'.|!
M-
M><
JW*«^,
i^^^'^^^^"*^^..^
,m;j?^^^
^f^^.^s-r'
^^-s:
%
wW^
r--
:^^^^^
;^
m^^^^^
'^^^
^^
.v.r-nl^^-/1
^
^^f "'■ ■■ ll
! '^H|
l#^-r^-.'
^
^^smm
^^%&^ kjf
^ : ■^•v>
V '^Ti^^ IT.
"^^^^^^fs^W;..- ■ /
^^K^;i:^^ '^■.
-*':mh^#*'
mmM:^^^
.**^^-:^^
.^if^%^^-^.
^T.r^^KAr:-j:
W^^
^'^^^Ot
H^r^
^;:'mrr^'
r^^
0^:
^^
^>-^>^
\*^,r>A^',
^;^^^'>^'^'^^
^^^^^^/\A^^tll^^'^-
:->^^rvr-r^.f^
-^^r''
■rrfi
,^^i->>
,^^:^:N^
i^|,^'^g^S;5;!^
^r<,Oo^.
^^^^^rS^fTf^
^^^^^Vv
:^.^^^'^1i:-v
K,'»rs;r\'n'
-r\rf ^^'^^^>f%^r^^^'^^ r'
i . t 3. JT
TRANSACTIONS
CAMBRIDGE
PHILOSOPHICAL SOCIETY.
ESTABLISHED November 15, 1819.
VOLUME THE FIFTH.
CAMBRIDGE:
PRINl'ED BY JOHN SMITH, PRtNTER TO THE UNIVERSITY:
AND SOLD BY
JOHN WILLIAM PARKER, WEST STRAND, LONDON;
J. & J.J. DEIGHTON; AND T.STEVENSON, CAMBRIDGE.
M.DCCC.XXXV.
CONTENTS OF THE FIFTH VOLUME.
Part I.
PAGE
N°. I. Mathematical Investigations concerning the Laws of the Equilibrium of Fluids
analogous to the Electric Fluid, with other similar Researches: by George
Green, Esq. Communicated by Sir Edward Ffrench Bromhead, Bart, M.A.
F.R.S.L. & E I
II. On Elimination between an Indefinite Number of Unknown quantities : by the
Rev. R. Murphy 65
III. On the General Equation of Surfaces of the Second Degree : by Augustus
De Morgan, Esq 77
IV. On a Monstrosity of the Common Mignionette : by the Rev. Professor Henslow. . . 95
Part II.
V. On the Calculation of Newton's Experiments on Diffraction: by Professor Airv... 101
VI. Second Memoir on the Inverse Method of Definite Integrals: by the Rev.
R. MUBPHY 113
VII. On the Nature of the Truth of the Laws of Motion : by the Rev. W. Whewkll 149
VIII. Researches in the Theory of the Motion of Fluids : by the Rev. James Challis US
IX. Theory of Residuo- Capillary Attraction; being an Explanation of the Pheno-
mena of Endosmose and Exosmose on Mechanical Principles: by the Rev.
J. PovfEH 205
X. On Aerial Vibrations in Cylindrical Tubes: by William Hopkins, M.A 231
XI. On the Latitude of Cambridge Observatory: by Professor Airy 271
IV CONTENTS.
Paet III.
PAAB
N° XII. On the Diffraction of an Object-glass with Circular Aperture: bt Professor
Airy 283
XIII. On the Equilibrium of the Arch: by the Rev. Hbnry Moseley 293
XIV. Third Memoir on the Inverse Method of Definite Integrals: by the Rev.
R. MuHPHY 315
XV. On the Determination of the Exterior and Interior Attractions of Ellipsoids
of Variable Densities : by George Green, Esq 395
XVI. On the Position of the Axes of Optical Elasticity in Crystals belonging to
the Oblique- Prismatic System: by W. H. Miller, Esq 431
ADVERTISEMENT.
The Society as a hody is not to he considered responsible for any
Jucts and opinions advanced in the several Papers, which must rest
entirely on the credit of their respective Authors.
The Society takes this opportunity of expressing its grateful
acknowledgments to the Syndics of the University Press, for their
liberality in taking upon themselves the expense of printing this
Part of its Transactions.
TRANSACTIONS
CAMBRIDGE
PHILOSOPHICAL SOCIETY.
Vol. V. Part I.
CAMBRIDGE:
PRINTED BY JOHN SMITH. PRINTER TO THE UNIATERSITY;
AND SOLD EV
JOHN WILLIAM PARKER, 445 WEST STRAND, LONDON;
J. & J. J. DEIGHTON, AND T. STEVENSON,
CAMBRIDGE.
M.DCCC.XXXIII.
d-?
^AL rtV
I. Mathematical Investigations concerning the Laws of the Equilibrium
of Fluids analagous to the Electric Fluid, with other similar Researches.
By George Green, Esq. Communicated hy Sir Edward Ffrench
Bromhead, Bart. M.A. F.K.S.L. and E.
[Read Nov. 12, 1832.]
Amongst the various subjects which have at different times occupied
the attention of Mathematicians, there are probably few more interesting
in themselves, or which offer greater difficulties in their investigation,
than those in which it is required to determine mathematically the
laws of the equilibrium or motion of a system composed of an infinite
number of free particles all acting upon each other mutually, and ac-
cording to some given law. When we conceive, moreover, the law of
the mutual action of the particles to be such that the forces which
emanate from them may become insensible at sensible distances, the
researches to which the consideration of these forces lead will be greatly
simplified by the limitation thus introduced, and may be regarded as
forming a class distinct from the rest. Indeed they then for the most
part terminate in the resolution of equations between the values of
certain functions at any point taken at will in the interior of the sys-
tem, and the values of the partial differentials of these functions at the
same point. When on the contrary the forces in question continue
sensible at every finite distance, the researches dependent upon them
become far more complicated, and often require all the resources of
the modern analysis for their successful prosecution. It would be easy
so to exhibit the theories of the equilibrium and motion of ordinary
fluids, as to offer instances of researches appertaining to the former
class, whilst the mathematical investigations to which the theories of
Electricity and Magnetism have given rise may be considered as in-
teresting examples of such as belong to the latter class.
