, or a
rigid body, or a fluid mass, or a set of solids, or portions of fluid, or a
system composed of a set of solids or portions of fluid, or of portions of
solid and fluid), I mean motion which at any and every time is precisely
similar to what itis at one time. By periodic motion I mean motion
which is perfectly similar, at all instants of time differing by a certain
interval called the period.
Example 1. Every possible adynamic motion of a free rigid body,
having two of its principal moments of inertia equal, is steady. So also
is that of a solid of revolution filled with irrotational inviscid incompres-
sible fluid.
Example 2. The adynamic motion of a solid of revolution filled with
homogeneously rotating inviscid incompressible fluid is essentially periodic,
and is steady only in particular cases.
Example 3. The adynamic motion of a free rigid body with three un-
equal principal moments of inertia is essentially periodic, and is only
steady in the particular case of rotation round one or other of the three
principal axes; so also, and according to the same law, is the motion ofa
rigid body having a hollow or hollows filled with irrotational inviscid
incompressible fluid, with the three virtual moments of inertia unequal.
Example 4. The adynamic motion of a hollow rigid body filled with
rotationally moving fluid is essentially unsteady and non-periodic, except
in particular cases. Even in the case of an ellipsoidal hollow and homo-
geneous molecular rotation the motion is non-periodic. The motion,
whether rotational or irrotational, of fluid in an ellipsoidal hollow is fully
investigated in a paper under this title published in the Proceedings of
the Royal Society of Edinburgh for December 7, 1885. Among other
results it was proved that the rotation, if initially given homogeneous,
remains homogeneous, provided the figure of the hollow be never at any
time deformed from being exactly ellipsoidal.
+ Communicated by the Author, having been read before the Royal
Society of Edinburgh on April 18, 1887.
:
460 Sir William Thomson on the Stability of
fluid will sometimes be considered as absolutely free in space
undisturbed by gravity or other force; and sometimes we
shall suppose it to be held absolutely fixed. But more fre-
quently we may suppose it to be held by solid supports of
real, and therefore viscously elastic, material ; so that it will
be fixed only in the same sense as a real three-legged table
resting on the ground is fixed. The fundamental philoso-
phic question, What is fixity? is of paramount importance
in our present subject. Directional fixedness is explained in
Thomson and Tait’s ‘ Natural Philosophy,’ 2nd edition, Part I.
§ 249, and more fully discussed by Prof. James Thomson in
a paper “On the Law of Inertia, the Principle of Chrono-
metry, and the Principle of Absolute Clinural Rest and of
Absolute Rotation.”’ For our present purpose we shall cut
the matter short by assuming our platform, the earth or the
floor of our room, to be absolutely fixed in space.
3. The object of the present communication, so far as it
relates to inviscid fluid, is to prove and to illustrate the proof
of the three following propositions regarding a mass of fluid
given with any rotation in any part of it :—
(I.) The energy of the whole motion may be infinitely in-
creased by doing work in a certain systematic manner on the
containing-vessel and bringing it ultimately to rest.
(II.) If the containing-vessel be simply continuous and be
of natural viscously elastic material, the fluid given moving
within it will come of itself to rest.
(III.) If the containing-vessel be complexly continuous and
be of natural viscously elastic material, the fluid will lose
energy ; not to zero, however, but to a determinate condition
of irrotational circulation with a determinate cyclic constant
for each circuit through it.
4. To prove 3 (I.) remark, first, that mere distortion of the
fluid, by changing the shape of the boundary, can increase
the kinetic energy indefinitely. For simplicity, suppose a
finite or an infinitely great change of shape of the containing-
vessel to be made in an infinitely short time; this will distort
the internal fluid precisely as it would have done if the fluid
had been given at rest, and thus, by Helmholtz’s laws of vor-
tex motion, we can calculate, from the initial state of motion
supposed known, the molecular rotation of every part of the
fluid, after the change. For example, let the shape of the
containing-vessel be altered by homogeneous strain ; that is
to say, dilated uniformly in one, or in each of two, directions,
and contracted uniformly in the other direction or directions,
of three at right angles to one another. The liquid will be
homogeneously deformed throughout ; the axis of molecular
Steady und of Periodic Fluid Motion. 461
rotation in each part will change in direction so as to keep
along the changing direction of the same line of fluid par-
ticles ; and its magnitude will change in inverse simple pro-
portion to the distance between two particles in the line of the
axis.
5. But, now, to simplify subsequent operations to the utmost,
suppose that anyhow, by quick motion or by slow motion, the
containing-vessel be changed to a circular cylinder with per-
forated diaphragm and two pistons, as shown in fig. 1. In
the present circumstances the motion of the liquid may be
supposed to have any degree of complexity of molecular rota-
tion throughout. It might chance to have no moment of
momentum round the axis of the cylinder, but we shall sup-
pose this not to be the case. If it did chance to be the case
(which could be discovered by external tests), a motion of the
cylinder, round a diameter, to a fresh position of rest would
leave it with moment of momentum of the internal fluid round
the axis of the cylinder. Without further preface, however,
we shall suppose the cylinder to be given, with the pistons as
in fig. 1, containing fluid in an exceedingly irregular state of
motion, but with a given moment of momentum M round the
axis of the cylinder. The cylinder itself is to be held absolutely
fixed, and therefore whatever we do to the pistons we cannot
alter the whole moment of momentum of the fluid round the
axis of the cylinder.
6. Suppose, now, the piston A to be temporarily fixed in
its middle position CC, and the
whole containing-vessel of cylinder
and pistons to be mounted on a
frictionless pivot, soas to be free to
turn round A A’ the axis of the
cylinder. If the vessel be of ideally
rigid material, and if its inner sur-
face be an exact figure of revolu-
tion, it will, though left free to turn,
remain at rest, because the pressure
of the fluid on it is everywhere in
plane with the axis. But now, in-
stead of being ideally rigid, let the
vessel be of natural viscous-elastic
solid material. The unsteadiness
of the internal fluid motion will cause deformations of the
containing-solid with loss of energy, and the result finally
approximated to more and more nearly as time advances is
necessarily the one determinate condition of minimum energy
with the given moment of momentum; which, as is well
Figs 2,
eno 3 aes
462 Sir William Thomson on the Stability of
known and easily proved, is the condition of solid and fluid
rotating with equal angular velocity. If the stiffness of the
containing-vessel be small enough and its viscosity great
enough, it is easily seen that this final condition will be
closely approximated to in a very moderate number of times
the period of rotation in the final condition. Still we must
wait an infinite time before we can find a perfect approxima-
tion to this condition reached from our highly complex or
irregular initial motion. We shall now, therefore, cut the
affair short by simply supposing the fluid to be given rotating
with uniform angular velocity, like a solid within the con-
taining-vessel, a true figure of revolution, which we shall now
again consider as absolutely rigid, and consisting of cylinder
with perforated diaphragm and two movable pistons, as repre-
sented in fig. 1. :
7. Give A a sudden pull or push and leave it to itself;
it will move a short distance in the direction of the impulse
and then spring back*. Keep alternately pulling and push-
* The subject of this statement receives an interesting experimental
illustration in the following passage, extracted from the Proceedings of the
Fig. 2.
Royal Institution of Great Britain for March 4, 1881; being an abstract
of a Friday-evening discourse on “ Elasticity viewed as possibly a Mode of
Steady and of Periodic Fluid Motion. 463
ing it always in the direction of its motion. It will not
thus be brought into a state of increasing oscillation, but the
work done upon it will be spent in augmenting the energy of
the fluid motion: so that if, after a great number of to-and-
fro motions of the piston with some work done on it during
each of them, the piston is once more brought to rest, the
energy of the fluid motion will be greater than in the begin-
ning, when it was rotating homogeneously like a solid. It
has still exactly the same moment of momentum and the same
vorticity* in every part; and the motion is symmetrical
round the axis of the cylinder. Hence it is easily seen that
the greater energy implies the axial region of the fluid being
stretched axially, and so acquiring angular velocity greater
than the original angular velocity of the whole fluid mass.
8. The accompanying diagram (fig. 3) represents an easily
performed experimental illustration, in which rotating water
is churned by quick up-and-down movement of a disk carried
on a vertical rod guided to move along the axis of the con-
taining-vessel which is attached to a rotating vertical shaft.
The kind of churning motion thus produced is very different
from that produced by the perforated diaphragm ; but the
ultimate result is so far similar, that the statement of § 7 is
equally applicable to the two cases. In the experiment, a
little air is left under the cork, in the neck of the containing-
vessel, to allow something to be seen of the motions of the
water. When the vessel has been kept rotating steadily for
some time with the churn-disk resting on the bottom, the sur-
face of the water is seen in the paraboloidal form indicated
(ideally) by the upper dotted curve (but of course greatly
distorted by the refraction of the glass). Now, by finger and
thumb applied to the top of the rod, move smartly up and down
several times the churn-disk. A hollow vortex (or column of
Motion,” and now in the press for republication along with other lectures
and addresses in a volume of the ‘ Nature Series.’ “A little wooden
ball, which when thrust down under still water jumped up again ina
moment, remained down as if imbedded in jelly when the water was caused
to rotate rapidly, and sprang back as if the water had elasticity like that
of jelly when it was struck by a stiff wire pushed down through the
a of the cork by which the glass vessel containing the water was
ed’
* The vorticity of an infinitesimal volume dv of fluid is the value of
dv. w/e, where w is its molecular rotation, and e the ratio of the distance
between two of its particles in the axis of rotation at the time considered,
to the distance between the same two particles at a particular time of
reference. The amount of the vorticity thus defined for any part of a
moving fluid depends on the time of reference chosen. Helmholtz’s fun-
damental theorem of vortex motion proves it to be constant throughout
all time for every small portion of an inviscid fluid.
464 Stability of Steady and of Periodic Fluid Motion.
air bounded by water), ending irregularly a little above the disk,
1s seen to dart down from the neck of the vessel. If, now, the
Fig. 3.
a - CS,
b
4
3
3
y
QW
churn-disk is held at rest in any position, the ragged lower
end of the air-tube becomes rounded and drawn up, the free
surface of the water taking a succession of shapes, like that
indicated by the lower dotted curve, until after a few seconds
(or about a quarter of a minute) it becomes steady in the
paraboloidal shape indicated by the upper dotted curve.
9. We have supposed the piston brought to rest after having
done work upon the fluid during a vast but finite number of
to-and-fro motions. But if left to itself it will not remain at
rest ; it will get into a state of irregular oscillation, due to
superposition of oscillations of the fluid according to an infi-
nite number of fundamental modes, of the kind investigated
in my article “ Vibrations of a Columnar Vortex,” Proc. Roy.
Soc. Hdinb., March 1, 1880, but not, as there, limited to being
infinitesimal! If the motion of the piston be viscously resisted
these vibrations will be gradually calmed down ; and if time
enough is allowed, the whole energy that has been imparted
to the liquid by the work done on the pistons will be lost, and
it will again be rotating uniformly like a solid, as it was in
the beginning.
[To be continued. ]
, 465 J
LIU. Notices respecting New Books.
A Treatise on Algebra. By Profs. OLIVER, Wart, and Jones.
(Ithaca, N. Y.: Dudley Finch, 1887; pp. viii+412.)
HIS is not an Elementary Textbook, and so is not a work for
ordinary school-use. It is a work very much of the same
high character as that by Prof. Chrystal which we had occasion
lately to notice in these columns, and, like it, this also is only a first
volume. With points of similarity there are numerous points of
dissimilarity. The motto of both is “Thorough.” Our present
Authors—an unusual combination, a triple chord—‘‘assume no
previous knowledge of Algebra, but lay down the primary definitions
and axioms, and, building on these, develop the elementary principles
in logical order; add such simple illustrations as shall make
familiar these principles and their uses.” Then as to form:
“ Make clear and precise definition of every word and symbol used
in a technical sense; make formal statement of every general
principle, and, if not an axiom, prove it rigorously; make formal
statement of every general problem, and give a rule for its solution,
with reasons, examples, and checks; add such notes as shall
indicate motives, point out best arrangements, make clear special
cases, and suggest extensions and new uses.” It will be gathered
from this outlme, and our Authors, we think, have kept close to
this chart, that here is about the same departure from ordinary
textbooks as in the case we have referred to above. Indeed, to
our mind we have almost too much logic and careful detail, but for
college students and mathematical teachers this elaboration is of
great service. Indeed the book has been written for the classes
which have been and are under the authors’ training. They them-
selves admit that the Work has so grown under their hands as to
embrace many topics quite beyond the range of ordinary college
instruction. The book fulfils their desire that it should be a
stepping-stone to the higher analysis. Having indicated the
nature of the work we give now some of the matters discussed in
the twelve chapters. ‘The first is on primary definitions and signs ;
the second is on primary operations (a valuable chapter); the third
on Measures, Multiples, and Factors ; the fourth on Permutations
and Combinations ; the fifth on Powers and Roots of Polynomials ;
the sixth on Continued Fractions; the seventh on Incom-
mensurables, Limits, Infinitesimals, and Derivatives ; the eighth
on Powers and Roots; the ninth on Logarithms; the tenth on
Imaginaries (with graphic representation and preparation for
Quaternions); the eleventh on Equations (Bezout’s method,
graphic representation of quadratic equations, application of
continued fractions to the same class of equations, maxima and
minima); and the last on Series (the elementary ones, convergence
and divergence, indeterminate coefficients, finite differences, inter-
polation, Taylor’s theorem, and the computation of logarithms).
We have come across much that is new to us and much of interest.
Phil. Mag. 8. 5. Vol. 23, No. 144. May 1887. 21
“ > ' See 23 7 es SAID Ne Pm aoe & 2
5 ETE RE A RE A
466 Geological Society :—
The work requires rather close reading in parts, and the arrange-
ment of the text, too crowded, militates in our opinion against an
enjoyable perusal of the text. But our view on these points must
go for what itis worth. The appearance of the work externally
and the type and apparently great accuracy in printing are all Al.
In an extra volume the Authors promise to treat of theory of
equations, integer analysis, symbolic methods, determinants and
groups, probabilities, and insurance, with a full index. Examples
accompany the text and conclude each chapter.
LIV. Proceedings of Learned Societies.
GEOLOGICAL SOCIETY.
[Continued from p. 222.]
January 26, 1887.—Prof. J. W. Judd, F.R.S., President,
in the Chair.
pee following communications were read :—
1. “On the Correlation of the Upper Jurassic Recs of the Jura
with those of England.” By Thomas Roberts, Esq., M.A., F.G.S.
The author described at length his observations on the rocks of
the Jurassic system, from the Callovian to the Purbeckian inclusive,
first in the Canton of Berne and then in the more southerly Cantons
of Neuchatel and Vaud. The sections in the former differed ma-
terially from those in the latter, and the following stages and sub-
stages were observed :—
Nort District. Souru District.
Purbeckian. Purbeckian.
Portlandian. Portlandian.
Virgulian.
Pterocerian. Pterocerian.
Astartian.
Astartian.
Calcaire a Nérinées. ——
Corallian. Oolithe Corallienne. ee
Terrain 4 chailles siliceux. Corallian.
Oxford Terrain 4 chailles marno-caleaire. Pholadomian. Oxford;
xoraian-*| Calcaire 4 Seyphies inférieur. Spongitian. On re
Callovj Le fer sous-Oxfordien. Supérieur. Gales,
aNovian. | Zone of Amm. macrocephalus. Inférieur. a
Dalle naerée, &c. Dalle nacrée.
Bathonian. {
Some of the lithological and paleontological differences between
these rocks and the English Oolites were noticed, and the views of
Oppel, Marcou, Waagen, Blake, and Renevier, as to the relations of
the beds in the two countries, were commented upon. The Author
then proceeded to compare the fossils of the Swiss Jurassic beds
with those of their English representatives, stage by ane and
finally suggested the following correlation :—
Upper Jurassic Rocks of the Jura and England.
ENGLAND.
Upper.
PURBECK: Middle.
Lower.
Portland stone.
sand, &e.
72
Upper Kimeridge Clay.
Clays with Exogyra virgula.
re 3 Ammonites alternans.
LOwER
KIMERIDGE.
Clays with Astarte supracorallina.
“ Ostrea deltoidea.
Kimeridge Passage-beds.
9?
467
SWISS JURA.
| Valangien.
Purbeckien.
|
|
|
}
Porilandien.
Ptérocérien.
Astartien.
{ Supracoralline.
Coral Rag.
| Calcaire & Nérinées.
s
Oolithe Corallienne.
Coralline Oolite.
CoRALLIAN. { Terrain 4 chailles siliceux. |
Middle Caleareous Grit.
aa SS
Hambleton Oolite. Ee Se eee aa Pholadomien. } ben Cee
\ Lower Calcareous Grit. SELES
eee |
Clays with cordatt Ammonites. | Le fer sous-Oxfordien.
ornati Ammonites.
Ee j
at i |
iZone of Amm. macroce-
| phalus.
33 39
\ Kelloway Rock.
Cornbrash. | Bathonien.
2. “The Physical History of the Bagshot Beds of the London
Basin.” By the Rev. A. Irving, B.Sc., B.A., F.G.S.
The Author, in reviewing the position taken up by him, attempted
to estimate the value of such palzontological evidence as exists, and
insisted on the importance of the physical evidence in the first place.
He gave reasons for considering the evidence of pebbles, pipe-clay,
derived materials, irony concretions, percentages of elementary
carbon (ranging in the more carbonaceous strata up to nearly 23°/.)
taken together with the evidence of carbon in combination, as ad-
duced in former papers, freshwater Diatoms (now, perhaps recorded
for the first time in the Middle and Lower Bagshot), and the micro-
scopic structure of the sands and clays, as furnishing such a cumu-
lative proof of the fluviatile and deita origin of the majority of the
Middle and Lower Bagshot Beds, as can hardly be gainsaid ; while
he regarded the wide distribution of the Sarsens, taken along with
the absence of such evidence as is quoted above, as indicating, along
with the fauna, a much greater areal range formerly of the Upper
Bagshot than of the strata below them.
‘NAITIVIOD
"NUTAOTTIVY
468 Intelligence and Miscellaneous Articles.
He referred to the evidence furnished by the Walton section (Q. J.
G. S. May, 1886), the Brookwood deep well (Geol. Mag. August,
1886), the contemporaneous denudation of the London Clay (Geol.
Mag. September, 1886) as affording further support to the view
which he has advocated; gave six new sections on the northern
side of the area, showing (1) the attenuation of the Lower Bag-
shots beneath the Middle Bagshot ciays, (2) the greater development
of clays towards the margin at the expense of the sands, (3) con-
temporaneous transverse erosion of the London Clay, (4) cases of
overlap, (5) the occurrence of massive pebble-beds at nearly the
same altitude along the northern flank underlying (as at Hast-
hampstead and Bearwood) Upper Bagshot sands, and resting either
immediately upon, or in near proximity to, the London Clay; and
added an account of his observations on the flank of St. Anne’s Hill,
Chertsey, which he takes to be nothing more than an ancient river-
valley escarpment, subsequently eroded by rain-water, the hollows
thus formed having been subsequently filled up and covered over by
pebbles and other débris of the beds in the higher part of the hill, —
these assuming the character of ordinary talus material. The con-
sideration of the southern margin of the Bagshot district is reserved
for a future paper.
The Author considered that his main position, resting as it does
upon physical evidence, remains untouched by the attempt of later
writers to disprove it; while the disproof breaks down even on
its own lines (the stratigraphical), the paper in which this dis-
proof is insisted upon being characterized by (1) an incomplete
grasp of the problem on the part of its authors, (2) equivocal data,
(3) omission of important evidence, (4) inconsistencies, (5) erro-
neous statements. 3
The Author (while correcting some errors of stratigraphical detail,
which appeared in his former paper, from insufficiency of data)
maintained that (though occasional intercalated beds with marine
fossils may be met with, as is commonly the case in a series of
delta- and lagoon-deposits) the view he has put forward is, in the
main, established ; and he proposed the following classification of
the Bagshot Beds of the London Basin :—
Old Reading. New Reading.
1. Upper Bagshot Sands =1. Marine-estuarine Series.
2. Middle Bagshot Sands
and Clays =2. Freshwater Series.
3. Lower Bagshot Sands
LV. Intelligence and Miscellaneous Articles.
ON THE INERT SPACH IN CHEMICAL REACTIONS.
BY OSCAR LIEBREICH.
A CCORDING to all previous observations, it has been assumed
that a chemical reaction in liquids which are perfectly mixed
takes place uniformly and simultaneously in all parts unless cur-
‘a i
~
Intelligence and Miscellaneous Articles. 469
rents are produced in consequence of inequalities of temperature.
In the reduction of copper sulphate by grape-sugar, on heating, the
suboxide is first perceived in the upper part. We know also that
in reducing liquids which contain certain metallic salts, the products
of reduction are deposited on the surfaces opposed to them. It has
never, however, been observed that in liquids, perfect mixture
being presupposed, certain parts are withdrawn from the reaction,
or show some retardation in the change.
J have succeeded in demonstrating the existence of a space in
mixtures in which a chemical reaction is not visible. I have called
it the ¢nert space (todter Raum). In introducing this idea as the
result of my experiments, I would define it as that space in a
uniformly mixed liquid in which the reaction occurs either not at
all, or is retarded, or takes place to a less extent than in the
principal liquid.
Reaciion-space and inert space can be most sharply separated
from each other in the experiments which Ladduce. The occurrence
of such an inert space is best demonstrated with hydrate of chloral,
which, when treated with sodium carbonate, decomposes into
chloroform, according to the following equation :
C,Cl,0,H, +Na,CO,=CHCl, + NaHCO, +NaHCo,,.