Vol. V. Pakt I. A
2 Mr green, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS.
It is not my chief design in this paper to determine mathematically
the density of the electric fluid in bodies under given circumstances,
having elsewhere* given some general methods by which this may be
effected, and applied these methods to a variety of cases not before
submitted to calculation. My present object will be to determine the
laws of the equilibrium of an hypothetical fluid analagous to the electric
fluid, but of which the law of the repulsion of the particles, instead of
being inversely as the square of the distance, shall be inversely as any
power n of the distance ; and I shall have more particularly in view
the determination of the density of this fluid in the interior of con-
ducting spheres when in equilibrium, and acted upon by any exterior
bodies whatever, though since the general method by which this is
effected will be equally applicable to circular plates and ellipsoids.
1 shall present a sketch of these applications also.
It is well known that in enquiries of a nature similar to the one
about to engage our attention, it is always advantageous to avoid the
direct consideration of the various forces acting upon any particle p of
the fluid in the system, by introducing a particular function V of the
co-ordinates of this particle, from the differentials of which the values
of all these forces may be immediately deduced f. We have, therefore,
in the present paper endeavoured, in the first place, to find the value
of V, where the density of the fluid in the interior of a sphere is given
by means of a very simple consideration, which in a great measure
obviates the difficulties usually attendant on researches of this kind,
have been able to determine the value F^, where p, the density of the
fluid in any element dv of the sphere's volume, is equal to the product
of two factors, one of which is a very simple function containing an
arbitrary exponent fi, and the remaining one J" is equal to any rational
* Essay on the Application of Mathematical Analysis to the Theories of Electricity and
Magnetism.
t This function in the present case will be obtained by taking the sum of all the molecules
of a fluid acting upon p, divided by the (n — 1)* power of their respective distances from^;
and indeed the function which Laplace has represented by F in the third book of the
Mecanique Celeste, is only a particular value of our more general one produced by writing
2 in the place of the general exponent n.
Mb green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. S
and entire function whatever of the rectangular co-ordinates of the element
dv, and afterwards by a proper determination of the exponent /3, have
reduced the resulting quantity ^ to a rational and entire function of
the rectangular co-ordinates of the particle p, of the same degree as
the function f. This being done, it is easy to perceive that the reso-
lution of the inverse problem may readily be effected, because the
coefficients of the required factor f will then be determined from the
given coefficients of the rational and entire function V, by means of
linear algebraic equations.
The method alluded to in what precedes, and which is exposed in
the two first articles of the following paper, will enable us to assign
generally the value of the induced density p for any ellipsoid, what-
ever its axes may be, provided the inducing forces are given explicitly
in functions of the co-ordinates of p ; but when by supposing these axes
equal we reduce the ellipsoid to a sphere, it is natural to expect that
as the form of the solid has become more simple, a corresponding degree
of simplicity will be introduced into the results ; and accordingly, as
will be seen in the fourth and fifth articles, the complete solutions both
of the direct and inverse problems, considered under their most general
point of view, are such that the required quantities are there always
expressed by simple and explicit functions of the known ones, inde-
pendent of the resolution of any equations whatever.
The first five articles of the present paper being entirely analytical,
serve to exhibit the relations which exist between the density p of our
hypothetical fluid, and its dependent function V; but in the following
ones our principal object has been to point out some particular appli-
cations of these general relations.
In the seventh article, for example, the law of the density of our
fluid when in equilibrium in the interior of a conductory sphere, has
been investigated, and the analytical value of p there found admits of
the following simple enunciation.
The density p of free fluid at any point p within a conducting sphere
A, of which O is the centre, is always proportional to the {n - 4)"' power
of the radius of the circle formed by the intersection of a plane per-
pendicular to the ray Op with the surface of the sphere itself, provided
A 2
4 Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS.
n is greater than 2. When on the contrary n is less than 2, this law
requires a certain modification ; the nature of which has been fully
investigated in the article just named, and the one immediately fol-
lowing.
It has before been remarked, that the generality of our analysis will
enable us to assign the density of the free fluid which would be induced
in a sphere by the action of exterior forces, supposing these forces are
given explicitly in functions of the rectangular co-ordinates of the point
of space to which they belong. But, as in the particular case in which
our formulae admit of an application to natural phenomena, the forces in
question arise from electric fluid diffused in the inducing bodies, we
have in the ninth article considered more especially the case of a con-
ducting sphere acted upon by the fluid contained in any exterior bodies
whatever, and have ultimately been able to exhibit the value of the
induced density under a very simple form, whatever the given density
of the fluid in these bodies may be.
The tenth and last article contains an application of the general
method to circular planes, from which results, analagous to those formed
for spheres in some of the preceding ones are deduced; and towards
the latter part, a very simple formula is given, which serves to express
the value of the density of the free fluid in an infinitely thin plate,
supposing it acted upon by other fluid, distributed according to any
given law in its own plane. Now it is clear, that if to the general ex-
ponent 11 we assign the particular value 2, all our results will become
applicable to electrical phenomena. In this way the density of the
electric fluid on an infinitely thin circular plate, when under the in-
fluence of any electrified bodies whatever, situated in its own plane,
will become known. The analytical expression which serves to repre-
sent the value of this density, is remai-kable for its simplicity ; and by
suppressing the term due to the exterior bodies, immediately gives the
density of the electric fluid on a circular conducting plate, when quite
free from all extraneous action. Fortunately, the manner in which
the electric fluid distributes itself in the latter case, has long since
been determined experimentally by Coulomb. We have thus had the
advantage of comparing our theoretical results with those of a very
Mit GREEN, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 5
accurate observer, and the differences between them are not greater
than may be supposed due to the unavoidable errors of experiment,
and to that which would necessarily be produced by employing plates
of a finite thickness, whilst the theory supposes this thickness infinitely
small. ]\Ioreover, the errors are all of the same kind with regard to
sign, as would arise from the latter cause.
1. If we conceive a fluid analogous to the electric fluid, but of
which the law of the repulsion of the particles instead of being in-
versely as the square of the distance is inversely as some power n of
the distance, and suppose p to represent the density of this fluid, so
that dv being an element of the volume of a body A through which
it is diffiised, pdv may represent the quantity contained in this element,
and if afterwards we write g for the distance between dv and any
particle /> under consideration, and these form the quantity
the integral extending over the whole volume of A, it is well known
that the force with which a particle p of this fluid situate in any
point of space is impelled in the direction of any line q and tending
to increase this line will always be represented by
(1).
I^\:
1-n \dq) '
?^, being regarded as a function of three rectangular co-ordinates of
p, one of which co-ordinates coincides with the line q, and (—7-)
being the partial differential of V, relative to this last co-ordinate.
In order now to make known the principal artifices on which the
success of our general method for determining the function V mainly
depends, it will be convenient to begin with a very simple example.
Let us therefore suppose that the body ^ is a sphere, whose centre,
is at the origin O of the co-ordinates, the radius being 1 ; and p is
such a function of x', y', %, that where we substitute for x', y', »' their
values in polar co-ordinates
6 Me green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS.
X = r' cos 0', y' = / sin 9' cos tst', %' = r' sin Q' sin tr',
it shall reduce itself to the form
P = (l-/y./(0;
f being the characteristic of any rational and entire function what-
ever: which is in fact equivalent to supposing
p = (1 - /' - y" - %'f.f{x" + y" + z'^).
Now, when as in the present case, p can be expanded in a series
of the entire powers of the quantities x, y', %', and of the various
products of these powers, the function V will always admit of a similar
expansion in the entire powers and products of the quantities x, y, %,
provided the point p continues within the body A*, and as moreover
V evidently depends on the distance Op — r and is independent of 6
and -sr, the two other polar co-ordinates of p, it is easy to see that the
quantity V when we substitute for x, y, z these values
x = r cos 9, y = r sin 9 cos w, z = r sin 9 sin tst
will become a function of r, only containing none but the even
powers of this variable.
But since we have
dv = r"dr d9' d-ur sin 0', and /> = (1 - ry.f{r'%
the value of V becomes
V= f-^, = jr'^dr'd9'd-w' sin 9' (1 - r''ff{r") .g'"",
J g"
the integrals being taken from tst' = to tr' = 2 tt, from 9' = to 9' = w,
and from r' = to r' = l.
* The truth of this assertion will become tolerably clear, if we recollect that V may be
regarded as the sum of every element pdv of the body's mass divided by the (n—l)"" power
of the distance of each element from the point p, supposing the density of the body A to be
expressed by p, a continuous function of x, y, z. For then the quantity V is represented
by a continuous function, so long as p remains within A ; but there is in general a violation
of the law of continuity whenever the point p passes from the interior to the exterior space.
This truth, however, as enunciated in the text, is demonstrable, but since the present paper
is a long one, I have suppressed the demonstrations to save room.
Mb green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 7
Now V may be considered as composed of two parts, one V due
to the sphere B whose centre is at the origin O, and surface passes
through the point p, and another V" due to the shell S exterior to B.
In order to obtain the first part, we must expand the quantity g^~"
t
T
in an ascending series of the powers of — . In this way we get
^1 -« _ ^1 ^2rr {cos 9 cosff -\- sin 9 sin 9' cos (^' - -sr)] + r'^]
l-n
2
= r' " " ,
If then we substitute this series for g^'" in the value of F", and
after having expanded the quantity (1 — r'^f , we effect the integrations
relative to r, 0', and w', we shall have a result of the form
r' = r*-'' [A-i-Br+Cf^ + Sic.]
seeing that in obtaining the part of V before represented by V, the
integral relative to r' ought to be taken from r =0 to r' = r only.
To obtain the value of F", we must expand the quantity g^-" in
an ascending series of the powers of — , and we shall thus have
l-n
2
g^-''={r^ — 2rr' [cos 6 cos 0' + sin 9 sin 6' cos (tst - -nr')] + r"')
the coefficients Qo, Qi, Q2, &c. being the same as before.
The expansion here given being substituted in P", there will arise
a series of the form
of which the general term T, is
T,= fd9'd^' sin ff QJr-'dr ^^^(l-ry.f{ry,
the integrals being taken from r' = r to r' = l, from 0' = O to & — it, and
from •z«r' = to 'ar' = 27r. This will be evident by recollecting that the
8 Mr green, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS.
triple integral by which the value of V" is expressed, is the same as
the one before given for V, except that the integration relative to r,
instead of extending from /=0 to r'=l, ought only to extend from
r =r to r= 1.
But the general term in the function J'{r'^) being represented by
Atr'^\ the part of T^ dependent on this term will evidently be
(2) Atr'fde'dw' sin 9'.QJr'''+^-^-''dr' {l-r'y-,
the limits of the integrals being the same as before.
We thus see that the value of T, and consequently of F'" would
immediately be obtained, provided we had the value of the general
integral
flr^dril-ry,
which being expanded and integrated becomes
^ 1 +Mzl). 1 _&c.
b + l l'b + 3 1.2 'b + 5
^+1 Q fj,+3 /3(/3-l) r'+»
+ T • i — ;r — ~ 7 r. • ~i = + <^c-
6 + 1 16 + 3 1.2 '6 + 5
but since the first line of this expression is the well known expansion of
(f)
r lf\ r li
or
nT
m'
when n = 2.p — h + \ and 5' = 2(/3 + l) we have ultimately,
By means of the result here obtained, we shall readily find the
value of the expression (2) which will evidently contain one term multi-
plied by r' and an infinite number of others, in all of which the quantity
r is affected with the exponent n. But as in the case under considera-
tion, n may represent any number whatever, fractionary or irrational,
Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 9
it is clear that none of the terms last mentioned can enter into V,
seeing that it ought to contain the even powers of r only, thence the
terms of this kind entering into V", must necessarily be destroyed by
corresponding ones in V. By rejecting them, therefore, the formula (2)
will become
m —^ ^-^ Air'.fde'd-Br' smd'Qs.