With a suitable concentration, and mixture in proper equivalents,
the chloroform separates not in thick oily drops, but as a fine mist
which gradually collects in drops at the bottom. The reaction
does not start at once, but depends on concentration and tempera-
ture. The concentration proper for the observations can be so
arranged that the commencement of the reaction varies between 1
and 25 minutes. ‘This time may even be considerably prolonged *.
If the reaction is made in an ordinary test-tube, there is a space
of 1 to 3 mm. below the meniscus, which is not affected by the
reaction ; that is, it remains perfectly clear; and the reaction-space
is bounded above with the sharpness of a hair, by a surface curved
in the opposite direction to that of the meniscus.
The upper space in the liquid which thus remains clear is the
inert space in the hydrate-of-chloral reaction.
Even after the tube has been left still for 24 hours this space is
visible ; for the boundary of the mert space can still be distinctly
recognized by minute spherules of chloroform which have not sunk.
Tf the test-tube is gently agitated, so that the chloroform-mist
passes into the inert space, after a few minutes the chloroform
settles to its former boundary, and the separation between the
inert space and the reaction-space is again reproduced.
Careful observation showed that the clear layer of liquid was
diminished by the ascent of the chloroform-mist, and was not
Increased by sinking.
I have observed the inert space in this reaction in differently
* I used equal volumes of aqueous solutions of 331 gr. hydrate of
chloral and 212 gr. sodium carbonate in the litre, which were diluted to
a corresponding extent, so as to prolong the duration of the reaction.
470 Intelligence and Miscellaneous Articles.
shaped vessels. If we take a glass box with parallel sides which
are ata distance of a centimetre apart, it is seen that the mert
space presents itself as a surface curved in the opposite direction to
the meniscus. It can moreover be observed that at the positions of
greatest curvature, a gradual equalization or a fresh reaction-zone
is formed. Ii a horizontal glass cylinder closed by parallel glass
plates is taken, the curvature of the active space is seen in great
sharpness and beauty.
If the reaction takes place between two glass plates which are
inclined to each other at an acute angle so that their line of contact
is vertical, the height of the meniscus is represented by a deeper
position of the inert space.
In capillary tubes which, after being filled, are placed horizontally,
the inert space is met with on each side. ven if the capillary
tubes are taken so fine, that the lumen must be examined by a
magnifying-power of 300 times, the active and the inert space can
be separately observed. The reaction occurs with separation of
small molecular drops of chloroform in the middle of the liquid
cylinder, while it remains clear at each end. With very small
drops in capillary tubes there is no reaction *.
If tubes closed at the top are filled with the active mixture so
that there is no air-bubble, the decomposition is uniform throughout
the entire liquid. If, however, tubes open atthe top are filled with
the liquid, and are closed by a small transparent animal membrane ~
stretched in a lead frame, it is possible by carefully raising it to
show here also the inert space.
If a glass tube open at both ends is placed on a fine membrane,
and is closed at the top also by a membrane, it is seen that when
the tube is held vertically an inactive space can be observed below,
in which the chloroform gradually settles as a cloud. I have not
been able to ascertain whether the reaction in this case is also
limited at the sides of the tube.
If a specimen of the liquid be taken from the inert space by
means of a capillary tube, and it be warmed, decomposition at once
sets in. ‘This isa proof that the two substances contain unaltered
hydrate of chloral and sodium carbonate. It is of course im-
portant to observe the phenomena of the inert space by other
reactions which take place slowly. The reaction which takes
place between iodic and sulphurous acids according to the following
equations :
380,+ H1IO,=380,+1H
51H +HI0,=3H,0+ 61
was found to be particularly suitable, since it has been found by
Landolt ? that by suitable dilution, and variation of the quantities,
it can be delayed at pleasure and in accordance with a definite law.
The occurrence of the iodine reaction is made manifest by the
* For this experiment it is necessary to free the liquid from absorbed
air by boiling. |
+ Berliner Sttzungsberichte, 1885, xvi., and 1886, x.
Intelligence and Miscellaneous Articles. 471
addition of soluble starch, which by the sudden blue coloration
indicates the liberation of iodine.
’ Solutions were used containing 0°25 gr. of iodic acid in a litre
of water, or the same quantity in the litre of a mixture of equal
parts of glycerine and water.
The sulphurous acid was used of such concentration that 5 cub.
cent. of its solution in water just decolorized 2 cub. cent. of a
one-per-cent. solution of potassium permanganate.
On mixing 10 cub. cent. of solution of iodic acid with 3 cub.
cent. of sulphurous acid, the reaction sets in in about 5 minutes, and
in the various glass vessels shows an inert space above, which lasts
for a time depending on the temperature.
The iodine reaction presents a phenomenon to which I shall
afterwards recur; that is, the occurrence of this reaction in the
centre of the tube. If a vertical glass tube 4 millim. in the clear
is filled by aspiration, and subsequent closing by an indiarubber
tube and clamp, trom the active liquid which is contained in a wide
glass cylinder, a fine blue thread is seen to form in the tube, while
the surrounding liquid remains clear and colourless. The blue
coloration extends gradually from the thread thronghout the entire
liquid column.
It could be observed in this phenomenon that the reaction in the
wider vessel set in sooner than in the narrow tube.
If either the hydrate-of-chloral or the iodic-acid mixture is placed
in a vessel in which the liquids can be drawn through fine glass
beads, no chemical reaction at all is produced.
It follows thus from these experiments :—
1. That in liquids the space of chemical action is bounded by an
inactive zone (the wert space), where the liquid is in contact with
the air, or is separated from it by a fine membrane.
2. That the reactions take place more slowly in narrow than in
wide tubes.
3. That capillary spaces can entirely suspend chemical reactions.
As lam engaged in continuing this investigation, I hope soon,
after a further extension of the experiments and the use of other
chemical reactions, to be able to report fresh results.—Berliner
Stizungsberichte, November 4, 1886.
—
APPARATUS FOR THE CONDENSATION OF SMOKE BY STATICAL
ELECTRICITY. BY H. AMAURY.
A glass cylinder is placed on a tripod perforated in the centre,
and below it a tin-plate box with an opening in the side and at
the top, in which touch-paper, tinder, or tobacco can be burned, and
thus the cylinder be filled with smoke. To the top of the cylinder
is fitted a small lid in which is a vertical tube. At half the height
of the cylinder are two diametrically opposite tubuli, through which
pass metal rods; these are connected with vertical rods parallel to
the sides and provided with points. If these combs are connected
with the conductors of an electrical machine, and the latter is
worked, the smoke is condensed.— Beiblatter der Physil:, No. 2, 1887.
FPF ee eae ee aan
Pri iecelels 48 Sa:
472 Intelligence and Miscellaneous Articles.
THE HEATING OF THE GLASS OF CONDENSERS BY INTERMITTENT
ELECTRIFICATION. BY J. BORGMANN.
The author takes two bundles of 30 cylindrical condensers, each
consisting of a glass tube 46 cm. in length and 5 mm. in diameter ;
each tube was coated externally with tinfoil, and filled with copper
filings, and a copper wire inserted, the ends being closed with
paratiin or shellac. Hach thirty tubes are formed into bundles, all
the outsides and insides being severally connected. One bundle was
also coated on the outside with tinfoil to improve the conductivity.
These two bundles of condensers were placed respectively in two
large air-thermometers. Hach reservoir consisted of a glass tube
- of about 50 cm. length and 4:5 cm. internal diameter, which was
surrounded by another tube of the same length and 7 cm. diameter.
Through the brass ends of the reservoirs passed on the one hand
the electrodes, and on the other the limb of the manometer. The
manometer filled with naphtha consisted of three limbs, of which
two were connected with the two reservoirs of the air-thermometer.
The charging was effected by means of a Kuhmkorff, and was
measured by a Siemens electrodynamometer. Notwithstanding its
better external conductivity, the bundle C was more heated than
the other, A.
If ¢ is the deflection of the electrodynamometer in divisions of
the scale, Aa and Ac the displacement of the naphtha in the mano-
meter in millimetres, which measure the quantities of heat, it was
found that
€ 345 280 147 101 je 343 159
Ac 11:3» 9:84... 4°84 2:9. f, Aa LOS aie
e/A 30° 284 82:4 348 e¢/A 317 306
It follows from this that the heatings of the condensers are
approximately proportional to the square of the difference of
potential of the coatings.—Beiblatter der Physik, 1887, p. 55.
ON THE CHEMICAL COMBINATION OF GASHS.
To the Editors of the Philosophical Magazine and Journal.
GENTLEMEN, Riga, April 8, 1887.
In the April Number of the Philosophical Magazine for this
year Prof. J. J. Thomson complains that [ have misunderstood his
theory of the Chemical Combination of Gases. After a repeated
study of the paper, I must confess that Prof. Thomson is in the
main right. As in my criticism I have done Prof. Thomson an
injustice which I am not able entirely to repair, I will not dwell
upon the injustice which he in the heat of his defence has done me
in his answer, the more so as it has no scientific, but a mere per-
sonal interest.
Have the kindness to insert the above explanation in the next
Number of your Magazine. Yours truly,
W. Ostwatp.
THE
LONDON, EDINBURGH, ann DUBLIN
PHILOSOPHICAL MAGAZINE
AND
JOURNAL OF SCIENCE.
[FIFTH SERIES.]
JUNE 1887.
LVI. The Laws of Motion. By Ropert FRaNKLIN MurrHeEaD,
B.A., of St. Catharine’s College, Cambridge*.
Preface.
HE aim of this Essay is to state in the clearest manner
possible the best eaisting conception of dynamical
science. The writer believes that the statement of dyna-
mical principles here given is to be found implicitly in
the reasonings of the best modern masters of the science, but
that it has never hitherto been stated explicitly. The general
statement indeed has sometimes been made that the proof of
a hypothesis or theory is its agreement with the facts, or that
the whole Principia is the proof of the Laws of Motion.
But I have pointed out in detail that the very conceptions
and definitions of Dynamics are unintelligible when taken
singly. I have endeavoured to free the science of Dynamics
from survivals from its childhood, in the shape of extra-
kinetic definitions of dynamical concepts, and @ priori
assumptions.
The Laws of Motion.
In view of the enormous development to which the science
of Dynamics has attained in modern times, of the simplicity
of its fundamental conceptions, and of the unquestioned
* Communicated by Professor James Thomson; being the Essay to
which the second Smith’s Prize was awarded in 1886.
Phil. Mag. 8. 5. Vol. 23. No. 145. June 1887. 2K
474 Mr. R. F. Muirhead on the Laws of Motion.
validity ot its processes and results, it may appear somewhat
strange that much difficulty has been found in stating its
principles in a satisfactory form.
,- “In the preface to the second edition of Tait and Steele's
_ ‘Dynamics of a Particle’ we read (referring to the chapter
on the Laws of Motion) :—“ These five pages, faulty and even
erroneous as I have since seen them to be, cost me almost as
much labour and thought as the utterly disproportionate
remainder of my contributions to the volume; and I cannot
but ascribe this result in part, at least, to the vicious system
of the present day, which ignores Newton’s Third Law, dc.”
And when we read Clerk Maxwell’s notice of the 2nd
edition of Thomson and Tait’s treatise in ‘ Nature,’ * we feel
that the reform introduced by Thomson and Tait, in ** return-
ing to Newton,” still leaves something to be desired. This
feeling is strengthened when we learn from the late Prof.
Clifford +, that “‘no mathematician can attach any meaning
to the language about force, mass, inertia, &c. used in current
text-books of Mechanics.”
It will then be worth while to clear up the logic of the
science, and, if possible, to state the laws of motion in a form
that shall be free from all ambiguity and confusion.
Let us cast a brief and partial glance over the history of
the development of dynamical first principles.
Though one region of the science of Dynamics, namel
Statics, was cultivated by the ancients, it was left for Galileo
Galilei to become the pioneer of dynamical science in its full
extent.
‘Before Galileo, the idea of force as something measurable
was attained to. The causes tending to disturb rest were
perceived to have a common kind of effect, so that for the
purposes of Statics they could be represented by the tension
of cords produced by suspending from them weights of
determinate magnitude. Galileo paved the way for the intro-
duction of the kinetic idea of force, %.e. that of the cause of
the acceleration of the motion of bodies. It is noteworthy,
however, that he approached the subject from a kinematical
standpoint. In his ‘ Dialogues,’ he treats of the science of
“ Local Motion,” not of the science of Force; and in his
investigations on the motion of Projectiles in that work, his
aim is to describe the properties of their motion, not to
speculate on causes.
Another stage was reached when Newton published the
* ‘Nature,’ yol. xx. p. 213, ff.
+ Ibid. vol. xxii. June 10th.
Mr. R. F. Muirhead on the Laws of Motion. A75
Principia. The Definitions and Axioms therein propounded
include all the principles underlying the modern science of
Dynamics. Subsequent progress has been either in the
direction of mathematical development or application to special
problems, or in attempts to improve the form of statement.
Let us now inquire whether Newton’s scheme of Definitions
and Axioms is satisfactory.
We are struck at once by the fact (noticed by many
writers) that the First Law of Motion is previously stated or
implied in the Definition of Inertia. This, however, may be
passed over as a mere awkwardness of arrangement.
Another defect which has been pointed out by several
writers, is the absence of any definition of equal times, which
renders the expression “ uniformiter” in Law I. perfectly
indefinite.
Of course the law implies that all bodies unacted on by
force pass through spaces in any interval of time whatever,
which are in the same proportion, so that taking any one such
body as chronometer, the First Law of Newton may be
affirmed of all the rest. We may, however, object to a form
of statement which does not directly state, but implies the
physical fact.
Again, “uniform rectilinear motion”? has no meaning
unless with reference to some base of measurement. And
the Law is not true except with reference to bases of a certain
type. [For instance, the “fixed stars describe not straight
lines, but circles, taking the Earth as base of measurement.”
Newton’s own statement is that the Laws of Motion are to
be understood with reference to absolute position and absolute
time.
The only explanation given of absolute time, is that in
itself and of its own nature, without reference to anything
else, it flows uniformly.
In explanation of the expressions “ absolute motion” and
“absolute position,” we have the statement that “ Absolute
and relative motion and rest are distinguished from one
another by their properties, causes, and effects. Itis a property
of rest that bodies truly at rest are at rest among themselves,
but true rest cannot be defined by the relative positions of
bodies we observe........ The causes by which true
and relative motion are distinguished from one another are
the forces impressed on the bodies to produce motion. ‘True
motion cannot change except by forces impressed.
“The effects by which absolute motion is distinguished
from relative are the centrifugal forces of rotation. For
2K2
a eS
—
Le
=. oh
——
476 Mr. R. F. Muirhead on the Laws of Motion.
merely relative rotation these forces are zero ; in true rotation
they exist in greater or less degree.” *
Thereafter comes the well-known experiment of the rotating
vessel of water.
Now the first criterion helps us only in a negative way, by
enabling us to deny the attribution of true rest to both of
two systems when they are moving relatively to each other.
The second criterion involves reasoning in acircle. Force is
defined as that which produces change of motion ; hence to
define unchanged or uniform motion as that which takes place
when no force acts does not carry us beyond the previous
definition, and is nugatory.
The third criterion, taken along with the first, implies a
physical fact, viz. that when two bodies severally show no
centrifugal force, they have no rotation relative to one
another.
Consider now Law II. It amounts merely to a definition
of force, specifying how it is to be measured.
This has been recognized by several writers. Some, how-
ever, have expanded it into the further assertion that when
two forces act simultaneously on a body, each produces its
own effect independently of the other, in accelerating the body’s
motion. But such a statement is entirely nugatory if we
keep by the kinetic definition of force. It is then simply an
identical proposition like “‘ A is A,” as will be seen by sub-
stituting in the statement “ acceleration of mass ” for ‘ force.”
We now perceive that even the residuum of meaning which
remained after our criticism of Law I. and the statements
regarding Absolute Motion seems to disappear. For we
were supposed to recognize a body absolutely at rest by the
absence of centrifugal force. But force is recognizable only
by its accelerative effect, while the acceleration must be
reckoned relative to a body absolutely at rest, which rest,
again, we cannot recognize until we know absolute motions.
We are thus reasoning in a circle.
Law III. This law at first sight undoubtedly seems to
express an experimental fact. We may therefore be sur-
prised to find that Newton deduces one case of it (viz. that of
two mutually attracting bodies) from Law I. (see Scholium
to the Awiomata).
This seeming paradox arises from the fact that in this
Scholium Newton makes Law I. apply to a body or system
of finite size, and not necessarily without rotation. This
assumes that there is some one point (centre of Inertia)
* Newton’s Principia, Scholium to the Definttiones.
Mr. R. F. Muirhead on the Laws of Motion. 477
whose motion may be taken to represent that of the system,
which implies that the 3rd Law is true so far as the parts of
such a system are concerned. Now it seems difficult to draw
a valid distinction between such a system and any mass-
system whatever ; in fact it seems quite as legitimate to
assume that every mass-system has a centre of Inertia.
But if this assumption were made, then clearly the first
Law could be deduced from the third in all its generality,
and vice versa.
We see that in this respect again Newton’s arrangement.
is defective. We find that the experimental fact is not
stated directly, but enplied in the assumption of the existence
of a mass-centre. In fact, strictly read, Newton’s Definitions
and Axioms abound in logical circles, nugatory statements,
and illusory definitions ; and what real meaning they imply
is not at all explicit.
The need for the removal of many obscurities which pertain
to the science of Dynamics as set forth in the Principia of
Newton, and in the writings of his successors, has been clearly
perceived by Professor James Thomson. In his paper on the
“ Law of Inertia, &c.,” * he propounds the following Law of
Inertia :—
“For any set of bodies acted on each by any force, a
Reference-Frame and a Reference Dial-traveller are kine-
matically possibie, such that relatively to them conjointly the
motion of the mass-centre of each body undergoes change
simultaneously with any infinitely short element of the dial-
traveller progress, or with any element during which the
force on the body does not alter in direction nor in magnitude,
which change is proportional to the intensity of the force act-
ing on that body, and to the simultaneous progress of the
dial-traveller, and is made in the direction of the force.”
For explanations of the terms used I refer to the paper
itself. At the end of this paper we have the assertion : ‘‘ The
Law of Inertia here enunciated sets forth all the truth which
is either explicitly stated, or is suggested by the First and
Second Laws in Sir Isaac Newton’s arrangement.”
Professor Thomson’s Law is doubtless, so far as order and
logic are concerned, an immense advance on the Newtonian
arrangement. Let us inquire whether it can be accepted as
absolutely satisfactory.
How are we to measure the “forces” referred to? If
kinematically, then we are again involved in a logical circle,
as may be seen by substituting in the Law, for the words
* Proc, R, S. E, 1883-4, p. 668,
478 Mr. R. F. Muirhead on the Laws of Motion.
“force acting on that body” the words ‘‘ rate of change of
motion of that body,’’ and for the words “ direction of force ”
the words “ direction of change of motion.”’ And we cannot
entertain any other measure of force, for reasons which will
be adduced later on. |
Again, Prof. Thomson, by not restricting his statement
to infinitesimal particles, has to assume the existence of
mass-centres. How is a mass-centre to be defined? We
shall give reasons later for rejecting any but a kinetic defini-
tion of mass and mass-centre. But it is impossible to arrive
at a kinetic definition when we start by assuming a know-
ledge of the measurement of mass in the Fundamental Law
of Motion, as is done by Professor Thomson.
While noting therefore that Professor Thomson has adopted
the right method of defining chronometry and “true rest,’
we cannot accept his Law as a satisfactory statement of the
fundamental principle of Dynamical science.
Let us endeavour to frame, after the manner of Professor
Thomson, a statement which shall be satisfactory. Taking
the definitions of dial-traveller and reference-frame, aS given
in the paper referred to, let us proceed thus :—
Let a material system be conceived divided into an infinite
number of particles whose greatest linear dimensions are all
infinitesimal. To each particle let us attribute a certain value
called its provisional-mass. Let us adopt a reference-frame
and dial-traveller. Let the acceleration of any particle multi-
plied into its provisional-mass be called the apparent-force on
the particle. Then it is possible so to choose the provisional-
masses, the dial-traveller, and the reference-frame, so that the
provisional-masses and the apparent-forces shall, within the
limits of error of observation, have relations expressible by
the laws of physical science, 7. e. the law of the Indestructi-
bility of Matter, the law of Hquality of Action and Reac-
tion the law of Universal Gravitation, the laws of electric,
magnetic, elastic, and capillary action, &., &e. Such a
system being chosen, the provisional-masses in it are masses
and the apparent-forces, forces. The dial-traveller indicates
“ absolute time,” and the reference-frame is absolutely without
rotation or acceleration.
We have thus kinetic definitions of force, mass, absolute
time-measurement, and of absolute rest so far as that is possible.
It is evident kinematically that any other reference-frame
which has no rotation or acceleration relatively to one chosen
as above would lead to exactly the same results; and that
this would not be the case if any reference-frame not fulfilling
this condition were chosen.
Mr. R. F. Muirhead on the Laws of Motion. 479
The above statement includes all in the First and Second
Laws of Newton that can concewwably be tested by experiment or
observation.
We observe that Newton’s Third Law appears classed
along with other laws of physics, and along with that of the
Indestructibility of Matter, which must be assumed as a
preliminary to the ordinary statement of Dynamical Laws
before the measurement of matter has received its definition.
In our statement of the fundamental principle of Dynamics,
neither of these Laws is assumed, and it could be modified so
as to be equally definite and intelligible were they untrue.
By dealing with infinitesimal particles, we have avoided
the necessity of assuming a priori the existence of mass-
centres ; for on the supposition that the angular motion of no
element is infinite (or, more generally, that there is no finite
relative acceleration or velocity between the parts of any
particle), the motion of any point of a particle might be taken
to represent the motion of that particle.