But as V ought to contain the even powers of r only, those terms
in which the exponent s is an odd number, will vanish of themselves
after all the integrations have been effected, and consequently the only
terms which can appear in V, are of the form
r(#+2-y-|) r(/3 + i)
(4) ^ Atr'^fde'dTir' sin ff Q,r,
2r(^ + /3 + 3-*'-|)
where, since s is an even number, we have written 2 s' in the place of
s, and as Qu- is always a rational and entire function of cos 9', sin 0'
cos w', and sin 9' sin -sr', the remaining integrations may immediately be
effected.
Having thus the part of T'a,- due to any term Atr'''* of the function
y(r'*) we have immediately the value of T'.^' and consequently of F'",
since
r"= u'+ t:+ t:+ t:+ T:+kc. -,
U' representing the sum of all the terms in F" which have been rejected
on account of their form, and T,' T,' T,' the value of T, Ty T„ &c.
obtained by employing the truncated formula (2) in the place of the
complete one (2).
But -v=v'+ V" = r'+ u^Ti^ r;+ t: + 7v+ &c.
or by transposition,
r-T:-T;-Ti-Ti-hc.=r-YU,
and as in this equation, the function on the left side contains none
but the even powers of the indeterminate quantity r, whilst that on
Vol. V. Part I. B
lO Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS,
the right does not contain any of the even powers of r, it is clear that
each of its sides ought to be equated separately to zero. In thi& way
the left side gives
(5) r=T:+T,'+T:+T:+kc.
Hitherto the value of the exponent /3 has remained quite arbitrary,
but the known properties of the function r will enable us so to
determine /3, that the series just given shall contain a finite number
of terms only. We shall thus greatly simplify the value of F) and
reduce it in fact to a rational and entire function of r*.
For this purpose, we may remark that
r(0)=«, r(-l)=oo, r { — 2) = CO, in infinitum.
If therefore we make — - + /3 = any whole number positive or
negative, the denominator of the function (4) will become infinite, and
consequently the function itself will vanish when s is so great that
1- /3 + i + 3 - *' is equal to zero or any negative number, and as
tit
the value of t never exceeds a certain number, seeing that f{i^^) is
a rational and entire function, it is clear that the series (4) will termi-
fMrte of itself, and V become a rational and entire function of r*.
(2) The method that has been employed in the preceding article
where the function by which the density is expressed is of the particular
form
may by means of a very slight modification, be applied to the far more
general value
P = (1 - ryf{^, i, a') = (1 - x" - y" - --'ff{x, y', z)
tvhere f is the characteristic of any rational and entire function what-
ever : and the same value of /3 which reduces V to a, rational and entire
function of r^ in the first case, reduces it in the second to a similar
function of x, y, % and the rectangular co-ordinates of p.
Ma GREEN, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS. H
To prove this, we may remark that the con-esponding value V will
beeoMie
F = fr"dr'de'd^' sin 6' (1 - ryf{x', y', «')^'-";
tJie integral being conceived to comprehend the whole volume of the
sphere.
Let now the function y be divided into two parts, so that
fi^, y, %') =/ ix', y', z') +f, ix', y', ^') ;
/i containing all the terms of the function J] in which the sum of the
exponents of af, y, %' is an odd number ; and ^ the remaining terms, or
those where the same sum is an even number. In this way we get
the functions F'l and V^a corresponding to^ andj^, being
V, = fr"dr'de'dvr' sin'0' {l~ryf, {x', y\z')g'-',
V^ = lr"dr'd&d-^ sin & (1 - ryf, {x', y% a') g^-\
"We will in the first place endeavour to determine the value J^j; and
for this purpose, by writing for x, y, %' their values before given in
r', ff, w', we get
f,{x',y,%')^rW')\
the coefficients of the various powers of r'^ in ^{r'^) being evidently
rational and entire functions of cos 0„ sin & cos w', and sin sin w.
Thus
V, = jr^dr'dffdTs' sin 6' (1 - ry />/.(/') ^'-";
this integral, like the foregoing, comprehending the whole volume of
the sphere.
Now as the density corresponding to the function Fi is -
p,=.{l-af^-y'^-^^ff,{x',^,%%
it is clear that it may be expanded in an ascending series of the entire
powers of x', y, »', and the various products of these powers consequently,
as was before remarked (Art. 1.), Fl admits of an analagous expansion
in entire powers and products of x, y, ■%. Moreover, as the density /i,
B 2
12 Mr green, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS.
retains the same numerical value, and merely changes its sign when
we pass from the element dv to a point diametrically opposite, where
the co-ordinates x, y , % are replaced by - x , - y' , —% \ it is easy to
see that the function V-^, depending vxpon /s,, possesses a similar property,
and merely changes its sign when x, y, %, the co-ordinates of p, are
changed into - x, —y, — as. Hence the nature of the function Vi is
such that it can contain none but the odd powers of r, when we sub-
stitute for the rectangular co-ordinates x, y, %, their values in the polar
co-ordinates r, 6, ■zs.
Having premised these remarks, let us now suppose Vx is divided
into two parts, one V^ due to the sphere B which passes through the
particle p, and the other V" due to the exterior shell aS*. Then it is
evident by proceeding, as in the case where p = (1 - r"^Yf{i% that Vi
will be of the form
the coefficients A, B, C, &;c. being quantities independent of the variable r.
In like manner we have also
F/' = fr'^dr'ae'dsr' sin ff {\-ry .r'>\,{r'')g^-'';
the integrals being taken from r' = r to r = l, from 6' = to 0' = 7r, and
from Gr' = to 'z<r' = 2 7r.
By substituting now the second expansion of g^" before used (Art. 1.),
the last expression will become
r," = t; + Ti + r. + ^3 + &c.
of which series the general term is
T, = fd9'dw' sin ff Q, fr"-dr' (1 - ry ^ x/. {r").