To define the expression force acting on a body, used in
Dynamics, we would require simply to define the centre of
mass by the usual analytical equations of the type pe
=m’
where the summation extends over all the particles of the body,
and then to define the mass of the body by =m, and the force
on the body as that acting on its whole mass supposed con-
centrated at its centre of mass.
What would be the meaning of “a force acting on a body
at a certain point’’? ‘This expression is appropriate only to
rigid bodies, or at least to such as retain their shape unaltered
while under consideration. The meaning would be that this
force, acting on the particle at the point referred to, together
with the forces between particles determined by the kinema-
tical conditions of rigidity, are the actual forces on the body.
One objection might be raised to the fundamental Law of
Dynamies, as above stated by us ; it seems awkward to imply
a knowledge of the whole of physical science in stating that
fundamental principle.
‘This objection leads us to cast aside Prof. James Thomson’s
type of statement, and to adopt another, which states exactly
the same thought in a different form. We shall propound as
preliminary a science of Abstract Dynamics, which shall be a
pure science to the same extent as Kinematics is a pure science.
It is as follows :-— |
In a dynamical system, each particle is credited with a certain
mass, and by coordinates with reference to a system of coordinate
anes its position and motion are determined. When a particle
a ~ A an oer ms* Oe rs SS ee 4 5 An ~ er: Sy hy Single ae is Oe _ -=. >< 2s >.
I Oa ST I SERRE NMS anomie eee SS +s 2 4
~ RSs RS SSS Seat Sas Se —s" me —eSee aoe “3 SRNL SME METS 5 no
Ss Se Ee Sea SS SS Ee EEE Oe eee
= ease
480 Mr. R. F. Muirhead on the Faws of Motion.
as accelerated, tt is said to have a force acting upon it in the
direction of the acceleration and of magnitude proportional to
the acceleration and mass conjointly.
The system of chronometry is arbitrary, as well as the
system of coordinate axes.
The expressions, mass of a body, centre of mass of a body,
force on a body, and force acting on a body at a point, are
defined in the same way as before.
This forms the subject of ‘‘ Abstract Dynamics,” which deals
only with mental conceptions, and which is a sort of Kine-
matics, but Kinematics enriched by the conceptions of force
and mass.
This being premised, then, in place of Newton’s Definitions
and his First and Second Laws of Motion, we have the
Physical Law or Theory that we can so choose the masses to be
assigned to our material particles, our coordinate axes, and our
system of chronometry, that the forces may be resolved by the
parallelogram of forces into such as are expressed by our
~ Physical Laws.
Perhaps we should keep more faithfully to the historical
conception of Dynamics were we to state our Law of Hxperi-
mental Dynamics as follows :—
It is possible to choose the masses of the solar system, the
axis, and the chronometry, so that the masses shall correspond
with those of Astronomy, and the forces shall be resolvable into
such as will be eapressed by the Law of Universal Gravitation,
and conformable to Newton’s 3rd Law of Motion and to the
Law of the Indestructibility of Matter (Conservation of Mass).
Then true time, absolute velocity, and mass-measurement
being defined from this system, there would be the further Law
of Physics, that the forces on the various particles composing
the different members of the solar system and others are expres-
sible by our various Physical Laws or Theories.
We have now arrived at the conclusion that the attempt to
state the Laws of Motion by means of a set of detached defini-
tions and axioms is futile. We have found that Newton’s
First Law of Motion cannot be stated until we have the con-
ception of a certain system of reference, whose definition
involves the knowledge of the First Law, as well as the defini-
tion of force, &e. We have therefore seen that the Experi-
mental Principle of Dynamics should be stated as an organic
theory or hypothesis. We have found it convenient to
formulate a science of Abstract Dynamics, which is an ex-
tended Kinematics, depending only on space and time-measure-
ments, but including the ideas of force and mass (abstract).
By means of this we can state in a succinct form the
Mr. R. F. Muirhead on the Laws of Motion. 481
experimental Law or Hypothesis of Dynamics (applied),
which enables us to give to time-measurement such a specifi-
cation that durations of time, as well as other dynamical
magnitudes, are made to depend ultimately for their measure-
ment solely on space-measurement and observations of coinci-
dences in time.
These conclusions we have arrived at by assuming that only
kinetic specifications for the measurement of force, mass, and
tune, and only a kinetic definition of “ true rest” are admissible.
Before attempting to justify these assumptions, it may be
expedient to devote a few paragraphs to a general considera-
tion of the idea of our method. A theory is an attempt to
dominate our experience ; it is a conception which may enable
us, with as little expenditure of thought as possible, to
remember the past and forecast the future.
The theory of Universal Gravitation is an example of a
very successful attempt, perfectly successful so far as it has
been tested. So with the Huclidean Geometry.
On the other hand, we have theories which have been found
useful to enable us to dominate one region of experience,
while they break down in certain directions. The Newtonian
Hmission Theory of Light is an example. There are others
which, if they do not break down absolutely, involve the mind
in difficulties hitherto unsolved; e. g. the ‘ elastic-solid”
Wave Theory of Light.
_ Theories which are found to break down when applied to
their full extent, as well as theories which have not been
sufficiently tested, are often called ‘ working hypotheses.”
The only merits or demerits a theory can have arise from
these two desiderata : (1) it must not be contradicted by any
part of our experience ; (2) it must be as simple as possible *.
Thus, for example, consider the two rival theories : (1)
that the earth has a certain amount of rotation about its
axis ; (2) that it has norotation. The latter will be found to
agree perfectly with our experience, provided we assume asa
new physical law that there is a repulsive force of magnitude
wr away from the Harth’s axis at every point of space, and
~ at right angles to the axis
of the Earth, and to the shortest distance of the point from
also at every point a force 2
* Since physical theories form an organic whole, of course these quali-
ties must be considered with reference to the body of physical theory as
a whole. Thus, of two theories, one may, taken by itself, be less simple
than another, and yet be preferred to it, because the whole body of
physical theory becomes simpler when it is adopted. Sometimes, too, a
theory may be preferred because it seems to promise better for the future.
i
482 Mr. R. F. Muirhead on the Laws of Motion.
the axis, where o is the angular velocity in the first theory,
and 7 is the distance of any point from the axis*. But we
reject the latter theory on account of its greater complexity.
It is incorrect to say that the one is true and the other false.
It follows that there is no essential difference between a
hypothesis and a theory, or what is called a law of nature.
One may be less exact than another, or less simple, or less
sufficiently tested, but the difference is one of degree.
Now there are two opposite methods of stating dynamical
principles ; the one employing independent definitions of the
various conceptions, the other that adopted in this Hssay.
Both, so far as observation has tested them, correspond equally
to the facts. The question is, then, Which is the simpler?
Which comprehends the various relations with the least
expenditure of mental energy ?
_ According to the former method, force, mass, time measure-
ment, and “true rest’ would be defined as preliminaries to
the science of Dynamics, and independently of that science.
According to the latter, these conceptions are defined by
means of one Law or Hypothesis.
Probably to learners unaccustomed to abstract reasoning,
who do not probe the processes of proof employed to the
bottom, the former method may be preferable because its
conceptions are more concrete ; but to one who has mastered
the essential relations of the subject, the latter will be found
superior.
Let us discuss the idea of force. What are the alternatives
to the kinetic definition of force and force-measurement ?
We might take some arbitrary standard, such as a spring-
balance having a graduated scale. ‘This would cbviously have
the disadvantage of want of permanence, or, to speak more
accurately, that of liability to invalidate all our other methods
of reckoning force, by reason of some physical change occur-
ring in the standard balance. Further, such a method would
be incapable of accuracy sufficient for many of our physical
problems, where we deal with forces so small as to be insen-
sible to our present observing powers on such a standard;
forces whose magnitude, therefore, we could not define, even
theoretically. And, besides, any such arbitrary definition of
force would be contrary to our whole tendency in modern
science. Suppose, for instance, experiment were to disclose
that Newton’s Second Law was untrue, the forces being thus
* We might either suppose these new forces not conformable to the
law of the equality of Action and Reaction, which would then have to be
modified ; or we might suppose the reactions to observed actions to exist
in the fixed stars, and to be beyond our present means of observation.
Mr. R. F. Muirhead on the Laws of Motion. 483
measured, should we hesitate between rejecting the law or
rejecting the method of force-measurement ? And it is certain
that we cannot find a spring-balance which would render this
event unlikely to happen.
A more promising method would be the definition of unit
of force as the weight of a certain piece of matter at a certain
place on the Harth’s surface. The force F would then be de-
fined as being equal to the weight of a body whose mass was
F times the standard mass. ‘This would involve an inde-
pendent method of mass-measurement, which we shall con-
sider later. In treating questions of the secular changes of
the Earth such a definition would be useless, unless we were
also to specify the date as well as the place of the weighing
supposed to be at the base of force-measurement; and this
could not be brought into connexion with measurements at
any other date without employing the whole science of Dyna-
mics, which would thus involve reasoning in a circle.
A modification of this method would be one in which force-
measurement would be made to depend on the gravitational
or astronomical unit of mass, as well as the theory of the
force of gravitation. But this also would be a system of
force-measurement, involving for its conception the whole
science of Dynamics, of which it would not be independent.
When Statics is treated as a science, independent of Kine-
tics, force is sometimes left undefined at first, while the mode
of procedure is as follows:—We are supposed to have a
certain idea of the nature of force, partly based on the sensa-
tions we experience when our body forms one of the two
bodies which exert force on one another”, and starting from
this, by the aid of &@ priori reasoning the idea of the measure-
ment of force is evolved. Then, with the help of certain
physical axioms and constructions (“ transmissibility of force,”
“superposition of forces in equilibrium,” &c.), the parallelo-
gram of forces is proved.
All this has a very artificial character, and would lead us
to prefer the simpler kinetic conception of force; but still
further argument is required before we get to Kinetics. The
“Second Law of Motion” is proved by means of experiments
which could not be accurately performed, and whose inter-
pretation generally involves a knowledge of the science whose
foundations we are laying. Then the proportionality of force
to mass is thus proved:—
Suppose two equal masses acted on by equal and parallel
* The so-called ‘“ sense of force” should be called ‘sense of stregs.”’
Our bodies subjected to forces, however great, if the force on each part is
proportional to its mass and in a common direction, feel nothing.
484 Mr. R. F. Muirhead on the Laws of Motion.
forces; they have the same acceleration. Next, suppose they
form parts of a single body; the acceleration will “ evidently ”
be the same as before, &e. (Third Law of Motion assumed.)
Hence accelerations being equal, force varies as mass.
This method has been discredited of late, chiefly through
the influence of Thomson and Tait’s ‘ Natural Philosophy,’ so
that we may omit further discussion upon it.
It may be remarked, however, that those who have most
emphatically declared against the statical measure of force do
not seem to perceive what is logically implied in that course.
(Cf. Professor Tait’s Lecture on Force.)
Consider next the idea of mass.
The definition based on the weight of bodies is open to the
same objections as the corresponding method in the case of
force.
If we define mass by reference to chemical affinity *, or to
volumetric observations, we in the first place lose the sim-
plicity of the kinetic method, and secondly we adopt a con-
ception of mass which is different from the actual conception
of modern science. This is demonstrated if we ask ourselves:
Supposing experiment to show a discrepancy between the
mass as measured kinetically and as measured otherwise,
which method should we call inexact? If the former, Kine-
tics could no longer be considered an exact science.
Consider next the reference system, and the idea of true rest.
The most obvious arbitrary definition of the system to which
the motions of bodies in Dynamics are to be referred is to look
on the centre of gravity of the Solar system as the fixed
point, and the directions of certain fixed stars as fixed direc-
tions. The objections are, first, this would be a very incon-
venient system in discussing the cosmical Dynamics ; second,
it is not the actual conception of the science of the present day.
If one of the stars chosen were found to have a motion com-
pared with the average position of neighbouring stars, we
should certainly conclude that its direction was not “ fixed ”’
in the dynamical sense.
It has been suggested to take as a fixed direction that of
the perpendicular to the ‘‘ invariable plane of the Solar system.”
This really is not an independent definition, and is open to
the objections we previously urged against such, when isolated
from the fundamental law of Hxperimental Dynamics.
The foregoing methods have been well criticised by
Streintz t, who propounds in their stead a method of re-
* See Maxwell’s ‘ Matter and Motion,’ art. xlvi.
+ Die physikalischen Grundlagen der Mechantk. Leipzig, 1883.
Mr. R. F. Muirhead on the Laws of Motion. 485
ference to a “ Fundamental Korper,”’ which is any body not
acted on by external forces and having no rotation. The
absence of rotation is to be determined by observations of
centrifugal force (as in Newton’s experiment of the rotating
bucket of water). Now as Streintz takes the kinetic definition
of force, it involves reasoning in a circle to speak at this stage
of a body “not acted on by forces.”’ Further, if the observa-
tions of centrifugal force are to be made with the whole re-
sources of Dynamics, and our knowledge of the laws of nature,
this is virtually the kinetic definition of force, but stated
in a form which involves reasoning in a circle. If, on the
other hand, want of rotation is to be defined as existing when
the surface of a bucket of water does not appear to deviate
from planeness, then our stock objections to such definitions
of dynamical ideas reappear.
A most instructive discussion relating to this subject is
given by Professor Mach in his book Die Mechanik in ihrer
Enitwickelung, historisch-kritisch dargestellt, pp. 214-222. Let
us quote a sentence on p. 218:—
“Instead of saying ‘the direction and velocity of a mass
# in space remain constant,’ we can say ‘the mean accelera-
tion of the mass u with reference to the masses m, m’, m"...
ge urge SPN spy 1
1Is=VU, or dz Sip =U. €
latter expression is equivalent to the former, so soon as
we take into consideration masses which are great enough,
numerous enough, and distant enough.”’
On the previous page, referring to Newton’s bucket ex-
periment, he remarks that no one can say how the experiment
would come out were we to increase the mass of the bucket
continually ; and, further, that we should be guilty of dis-
at the distances 7, 7’, 7’...
honesty, were we to maintain that we know more of the motion
of bodies than that their motion relative to the very distant
stars appears to follow the same laws as Galileo formulated for
terrestrial bodies relative to the Harth. |
Of course this charge of dishonesty cannot be urged against
the method of this Essay, as explained in our paragraphs on
the nature of theories. And our definition of “ true rest”’
being based entirely on experiment and observation, is not
affected by Prof. Mach’s strictures on the use of the terms
absolute rest, absolute space, Kc.
Though on the principles of this Essay no exception in
principle can be taken to Prof. Mach’s substitute for the
“First Law of Motion above quoted,” we reject it because it
is not the actual conception which has been historically evolved
in Dynamics.
486 Mr. R. F. Muirhead on the Laws of Motion.
Lastly, let us consider the conception of time-measurement.
The only rival definition of equal times that need be con-
sidered is that adopted by Streintz, and ascribed by him to
D’Alembert and Poisson, viz. ‘Times are equal in which
identical processes take place.’’ The difficulty here would be
to distinguish when we have identical processes going on.
We find that practically this will reduce to assuming each
rotation of the Harth with reference to the fixed stars a pro-
cess identical with all the others. For the ‘‘ processes ’’ must
consist in movements of matter, of which the Earth’s rotations
are the most ‘ identical ’’? we have experience of.
But even these we know are not absolutely identical, so
that our definition is not practicable. With this definition,
what should we mean by saying that the rotation period of
the Harthis altering? We should mean that if identical pro-
cesses happened at different dates, their durations measured by
sidereal time would differ. But the only identical processes
actually available are wrapped up in the general dynamical
theory of the Solar system; so that this theoretically inde-
pendent definition of time turns out to involve all our Dynamics
implicitly when we try to give it physical meaning.
In seeking to justify our preference of kinetic definitions
over non-kinetic definitions of our fundamental dynamical
conceptions, we have found that the latter, besides being
theoretically inconvenient, very often have only an illusory
independence of Dynamics.
In fact no one has ever built up a science of Dynamics
from independently formed conceptions ; and to do so in a
strictly logical manner would require expositions whose length
would render them tedious in the extreme.
We have hitherto made no reference to any scheme of
dynamical principles apart from that of Newton, and those
various modifications of it proposed by later writers. This
course has been adopted in order to concentrate attention upon
the principle at issue.
Systems of Dynamics founded on such principles as Mau-
pertius’s “ Principle of Least Action,” or Gauss’s “ Principle
of Least Coercion ” (Kleinsten Zwanges), may be treated from
exactly the same point of view, and will not be further re-
ferred to.
Note A.—On Theories and Hypotheses,
In the preceding Essay we have assumed as known the science of
Geometry ; but of course the views put forward in this Essay con-
cerning the nature of physical theories apply equally to geometri-
NS = ks
Mr. R. F. Muirhead on the Laws of Motion. 487
eal theories. This is the standpoint adopted by Riemann in his
epoch-making paper, “ Ueber die Hypothesen welche der Geometrie
zu Grunde liegen.” That space is infinite and that one and only
one parallel to a straight line can be drawn through any point, are,
it is true, the simplest hypotheses which serve to express our ex-
perience ; but, as Helmholtz points out in his tract Ueber che
Erhaltung der Kraft, at page 7, the task of theoretical science is
only completed when we have proved that our theories are the
only ones by which the phenomena can be explained. “Dann
ware dieselbe als die nothwendige Begriffsform der Naturauffas-
sung erwiesen; es wirde derselben alsdann also auch objective
Wabrheit zuzuschreiben sein.”
In his critique of the second edition of Thomson and Tait’s
treatise on Natural Philosophy (‘ Nature,’ vol. xx. p. 213), Clerk
Maxwell clearly indicates the hypothetical nature of abstract Dy-
namics. On p. 214 we read :—‘‘ 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......
We have thus vindicated for figures with mass, and, therefore, for
force and stress, impulse and momentum, work and energy, their
place in abstract science beside form and motion.”
“The phenomena of real bodies are found to correspond so
exactly with the necessary laws of dynamical systems that we can-
not help applying the language of Dynamics to real bodies,” &c.
It will be seen that, so far as they go, the above extracts are in
complete harmony with the views in this Essay. It is to be re-
eretted that these views are not consistently followed out in Clerk
Maxwell’s book ‘ Matter and Motion.’ In that book, while there are
very many clear expositions of particular points, the arrangement is
in many parts highly illogical. This has been pointed out to a
certain extent by Streintz in his aforementioned book, and the
reader of the foregoing Essay will have little difficulty in making
further criticisms.
One point in Maxwell’s book (‘ Matter and Motion’) calls for
special notice, viz., his @ prior proof of the first law of Motion.
This proof rests on the assumption of the impossibility of defining
absolute rest. ‘‘ Hence,” he says, “the hypothetical law is with-
out meaning unless we admit the possibility of defining absolute
rest and absolute velocity.” But itis obvious that if the “ hypo-
thetical law” spoken of (velocity diminishing at a certain rate)
corresponded with experience, we should then have, by that very
fact, a conception of absolute rest and absolute velocity which
would be perfectly intelligible, so that the assumption “absolute
rest unintelligible” would not be justified. Thus, Maxwell’s con-
clusion, “‘ It may thus be shown that the denial of Newton’s law is
in contradiction to the only system of consistent doctrine about
488 Mr. R. F. Muirhead on the Laws of Motion.
space and time which the mind has been able to form” is unwar-
ranted.
Kirchhoff in his Mechamk appears to adopt a view somewhat
similar to that set forth in this Essay. In his preface we find him
stating as the problem of Mechanik, “die in der Natur vor sich
gehenden Bewegungen vollstandig und zwar auf die einfachste
Weise zu beschreiben.”
This author uses the term force only as a convenient means of
expressing equations shortly in words. Mass appears as a coeffi-
cient in the equations of motion, and thus receives a kinetic defi-
nition. Butno explanations are given as to time-measurement, or
as to the axes of reference.
Nore B.—Newton’s Absolute Space and Time.
My criticisms of the Newtonian scheme of Definitions and Axioms
have been directed not so much against what I suppose to be
Newton’s meaning, as against the form in which it is put, especially
as against that form on the supposition that force is to be measured
kinetically.
Thus, instead of looking on the Second Law as a mere definition
of force-measurement, we might suppose that Newton had in his
mind some non-kinetic conception of force-measurement ; in which
case the Second Law would be a real and not an illusory statement
of physical fact, though imperfect through the want of any speci-
fication of how force was to be measured.
Again, take the question of absolute space and time, with respect
to which Newton’s laws are stated.
There are three ways of looking at it. Some characterize these
terms as mere metaphysical nonsense (Mach, p. 209). Streimtz*
quotes the Hypothesis I. from the third Book of Newton’s Prin-
cypia to show that by absolute rest Newton means rest relative to
the centre of gravity of the universe. But Newton evidently places
this Hypothesis in a different category from his laws of motion.
I think the meaning of the terms amounts simply to this, that
Newton looked on Dynamics as an abstract science. ‘“ In rebus
philosophicis abstrahendum est a sensibus” 7, “loca primaria moveri
absurdum est” +. And an abstract science is one which deals
with a certain body of conceptions, every relation in which holds
with absolute exactness. ‘The point at which considerations as to
degree of exactitude may arise, is its application to experience. »
If this be the correct view of Newton’s meaning, then the fore-
going Hssay has been simply the explicit and developed statement of
that meaning.
Thomson and Tait, while in various ways improving the form in
which they state the Newtonian theory, entirely ignore his idea of
“absolute space and time,” which, as I have tried to show, is the
germ of the true theory.
* Physikalische Grundlagen, p. 10.
+ Scholium to Definitiones.
ah ih
Production, Properties, and Uses of the Finest Threads. 489
The late C. Neumann, in his pamphlet Ueber die Principien der
Galilei-Newtonschen Theorie (Leipzig, 1870), like Newton, postu-
lates an “absolute rest.” He does so by assuming that there is a
*“Korper Alpha,” an ideally existing body which is absolutely at
rest and absolutely rigid, with respect to which the First Law of
Newton holds good.
Streintz criticises this rather unintelligently, I think, for it is
evident in reading Neumann’s essay that this is merely an. awk-
ward and metaphorical way of stating the theory of an “ Abstract
Dynamics.”
Nore C.—The Parallelogram of Force.
Force being defined kinetically, it is hardly necessary to demon-
strate this proposition. It follows as easily from the parallelogram
of accelerations as that does from the parallelogram of velocities, or
the parallelogram of velocities from the parallelogram of steps.
This applies primarily to forces acting on a particle, but it is easy
to extend the theorem to “ forces acting on a body,” as defined in
the Essay.
LVII. On the Production, Properties, and some suggested Uses
of the Finest Threads. By ©. V. Boys, Demonstrator of
Physics at the Science Schools, South Kensington”.
HAVE lately required for a variety of reasons to have
fibres of glass or other material far finer than ordinary
spun glass; I have therefore been compelled to devise means
for producing with certainty the finest possible threads. As
these methods may have some interest, and as some results
already obtained are certainly of great importance, I have
thought it desirable to bring this subject under the notice of
the Physical Society, even though at the present time any
account must of necessity be very incomplete.
The subject may be naturally divided, as in the title, into
; three parts.
1. Production.
The results of the natural methods of producing fibres by
living things, as spiders, caterpillars, and some other creatures,
are well known ; but it is useless to attempt to improve on
Nature in this direction by our own methods.
Fibres are also produced naturally in volcanoes by the
rushing of steam or compressed gases past melted lava, which
is carried off and drawn out into the well-known Pelés hair.
The same process is employed in making wool from slag, for
* Communicated by the Physical Society : read March 26, 1887.
Phil. Mag. 8. 5. Vol. 23. No. 145. June 1887. 2 L
490 Mr. 0. V. Boys on the Production, Properties,
clothing boilers, &c. ; but in each of these cases the fibres are
matted together, they are not adapted to the requirements of a
Physical Laboratory. By drawing out glass softened by heat
by a wheel we obtain the well-known spun glass.
There is a process by which threads may be made which is
natural in that natural forces only are employed, and the thread
is not in any way touched during its production. This is the
old, but now apparently little-known experiment of electrical
spinning. Ifa small dish be insulated and connected with an
electrical machine and filled with melted rosin, beeswax, pitch,
shellac, sealing-wax, Canada balsam, guttapercha, burnt india-
rubber, collodion, or any other viscous material, the contents
will, if they reach one edge of the dish, at once be shot out in
the most extraordinary way in one, two, or it may be a dozen
threads of extreme tenuity, travelling at a high speed alon
“lines of force.” If the material is very hot, the liquid
cylinders shot out are unstable and break into beads, which
rattle like hail on a sheet of paper a few feet off. As the
material cools, the beads each begin to carry long slender
tails, and at last these tails unite the beads in twos and threes ;
but the distance between the beads is far greater than that
due to the natural breaking of a cylinder into spheres, as
after the first deformation of the surface occurs which deter-
mines the ultimate spheres the repulsive force along the thread
continues, and drags them apart many times their natural
distance. As the temperature continues to fall and the
material to become more viscid, the beads become less
spherical, and the tails less slender, and at last a perfectly
uniform cylindrical thread is formed. If sealing-wax is
employed, and a sheet of paper laid for it to fall on, the paper
becomes suffused in time with a delicate rosy shade produced
by innumerable fibres separately almost invisible. On placing
the fingers on the paper, the web adheres and can be raised
in a sheet as delicate and intricate as any spider’s-web.
It is interesting to see how these fibres fly to any conduct-
ing body placed in their path. If the hand is held there it is
quickly surrounded by a halo of the finest threads. If a
lighted candle is placed in the way, the fibres are seen by the
light of the candle to be rushing with the greatest velocity
towards it, but when a few inches off they are discharged by
the flame, they stop, turn round, and rush back as fast into
the saucer whence they came. ‘The conditions for the success
of this beautiful experiment are not very easily obtained*.
Fibes spun by the electrical method are so brittle that they
* If the wick of the candle is connected with the opposite pole of the
machine, the threads at one stage are sure to return to the saucer.
Se
= _
——— a
and some suggested Uses of the Finest Threads. 491
do not seem to be’of any practical use. It is possible, how-
ever, that this method might be available for reducing to a
fine state of division such of the rosins or other easily fusible
bodies as cannot readily be powdered mechanically.
On returning to bodies which, like glass, require a high
temperature for their fusion, to which the electrical method is
inapplicable, we find that the only method practically available
is that of drawing mechanically. It would seem that if finer
threads than can be formed by the ordinary process of glass-
spinning were required, it would be necessary to obtain a
higher speed, to have the glass hotter, and to have as small a
quantity as possible hot. I put this idea to a test by mounting
at the back of a blowpipe-table a pair of sticks which could be
suddenly moved apart by a violent pull applied to each near
their axes. By these means the upper ends were separated
about 6 feet, and the motion was so rapid that it was impos-
sible to follow it. A piece of glass drawn out fine was
fastened to the end of each stick, and the ends of these heated
by a minute blowpipe-flame. They were immediately made to
touch and allowed to fly apart. In this way I obtained threads
of glass about 6 feet long, finer than any spun glass I have
examined. By using the oxyhydrogen jet with the same
apparatus, still finer threads were produced. It was evident
then that the method was right ; but some more convenient
device which also would make long threads would be prefer-
able.
There are several ways of obtaining a high speed, the most
usual depending on an explosive ; but it would be difficult to
arrange in a short time a gun which could be used to shoot
a projectile carrying the thread which would not also destroy
the thread by the flash. It is possible that an air-gun could be
so:arranged. Rockets when at the period of most rapid com-
bustion have an acceleration which is enormous. ‘Thus a well-
made 2-oz. rocket is at one part of its flight subject to a
force of over 3 lb. in gravitative measure. ‘This force, acting
on such a body for 10 seconds only, would, neglecting atmo-
spheric resistance, starting from rest, carry it more than 6 miles.
The acceleration is about 28 times that due to gravity on the
earth, or about the same as that on the sun. Anyone who
will stay in a room with a lighted two-ounce rocket, having
no stick or head, will obtain a more vivid notion of the value
of gravity on the sun than in any other way I know.
A rocket is perhaps more available for thread-drawing than
a gun, but it does not seem altogether convenient. One
other method, however, is so good in every respect, that there
seems no occasion to try a better. The bow and arrow at
2L2
-
492 Mr. C. V. Boys on the Production, Properties,
once supply a ready means of instantly producing a very high
velocity, which the arrow maintains over a considerable dis-
tance. For the special purpose under consideration, the
lightest possible arrow is heavy enough. I have made arrows
of pieces of straw, which may be obtained from wool-shops,
a few inches long, having a needle fastened to one end for
a point. Arrows made in this way travelled the length of
the two rooms in which I made these experiments—about
90 feet—in what seemed to be under half a second. They
completely pierced a sheet of card at that distance, which I
put up thinking that a yielding target might damage them
less than the wall, and were then firmly stuck unharmed in the
wall behind; in every way they behaved so well that I do
not think a better make of arrow possible.
The bow I used was a small cross-bow held in a vice
with a trigger that could be pulled with the foot. The first
bow was made of oak, the first wood that came to hand. I
then made some bows of what was called lance-wood (it was
unlike any lance-wood I have seen) ; but the trajectory was
at once more curved, the arrow took perceptibly longer to
travel, and the threads produced were thicker. As the arrow
is so light, the only work practically that the bow has to do
is to move itself; that wood then which has the highest
elasticity along the fibres for its mass is most suitable ; in
other words, that wood which has the greatest velocity of
sound is best. I therefore made bows of pine, and obtained
still higher velocities and finer threads than I could obtain
with oak bows.
With a pine bow and an arrow of straw I have obtained a
glass thread 90 feet long and j5} 9 inch in diameter, so
uniform that the diameter at one end was only one sixth more
than that at the other. Pieces yards long seemed perfectly
uniform.
A fragment of drawn-out glass was attached to the tail of
the arrow by sealing-wax, and heated to the highest possible
temperature in the middle, the end heing held in the fingers.
With every successful shot the thread was continuous from
the piece held in the hand to the arrow 90 feet off. The
manipulation is, however, difficult, but another plan equally
successful has the advantage of being quite easy. It is not
necessary to hold the tail of glass at all; if the end of the
tail only be heated with the oxyhydrogen jet until a bead
about the size of a pin’s head is formed, and the arrow shot,
this bead will remain behind on account of its inertia, and
the arrow go on, and between them will be pulled out the
thread of glass.
,
and some suggested Uses of the Finest Threads. 493
Prof. Judd has kindly given me a variety of minerals
which I have treated in this manner. Some behave like glass
and draw readily into threads, some will not draw until below
a certain temperature, and others will not draw at all, being
_ perfectly fluid like water, or when a little cooler perfectly
ard.
Among those that will not draw at all may be mentioned
Sapphire, Ruby, Hornblende, Zircon, Rutile, Kyanite, and
Fluorspar.
Hmerald and Almandine will draw, but care is required to
obtain the proper temperature. In the case of the Garnet
Almandine, if the temperature is too high, the liquid cylinder,
if formed, breaks up, and a series of spheres fall on the table
in front of the bow. At a slightly lower temperature the
thread is formed, but it is beaded at nearly regular intervals
for part of its length.
Several minerals, especially complex silicates as Orthoclase,
draw very readily, but that which surpasses all that I have
tried at present is Quartz, which, though troublesome in many
ways at first, produces threads with certainty. It required
far more force to draw quartz threads than had been previously
experienced. The arrow, instead of continuing its flight,
hardly disturbed by the drag of the thread, invariably fell very
low, and was not in general able to travel the whole distance.
So great is the force required that I split many arrows before
I succeeded at all. I have obtained threads of quartz which
are so fine that I believe them to be beyond the power of any
possible microscope. Mr. Howes has lent me a +4,-in. Zeiss of
excellent definition, and though, on looking at suitable objects,
definite images appear to be formed on which are marks
corresponding according to the eyepiece-micrometer to toq/o90
inch, yet these threads are hopelessly beyond the power of the
instrument to define atall. On taking one that tapers rapidly
from a size which is easily visible, the image may be traced
until it occupies a small fraction of one division, of which 13°4
correspond to yo/9 inch on the stage ; then the diffraction
bands begin to overlap the image until it is impossible to say
what is the edge of the image. Having reached this stage,
the thread may be traced on and on round the most marvellous
convolutions, the diffraction-fringe now alone appearing at all,
but getting fainter and apparently narrower until the end is
reached. ‘That a real thing is being looked at is evident, for
if the visible end is drawn away the convolutions of fringes
travel away in the same direction. It is impossible to say
what is the diameter of these threads ; they seem to be certainly
less than yg !990 inch for some distance from the end.
494 Mr. C. V. Boys on the Production, Properties,
It might be possible to calculate what would be the appear-
ance presented by a cylinder of given refractive power, and
1, 2, 3, &e. tenths of a wave-length of any kind of light
in diameter, when seen with a particular microscope. By no
other means does it seem possible to find out what the true
size of the ends of these threads really is.
2. Properties.
I can at present say very little of the properties of these
very fine fibres ; I am now engaged with Mr. Gregory and
Mr. Gilbert in investigating their elasticity. The strength
goes on increasing as they become finer, that is, when due
allowance is made for their reduced sectional area, and it
seems to reach that of steel, about 50 tons to the inch in
ordinary language ; but on this point I have not yet, made
any careful experiments.
The most obvious property of these fibres is the production
of all the colours of the spider-line when seen in a brilliant
hight. The most magnificent effect of this sort I have seen,
was produced by a thread of almandine. One of these the
length of the room, even though illuminated with gas-light
only, was glistening with every colour of the rainbow. In
attempting, however, to wind it up, it vanished before me.
It is of course only visible in certain directions.
The chief value of threads to the physicist lies in their
torsion. Spun glass, as is now well known, cannot be used
for instruments of precision, because its elastic fatigue is so
great that, after deflection, it does not come back to the original
position of rest, but acquires a new position which perpetually
changes with every deflection. If left alone, this position
slowly works back towards a definite place more rapidly as it
is further from it. |
To compare threads made of different materials, I made a
flat cell in which a galvanometer-mirror, made by Elliot Bros.,
might hang, being attached to the lower end of the thread.
The upper end was secured to a fixed support, and a fixed tube
protected the length of the fibre from draught. The cell,
which could be moved independently of the rest, was protected
by a cover. By means of a lamp and scale, the exact position
of rest of the mirror could be determined with great accuracy.
On turning the cell round as many times as might be desired,
the mirror was turned with it, and could be left any time in any
position. On turning the cell back again, the mirror was
allowed to come to its new position of rest, air-resistance of
the cell bringing about this result in a few swings. By this
means I hoped quickly and accurately to determine the fatigue
— ee
and some suggested Uses of the Finest Threads. , 495
in a variety of threads, but an unforeseen difficulty arose which
I cannot yet explain. When the cell was moved round slightly
so as not to touch the mirror, the mirror moved at first in the
same direction as was to be expected, but it came to rest in a
new position, to reach which it had to move in the opposite
direction to the movement of the cell. Whichever way the
cell was shifted, the mirror always went the other to find its
position of rest. Thinking that it or the cell were electrified,
I damped both by breathing on them, but with no result, and
the next day the same effect was observable. So great was this
effect that I could set the cell with greater accuracy by
watching the spot of light than by the pointer carried by the
cell working over a 4-inch circle.
Thinking that magnetism might have something to do with
this effect, I brought a horseshoe-magnet near the mirror,
when it was instantly deflected through a large angle. An
examination of the cement used (Loudon’s bicycle cement)
showed that it was magnetic. Of many cements examined,
sealing-wax was more nearly neutral than any other. Bicycle
cement and electrical cement were strongly magnetic; all
others except sealing-wax strongly diamagnetic. The appa-
ratus was therefore taken to pieces and carefully cleaned. It
was put together with as small a quantity of sealing-wax as
possible, and the mirror was attached to a fragment of thin
pure copper wire, which again was fastened by a speck of
sealino-wax to the thread. Hven then the same kind of
effect as that already described occurred. Still a magnet
deflected the mirror, but not so much, and the cell was
practically neutral; yet, when the cell was turned a little,
the mirror changed its position of rest.
Without pursuing this question further, I put a window in
the protecting tube and turned the mirror by means of a
small instrument passed up from below. Thus neither window
nor support were moved. A piece of spun glass nearly 9
inches long gave a period of oscillation to the mirror of 2°3
seconds about. A lamp and millimetre-scale were placed 50
inches from the mirror. As all the observations were
expressed in tenths of a millim., to about which extent they
can be trusted, it is convenient to employ one scale of numbers
of which one tenth millim. is the unit. One complete turn of
the mirror is very nearly 160,000 on thisscale. If the mirror
is moved through 160,000 in either direction and held for one
minute, and then allowed to take its new position, the change
in the position of rest is as soon as it can be read about 370.
This is reduced in about three minutes to 110. If the mirror is
moved through three turns, 480,000 of the scale, and held one
496 Mr. C. V. Boys on the Production, Properties,
minute, the position of rest is at first moved about 1100, which
falls in three minutes to about 400. I have given these figures,
not because the effect is not perfectly well known, but to serve
as comparison figures to those that are to follow. They can —
only be properly represented on a time-diagram.
A piece of the same fibre that was used in the last experi-
ment was laid in a box of charcoal and heated in a furnace
to a dull red heat and allowed to cool slowly. This was
examined in the same way as the last. The effect of a
movement of 160,000 for one minute was now only about 60,
which was reduced to about 45 inthree minutes. The change
for 480,000 lasting one minute was at first about 250, which
fell to about 180 in three minutes.
Annealed spun glass then shows far less of this effect than
spun glass not annealed, but it is slower in recovering. It is
possible that if time were given, it would show as great an
effect as plain glass. The only mineral from which at the
present time I have obtained any valuable results in this
direction, is quartz. Here the effect of the usual minute at
160,000 was only 7, in the place of 370 for glass, at 820,000
only 17, and 640,000 only 32, which in four minutes fell to
22. This fibre was as usual fastened at each end by sealing-
wax. When this experiment was made, the thread had only
just been fastened. The same fibre treated previously in the
same way, but some days after fastening, did not even show
this effect; but as this was before I had completed the
proper cell, the observations cannot so well be trusted. After
a complete turn, there was not a movement of one tenth of a
millim., nor had the position changed this much in 16 hours.
It is as yet too soon to be sure, but this seems to point to the
possibility of the very slight effect observed being largely due
. to the sealing-wax. Whether this is so or not does not much
matter, the behaviour of the quartz thread approaches suffi-
ciently near to that of an ideal thread, to make it of the
utmost value as a torsion-thread. I hope shortly to be able to
bring results of carefully conducted experiments on the
elastic fatigue of quartz and other fibres before the notice of
this Society. }
A thread of annealed quartz behaves like a thread of quartz
not annealed. That it was affected by the process of annealing
is evident, because in the first place it was very rotten and
difficult to handle, and in the second a piece of quartz fibre,
which was wound up, retained its form. By this test, quartz
can only be partly annealed in a copper box, as any form is not
retained perfectly ; at a temperature above that of melting
copper, quartz seems to perfectly retain any form given to it.
and some suggested Uses of the Finest Threads. A97
It is probable that a body hung by a fibre of quartz and
vibrating in a perfect vacuum would remain twisting back-
wards and forwards for a far longer time than a similar body
hung by a glass thread, also that the most perfect balance-
spring for a watch would be one of quartz. I have a piece of
quartz drawn out to a narrow neck which just cannot hold up
its head ; this keeps on nodding in all directions for so long a
time, even in the air, as to make it evident that the material
has very unusual properties.
3. Uses.
As torsion-threads these fibres of quartz would seem to be
more perfect in their elasticity than any known ; they are as
strong as steel, and can be made of any reasonable length
perfectly uniform in diameter, and, as already explained,
exceedingly fine. The tail ends of those that become invisible
must have a moment of torsion 100 million times less than
ordinary spun glass ; and though it is impossible to manipulate
with those, there is no difficulty with threads less than yody5
inch in diameter.
I have made a spiral spring of glass of about 30 turns
which weighs about one milligram; this, examined by a
microscope, would show a change in weight of a thing hung
by it of one 10 millionth of a gram. Since this has been
annealed its elastic fatigue is that of annealed glass, and
therefore very small. J have succeeded in doing the same
thing with a quartz fibre, but the difficulties of manipulation
are very great in consequence of the rottenness of annealed
quartz. The glass spring can be pulled out straight, and
returns perfectly to its proper form.
Since these fibres can be made finer than any cobweb, it is
possible that they may be preferable to spider-lines in eye-
pieces of instruments ; they would in any case be permanent,
and not droop in certain kinds of weather.
Those who have experienced the trouble which the shifting
zero of a thermometer gives, might hope for a thermometer
made of quartz. When made, it would probably be more
perfect in this respect than a glass thermometer, but the
operation of making would be difficult.
These very fine fibres are convenient for supporting small
things of which the specific gravity is required, for they weigh
nothing, and the line of contact with the surface of the water
is so small, that they interfere but little with the proper swing
of the balance.
It seemed possible that a diffraction-grating made of fibres
side by side in contact with one another would produce
498 Production, Properties, and Uses of the Finest Threads.
spectra which would be brighter than those given by a
corresponding grating of ordinary construction, because not
only is all the light which falls on the surface brought to a
series of linear foci forming the bright lines instead of being
half removed, as is usually the case, but the direction of the light
on reaching these lines is not normal to the grating as usual,
and therefore in the direction of the central image, but
spreading, and thus in the direction of all the spectra. I
picked out a quantity of glass fibre not varying in diameter
more than one per cent., and made a grating in this way
covering about:one eighth of an inch in breadth. This not
only showed three spectra on each side, and a quantity of
scattered light, but all the spectra were closely intersected by
interference-bands, such as are seen when a Newton’s ring of
a high order is seen ina spectroscope. This is probably due to
a cumulative error in the position of the fibres, for they .
were spaced by being pushed up to one another with a
needle-point, or to light passing between the fibres in a few
places where dust particles keep them apart.
A. diffraction-grating made of these fibres, spaced with a
screw to secure uniformity, and of a thickness equal to the
spaces between them (and one of 1000 lines to the inch could
be easily made) would be far more perfect for the number of
lines than any scratched on a surface ; that is, for investigation
on the heat of a spectrum, sucha grating would be preferable
to a scratched one, as there is no uncertainty as to the grating
_ or to the substance of which it is made*. Ifthe transparency
“of the fibres interfered they could be rendered opaque by
metallic deposit without visibly increasing their diameter.
There is one use to which the fibres of quartz tailing-off to
a mere nothing might be applied, namely as a test-object for
a microscope. Theory shows that no microscope can truly
show any structure much less than +5,/559 inch, or divide two
lines much less than this distance apart. Natural bodies such
as Diatoms &c. have this advantage, that they can be ob-
tained in any quantity alike, but no one knows what the real
structure of these may be. Nobert’s bands,are good in that |
we know the number of lines in any band, but as to the indi-
vidual appearance of the lines and spaces it is impossible to
say anything. These fibres have the advantage that we have
a single thing of known form, which tapers down from a
definite size to something too small even to be seen. Though
it may be possible to calculate the size from the appearance
of the fringes, yet whether the size is known or not, at each
* See ‘ Heat,’ by Prof. Tait, p. 268.