Moreover, the general term of the function \l^ {r'-) being represented by
Air'^\ the portion of 1\ due to this term, will be
(a) r fdffdw' smO' Q,Atjr''-''^''-Ulr' {l-ry-,
•the limits of the integrals being the same as before.
Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. IS
If now we effect the integrations relative to r' by means of the for-
mula (3), Art. 1, and reject as before those powers of the variable /•,
in which it is affected, with the exponent w, since these ought not enter
into the function Fi, the last formula will become
(^^"F^V(3+i)
(a) ;r-7rrj ^-rr: r'fde'diir' sin 6' Q,A„
2 r C ^ + ^/J-w + g^-jy x •' ^'
and as F, ought to contain none but the odd powers of r, we may make
* = 2*' + l, and disregard all those terms in which s is an even number,
since they will necessarily vanish after all the operations have been
effected. Thus the only remaining terms will be of the form
t^''^' fde'dsr' sin 9' Q,,.^, A,;
2.T '
)
where, as At and 02/+ 1 are both rational and entire functions of cos 0',
sin ff cos •ht', sin ff sin -sr', the remaining integrations from 6' = to 9' = tt,
and Tsr' = to tjt' = 2 tt, may easily be effected in the ordinary way.
If now we follow the process employed in the preceding article, and
suppose To', Ti, T2, &c. are what T^, Ti, 71, &c. become when we use
the truncated formula («') instead of the complete one (a), we shall
readily get
F, = t: + t: + t: + r/ + &c.
In like manner, from the value of V^ before given, we get
r," = fr'dr'd&dsr' sin &{1- ry(p{r")g' -" ;
the integrals being taken from r' = r to r = l, from 9' = to 9' = ^, and
from -ar = to tsr = 2 tt.
Expanding now g^'" as before, we have
r;'= t;-„+t7,+ z7.+ j7, + &c.
where
U. = fdffd-sr' sin ■sr'Qjy'-'^dr'il-ry ^ (/*),
14 Mr green, ON THE LAWS OF THE EQUILIBinUM OF FLUIDS.
acnd the part of U, due to the general term i?(/-''^' in (/*), will be
(J) r'fd&diir' sm 9' Q^Bt/lr'^-"^^'-' dr' {l-ry-,
which, by employing the formula (3') Art. 1., and rejecting the inad-
missible terms, gives for truncated formula
[ 2 j
By continuing to follow exactly the same process as was before
employed in finding the value of Fl, we shall see that * must always
be an even number, say 2 s'; and thu« the expression immediately {Br-
eeding will become
,, l4!-n + 2t-2s
^ (6-n + 23 + 2t-2s'
2r I
; — r'^' fdO'dw sin d'^2.- B,.
2 J
Moreover, the value erf V^ will be
r, = u: + u: + u: + u: + &c. ;
U^, Ui, Ui, U3, &c. being what Uo, Ui, Ui, &c. become when we use
the formula {b') instead of the complete one (h).
The value of V answering to the density
p = p, + p, = (l-ry. /{:>/, y',z'),
by adding together the two parts into which it was originally divided,
therefore, becomes
r = r,+r,= t: + t: + t: + t/ + &c.
+ £/■„' +t4'+C7;'+t7e' + &c.
When /3 is taken arbitrarily, the two series -entering into V extend
in infinitum, but by supposing as before. Art. 1.,
— n n
Ma GREEN, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 15
w representing any whole number, positive or negative, it is clear from
the form of the quantities entering into JLs'+i and U2/, and from the
known properties of the function F, that both these series wiU terminate
of themselves, and the value of F' be expressed in a finite form ; which,
by what has preceded, must necessarily reduce itself to a rational and
entire function of the rectangular co-ordinates x, y, ss. It seems needless,
after what has before been advanced, (Art. 1.) to offer any proof of this:
we will, therefore, only remark that if 7 represents the degree of the
function f{x', y\ &'), the highest degree to which V can ascend will be
7 + 2 a> + 4.
In what immediately precedes, w may represent any whole number
whatever, positive or negative ; but if we make w= —2, and consequently,
^ = ^ the degree of the function J^ is the same as that of the factor
A^\ y', ^),
comprised in p. This factor then being supposed the most general of
its kind, contains as many arbitrary constant quantities as there are
terms in the resulting function V. If, therefore, the form of the rational
and entire function V be taken at will, the arbitrary quantities contained
in fkpd, y, %') will in case w = — 2 always enable us to assign the corres-
ponding value of p, and the resulting value of J'{a;', y, %') will be a rational
and entire fimction of the same degree as T-^. Therefore, in the case
now under consideration, we shall not only be able to determine the
value of F' when p is given, but shall also have the means of solving
the inverse problem, or of determining p when V is given ; and this
determination will depend upon the resolution of a certain number of
algebraical equations, all of the first degree.
3. The object of the preceding sketch has not been to point out
the most convenient way of finding the value of the function ^, but
merely to make known the spirit of the method ; and to show on what
its success depends. Moreover, when presented in this simple form,
it has the advantage of being, with a very slight modification, as ap-
plicable to any ellipsoid whatever as to the sphere itself. But when
16 Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS.
spheres only are to be considered, the resulting formula?, as we shall
afterwards show, will be much more simple if we expand the density p
in a series of functions similar to those used by Laplace {Mec. Cel.
Liv. iii.) : it will however be advantageous previously to demonstrate
a general property of functions of this kind, which will not only serve
to simplify the determination of F, but also admit of various other
applications of dcr.
Suppose, therefore, J^''' is a function of 9 and trr, of the form con-
sidered by Laplace {Mec. Cel. Liv. iii.), r, 9, -zs- being the polar co-ordi-
nates referred to the axes JT, Y, Z, fixed in space, so that
ar = r cos 0, y = r %\w9 cos Tsr, x = /• sin sin vr ;
then, if we conceive three other fixed axes Xi, Y^, Z,, having the same
origin but different directions, P'^'^ will become a function of 0, and •zjti,
and may therefore be expanded in a series of the form
(6) r <^> = r/"' + F.*'> + F/^' + F/^' + &c. .