Electrical Resistance of Vertically-suspended Wires. 499
point we have a definite thing of known form which can be
examined by a series of microscopes, and the point up to
which it can be clearly seen observed for each.
I have thought it worth while to bring this subject forward
in this very incomplete form, because there are already
results of interest and there is so much prospect of more, that
it is likely that Members may be glad to investigate some of
the questions raised.
LVIII. On the Electrical Resistance of Vertically-suspended
Wires. By SHELFORD BipweE tt, J.A., F.R.S.*
ROM the experiments to be described in this paper, it
appears probable that the electrical resistance of verti-
cally-suspended copper and iron wires alters to a small extent
with the direction of the current traversing them. Ifthe wire
is of copper, the resistance is slightly greater when the cur-
rent goes upwards than when it goes downwards ; while, on
the other hand, the resistance of an iron wire is apparently
greater for downward than for upward currents.
1 eg
The arrangement employed for exhibiting this effect is
shown in the annexed diagram. A wire, A B, of the material
* Communicated by the Physical Society: read March 12, 1887.
T Venturing to imitate the fanciful analogy used by Sir William
Thomson, who, in discussing the thermoelectric effect now universally
associated with his name, speaks of the “specific heat” of electricity, we
may perhaps also speak of the “specific gravity” of electricity, and say
that (like its specific heat) it is positive in copper and negative in iron.
500 — Mr. S. Bidwell on the Poaticdl Resistance
to be tested is suspended at its middle point, P, from a support
10-5 metres above a metre-bridge, to the terminals, TT’, of
which the ends of the wire are connected. Another wire, C,
is soldered at one end to P, and connected through the gal-
vanometer, G, with the slider, 8. A resistance of 100 ohms
is inserted in each of the gaps, R R’, and a commutator, K, is
interposed between the two-cell battery, D, and the bridge.
With this arrangement, supposing that the two halves of
the wire A B are of uniform sectional area and in the same
physical condition, and that the various parts of the apparatus
are in fair order and adjustment, there will be a balance when
the slider is near the middle division of the scale. And if
the resistances in the circuit are independent of the direction
of the current, it is clear that the balance will be maintained
notwithstanding that the commutator K be reversed. But
this is found not to be the case.
A series of experiments was made with a copper wire
‘A millim. in diameter (No. 28 B.W.G.), and having a total
- resistance of 2:11 ohms. The commutator was first set so .
that the current through the wire passed up the portion B
and down the portion A (7. e. in the direction BPA), and a
balance was obtained by adjusting the slider. The commu-
tator was then reversed and the current made to pass up A
and down B. This at once destroyed the balance, and in
order to restore it, it was necessary to move the slider several
divisions towards the right. Assuming that the total resist-
ance of the wire remains constant, this result may be explained
by supposing that the reversal of the current is accompanied
by increased resistance in the portion A, and diminished re-
sistance in the portion B. Owing to its vertical suspension,
the resistance of that portion of the wire in which the current
travels upwards is greater than it would be if the wire were
placed in a horizontal position, while the resistance of the
portion in which the current travels downwards is less.
The experiment was repeated with an iron wire of larger
size, its diameter being ‘8 millim. (No. 22 B.W.G.). With
this the effect of reversal was smaller ; but it was well marked,
and of the opposite nature to that observed in the former case.
The readings obtained in the two series of experiments are
given in the following Table :—
of Vertically-suspended Wires. 501
Copper Wire.
Scale-readings.
Number of
experiment. Difference.
Current direct. | Current reversed.
1 569 633 — 64
2 567 | 637 | —70
3 595 | 651 —56
Mean difference ...... —63'°3
|
|
|
|
|
1 780 770 | 410
2 760 | 748 | p12
5 759 | 748 Naess
Mean difference ...... +11 |
I believe these effects are associated with certain thermo-
electric phenomena discovered by Sir William Thomson. In
his famous Bakerian lecture, published in the Philosophical
Transactions for 1856, he showed that if a stretched copper
wire is connected with an unstretched wire of the same metal
and the junction heated, a thermoelectric current will flow
from the stretched to the unstretched wire through the hot
junction ; while, if the wires are of iron, the direction of the
current will be from unstretched to stretched. It follows,
therefore, from the laws of the Peltier effect, that if a battery-
current is caused to flow from a stretched to an unstretched
wire, heat will be absorbed at the junction when the metal is
copper, and will be developed at the junction when the metal
is iron: and if the direction of the current is reversed the
thermal effects will also be reversed.
Now a vertically suspended wire is unequally stretched by
its own weight, the stress gradually increasing from zero at
the lowest point to a maximum at the highest. Any small
element of the wire is more stretched than a similar element
immediately below it, and less stretched than a neighbouring
502 «= Mr. S.. Bidwell on the Electrical Resistante
element just above it. Thus a current of electricity, in pass-
ing from the lowest to the highest point of such a wire,
is always flowing from relatively unstretched to relatively
stretched portions. If, then, the wire were of copper, heat
would be evolved throughout its whole length ; the tempera-
ture of the wire would rise, and its resistance would conse-
quently be increased. With a current flowing from top to
bottom, the temperature of the wire would fall and its resist-
ance diminish. So also an iron wire would be cooled and
and have its resistance lowered by an upward current, while
a downward current would heat it and increase its resistance.
The changes of resistance are thus, as I believe, proximately
due to changes of temperature. |
The resistance of the bridge-wire used in my experiments
was ‘244 ohm, and, as already mentioned, an additional resist-
ance of 100 ohms was placed in each of the gaps adjoining
the bridge-wire. Denoting the resistance of the half A of
the suspended wire by a, and that of B by 6, we have, from
the first experiment with the copper wire (the result of which ~
agrees closely with the mean) :—
For direct current,
a 100° + °569 x *244°
O 100% AS 1 x c244e
Dis 1001389
~ 100105
Also
a ab — alien
Hence
aia 2:
b=1:0548208°.
For reversed current,
a_ 100°+°633 x :244°
b 100° 4-367 x 244°
_ 100154
— 100090
* Ofcourse the resistances are not really measured to the high degree
of accuracy suggested by these figures; but any small error of excess or
defect would be approximately the same for the two values of a (with
direct and reversed currents) and would not materially affect their differ-
ence, to which alone importance is attached.
a
| of Vertically-suspended Wires. 508
And, as before, |
a+b=2:11".
Hence
a=1-0553372° *,
b=1°0546628".
When therefore the current was reversed, the value of a
was increased by
1:0553372 —1:0551792 ohm
='(00158 ohm.
This is equivalent to about 16 thousandths per cent.
Assuming that a change of temperature of 1° C. produces
an alteration of -4 per cent. in the resistance, it follows that
the temperature of the copper wire was = degree C. higher
with an upward than with a downward current.
The current traversing the wire was not measured, but it
was probably about 1 ampere.
It will be seen from the figures in the Table, that the
changes which occurred in the resistance of the iron wire
were considerably smaller than those observed in the case of
copper. This was unexpected, since the thermoelectric effects
are, I believe, somewhat greater withiron. But the apparent
anomaly is obviously to be accounted for, at least in part, by
the higher specific resistance of iron. With the same electro-
motive force the current per unit of sectional area would be
six or seven times greater in copper than in iron, and the
Peltier effect is proportional to the current. To render the
results in the two cases strictly comparable, other less impor-
tant differences, such as those of specific heat and radiating-
power, would have to be taken into account.
If a convenient opportunity offered it would be satisfactory
to repeat the experiments with much longer wires, such as
might be suspended in the shaft of a coal-pit or in a shot-
tower. The effects hitherto observed are so small that they
might possibly be due to accidental causes, and I publish this
account of them with some diffidence.
* See note in preceding page.
Fo goa 4
LIX. The Evolution of the Doctrine of Affinity.
By Professor Lornar Meyer, of Tiibingen”.
aL may not be amiss, on the issue of a Journal { specially
devoted to the theoretical and physical aspects of
Chemistry, to take a rapid survey of the development of the
doctrine of chemical affinity, a correct knowledge of which is,
and must ever remain, the most important object of the theory
of our science.
The former doctrines of affinity, conceived without know-
ledge of the laws of chemical combination, reached their acme
in Berthollet’s teaching, which united all previous investiga-
tions and speculations into a compact theory.
The basis of Berthollet’s conception was his statement that
the chemical action of every substance must be proportional
to its active mass and to a constant depending on its nature,
and named by him Affinity, except in so far as external con-
ditions (e.g. temperature, state of aggregation, solubility,
volatility, and so on) acted as retarding or accelerating causes.
The doctrine of Berthollet is now fully recognized, although
it “was for long ignored or forgotten. This unfortunate
neglect is explicable when it is remembered that, along with
the most illustrious of his contemporaries, he committed the
error of supposing that the capacity for saturation was a mea-
sure of affinity. Sir Humphrey Davy had, indeed, shown that
this assumption led to not a few difficulties ; but it was first dis-
proved by the brilliant experimental development by Berzelius
of Richter’s ‘‘ Stoichiometry ”’ and Dalton’s Atomic Theory.
That, in consequence of this disproof of an unimportant and
incidental addition to the experimentally correct doctrine of
Berthollet, his doctrine should have almost been forgotten, and
have been completely neglected, would appear inconceivable,
if we did not consider the enormous influence exercised by
Berzelius on the growth of Chemistry. He united to an acute
perception of the most minute peculiarities in the behaviour of
chemical substances, and the most refined choice of analytical
and synthetical methods, a special talent for systematic
arrangement of facts discovered from day to day by himself
and by his students. All theoretical views were employed by
him in support of his system ; indeed, he accepted none unless
it proved of assistance in his endeavour to perfect his mar-
vellous arrangement of the chemical elements and their com-
pounds. He was indifferent to theoretical speculations which
* Translated and communicated by Professor William Ramsay.
' + Zeitschrift fiir physikalische Chenue, edited by W. Ostwald and
J. van’t Hoff (Riga and Leipsig).
Evolution of the Doctrine of Affinity. 505
did not seem to further his great work; while he offered a
most strenuous opposition to all those which he conceived
would bring disorder into his classification. He even disputed
for half a generation Davy’s discovery of the elementary cha-
racter of chlorine, simply because he could not reconcile it
with his views.
But the discovery in the earlier part of this century of the
relations between the electrical and the chemical behaviour of
elements and compounds appeared to him to afford great
assistance in the development of his system; and hence he
based his whole classification on positive and negative cha-
racters of substances, manifested electrically ; and for a time,
at least, he identified affinity with electrical attraction.
Alongside of this electrochemical hypothesis, every other
doctrine of affinity appeared superfluous; and, as a con-
sequence, Berthollet’s teachings were forgotten, although they
were by no means contrary to the newer views. The electro-
chemical theory of Berzelius, however, was never fully deve-
loped in detail. Hven though he laid great stress on it, though
he often referred to it, and insisted on its fundamental nature,
yet there is not to be found in any of his numerous memoirs
in which it is mentioned, nor in his Jahresbericht, in which
he criticised the electrochemical theories of other investi-
gators, nor even in any one of the numerous editions of his
Textbook, an attempt at a complete exposition of his theory.
In actual fact, the electrochemical theory never rose above
the general conception that the chemical and electrical
behaviour of bodies are closely connected. The explanation
was only an apparent one: it consisted only in ascribing to
electrical causes observed chemical facts. An attempt to
measure affinities on such a basis failed, owing either to the
lack of experimental data or to its being contradicted by
them.
Hrroneous deductions from his theory misled Berzelius,
not only in causing him to disbelieve Davy’s proof of
the elementary nature of chlorine, but also in leading him
vigorously and persistently to dispute Faraday’s electrolytic
law. While he withdrew from his opposition to Davy after
a sixteen years’ struggle (1826), when the analogy between
hydrogen chloride and hydrogen sulphide had been fully
recognized, he continued to reject until his death that most
important of all electrochemical discoveries, Faraday’s law.
These two facts serve sufficiently to show that Berzelius’s
theory was unable to yield a thorough explanation of affinity.
That in spite of such weak points, ‘sufficiently evident today,
a man of Berzelius’s great power could hold fast to them
Phil. Mag. 8. 5. Vol. 23. No. 145. June 1887. 2M
506 Prof. L. Meyer on the
throughout his whole life, and, moreover, impress others with |
their truth for many years, was a consequence of the enormous
benefit which his systematic arrangement conferred on che-
mical science. Hven his system died, as soon as its incom-
petence to classify organic compounds became manifest ; and
the chemical world looked on its departure with indifference.
From this period, however, the speculations of chemists
ran in quite a new channel. It was not so necessary to
investigate the mode of action of the forces of affinity, as to
prepare and examine the wonderful forms of combination
which by its influence the atoms could be induced to assume
in organic compounds. _ This work for long absorbed the
attention of chemists. The obstinate battles fought over the
laws governing the linkage of atoms are still so fresh in the
minds of at least the older of the present generation of
chemists, that they need not here be more than mentioned.
During this contest regarding the constitutional formule
of organic compounds, a complete revolution of the doctrines
of affinity was in progress, which was prompted chiefly by
facts in inorganic Chemistry. Chemists had persistently clung
to the assumption, long before proved untenable, that heat
was a form of matter capable of entering into combination with
other forms of matter to produce chemical compounds. Al-
though Rumford, at the end of last century, had proved that
heat is a mode of motion (a view held even in the 16th and
17th centuries*), and Davy had furnished a brilliant con-
firmation of his proof, yet even up to the middle of this century
it was stated in the most widely-read textbooks of Chemistry
that heat, light, and electricity are to be regarded as impon-
derable forms of matter. It must be noted that, though
rejected by all journals of Physics, Julius Robert Mayer’s
treatise received compensation, oddly enough, by finding
a resting-place in Liebig’s ‘ Annals of Chemistry’. With
the recognition of the importance of the mechanical theory
of heat arose the hope that, by its help, our knowledge of the
doctrine of affinity might be materially advanced.
The view was at once suggested that, just as a heavy body,
in consequence of the mutual attraction between it and the
earth, moves towards the earth with accelerated velocity,
thereby converting potential energy, due to its elevated
position, into kinetic energy, so the atoms, as a consequence
of their affinity, move towards each other, converting their
affinity into energy of motion, which, as a rule, is manifested
in the form of heat. According to this doctrine, the heat
* Bacon, Novum Organum, Lib. ii. Aph. xx.
+ Ann. Chem. Pharm, 1842, vol. xlii. p. 233.
Evolution of the Doctrine of Affinity. 507
evolved by the sum of the impacts should bea measure of the
affinity. But, from the first, difficulties have to be surmounted
in accepting this view. One of the greatest is that it seldom
occurs that a compound is formed through the union of isolated
atoms ; in almost all cases the atoms themselves have to be
liberated from compounds iu which they have existed in a
state of combination. As this liberation must be accompanied
by gain of energy (i.e. by absorption of heat), while the
formation of the new compound gives rise to loss of energy
(i.e. a heat-evolution), it happens that, as a rule, only the
difference between the absorption and emission is manifested
externally. Hence this doctrine of affinity can deduce from
observations, not the absolute value of either affinity, but only
the amount by which the one is greater than the other.
Many other difficulties present themselves; especially the
fact that, along with chemical changes, physical changes
(alteration of the state of aggregation, of the volume, and so
‘on) occur simultaneously, and are themselves accompanied by
an emission or by an absorption of heat. Moreover, thermo-
chemical experiments are by no means easy of execution, and
are subject to many sources of error; hence it is not to be
wondered that slow progress was made in developing the
thermochemical doctrine of affinity. The more numerous
the observations, and the greater their accuracy, the greater
the number of instances in which theory and experiment
failed to display coincidence. It has been frequently observed
that a much larger evolution of heat occurs on neutralizing an
evidently weak acid than a strong one capable of expelling
the weak one more or less completely from its compounds
with bases. ‘The expulsion, in such cases, is attended by an
absorption, not an evolution, of heat. Similar facts have also
been noticed in nota few other chemical reactions, which must
be, and have been, regarded as produced by the action of
affinity; e.g., the formation of the ethereal salts of organic
acids by their action on the alcohols*. Many attempts have
been made to explain reactions which are accompanied by
negative heat-changes, and to bring them into unison with
the thermal doctrine of affinity. But as all such attempts
have been unsuccessful, the fundamental hypothesis of the
doctrine loses much of its probability. And consequently its
most ardent supporter, M. Berthelot, has relinquished his
assertion that, by the action of the affinities of all the substances
_ partaking in a reaction, those compounds are always formed
which are accompanied by the greatest evolution of heat. He
has modified his statement to this: that there is present “a
* J. Thomsen, Thermoch. Untersuch. iv. p. 388,
2M 2
508 Prof. L. Meyer on the
tendency towards (tends vers la) the production of such states
of combination.””* But even in order to reconcile this state-
ment with reactions distinctly accompanied by a negative
heat-change, far-fetched and artificial, explanations of an
unsatisfactory nature are necessary. Nevertheless, the funda-
mental hypothesis—that the heat of combination is, in reality,
affinity transformed into kinetic energy—might have passed
as true for a much longer time, had not the progress of
thermo-chemical research shown it to be thoroughly un-
tenable.
It will be remembered that Julius Thomsen} made use of
the positive or negative heat-change accompanying a chemical
reaction to determine the extent to which the reaction had
proceeded ; and this was legitimate, owing to the fact that the
heat-change is proportional to the quantity of matter which
has altered its form of combination. While investigating
the expulsion of acids from their salts in dilute aqueous solution
by other acids, the very remarkable observation was made that
that acid is by no means always the stronger which evolves
the greatest heat on neutralization. For example, although
sulphuric acid when neutralized in dilute aqueous solution
gives rise to an evolution of heat surpassing by three thousand
units that furnished by an equivalent amount of nitric or
hydrochloric acid, yet it is only half as strong an acid as the
latter ; that is, if equivalent amounts of nitric and sulphuric
acids be mixed with an amount of caustic soda equivalent to
one of them, the sulphuric acid enters into combination with
only half as much soda as the nitric acid ; so that one third of
the nitric acid remains in the free state, while two thirds of
the sulphuric acid is free. There can be absolutely no doubt —
that nitric acid is by far the stronger acid, although, judging
from the thermal theory of affinity, sulphuric acid should be
the stronger.
While Thomsen was prosecuting his researches, it was
generally held that the evolution of heat was an absolute mea-
sure of affinity ; hence Thomsen devised the term “ avidity ”’
to express the “ tendency of an acid towards neutralization.”
But this is nothing else than the real affinity of the acid
towards the base, labelled with a special name to avoid con-
fusion. Ostwald t, who confirmed and extended Thomsen’s
researches by wholly different methods, named this quantity
“relative affinity.” :
It appeared, from the investigation of a great number of
* M. Berthelot, Essaz de mécunique chimique, i. p. 421.
+ Thermoch. Unters. 1. p. 97 et seg.
{ T. pr. Chem. 1877, xvi. p. 385; xviii. p. 328,
Ewolution of the Doctrine of Affinity. 509
acids, that there was absolutely no connexion between avidity
or relative affinity and heat of neutralization. Hven the order
of magnitude of the two, when a number of instances is com-
pared, is entirely different. The strongest of all acids—nitric
acid—occupies only the nineteenth place among forty acids,
when they are arranged in the order of their heats of neu-
tralization ; while hydrofluoric acid evolves most heat on
neutralization, although its avidity is only one twentieth of
that of nitric acid ; and so with other instances. As it would
be absurd to ascribe the greatest affinity to a base to an acid
which is in great part expelled by another, it must be acknow-
ledged that the fundamental hypothesis of the thermal doctrine
of affinity is not justified by fact.
But a further conclusion follows from Thomsen’s investi-
gations, namely, that, while the heat of formation of compounds
depends on the nature of their constituents, it does not, at
least in many cases, depend on any special change, attraction,
or affinity in which both constituents are concerned. This is
most easily seen when the heat of formation of salts from strong
bases and acids is considered. If, as Thomsen has experimen-
tally shown, the total amount of heat evolved during the process
of formation of a salt from the elements which it contains, and
its solution in a large quantity of water be measured, the
extremely remarkable fact is to be noticed that a definite dif-
ference in composition involves a similarly definite difference
in heat of formation, varying only within very narrow limits*.
The heat of formation of a salt of lithium is, for example, in
round numbers, always 11,400 calories greater than that of a
salt of sodium with the same acid, and about 2000 calories
greater than that of a salt of potassium; and so for other
metals. This has been proved for the chlorides, bromides,
iodides, hydrates, hydrosulphides, sulphates, dithionates, and
nitrates of nineteen metals. If, on the other hand, the metal
remain the same, but the acid radical be varied, a definite
difference in the heat of formation is again to be observed for
each acid radical. That of the bromides is always about
21,800 calories less than that of the chlorides, and that of the
iodides 52,300 less; while the chlorides invariably evolve
during their formation about 200,000 calories less than the
corresponding sulphates. The heat of formation of every salt
may therefore be represented as the sum of certain numbers,
each of which numbers is peculiar to one constituent group or
element, and remains constant into whatever form of combi-
nation that element or group enters. It is therefore possible,
* Thomsen, Thermoch. Untersuch. iii. pp. 290, 456, 545; see also
Lothar Meyer’s Moderne Theorien der Chemie, 5th edition, p. 448.
|
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510 Prof. L. Meyer on the
so soon as these constants have been definitely determined for
each constituent group or element, to calculate the heat of
formation of the salt in a manner similar to that by which the
molecular weight of a compound may be deduced from the
atomic weights of its constituents.
This simple relation could not hold were the heats of for-
mation of salts affected by the affinity of the constituents for
each other ; for in such a case each constituent would contri-
bute so much the more to the heat of formation the greater
the affinity between it and the other constituent with which
it combined. We must therefore believe that the evolution of
heat, developed by the formation of a compound, results solely
from the change of condition of the constituent elements. Similar
laws, as Ostwald has shown, apply also to other changes of
condition, e. g. to expansion or contraction, or to change of
optical properties which accompany the formation of salts.
These also can be calculated by simple addition; for their
constituent numbers are constants belonging to each one of the
reacting substances, and are independent of the nature of the
other.
The thermal theory of affinity has a changed complexion
owing to this discovery. What was formerly attributed to
the mutual action of several substances must now be regarded
as change of condition of each individual substance, each being
wholly independent of all others with which it may combine.
The heat-change accompanying a chemical change must no
longer be regarded as the conversion of potential energy into
kinetic energy, owing to the mutual attraction of the atoms ;
but it must be concluded that each substance, each atom, each
compound, possesses its own peculiar store of available energy,
capable of being increased or diminished by its entering into
reaction, or by any change of condition. But this store of
energy and its changes must in nowise be confounded with
affinity—that is, with the reason of chemical change. For
the amount of energy lost by a substance during reaction
depends solely on its own nature, and on the kind of change
which it undergoes; not on the nature of the substance
through whose influence this change is produced. ‘To employ
an old simile, affinity acts like the spark to the powder ; like
the trigger which releases the weight by which, under the
action of gravity, the pile is driven in. Just as in these and
similar cases there is no proportionality between the effective
cause and the final result, so also with chemical changes.
Such considerations lead naturally to the old question re-
garding the necessity of imagining any special affinity or
attracting-power between atoms. And the more proofs are
Evolution of the Doctrine of Affinity. 511
furnished that not only the supposed affinity, but even the
actually measured avidity, is an inherent property of each
separate kind of matter, independent of any reaction with any
other kind of matter, the more doubtful is the necessity. The
more recent investigations of Ostwald* have shown that the
most heterogeneous actions of acids, in influencing the chemical
changes of various substances,—as, for example, in decompo-
sing the amides, in forming the ethereal salts, in inverting the
sugars, and moreover in influencing the electrical conduc-
tivity,—are all dependent on the same constant, the affinity or
avidity. If the ability of acids to act remain the same in rela-
tion to so many different phenomena, the assumption appears
justified that it is caused, not by mutual action, by attraction
of one kind or another, but is in reality something peculiar to
the nature of the acids themselves.
There might be a temptation to believe that, in relinquish-
ing the hypothesis of an attractive force between the atoms,
we must also relinquish the possibility of any definite con-
ception of the influences of the nature of reacting bodies in
determining chemical changes. But this is by no means
the case. For just as it was formerly supposed that the heat
liberated during an act of combination was the equivalent in
kinetic energy of so much potential energy due to the attrac-
tion of the atoms, it is open to regard the atoms as particles
in rapid motion, but devoid of attracting-power, the whole of
whose store of energy consists in this motion, and is therefore
kinetic; and it may therefore be assumed that such atoms
may unite to form molecules, or that such molecules may
otherwise react, owing to some as yet undiscovered relation
between their modes of motion and velocities. Itis of course
unnecessary to picture to ourselves attractive forces. They
may or may not be conceived, but they are of no great im-
portance to science. For my part I believe that a less
restricted and prejudiced view of the facts is to be attained
by abandoning the hypothesis of mutual attraction between
atoms, and avoiding all reference to the unnecessary distinc-
tion between the potential and the kinetic energy of the atoms.
It may fall hard on many who have devoutly believed in
the thermal theory of affinity, exalting it high above all facts,
to see it dethroned ; perhaps here and there some will refuse
to abandon it, like Berzelius with his electrochemical theory,
chiefly prompted by the fear that when it is gone the kingdom
of chaos, so painfully conquered, may againarise. Yet things
* “Studien zur chemischen Dynamik,” Journ. prakt. Chem. xxvii. p.1;
xxvii. p. 449; xxix. p. 385; xxxi. p. 807. ‘ Hlectrochemische Studien,”
Ibid. xxx. p. 225; xxxi, p. 483; xxxil. p. 300; xxxiii. p. 352,
i)
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512 Ewolution of the Doctrine of Affinity. ‘
are not in so dangerous a plight as might at first appear.
The thermal doctrine of affinity has had no more real influence
on the steady experimental evolution of Chemistry than had
the electrochemical theory of Berzelius. Both long held
honourable places as great general truths ; attempts have been
made with both to apply them to experimental details ; but,
as frequently happens, theory and experiment did not agree,
the theory has been calmly ignored, and we must trust to the
future to make things plain. If, once again, a theory has
unexpectedly proved untenable, once again the old course of
events will be repeated ; attempts at generalization have been
too soon made: “gestit enim mens exsilire ad magis gene-
ralia ut acquiescat’’ (Bacon). Theory has attempted to
precede fact; it has pursued a false path, and must wait
until fact with quiet progress shows the way.
Such is our present state of knowledge. But long before
the thermochemical doctrine of affinity became untenable, the
efforts of investigators had been directed to ascertaining the
conditions on which chemical reactions depend, such as the
influence of time, of temperature, of mass, and of solvents ;
and to the measurement of the resulting changes in volume,
in evolution of heat, and similar phenomena. Now that ex-
periment has shown the fallacy of an attempt to deduce
chemical change from the fundamental principles of thermo-
chemistry, we hail with joy the appearance of a new, really
kinetic, doctrine of affinity, which, quietly and unostenta-
tiously making its way along the road of induction, holds out
to us the prospect of a real knowledge of the essential nature —
of chemical change. By its help also those numerous thermo-
chemical observations, which were unable to lend support to a
onesided theory, for which they furnished the sole basis,
acquire for the first time their true meaning when viewed in
connexion with all other phenomena accompanying chemical
change. Thus, although one illusion more has been dissi-
pated by this new evolution of the doctrine of affinity, yet
science is enriched by the acquisition of a less hypothetical
and more far-reaching and inclusive conception of the nature
of chemical combination.* |
* A short paper by Mr. Clarence A. Seyler, ‘On the Thermal Equi-
valents” of some elements and groups, has been published in the ‘Chemical
News’ of April 1, vol. lv. p. 147.
[ 513 ]
LX. Contributions to the Theory of the Constitution of the
Diazoamido-Compounds. By RapHaEL Meupota, F.R.S.,
F.I.C., F.C.S., Professor of Chemistry in the City and Guilds
of London Institute, Finsbury Technical College™.
ie a series of investigations upon the diazoamido-compounds
which have been carried out by me in conjunction with
Mr. F. W. Streatfeild{, certain facts have been brought to
light which are quite inexplicable by any of the formule at
present in use; and it has therefore become necessary to
reconsider the whole question of the chemical constitution of
these interesting derivatives, which have taxed to the utmost
the ingenuity of all those chemists who have concerned them-
selves with their investigation.
The formula which up to the present time has been generally
adopted is due to Kekulé{, and is thus expressed in general
terms :— |
x) N,ANE CY,
X and Y being similar or dissimilar radicals. The chief
objection to this formula has hitherto been its asymmetrical
character, which renders it incapable of explaining the remark-
able observation of Griess§, which is now confirmed from
many sides, viz. that the mized diazoamido-compounds, in
which X and Y are dissimilar, are always identical whichever
radical is first diazotized. This difficulty has been to some
extent met by the suggestion of Victor Meyer||; and ina
former paper by Mr. Streatfeild and myself{] it was shown
that the results of our investigations, as far as these had been
carried, could be explained by means of this hypothesis of the
formation of intermediate additive compounds with a consi-
derable show of reason.
The extension of our work has, however, forced me to the
conclusion that Kekulé’s formula does not adequately express
all the known facts concerning the diazoamido-compounds ;
and if this formula is, as I believe it must be, abandoned, the
supplementary hypothesis is also rendered unnecessary.
The evidence which has led to the present theoretical
* Communicated by the Author.
+ Journ. Chem. Soc., Trans. 1886, p. 624; and 1887, p. 102.
{ Lehrbuch d. org. Chem. vol. ii. pp. 689,715, and 741; Zeit. f. Chem.
1866, pp. 308, 689, and 700.
§ Ber. deut. chem. Gesell. vii. (1874), p. 1619.
| Ibid. xiv. (1881), p. 2447, note.
4] Journ, Chem, Soc., Trans. 1887, p. 116.
514 Prof. R. Meldola on the Theory of the
discussion is briefly as follows :—By the action of diazotized
metanitraniline upon paranitraniline an unsymmetrical diazo-
amido-compound is obtained, which has a melting-point of
211°. The same compound is obtained by reversing the order
of combination, % e. by acting upon metanitraniline with
diazotized paranitraniline. According to Kekuleé’s view, this
substance could have only one of the two formule :-—
I,
(p) NO, e C,H, ° N, .NH. C,H, ° NO, (m),
II.
(m) NO,.C,H,.N,.NH.C,H,. NO, (p).
By replacing the H-atom of the NH-group by ethyl an
ethyl-derivative (of m.p. 148°) is formed; and this, on the
same theory, could have only one of the two corresponding
formulee :-—
TI, ‘
(p) NO,.C,H,.N,.N(C,H;) . C,H, . NO, (m),
IV.
(m) NO, ° C,H, ° N, . N(C.H;) ° C,H, ° NO, (p).
If the unsymmetrical compound had the formula L., its
ethyl-derivative (II1.) might have been expected to be iden-
tical with the compound produced by the action of diazotized
paranitraniline upon ethylmetanitraniline ; if it had the for-
mula II., its ethyl-derivative (1V.) might have been expected
to be identical with the compound produced by the action of
diazotized metanitraniline upon ethyl-paranitraniline. As -
a matter of fact, it has been found that the ethyl-derwative of
the unsymmetrical compound is identical with neither of the
compounds prepared by the action of the diagotized nitranilines
upon the ethyl-nitranilines. We have therefore to allow the
existence of three isomeric ethyl-derivatives containing para-
and metanitraniline residues, a fact which cannot be repre-
sented by Kekulé’s formula. The properties of these and all
the allied compounds prepared by us in the course of the
inquiry are summarized in the following Table :—
515
Constitution of the Diazoamido-Compounds.
Compound. Melting-point. Decomposed by cold HCl into
1. Action of diazotized p-nitraniline A mixture of p- and m-nitrodiazo-
upon m-nitraniline, or of diazotized 211° benzene-chlorides and m- and
m-nitraniline upon p-nitraniline ...... | p-nitranilines.
2. Action of diazotized p-nitraniline 993° p-nitrodiazobenzene-chloride and
WPOM! P=OUCAMUMMNG ws ene ence scse ieriell p-nitraniline.
8. Action of diazotized m-nitraniline 194° m-nitrodiazobenzene-chloride and
upon m-nitraniline.........ceceeseeeees és m-nitraniline.
; : “( A-mixture of p- and m-nitrodiazo-
4, Prepared by the ethylation of com- 148° bézené-chlorides “acd sp-” andl
POCOUL CHUN Cpa leenrancgrererrr rrr pricnatce aiden ; na
m- ethylnitranilines.
5. Prepared by the ethylation of com-
pound No. 2, or by the action of di- 191°-192° p-nitrodiazobenzene-chloride and
azotized p-nitraniline upon p-ethyl- p-ethylnitraniline.
valli halla: Bea Sneha pe Un Habe hoen ase nee
6. Prepared by the ethylation of com-
pound No. 38, or by the action of 119° m-nitrodiazobenzene-chloride and
diazotized m-nitraniline upon m- m-ethylnitraniline. :
ethylnitraniline ......... eaigthinwsyuveeiees
7. Prepared by the action of diazotized 11749-1759 { m-nitrodiazobenzene-chloride and.
m-nitraniline upon p-ethylnitraniline i p-ethyInitraniline.
8, Prepared by the action of diazotized 187° p-nitrodiazobenzene-chloride and
p-nitraniline upon m-ethylnitraniline* m-ethylnitraniline.
Decomposed by hot HCl into
A mixture of p- and m-nitrochlor-
benzenes and p- and m-nitranilines,
p-nitrochlorbenzene and p-nitrani-
line,
m-nitrochlorbenzene and m-nitrani-
line.
A mixture of p- and m-nitrochlor-
benzenes and p- and m-ethylnitra-
nilines.
—
p-nitrochlorbenzene and p-ethylni-
traniline.
m-nitrochlorbenzene and m-ethylni-
traniline.
—_—— —_—
m-nitrochlorbenzene and p-ethylni-
traniline.
. ee
p-nitrochlorbenzene and m-ethylni-
traniline.
* Owing to the fact that this compound was distinct in appearance from the other ethyl-derivatives (Nos. 4, 5, 6, and 7) we were at first
led to suppose that it was an amidoazo-compound,
and that the unsymmetrical compound (No. 1) accordingly had
the formula I., the iso-
meric transformation when the metauitraniline was first diazotized being explained by Victor Meyer’s hypothesis (Journ. Chem. Soc. ‘Trans.
1887, p. 116). A more searching investigation has, however, shown that the ethyl-derivative of m.p.
diazo-compound.
187° has all the characters of a true
516 Prof. R. Meldola on the Theory of the
If the ethyl-derivatives Nos. 7 and 8 are formulated on
Kekulé’s type they would have the formule IV. and III.
respectively, and thus no other expression is left for the ethyl-
derivative No. 4.
The conditions to be fulfilled by any formula proposed for
the diazoamido-compounds are, therefore, (1) that it should
be symmetrical so as to represent the identity of mixed diazo-
amido-compounds, and (2) that it should be capable of repre-
senting more than two isomeric alkyl-derivatives of mixed
compounds. These conditions are certainly not met by the
formula now in use ; and the objections which apply to this
apply also to the alternative formula proposed by Strecker*:-—
X.N.NH.Y
This formula fails to explain the existence of more than two
isomeric alkyl-derivatives of the unsymmetrical (mixed) com-
pounds ; and is even less able than Kekulé’s of representing
the identity of mixed compounds, since it is incapable of the
rearrangement suggested by Victor’ Meyer.
The first symmetrical formula proposed to explain the iden-
tity of mixed diazoamido-compounds is due to Griesst, the
discoverer of these compounds, who suggested that diazoamido-
benzene and its analogues should be written according to the
type :—
uf C,H. —N—N—N—C,H,
lobe oleae el
This formula certainly explains the identity of mixed com-
pounds, but is otherwise open to certain objections; since in
the first place it represents diazoamidobenzene as a phenylene
derivative, and in the next place it shows the presence of
three N H-groups containing three replaceable hydrogen atoms.
All our experiments upon the salts and alkyl-derivatives of
the dinitrodiazoamido-compounds have shown, however, that
only one replaceable H-atom is presentt. This formula,
moreover, is not capable of explaining the easy resolution of
* Ber. deut. chem. Gesell. iv. (1871), p. 786 ; Erlenmeyer, 2bed. vii. (1874),
p- 1110, and xvi. (1883) p. 1457. Also Blomstrand, ibzd. viii. tasty
Ol,
, + Ber. deut. chem. Gesell. x. (1877), p. 528.
{ These compounds give only monalkyl-derivatives; and the same
appears to be the case with diazoamidobenzene, according to Messrs.
Friswell and Green (Journ. Chem. Soc., Trans. 1886, p. 748), to whom I
communicated the method of alkylization in the course of conversation,
and who applied it to this compound successfully.
Constitution of the Diazoamido- Compounds. 517
diazoamido-compounds by acids, nor the production of mixed
products from mixed compounds (see the foregoing Table).
Another symmetrical formula has been proposed by Victor
Meyer™, viz.:—
X—N——N—Y
AN
H
but this was abandoned by him as having but little probability.
One of the greatest objections to this formula is that it fails to
represent the N-atom which is attached to the replaceable
H-atom as being also directly attached to one or the other of
the aromatic radicals. The decomposition of the ethyl-
derivatives of the dinitrodiazoamido-compounds by acids
shows that this mode of attachment of the NH-group certainly
exists (see the foregoing Table).
In the course of the present investigations another symme-
trical formula has suggested itself, which may be here given:—
H ui
KX _N—y or %—N—-Y
aS N
N
This formula does not, however, appear to me to have any
probability, as it fails to explain the decomposition of the
diazoamido-compounds by acids, or the existence of isomeric
alkyl-derivatives. Moreover, the formula of diazoamidoben-
zene written on this type :—
H
C,H; e N e C,H;
eX
N=N
would indicate a close relationship between this substance and
the remarkably stable diphenylamine. The latter is not
found, however, among the reduction-products of diazoamido-
benzene ; and there is no experimental evidence of any kind
in favour of such a relationship.
Before proceeding to put forward my own views upon the
constitution of these compounds it will be desirable to take a
* Ber. deut. chem. Gesell. xiv. (1881), p. 2447, note.
518 Prof, R. Meldola on the Theory of the
general view of their characters, so as to gain a clear notion
of all the conditions whicb have to be fulfilled by any proposed
formula. These characters are summarized below, those com-
pounds containing similar radicals being spoken of as “normal”
compounds, and those containing dissimilar radicals as “‘mixed”’
compounds :—
(1) Normal compounds are prepared by diazotizing an
amine, X.NHb., and acting with the diazo-salt upon another
molecule of the same amine, X . NH4, or, what amounts to the
same thing, one molecule of nitrous acid may be made to act
upon two molecules of X . N Hg.
(2) Mixed compounds are obtained by diazotizing an amine,
X .NH,, and acting with the diazo-salt upon one molecule of
another amine, Y.NH,. The same compound results if the
order of combination is reversed.
(3) The diazoamido-compounds, both mixed and normal,
contain one atom of hydrogen easily replaceable by metals
and alkyl radicals. If the aromatic radicals contain strongly
acid groups (such as NO,), the resulting diazoamido-com-
pounds may be distinct monobasic acids.
(4) Normal compounds are resolved by acids into their
constituents, the diazo-salt and amine.
(5) Mixed compounds are resolved by acids into a mixture
of the two bases from which they are derived, and a mixture
of the two diazo-salts corresponding to these two bases.
(6) Alkyl derivatives of normal compounds may be pre-
pared in two ways:—
a. By the action of a diazotized amine, X.NH,, upon
the alkylamine of the same base, X. NHR.
8. By the direct alkylization of the normal diazoamido-
compound.
(7) The alkyl-derivatives of normal compounds are decom-
posed by acids into their constituents, the diazo-salt and
alkylamine.
(8) Alkyl-derivatives of mixed diazoamido-compounds are
formed by the direct alkylization of these compounds (see
group 2).
(9) Another group of mixed alkyl-derivatives can be pre-
pared by the action of a diazotized amine, X . NHg, upon the
alkyl-derivative of a dissimilar amine, Y. NHR. These com-
pounds are isomeric with those of the preceding group.
(10) Mixed alkyl-derivatives of group (8) are resolved by
the action of acids into a mixture of the two diazo-salts and
the two alkylamines.
pe I ce tsi ci
Constitution of the Diazoamido- Compounds. 519
(11) Mixed alkyl-derivatives of group (9) are resolved by
acids into their constituents, the diazo-salt and the alkylamine,
but not into a mixture of diazo-salts and alkylamines, as is
the case with the compounds of group 9*.
(12) Normal compounds, by the action of weak reducing
agents, are reduced to the original amine X.NH,, and the
hydrazine X.N.H;. Mixed compounds give, on reduction,
the base X .NH, and the hydrazine Y.N.,H;, or the base
Y .NH, and the hydrazine X . N,H3f.
(13) Alkyl-derivatives of normal compounds reduce to the
hydrazine X.N.H; and the alkylamine X.NHRf. Alkyl-
derivatives of mixed compounds give, on reduction, the
hydrazine X- or Y . N.H3;, and the alkylamine Y- or X . NHR.
This production of alkylamines indicates that the N-atom
which is in combination with the alkyl-radical is also attached
to the aromatic nucleus ||.
From the foregoing summary it will be seen that the mixed
diazoamido-compounds and their alky!-derivatives display the
most striking characters, and are of special importance to the
present discussion, because it is in the attempt to formulate
these compounds on Kekulé’s plan that the greatest difficulties
are encountered. In view of the objections which apply to
all the formulz hitherto proposed it has been no easy matter
to suggest any alternative formula ; but I believe that the true
solution of the problem will be arrived at by regarding phenyl
as a triatomic radical, C,H;!, instead of monatomic, as has
always been assumed in previous formule. This suggestion
is in accordance with Fittig’s theory of the constitution of
* To this class, in addition to the ethyl-derivatives of m.p. 174°-175° and
187° (Nos. 7 and 8 in the Table), belong the two following compounds :—
(1) produced by the action of diazotized p-toluidine upon ethylaniline,
and (2) prepared by the action of diazotized aniline on ethyl-p-toluidine.
These two compounds are zsomeric; the first being decomposed by acids
(hot) into p-cresol and ethylaniline, and the second into phenol and
ethyl-p-toluidine (Nolting and Binder, Bull. Soc. Chim. vol. xlii. p. 341 ;
Gastiger, id. p. 342). . This pair of isomerides is completely analogous
to our two ethyl-derivatives (Nos. 7 and 8), which they resemble in their
mode of decomposition.
t+ Thus the compound produced by the action of diazotized aniline
upon p-toluidine or the reverse gives, on reduction, phenylhydrazine and
p-toluidine (Nolting and Binder, loc. cit. p. 336).
¢ Thus the compound obtained by the action of diazotized aniline on
methylaniline reduces to phenylhydrazine and methylaniline (doc. ctt.).
§ Thus the compound prepared by Gastiger by the action of diazotized
aniline on ethyl-p-toluidine reduces to phenylhydrazine and ethyl-p-
toluidine (Joc. cit. p. 342).