Suppose now we take any other point p and mark its various co-ordinates
with an accent, in order to distinguish them from those of p ; then, if
we designate the distance pp by {p, p), we shall have
^ - = f r' - 2rr' [cos 9 cos ff + sin 9 sin & cos {tn- - •sr')] + r'^\ "*
as has been shewn by Laplace in the third book of the Mec. Cel., where
the nature of the different functions here employed is completely ex-
plained.
In like manner, if the same quantity is expressed in the polar co-
ordinates belonging to the new system of axes X-,, F„ Z,, we have,
5ince the quantities r and r' are evidently the same for both systems,
{^p, p) r \^ r r IT I
^nd it is also evident from the form of the radical quantity of which
Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 17
the series just given are expansions, that whatever number i may re-
present, Qi*** will be immediately deduced from Q*'> by changing 9, sr,
9', -sr' into 0„ "sr,, 9/, ^r,'. But since the quantity - is indeterminate,
and may be taken at will, we get, by equating the two values of . ,
. f
and comparing the like powers of the indeterminate quantity -,
If now we multiply the equation (6) by the element of a spherical
surface whose radius is unity, and then by Q<*' = Q/*>, we shall have,
by integrating and extending the integration over the whole of this
spherical surface,
fdf.dwQ"^ r® = fdfx, d-ar, Q/** { F/"' + Y^ + F*^' + &c. } .
Which equation, by the known properties of the functions Q**' and Y^^\
reduces itself to
when h and i represent different whole numbers. But by means of a
formula given by Laplace {Mec. Cel. Liv. iii. No. 17.) we may imme-
diately effect the integration here indicated, and there wiU thus result
"-2^ + 1-^^ '
F;'<*> being what Fi''" becomes by changing 9^, tsti into 0,', •ar/, and as
the values of these last co-ordinates, which belong to p, may be taken
arbitrarily like the first, we shall have generally F,**', except when
h = i. Hence, the expansion (6) reduces itself to a single term, and
becomes
F® = F®.
We thus see that the function F<'' continues of the same form even
when referred to any other system of axes X„ F„ Z„ having the same
origin O with the first.
This being established, let us conceive a spherical surface whose center
is at the origin O of the co-ordinates and radius r', covered with fluid.
Vol. V. Part I. C
18 Mr green, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS.
of which the density p = P''*'' ; then, if d<r' represent any element of
this surface, and we afterwards form the quantity
the integral extending over the whole spherical surface, g being the
distance p, da and y\f the characteristic of any function whatever. I
say, the resulting value of V will be of the form
V= Y^B;
R being a function of r, the distance Op only and K<'' what Y'^^ becomes
by changing 9', w, the polar co-ordinates, into 9, tit, the like co-ordinates
of the point p.
To justify this assertion, let there be taken three new axes JT,, I^„ Z„
so that the point p may be upon the axis Xx ; then, the new polar
co-ordinates of da' may be written r', ff, tjt', those of p being r, 0, •sr.
and consequently, the distance will become
g = ^{r" - 2 rr' cos 9,' + r^) ;
and as da^' = r'^d9i'd'sri sin 9,', we immediately obtain
r = fY'^'Vde.d-sr, sin 9, f (/•-- 2rr' cos d,' + O
= r'^SZd9; sin 0/ ^{r'-^rr' cos 0/ -f r'^)f^Zd-ur( Y' <".
Let us here consider more particularly the nature of the integral
In the preceding part of the present article, it has been shown that
the value of Y'^'^, when expressed in the new co-ordinates, will be of
the form P'/*'' ; but aU functions of this form (Vide Mec. Cel. Liv. iii.)
may be expanded in a finite series containing 2 « + 1 terms, of which
the first is independent of the angle "sr,', and each of the others has
for a factor a sine or cosine of some entire multiple of this same angle.
Hence, the integration relative to ro-/ will cause all the last mentioned
terms to vanish, and we shall only have to attend to the first here.
But this term is known to be of the form
, / ,. i.i — \ ,. „ i.i—l.i-2.i — S ,,• , « N
mh green, on the laws of the equilibrium of fluids. 19
and consequently, there will result
where ni = cos 9^ and ^ is a quantity independent of 6/ and tr/, but
which may contain the co-ordinates 9, -ar, that serve to define the
position of the axis JCi passing through the point p.
It now only remains to find the value of the quantity k, which may
be done by making 0i' = O, for then the line r coincides with the axis
JTi, and K*'' during the integration remains constantly equal to Y^\
the value of the density at this axis. Thus we have
^ ^rin ^ 7 [-. ii—l i.i — l.i—2.i — 3 „ \
V 2.2?— 1 2.4.2«— 1.2^ — 3 I
or, by summing the series within the parenthesis, and supplying the
common factor 2 7r, -
•jr(i) _ ^-^-^ ^ J,
1.3.5 2«-l '
and, by substituting the value of k, draAvn from this equation in the
value of the required, integral given above, we ultimately obtain
If now, for abridgement, we make
^^> = ^' - 2:27:11^' + 2.4.2i-1.2i-3 ^' -^^-
we shall obtain, by substituting the value of the integral just found in
that of V before given,
r= r(^27rr'%i44^^^-^^^^^/_}r?^/(^H(^-2rr'M/ + r'^);
which proves the truth of our assertion.
From what has been advanced in the preceding article, it is likewise
very easy to see that if the density of the fluid within a sphere of
any radius be every where represented by
c 2
20 Mr green, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS.
<p being the characteristic of any function whatever; and we afterwards
form the quantity
where dv represents an element of the sphere's volume, and g the dis-
tance between dv and any particle p under consideration, the resulting
value of V wiU always be of the form
V^^ being what I^'*" becomes by changing 9^, nr , the polar co-ordinates
of the element dv into Q, w, the co-ordinates of the point p; and R
being a function of r, the remaining co-ordinate of p, only.
4. Having thus demonstrated a very general property of functions
of the form P"*'', let us now endeavour to determine the value of F"
for a sphere whose radius is unity, and containing fluid of which the
density is every where represented by
p = {l-x''-y"-zyf{x',y',z');
on', y , z' being the rectangular co-ordinates of dv, an element of the
sphere's volume, and Jl the characteristic of any rational and entire
function whatever.
For this purpose we will substitute in the place of the co-ordinates
x', y , z' their values
x = r cos &\ y = r sin & cos w'. z = r' sin ff sin -bt' ;
and afterwards expand the function/(a;', y', s) by Laplace's simple method,
{Mec. Cel. Liv. iii. No. 16.). Thus,
(7) /{x, y, z) =/<«>+/'" +/'<^> + &c +/'«;
s being the degree of the function /{x, y', z').
It is likewise easy to perceive that any term /'■''' of this expansion
may be again developed thus,
/'(•■) =/;(•■>/* +/'«/-= +^'<V^+^ + &c.;
and as every coefficient of the last developement is of the form [/'",
(Mec. Cel. Liv. iii.), it is easy to see that the general term y'''V'+^' may
always be reduced to a rational and entire function of the original
co-ordinates x, y', »'.
Mb GREEN, ON THE LAWS. OF THE -EQUILIBRIUM OF FLUIDS. 21
If now we can obtain the part of ^ due to the term
we shall immediately have the value of V by summing all the parts
corresponding to the various values of which i and t are susceptible.
But from what has before been proved (Art. 3.), the part of V now
under consideration must necessarily be of the form F"*'^; representing,
therefore, this part by F"/'', we shall readily get
r/" =/J/'+^'+^</r' (1 - ryjtl-nr'de' sin ^/'<'^^'-".
Moreover from what has been shown in the same article, it is easy
to see that we have generally
fV'^'^clu'de' sin e'^ig"-) = Stt F« ^f'f •:'^'~^ /-i'c?Mi' (i) yl^{r'-2rr',x,' + r") ;
\(/ being the characteristic of any function whatever, and P'^'' what P''"*
becomes by substituting 9, w the polar co-ordinates of p in the place
of 6', TB-', the analogous co-ordinates of the element dv. If therefore
in the expression immediately preceding, we make
F'«=/'« and fig^) =^'.- = (^^)^,
and substitute the value of the integral thus obtained for its equal in
Vf-'^ there will arise
where yj® is deduced from Jl'^^ by changing 9", %r' into 6, w, and (i), for
abridgement, is written in the place of the function
,; i.i-1 ,;_; , i.i-l.i-2.i-3 ,^_^ .
'"~2.2e-l'" ^2.4.2i-1.2«-3'" *'''•
As the integral relative to n\ which enters into the expression on
the right side of the equation (8) is a definite one, and depends therefore
on the two extreme valvies of fj.\ only, it is evident that in the deter-
mination of this integral, it is altogether useless to retain the accents
22 Mr green, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS.
by which n\ is affected. But by omitting these superfluous accents,
we shall have to calculate the value of the quantity
I-n
2 ,
fj.dfi. (^) . (r' - 2 rr'/m + r")
where
,. . i.i — 1 . i.i-i,i-2.i-3 ; , -
^'^ = ''-2:2^:1'' + 2.4.2i-1.2i-3 -^ -^^-
The method which first presents itself for determining the value of
l-n
the integral in question, is to expand the quantity {r^ — 2rr'/u. + r'^) ^ by
means of the Binomial Theorem, to replace the various powers of m by
their values in functions similar to (i) and afterwards to effect the in-
tegrations by the formulee contained in the third Book of the Mec. Cel.
For this purpose we have the general equation
.-s i ... , i.i — 1 ,. „. , i.i—l.i — 2.i — 3,. ,,
^^^ '^ =^^)+ 2:271:1 (^-^^-^2X2I33:27::5(*-*)
i.i-l.i-2.i-3.i-4!.i-5 .
2.i.6.2i-5.2i-7.2i-9 ^' '") + ^^'
To remove all doubt of the correctness of this equation, we may
multiply each of its sides by (i, and reduce the products on the right
by means of the relation
which it is very easy to prove exists between functions of the form (?).
In this way it will be seen that if the equation (9) holds good for any
power fx' it will do so likewise for the following power ^'+^ and as it
is evidently correct when i='l, it is therefore necessarily so, whatever
whole number i may represent.
Now by means of the Binomial Theorem, we obtain when r^r'
= 2,
r"'-K(r'-2rr'^ + r"y = (i-2m J + ^,)
y>n—l.n + l.n-\-3 n + 2s — 3
l—n
3
2s
If now we conceive the quantity (2ju- rj to be expanded by
Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 23
the same theorem, it is easy to perceive that the term having f— |
for factor is
i + 2t'
7i-l.n-\-l.n + 3 « + 2^• + 4^'-3 ,^,„ ,^„, /r\
2.4.6 2e + 4r '^ W)
i + iV
n-l.n + 1 n + 2i + 4>t'-5 ,^,,_, ,+,„_, fn'^"''!! i+^t-1
2 . 4 2? + 4#'-2 '" """ ' \r')
2.4 2« + 4ir-4 ^ '^^ VJ /•''■ 1.2
— &c &c &c
/^\ i + 2('
and therefore the coefficient of I — I in the expansion of the function
will be expressed by
v "-^-^ + l ^ + 2^'+4^'-2^- 3 ,,,,.,, ..
2.4 2« + 4^'-2* ^""^^ '^ ^
« + 2#'-*.^ + 2^'-«-l ^■ + 2^-2* + l
Hence the portion of this coefficient containing the function (i), when
the various powers of /i have been replaced by their values in functions
of this kind agreeably to the preceding observation will be found, by
means of the equation (9), to be
. .X „ n — l.n + 1 n + 2i+4<t'~2s — 3
^^^ 2 . 4 2? + 4^'-2*
^^ + 2/-2^.^• + 2^'-2,y-l i + 1
"^ 2.4 2^-2*x2^■ + 2if'-2* + 1.2^ + 2f-2#-1...2^■ + 3
i + 2t'-s.i+2t'-s-l i + 2^-2#+l
.2*+^''-=»(-l)»x-
1.2.3
n-l.n + 1. n + S n + 2i+it' -2s-3 ,,,,.,.,..