__ || The presence of substituents in one or both aromatic radicals may
interfere with the formation of hydrazines; in such cases the correspond-
ing substituted amines are formed, or, if the substituent is NO,, the cor-
responding diamines.
520 Prof. R. Meldola on the Theory of the
quinone, this compound being regarded by him as a double
ketone :—
Phenylene, according to this view, must be regarded as a
tetratomic radical, and a slight extension of the same view
enables us to consider phenyl as triatomic:—
H
ue ee
&
HC ‘cH HCO ‘OH
lene ens
HC OH HC OH
Ww NA
If this assumption be made, it then becomes possible to
construct formule for the mixed diazoamido-compounds which
meet all the requirements of the case, and which must, there-
fore, commend themselves to the notice of all chemists who,
like myself, have been puzzled to explain the behaviour of
these compounds in accordance with the existing theoretical
notions. The formula now proposed may be written in two
ways:—
N:N N.N
a
mei
Of these two formule I am disposed to attach the greater
weight to the first because it indicates the presence of the
very stable azo-group, —N : N—; and this is in accordance
with the general character of the compounds, which decom-
pose under the influence of acids or of reducing agents in
such a manner that the N-atoms of the azo-group always
remain in combination, either in the form of a diazo-salt or a
hydrazine. On the other hand, the second formula indicates
the presence of the group =—C—N . N=C=; and this might
Constitution of the Diazoamido-Compounds. 521
be expected to split asunder between the N-atoms more
readily on reduction or on decomposition by acids than is
shown to be the case by experiment.
According to the proposed formula the unsymmetrical
compound cf m.p. 211° (No. 1 in the table) and its ethyl-
derivative of m.p. 148 (No. 4 in the table) would be thus
written :—
vee : aN
(p) NO, . CoH, C.H,.NO, (m),
Sy
H
v7 2 NK
(p) NO,. CeHy’ >C.Hy . NO, (m).
<3 Va
CH,
It may now be pointed out how far these formule are in
harmony with the known characters of the mixed diazoamido-
compounds. In the first place it is obvious that the formula
is symmetrical, and thus explains the identity of the com-
pounds irrespective of the order of combination. Putting P
for the p-nitraniline residueand M for the m-nitraniline residue,
this fact may be thus represented without assuming the for-
mation of any intermediate additive compound:—
ae : tine
Reh
H
Consider in the next place the decomposition by hydro-
chloric acid :—
N:N a cc et
B x Si eg ee M
oN N ye N d
H H
If separation took place along the line ab, the products would
Phil. Mag. 8. 5. Vol. 23. No. 145. June 1887. 2N
522 —-~ Prof, R. Meldola on the Theory of the
be p-nitrodiazobenzene-chloride and m-nitraniline ; along cd,
the products would be m-nitrodiazobenzene-chloride and p-
nitraniline. As a matter of fact all four products are obtained,
so that decomposition must take place in both directions. The
same explanation obviously applies to the mixed decompo-
sition-products of the ethyl-derivative. Again, the formula
shows the presence of the one replaceable H-atom in combi-
nation with the N-atom which is attached to the aromatic
radicals. The question of the existence of more than two
isomeric alkyl-derivatives will be considered subsequently.
The formula which has now been suggested for the mixed
diazoamido-compounds derived from p- and m-nitraniline
can be applied with equal success to all other mixed com-
pounds. Thus, to take examples of those compounds whose
products of decomposition have been studied :—
N:N.
Both these compounds were discovered and their decompo-
sition products studied by Griess : the first is obtained by the
action of diazotized aniline upon p-toluidine, or of diazotized
p-toluidine upon aniline; the second is similarly produced
from aniline and amidobenzoic acid. When heated with acids
the first compound gives a mixture of aniline, p-toluidine,
phenol, and p-cresol (Nolting and Binder)* ; and the second
compound under similar circumstances gives aniline, phenol,
oxybenzoic, and amidobenzoic acid (Griess). The products
in these cases indicate separation along both planes of decom-
position ab and cd.
The explanation of the existence of more than two isomerie
alkyl-derivatives of mixed diazoamido-compounds is closely
connected with the question whether the new formula can be
applied to the normal diazoamido-compounds. The following
considerations will show that these compounds cannot be for-
mulated on the new type :—
As a type of the normal compounds let us consider that
derived from p-nitraniline (No. 2 in the table). If this had
the new formula it (and its ethyl-derivative) would have to
be written :—
* This compound reduces to phenyl-hydrazine and p-toluidine, thus
indicating a preferenco of the N,- (and therefore the NH-NH,) group to
remain attached to the more positive radical. The separation is in this
case along ab only.
a —_
Constitution of the Diazoamido- Compounds. 523
N ° oN
(p) NO,. C,H,
Ne
H
Ne N
(p) NO,. on Nott . NO; (p).
ie.
CoH;
Now the ethyl-derivative (No. 5 in the table) is prepared
by the direct ethylation of the compound itself, and also by
the aetion of diazotized p-nitraniline upon p-ethylnitraniline.
Analogy would therefore lead us to suppose that if the ethyl-
derivative had the above constitution, the other ethyl-deriva-
tives, prepared by the action of diazotized p-nitraniline upon
m-ethylnitraniline (m.p. 187°) and of diazotized m-nitraniline
upon p-ethylnitraniline (m.p. 174°-175°), would have a similar
C,H. NOs (p),
constitution :—
N:N
(p) NO, Hewes eet NO,\ (mn),
Se i seh
C,H,
N
OH...
But these two formule are identical with one another and
with that of the ethyl-derivative of m.p. 148°, whereas their
melting-points and mode of decomposition show most con-
clusively that the three compounds are isomeric and not
identical. It must, therefore, be concluded that the formula
how proposed does not apply to the normal compounds, and
the suggestion at once arises whether these may not be the
2N2
524 Prof. R. Meldola on the Theory of the
true representatives of Kekulé’s type. In answer to this I
may point out that, as far as the experimental evidence at
present goes, the normal compounds and the analogous ethyl-
derivatives may be written on Kekulé’s type:—
(p) NO,. CoH. Np. NH . OgH, . NO, (p),
m.p. 223°,
(m) NO, . C,H, .Ny..NH. C,H, . NO, (m),
m.p. 194°,
(p) NO, . C5H,. Nz .N(C,H,) . C,H, . NO, (p),
m.p. 191°-192°,
(m) NO, . CsH, . N;./N(C,H;) . CoH, . NO; (m),
m.p. 119°.
(p) NO, . C,H, . N; - N(C2H;) . C,H, . NO, (m),
mp. VSi?;
(m) NO, . CH, .. Ny ..N(Q,Hg) . CH. NOz (p).
m.p. 174°-175°.
The modes of decomposition of these compounds are ex-
plained by the above formule by supposing the planes of
separation to be along the dotted lines ; and it further appears
that the isomerism of the three ethyl-derivatives of m.p’s 148°,
174°-175°, and 187° (Nos. 4, 7 and 8 in the table) may be
explained by the different formule ascribed to these com- ,
pounds respectively. |
But although Kekulé’s formula may pass muster for the
normal compounds, we are not necessarily reduced to this as
a final expression ; and I am strongly inclined to the belief —
that it will have to be abandoned also in the case of these
compounds. In the first place, as there is a great resemblance
in character between the normal and the mixed compounds,
analogy leads me to suppose that their constitutions are not
so widely different, as appears from the two modes of formu-
lation :—
N:N
Lo ae XN Na
Pe and X.N>,. New
H
Constitution of the Diazoamido- Compounds. 525
In the next place the group —N=N—NH— assumed to
be present, according to the prevailing view, has always
seemed to me to be a most improbable arrangement of N-
atoms, and without any analogy among chemical compounds.
Those compounds in which three combined N-atoms are pre-
sent are only stable when the N-atoms form a closed chain,
as in Griess’s benzeneimide :—
or in the azimidobenzene of this same author*:—
N
H
From these considerations I am led to conclude that an
open chain of three nitrogen atoms does not exist in any of
the diazoamido-compounds, and the remarkable stability of
the dinitrodiazoamido-compounds in the presence of alkalis Tf
certainly supports this view.
The formula which I now venture to suggest for the normal
compounds is, as far as I can see, at any rate as equally
capable as Kekulé’s of representing the characters of these
compounds, and at the same time indicates the analogy of
these to the normal compounds. It has moreover the advan-
tage of doing away with the assumption of the open chain of
N-atoms :—
<
NH.X
According to this formula the preceding compounds would
be written :—
* I have accepted this formula rather than the alternative one
N
eu | \NH, because, according to Boessneck (Ber. xix. 1886, p. 1757),
\n~Z
acetorthotoluylene-diamine yields acetazimidotoluene by the action of
nitrous acid. The N.C,H,O-group must, therefore, be attached to the
aromatic nucleus, and this acetyl compound gives azimidotoluene by
hydrolysis, so that the NH-group must also be attached to the aromatic
nucleus.
Tt These compounds can be boiled with strong potash solution for days
without undergoing any alteration (Journ. Chem. Soc., Trans. 1886,
p- 627). Even the simpler compounds like diazoamidobenzene are much
more stable in neutral or alkaline solutions than is generally supposed.
526 Prof. R. Meldola on the Theory of the
N:N N:N
(p or m) x0.00n Kf (p or m) nose.
SS NE.C,H,.NO, (p or m)
N
N:N N:N
(p) No.0 (m) NO,.C,H, <
SSC ee alle / CH;
N\C-H..NO, (m) N\C,Hi-NO, (p)
m.p. 187°. m.p. 1749-175°.
[The planes of decomposition are represented by the dotted
lines.
Kt sl readily be seen that these formule are in harmony
with the characters of the compounds which they represent.
Thus, taking the products of decomposition of the last pair of
isomeric ethyl-derivatives, the 187° m.p. modification is re-
solved into p-nitrodiazobenzene-chloride and m-ethy]nitrani-
line, while the other modification yields m-nitrodiazobenzene-
chloride and p-ethylnitraniline.
The corresponding pair of isomeric ethyl-derivatives con-
taining aniline and toluidine residues, prepared by Noélting
and Binder and by Gastiger, may be similarly formulated :—
N:N N:N
[e- (p) ome
ni/ Ooi = é C,H,
\C_H, (p) MY C,H, f
These compounds, which have already been referred to, are
decomposed by acids (hot); the first into phenol and p-ethyl-
toluidine, and the second into p-cresol and ethylaniline.
The mixed compounds containing both aromatic and fatty
radicals are in all respects analogous to the normal compounds,
and, according to Wallach*, behave like these on decompo-
sition. Thus the typical compounds of this group, first pre-
pared by Baeyer and Jager} by the action of diazobenzene
salts upon ethylamine, dimethylamine, and piperidine, may be
written :—
N:N N
N:D N:N
Hs: we C,H; <| C,H; Gees -
7 NE.C,H, i N(CH). = Ne
* Ineb. Ann, vol. ccxxxv. p. 2388.
+ Ber. deut. chem. Gesell. viii. (1875) pp. 148, 898.
Constitution of the Diazoamido-Compounds. 527
A few remarks may here be made in connexion with the
transformation of diazoamido- into amidoazo-compounds. It
has always been supposed hitherto that this transformation is
preceded by a resolution of the diazoamido-compound into its
constituents*. This may be the case in the presence of excess
of acid; but it is doubtful whether such a resolution occurs
when the diazoamido-compound (say diazoamidobenzene) is
allowed to stand in the presence of excess of aniline containing
only a small quantity of aniline hydrochloride. Itis well known,
however, that such a mixture will effect the complete conver-
sion of diazoamido- into amidoazobenzene in the course of a
few hours, especially if aided by heat. If the formula of
diazoamidobenzene is written according to the present view,
it will be seen ‘that a slight rearrangement of the ‘ bonds”
would convert it into a symmetrical compound of the type
already proposed for the mixed diazoamido-compounds ;
thus :—
N:N N:N.
C,H; in | C,H; a > C,H;
NH.O,H; it
It seems not improbable that such a symmetrical compound
may precede the formation of amidoazobenzene, the separation
(accompanied by the migration of the H-atom to the NH-
group ){ occurring along one of the dotted lines.
Although the formule now proposed for the normal diazo-
amido-compounds appear capable of meeting all the require-
ments of the case, it will be of interest to point out that other
molecular arrangements in which phenyl functions as a tri-
atomic radical are possible :—
at Be
xX ue
SN ye SN :N.X
iF i; Kil.
* See the last contribution to this question by Wallach (Zeb. Ann.
vol. eexxxy. p. 238). Iam bound to express the opinion, however, that
the suggestion there thrown out does not materially add to the solution
of the problem (Proc. Chem. Soc. 1887, p. 27).
+ Such a transference of hydrogen is analogous to that which takes
place when hydrazobenzene, C,H;. NH.NH.C,H.,, is converted into ben-
zidine, NH,.C,H,.C,H,.NH,, by the action of acids. No previous
resolution into constituents has ever been supposed in this case. The
transformation appears rather to be of the nature of a rotation of the ben-
zene rings, and there is reason for believing that a similar rotation takes
place in the decomposition of mixed diazoamido-compounds by acids.
This point cannot be discussed, however, until the evidence is more
complete.
928 Constitution of the Diazoamido- Compounds.
All these formule are, however, more or less open to objec-
tion, and need not be further discussed at present. It will
suffice to mention that No. ILI., which at first sight might
appear the most probable of the three, is incapable of repre-
senting such compounds as diazobenzenedimethylamide.
The views now advanced concerning the constitution of this
interesting group of compounds open up suggestive lines of
investigation i in the direction of isomerism as connected with
position. In the formula representing phenyl as a triad
radical previously given, the free bonds have been represented
in the para-position, because the ortho-quinone of the benzene
series does not appear to be capable cf existence. But the
formula obviously allows the possibility of such an ortho-
arrangement :—
a
hes Cus
aN
HO cs. He e2
ip a |
HC CH HO CH
WA 77,
(‘ | (
i tl
The para-position of the substituents is, however, in har-
mony with the known behaviour of diazoamidobenzene, the
isomeric amidoazobenzene having its substituents in the para-
position. It seems probable, therefore, if there is anything
in the previously expressed view concerning this isomeric
transformation, that at least in this diazoamido-compound
the substituents —N:N— and =NH= are in the para-
position.
Summing up ihe general results of the present discussion,
it appears to me that the formula now proposed for the mixed
diazoamido-compounds is the only one that has hitherto been
in harmony with all the known characters of these compounds,
and as such it is at any rate worthy of serious consideration
by chemists. Analogy leads to the belief that the normal
compounds have a similar, or at least a partly similar con-
stitution ; but the evidence is not so satisfactory in these
cases, owing to the similarity of the two halves of the mole-
cule, which renders it impossible to follow the course of
decomposition with the same certainty as in the mixed com-
pounds. Before, however, the whole question of the consti-
tution of the diazoamido-compounds can be completely worked
out on the lines here suggested, it will be necessary to have a
Maximum and Minimum Energy in Vortex Motion. 529
much larger body of experimental evidence. Further inves-
tigations in the required direction are now in progress in the
laboratory of the Finsbury Technical College.
Postscript.—Since writing the foregoing paper, the detailed
evidence which has led to the conclusion that the ethyl-
derivative of m.p. 187° is a diazo-compound has appeared in
a communication to the Chemical Society (Journ. Chem. Soc.,
Trans. 1887, p. 434). Additional evidence of the production of
mixed compounds on the decomposition of mixed diazoamido-
derivatives is given in a recent paper by Heumann and Oeco-
nomides (Ber. 1887, p. 904). These authors find that the
mixed compound C,H;.N;H.C,H;, on being heated with
phenol, gives a mixture of aniline and p-toluidine together
with oxyazobenzene and p-tolueneazophenol.
LXI. On the Stability of Steady and of Periodic Fluid Motion
(continued from May number).—Mazimum and Minimum
Energy in Vortex Motion*. By Sir Wituiam Taomson,
1A ia ep
10. ge condition for steady motion of an incompressible
inviscid fluid filling a finite fixed portion of space
(that is to say, motion in which the velocity and direction of
motion continue unchanged at every point of the space within
which the fluid is placed) is that, with given vorticity, the
energy is a thorough maximum, or a thorough minimum, or
aminimax. The further condition of stability is secured, by
the consideration of energy alone, for any case of steady
motion for which the energy is a thorough maximum or a
thorough minimum ; because when the boundary is held fixed
the energy is of necessity constant. But the mere consi-
deration of energy does not decide the question of stability
for any case of steady motion in which the energy is a
minimax.
11. It is clearf that, commencing with any given motion,
the energy may be increased indefinitely by properly-designed
operation on the boundary (understood that the primitive
boundary is returned to). Hence, with given vorticity, but
with no other condition, there is no thorough maximum
of energy in any case. There may also, except in the case
of irrotat‘onal circulation in a multiplexly continuous vessel
* Being a communication read before the British Association, Section A,
at the Swansea Meeting, Saturday, August 28, 1880, and published in the
Report for that year, p.473; and in ‘ Nature,’ Oct. 28, 1880. Reprinted
now with corrections, amendments, and additions. .
+ See also §§ 3 to 9 above.
#
Fae Sir W. Thomson on Maximum and
referred to in § 3 III. above, be complete annulment of the
energy by operation on the boundary (with return to the pri-
mitive boundary), as we see by the following illustrations :—
(a) Two equal, parallel, and oppositely rotating, vortex
columns terminated perpendicularly by two fixed parallel
planes. By proper operation on the cylindric boundary, they
may, in purely two-dimensional motion, be thoroughly and
equably mixed in two infinitely thin sheets. In this condition
the energy is infinitely small.
(b) A single Helmholtz ring, reduced by diminution of its
aperture to an infinitely long tube coiled within the enclosure.
In this condition the energy is infinitely small.
(c) A single vortex column, with two ends on the boundary,
bent till its middle infinitely nearly meets the boundary; and
further bent and extended till it is broken into two equal and
opposite vortex columns, connected, one end of one to one end
of the other, by a vanishing vortex lgament infinitely near
the boundary ; and then further dealt with till these two
columns are mixed together to virtual annihilation. }
12. To avoid, for the present, the extremely difficult general
question illustrated (or suggested) by the consideration of such
cases, confine ourselves now to two-dimensional motions in a
space bounded by two fixed parallel planes and a closed
cylindric, not generally circular cylindric, surface perpen-
dicular to them, subjected to changes of figure (but always
truly cylindric and perpendicular to the planes). Also, for
simplicity, confine ourselves for the present to vorticity either
positive or zero, in every part of the fluid. It is obvious that,
with the limitation to two-dimensional motion, the energy
cannot be either infinitely small or infinitely great with any
given vorticity and given cylindric figure. Hence, under
the given conditions, there certainly are at least two stable
steady motions—those of absolute maximum and absolute
minimum energy. ‘The configuration of absolute maximum
energy clearly consists of least vorticity (or zero vorticity, if
there be fluid of zero vorticity) next the boundary and greater
and greater vorticity inwards. The configuration of absolute
minimum energy clearly consists of greatest vorticity next
the boundary, and less and less vorticity inwards. If there
be any fluid of zero vorticity, all such fluid will be at rest
either in one continuous mass, or in isolated portions sur-
rounded by rotationally moving fluid. For illustration, see
figs. 4 and 5, where it is seen how, even in so simple a case as
that of the containing vessel represented in the diagram, there
can be an infinite number of stable steady motions, each with
maximum (though not greatest maximum) energy ; and also
Minimum Energy in Vortex Motion. 531
an infinite number of stable steady motions of minimum
(though not least minimum) energy.
13. That there can be an infinite number of configurations of
stable motions, each of them having the energy of a thorough
minimum (as said in § 12), we see, by considering the case
in which the cylindric boundary of the containing canister
consists of two wide portions communicating by a narrow
passage, as shown in the drawings. If such a canister be
completely filled with irrotationally moving fluid of uniform
vorticity, the stream-lines must be something like those indi-
cated in fig. 4.
Fig. 4.
- Hence, if a not too great portion of the whole fluid is irro-
tational, it is clear that there may be a minimum energy, and
therefore a stable configuration of motion, with the whole of
this in one of the wide parts of the canister ; or the whole in
the other ; or any proportion in one and the rest in the other.
Hig, OD.
Single intersection of stream-lines in rotational motion
may be at any angle, as shown in fig. 4. It is essentially
at right angles in irrotational motion, as shown in fig. 9,
representing the stream-lines of the configuration of maai-
mum energy, for which the rotational part of the liquid is
in two equal parts, in the middles of two wide parts of the
enclosure. There is an infinite number of configurations of
532 Sir W. Thomson on Maximum and
maximum energy in which the rotational part of the fluid is
unequally distributed between the two wide parts of the
enclosure. sVeLuM
14. In every steady motion, when the boundary is cir-
cular, the stream-lines are concentric circles and the fluid is
distributed in co-axial cylindric layers of equal vorticity. In
the stable motion of maximum energy, the vorticity is greatest
at the axis of the cylinder, and is less and less outwards to the
circumference. In the stable motion of minimum energy the
vorticity is smallest at the axis, and greater and greater out-
wards to the circumference. To express the conditions sym-
bolically, let T be the velocity of the fluid at distance r from
the axis (understood that the direction of the motion is per-
pendicular to the direction of r), and let a be the radius of the
boundary. ‘The vorticity at distance r is
(742)
*\ pr dr}
If the value of this expression diminishes from r=0 to r=a,
the motion is stable, and of maximum energy. If it increases
from r=O0 to r=a, the motion is stable and of minimum
energy. If it increases and diminishes, or diminishes and
increases, as 7 increases continuously, the motion is unstable”.
15. As a simplest subcase, let the vorticity be uniform
through a given portion of the whole fiuid, and zero through
the remainder. In the stable motion of greatest energy, the
portion of fluid having vorticity will be in the shape of a cir-
cular cylinder rotating like a solid round its own axis, coin-
ciding with the axis of the enclosure ; and the remainder of
the fluid will revolve irrotationally around it, so as to fulfil the
condition of no finite slip at the cylindric interface between
the rotational and irrotational portions of the fluid. The
expression for this motion in symbols is
c= Cro noms — to 2p:
meee
r
and from r=b to r==a.
* This conclusion I had nearly reached in the year 1875 by rigid mathe-
matical investigation of the vibrations of approximately circular cylindric
vertices ; but 1 was anticipated in the publication of it by Lord Rayleigh,
who concludes his paper “ On the Stability, or Instability, of certain Fluid
Motions” (‘ Proceedings of the London Mathematical Society,’ Feb. 12,
1880) with the following statement :—“ It may be proved that, if the fluid
move between two rigid concentric walls, the motion is stable, provided
that in the steady motion the rotation either continually increases or
continually decreases in passing outwards from the axis,’—which was
unknown to me at the time (August 28, 1880) when I made the com-
munication to Section A of the British Association at Swansea.
Minimum Energy in Vortex Motion. 533
16. In the stable motion of minimum energy the rotational
portion of the fluid is in the shape of a cylindric shell, en-
closing the irrotational remainder, whichin this case is at rest.
The symbolical expression for this motion is
T=0, when r< /(a?—0?),
a? — §?
and T=E(r— ), when r> 4/(a?—6?).
17. Let now the liquid be given in the configuration (14)
of greatest energy, and let the cylindric boundary be a sheet
of a real elastic solid, such as sheet-metal with the kind of
dereliction from perfectness of elasticity which real elastic
solids present ; that is to say, let its shape when at rest be a
function of the stress applied to it, but let there be a resist-
ance to change of shape depending on the velocity of the
change. Let the unstressed shape be truly circular, and let
it be capable of slight deformations from the circular figure
in cross section, but let it always remain truly cylindrical.
Let now the cylindric boundary be slightly deformed and left
to itself, but held so as to prevent it from being carried round
by the fluid. The central vortex column is set into vibration
in such a manner that longer and shorter waves travel round
it with less and greater angular velocity*. These waves cause
corresponding waves of corrugation to travel round the cylin-
dric bounding sheet, by which energy is consumed, and
moment of momentum taken out of the fluid. Let this pro-
cess go on until acertain quantity M of moment of momentum
has been stopped from the fluid, and now let the canister run
round freely in space, and, for simplicity, suppose its material
to be devoid of inertia. The whole moment of momentum
was initially—
mE D(a? 40?) ;
it is now
me b? (a? xa 3b”) —
and continues constantly of this amount as long as the
boundary is left free in space. The consumption of energy
still goes on, and the way in which it goes on is this: the
waves of shorter length are indefinitely multiplied and exalted
till their crests run out into fine laminz of liquid, and those
of greater length are abated. Thus a certain portion of the
irrotationally revolving water becomes mingled with the
central vortex column. The process goes on until what may
* See ‘Proceedings of the Royal Society of Edinburgh’ for 1880,
or ‘ Philosophical Magazine’ for 1880, vol. x. p. 155: “ Vibrations of a
Columnar Vortex :” Wm. Thomson.
534 Sir W. Thomson on Maximum and
be called a vortex sponge is formed; a mixture homogeneous*
on a large scale, but consisting of portions of rotational and
irrotational fluid, more and more finely mixed together as
time advances. The mixture is altogether analogous to the
mixture of the white and yellow of an egg whipped together
in the well-known culinary operation.’ Let b’ be the radius
of the cylindric vortex sponge, and @ its mean molecular
rotation, which is the same in all sensibly large parts.
Then, b being as before the radius of the original vortex
column, we have
TOs trom res0 tone
and
T= C0" ie from r=b' to r=a%
where
C= Cb? /b”,
and
Pee iD
Pa cae
18. Once more, hold the cylindric case from going round
in space, and continue holding it untilsome more moment of
momentum is stopped from the fluid. Then leave it to itself
again. ‘The vortex sponge will swell by.the mingling with it
* Note added May 13, 1887.—I have had some difficulty in now proving
these assertions (§§ 17 and 18) of 1880. Here is proof. Denoting for
brevity 1/2 of the moment of momentum by p, and 1/2m of the energy
by e, we have |
a
=\ Tr.rdr, and e=i("T?. rdr.
B i ; ai A
The problem is to make e least possible, subject to the conditions: (1) that
p has a given value; (2) that
T =:
(5+ Z)Ee, and 20 ;
and (3) that when r=a, T= (6?/a; this last condition being the resultant of
ms iS ae
\3 - == =) r di =J Srar,
which expresses that the total vorticity is equal to that of ¢ uniform within
the radius 6. The configuration described in the last three sentences of
§ 17 and the first three of § 18 clearly solve the problem when
M <3n(b?(a?—b?); or p>2 (67a.
The fourth sentence of § 18 solves it when
M = 30 (0°(a? —0?); or p= 1 ¢07a2.
The second paragraph of § 18 solves it when
M> 37¢b?(a?— 6"); or p <1 ¢Ba?.
Minemum Energy in Vortex Motion. 5385
of an additional portion of irrotational liquid. Continue this
process until the sponge occupies the whole enclosure.
After that continue the process further, and the result will
be that each time the containing canister is allowed to go
round freely in space, the fluid will tend to a condition in
which a certain portion of the original vortex core gets filtered
into a position next to the boundary, (within a distance from
the axis which we shall denote by c), and the fluid in this space
tends to a more and more nearly uniform mixture of vortex
with irrotational fluid. This central vortex sponge, on repe-
tition of the process of preventing the canister from going
round, and again leaving it free to go round, becomes more
and more nearly irrotational fluid, and the outer belt of pure
vortex becomes thicker and thicker. The resultant motion is
“ii 2 2 2
ie FOP IG,
272
Tatr—2—, for r>c;
and the moment of momentum is
eTCY tae) (ere yt
The final condition towards which the whole tends is a belt
constituted of the original vortex core now next the boundary ;
and the fluid which originally revolved irrotationally round it
now placed at rest within it, being the condition (16 above)
of absolute minimum energy. Begin once more with the con-
dition (15 above) of absolute maximum energy, and leave the
fluid to itself, whether with the canister free to go round some-
times, or always held fixed, provided only it is ultimately held
from going round in space ; the ultimate condition is always
the same, viz. the condition (16) of absolute minimum energy.
The enclosing rotational belt, being the actual substance of the
original vortex, is equal in its sectional area to wb”; and
therefore c’=a’—l”. The moment of momentum is now
47fb*, being equal to the moment of momentum of the
portion of the original configuration consisting of the then
central vortex.
19. It is difficult to follow, even in imagination, the very
fine—infinitely fine—corrugation and drawing-out of the
rotational fluid; and its intermingling with the irrotational
fluid; and its ultimate re-separation from the irrotational
fluid, which the dynamics of §§ 17, 18 have forced on our
consideration. This difficulty is obviated, and we substitute
536 Sir W. Thomson on Maximum and
for the “ vortex sponge”’ a much easier (and in some respects
more interesting) conception, vortex spindrift, if (quite arbi-
trarily, and merely to help us to understand the minimum-
energy-transformation of vortex column into vortex shell) we
attribute to the rotational portion of the fluid a Laplacian*
mutual attraction between its parts “‘insensible at sensible
‘distances’? and between it and the plane ends of the con-
taining vessel of such relative amounts as to cause the inter-
face between rotational and irrotational fluid to meet the end
planes at right angles. Let the amount of this Laplacian
attraction be exceedingly small—so small, for example, that
the work required to stretch the surface of the primitive
vortex column to a million million times its area is small in
comparison with the energy of the given fluid motion.
Everything will go on as described in §§ 17, 18 if, instead of
“run out into fine lamine of liquid” (§ 17, line 31) we sub-
stitute break off into millions of detached fine vortex columns ;
and instead of “sponge”? (passim) we substitute “ spin-
drift.”
20. The solution of minimum energy for given vorticity
and given moment of momentum (though clearly not unique,
but infinitely multiplex, because magnitudes and orders of
breaking-off of the millions of constituent columns of the
spindrift may be infinitely varied) is fully determinate as to
the exact position of each column relatively to the others ; and
the cloud of spindrift revolves as if its constituent columns
were rigidly connected. The viscously elastic containing
vessel, each time it is left to itself, as described in §§ 17, 18,
flies round with the same angular velocity as the spindrift
cloud within ; and so the whole motion goes on stably, without
loss of energy, until the containing vessel is again stopped or
otherwise tampered with.
21. It might be imagined that the Laplacian attraction
would cause our slender vortex columns to break into detached
drops (as it does in the well-known case of a fine circular jet
of water shooting vertically downwards from a circular tube,
and would do for a circular column of water given at rest in
a region undisturbed by gravity), but it could not, because the
energy of the irrotational circulation of the fluid round the
vortex column must be infinite before the column could
break in any place. The Laplacian attraction might, how-
ever, make the cylindric form unstable; but we are excluded
* So called to distinguish it from the “ Newtonian ” attraction, because,
I believe, it was Laplace who first thoroughly formulated “ attraction in-
sensible at sensible distances,” and founded on it a perfect mathematical
theory of capillary attraction.
Minimum Energy in Vortex Motion. 537
from all such considerations at present by our limitation (§ 12)
to two-dimensional motion.
22. Annul now the Laplacian attraction and return to our
purely adynamic system of incompressible fluid acted on only
by pressure at its bounding surface, and by mutual pressure
between its parts, but by no “applied force’’ through its
interior. For any given momentum between the extreme
possible values {b7(a?—ib”) and 47€0*, there is clearly,
besides the §§ 17, 18 solution (minimum energy), another
determinate circular solution, viz. the configuration of circular
motion, of which the energy is greater than that of any other
circular motion of same vorticity and same moment of
momentum. This solution clearly is found by dividing the
vortex into two parts—one a circular central column, and the
other a circular cylindric shell lining the containing vessel ;
the ratio of one part to the other being determined by the con-
dition that the total moment of momentum have the prescribed
value. But this solution (as said above, § 14 and footnote)
may be proved to be unstable.
I hope to return to this case, among other illustrations of
instability of fluid motion—a subject demanding serious con-
sideration and investigation, not only by purely scientific
coercion, but because of its large practical importance.
23. For the present I conclude with the complete solution,
or practical realization of the solution (only found within ‘the
last few days, and after §§ 10-18 of the present article were
already in type) of a problem on which I first commenced trials
in 1868: to make the energy an absolute maximum in two-
dimensional motion with given moment of momentum and given
vorticity in a cylindric canister of given shape. ‘The solution is,
in its terms, essentially unique ; “absolute maximum ” mean-
ing the greatest of maximums. But the same investigation
includes the more extensive problem : To find, of the sets of
solutions indicated in § 12, different configurations of the
motion having the same moment of momentum. For each
of these the energy is a maximum, but not the greatest
mavinuum, for the given moment of momentum. The most
interesting feature of the practical realization to which I have
now attained is the continuous transition from any one steady
or periodic solution, through a series of steady or periodic
_ solutions, to any other steady or periodic solution, produced by
a simple mode of operation easily understood, and always under
perfect control. The operating instrument is merely a stirrer,
a thin round column, or rod, fitted perpendicularly between
the two end plates, and movable at pleasure to any position
Phil. Mag. 8. 5. Vol. 23. No. 145. June 1887. 20
538 Sir W. Thomson on Maximum and
parallel to itself within the enclosure. Itis shown, marked §,
in figs. 6, 7,8, 9: representing the solution of our problem
Fig. 6. Fig. 7.
C
Still water
Still water.
Fig. 6.2Dotted circle with arrowheads refers to the velocity of the stirrer
and of the dimple, not to the velocity of the fluid.
Fig, 7. Arrowheads in the vortex refer to velocity of fluid. Arrowheads
in the irrotational fluid refer to the stirrer and dimple. Arrow-
heads in abcirefer to motion of irrotational fluid relatively to the
dimple.
Fig, 8.
weocso oe
a
44
e S 2
Almost motionless
\
\
XN
Fig. 8. Arrowheads refer to motion of the stirrer, and of the vortex as a
whole.
Fig. 9. Arrowheads on dotted circle refer to orbital motion of ec, the
centre of the vortex. Arrowheads on full fine curves refer to
absolute velocity of fluid.
Minimum Energy in Voriex Motion. 539
for the case of a circular enclosure and a small part of its
whole volume occupied by vortex, to which exigency of time
limits the present communication.
24. Commence with the vortex lining uniformly the en-
closing cylinder, and the stirrer in the centre of the still
water within the vortex. The velocity of the water in the
vortex increases from zero at the inside to &?/a at the out-
side, in contact with the boundary ; according to the notation
of §§ 14 and 15. Now move the stirrer very slowly from its
central position and carry it round with any uniform angular
velocity <¢b/a and >4$¢b/a. A dimple, as shown in fig. 6,
will be produced, running round a little in advance of the
stirrer, but ultimately falling back to be more and more nearly
abreast of it if the stirrer is carried uniformly. If now the
stirrer is gradually slowed till the dimple gets again in advance
of it as in fig. 6, and is then carried round in a similar relative
station, or always a little behind the radius through the middle
of the dimple, the angular velocity of the dimple will decrease
gradually and its depth and its concave curvature will increase;
till, when the angular velocity is }¢b/a, the dimple reaches the
bottom (that is, the enclosing wall) with its concavity a right
angle, as shown in fig. 7, and the angular velocity of propa-
gation becomes 4 ¢b/a.
25. The primitively endless vortex belt now becomes divided
at the right angle, and the two acquired ends become rounded ;
provided the stirrer be carried round always a little rearward,
or considerably rearward, of abreast the middle of the gap.
Figs. 8 and 9 show the result of continuing the process till
ultimately the vortex becomes central and circular (with only
the infinitesimal disturbance due to the presence of the stirrer,
with which we need not trouble ourselves at present).
26. Suppose, now, atany stage of the process, after the for-
mation of the gap, the stirrer to be carried forward to a station
somewhat in advance of abreast of the middle of the gap ; or
somewhat rearward of the rear of the vortex (instead of some-
what in advance of the front as shown in fig. 8). The velocity
of propagation will be augmented (by rearward pull!), the
moment of momentum will be diminished : the vortex train
will be elongated till its front reaches round to its rear, each
then sharpened to 45° and brought into absolute contact with
the enclosing wall: the front and rear unite in a dimple
gradually becoming less; and the process may be continued
till we end as we began, with the vortex lining the inside of
the wall uniformly, and the stirrer at rest in the middle of
the central still-water.
[To be continued. |
[40.4
LXII. On the Variations in the Electrical Resistance of Anti-
mony and Cobalt in a Magnetic Field. By Dr. G. Fas,
Assistant in the Physical Institute of the Royal Unwersity of
Padua*. :
N a series of researches, still in progress, on the variations
in the electrical resistance of different bodies when
brought into a magnetic field, I arrived at some results with
antimony and cobalt which I believe to be new and interesting.
Reserving for a future occasion a fuller account of my inves-
tigations and also of the methods and instruments employed,
I think it may not be useless to publish a preliminary notice.
It is well known, particularly from the experiments of Sir
W. Thomson and M. A. Righi, that magnetism has a distinct
influence on the electrical resistance of iron and nickel, and
much more upon that of bismutht. Looking at the coefii-
cients of rotation found by Hall, Righi, and others, and at the
explanations given, the idea suggests itself that a connexion
may exist between these coefficients and the variation of the
electrical resistance in a magnetic field. On the other hand,
the difference in the behaviour of iron and bismuth in a mag-
netic field seems to be connected with the fact that the first of
these two metals is paramagnetic and the second diamagnetic.
From these considerations and from others, which I shall not
now enter upon, | have undertaken to examine various sub-
stances. The results of my experiments agree with my
previsions.
First of all I thought that cobalt and antimony would in par-
ticular be worthy of investigationt. Cobalt, asis well known,
occupies the third place in the list of paramagnetic metals,
whilst antimony is found immediately after bismuth in the
list of diamagnetic metals.
I have examined antimony in the form of very small cylin-
ders, which I prepared by melting the metal in a crucible
and drawing it into thin glass tubes. The glass was after-
wards broken and taken away by alternately cooling and
heating. Two thick wires of copper were soldered in the
ends of the cylinders in order to connect them up in the elec-
trical circuit. As I shall describe on another occasion the
* Communicated by the Author.
+ From experiments repeated by myself, I have obtained results agree-
ing with those of Sir W. Thomson for nickel and with those of M. Righi
for bismuth.
{t And manganese also; but I have not yet been able to get it pure and
in a convenient form,
Electrical Resistance of Antimony and Cobalt. 5A
method of measurement employed by me, I will only observe
that it was like that of Matthiessen and Hockin, with the
exception of some modifications suggested by the special cir-
cumstances and object of my investigation. The magnetic
field was formed by a large Ruhmkorff electromagnet, ex-
cited by a number of Bunsen’s elements or by a dynamo
machine.
My experiments on antimony showed that, when brought
into a magnetic field, there was an increase in its electrical
resistance, both across and along the lines of force. It ap-
peared to me moreover that, with the same intensity of the
magnetic field, the increase across was greater than that along
the lines of force.
Cobalt I investigated in the form of a small thin plate, pre-
pared by electrolysis of the chloride, or by depositing the metal
on a plate consisting of a mixture of graphite and stearine, as
indicated by M. Righi*. I soldered two thick copper wires
at the two ends of this small plate of cobalt, before detaching
it from the plate of graphite and stearine. ‘These wires were
rigidly connected by means of a piece of ebonite, and served
to make the connexions in the circuit. By a movable support
I could very easily adjust the small plate in any position in
respect to the lines of force. By a long series of observations
I found that :—
(a) When the plate of cobalt was arranged in the magnetic
field with its plane perpendicular to the lines of force, a dimi-
nution of its electrical resistance was observed.
(6) When the plate was arranged parallel to the lines of force,
and the current also had the same direction, an increase in its
electrical resistance was observed.
Judging therefore from the intensity of the effects, the
behaviour of antimony is the same as that found by M.
Righi for bismuth ; and the behaviour of cobalt the same as
that found by Sir W. Thomson for iron and nickel.
I will not dwell on similar experiments on other substances,
because the results are not yet definitive.
In the meantime I must express my obligations to Prof.
Righi for his encouragement in these experiments, and for
also giving me the means of making them in the Physical
Institute under his direction.
Padua, December 12, 1886.
* Mem. dell’ Ace. di Bologna, (4) v. 1883, p. 122; WN. Cimento, (3)
xv. 1884, p. 140,
a sy LF
LXII. The Differential Equation of the most general Substi-
tution of one Variable. By Captain P. A. MacManon, &.A.*
[ the Philosophical Magazine for February 1886, Dr. T.
Muir considers the differential equations of the general
conic and cubic curves by a perfectly general method.
The general linear substitution
__(a,0)(@,1)
2 EVV (@,1)
leads, as is well known, to the differential equation
dy dy genet.
2 oe db ( dat) =0
wherein the expression on the left has been called the
Schwarzian derivative : this is a reciprocant; but it isalsoan
invariant, as may be seen by writing
dy d’y d’y
~ —|]! —~ =9!) —_% —3!
dig ORT TOT gna aan ae
when it assumes the form :
12(tb—a’).
In the case of the general substitution of order n, the
resulting expression is no longer a reciprocant, but it is an
invariant (catalecticant) of a certain binary quantic fT.
For, writing
jo See ES _ Un
(agp. ical) | Na
we have
Nn;
Differentiating this equationn + 1,n+2,n+3,...2n+1 times
successively by Leibnitz’s theorem, and putting
CN av,
dae 9” dex
there results the set of equations :—
= Vo,
* Communicated by the Author.
+ The formation of the differential equation was recently set as a ques-
tion in an examination for Fellowship at Trinity College, Dublin; but
I am not aware that its connexion with the theory of Invariants has been
before noticed.
The Differential Equation of the general Substitution. 543
. | ,, Va) (n+ 1)! (2) Gays
mtiVat (n+1)ynVn + QW (n—1)! 7") aN, +s s Bete Al T! YWVn =(0,
: } n+2)! n+2)! x
ntaVnt (W+2)YnsiVa + 4 2 Yn VO 4. + i a yoVS=0,
n+)! n+3)! A
na V nt (0+ 8)ynaVO tone) YntiVn beeet a 2 ysVn=0,
a, (n+)! 5 (2n+1)! a
ee ones tt ald
or writing
Y=llt, y=2!a, yy=d!la,,... ¥,=p!
— are
= (n) (n—1) (m—2)
cs ~ V5 gee eae + ay a a Va. +.-
lL yam (n=1) (n—2)
do Vn +1 V + Ag (n—D)! 7 V; +..
i VO + 1 Vey Na
oa? "GT cee 06 = iit
1 (n) 1 (n—1) 1 (n—2)
Gn Vn iene + dnt1 Tm —9yI Vm +..
Hliminating the n+1 Sil
| = (n) 2),
pVn A Gain “i Mi Sas (cae
(n+1)!
A599
+ On-1V in =.
bd, oN =O
oor OnsiVn =0,
sick: Gon—1Vn= 0.
between aie n-+1 equations, we find that the desired differ-
ential equation is
Ao ay Ag eee an— 1 Qn
On—1 Un OAntierrs Agn—2 Aen-1
=().
This determinant is the catalecticant of the binary quantic
Gi \CR YT
(& digs G5 «\
— 25) so 25
10 90 80 70 60 50) 40. 30 2 10 O
DRYER: ERE EN Per centage diffused .
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Phil. Mag. 5.0. Vol. 28. PI. IL.
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