■^^2.4.6 2« + 4^-2« ^ ^
24 Mr GREEN, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS.
^^ + l.^^ + 2.^^ + 3.^' + 4 i + ^t' -s
^ 1.2.3 *x2.4.6 2^-2* X 2? + 2^'- 2* + 1 2e + 3
_ . (-1V.W-1.W + 1.W + 3 w + 2? + 4^' -2.9 -3
-2'-W-2g^ 2?x2.4 2*x2.4 2#'-2* x 2^■ + 2^'-2*+1...2^■ + 3
3.5.7 2« + l ... n-\.n + \.n + S ?« + 2m2^'-3
f^(0
X
1.2.3 i ^^ 3.5.7 2« + 2r + l
r - 1 V ^ + ^^' + ^^-1 w + 2? + 4/-2.y-3
^ "^ ^ 2.4.6 2^-2*
2t + 2^'-2.y + 3 2? + 2/' + 1
^ 2.4.6 2*
where all the finite integrals may evidently be extended from * =
to * = 00 , and it is clear that the last of these integrals is equal to the
coefficient of a^ in the product
(, w + 2i + 2#'-l » + 2i + 2/-l.w + 2i + 2/' + l „ „ . . .,
{1+ x + ^-^ oi? + hc.tninf.\
,, 2? + 2^ + l , 2i + 2r + 1.2? + 2#'-l „ . . . ^,
X {1 ~ x+ ^-^ af-kc. intnf.]
If now we write in the place of the series their known values, the
preceding product will become
n + 2i+2f-l 2i+2t' + l i-n
(l-or) " x(l-;r) ' ={\-x)\
and consequently the value of the required coefficient of af^ is
« — 2.W.W + 2 w + 2/' — 4
2 .4. 6 2^'
This quantity being substituted in the place of the last of the finite
integrals gives for the value of that portion of the coefficient of
which contains the function (i) the expression
3.5.7 2? + l n-l.n + 1 w+2?'+2^'-3 n-2.n m+2^ -4 , .,
1.2.8 i ^3.5 2e + 2^ + l ^ 2.4 2t' ^*^'
Mb green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 25
By multiplying the last expression by ( — ) , and taking the sum
of all the resulting values which arise when we make successively
^ = 0, 1, 2, 3, 4, 5, 6, &c. in infinitum,
we shall obtain the value of the term I^<'> contained in the expression
, l-n
(l - 2m ^ + ^,) ~ = Y^'^ + rc> + F(=> + F^^' + &c.
Hence,
1.2 i ^^^ 3.5 2i + 2f+l
n-2.n n + ^t'-i /rV*^*'
"^ 2.4 2t' [?) '
the finite integral extending from t' = to t' = oc.
But by the known properties of functions bf this kind, we have
by substituting for F'"' its value
/_\d^ (i) (l - 2m p + ^^~=/-\d^ (i) . F«
3.5.7 2« + l ., ,.,„ ^n-l.n + l n + 2i + 2t'-3
= 1.2.3 i /^^(O^x^ 3 . 5 2e + 2^' + l
n-2.n « + 2/-4/rV*"'
2.4 2t'
(p)
^ 1.2.3 i n-l.n + 1 n + 2i + 2t'-3
~ 1.3.5 2?-l 3 . 5 2i + 2t' + 2
71 — 2. n n + 2t' — ^! fr'
t' ~ \r'} '
2.4 2t
since by what Laplace has shown {Mec. Cel. Liv. iii. No. 17.)
■^^^ W^= 2m I 1.3.5 2e-l j •
Vol. V. Paet I. D
26 Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS.
If now we restore to n. the accents with which it was originally
affected, and multiply the resulting quantity by r'""\ we shall have when
r<r''
(10) /Ad^\(i) if-^rr't^, + ■' =/»-y_;</M'i ii) (l - UJ-. + ^-.)
_ ,j_„ 1.2.3 i w-l.w + 1 ?^ + 2^^ + 2^-3
~ '1.3.5 2«-l 3 . 5 2e + 2#'+l
;« — 2.W « + 2#'-4 /r
2.4 2jf'
9
j + 21'
and in order to deduce the value of the same integral when /•' /. r, we
shall only have to change r into /, and reciprocally, in the formula
just given.
We may now readily obtain the value of Vi^ by means of the
formula (8). For the density corresponding thereto being
:/;«/•'+=' (l-r"7, ■
it follows from what has been observed in the former part of the
present article, that ^'®r'^^' may always be reduced to a rational and
entire function of icf, y, %' the rectangular co-ordinates of the element
dv, and therefore the density in question will admit of being expanded
in a series of the entire powers of x, y', %' and the various products of
these powers. Hence (Art. 1.) F/'' admits of a similar expansion in
entire powers, he. of x, y, z the rectangular co-ordinates of the point p,
and by following the methods before exposed Art. 1 and 2, we readily
get
^t ^-^J J r' ui y>. , ) .^ 3.5 2t + 2t' + l
n-2.n.n + 2 » + 2#' — 4 /r\'+"'
X
2 .4. 6 2t'
-4 /r^y^-""
" V'j '
and thence we have ultimately,
(,ii; rt Airj, ^33 2i + 2t' + 1 2.4 2/'
Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 27
r r-%^"-'' )r(3+i) r{/3 + i)r(i^)
,2r^*'
=2Trf,^\ ^^ - — -r'
/ 2t-2t' + 2fi + 6-n \ ■^' / 6 + 2/3-w\
4-n.6-n 2t~2t'+2-n n-2.n n + 2t'-4!
6 + 2fi-n 2t-2t'+2l3 + 4>-n ^ 2.4 W
n—1 .n + 1 n + 2i + 2t'-3 ^
"^ 3 . 5 2i + 2t' + l '
the finite integrals being taken from t' =0 to t'=cD and r being the
characteristic of the well known function Gamma, which is introduced
when we effect the integrations relative to r' by means of the formula
(3), Art. 1.
Having thus F"/" or the part of F corresponding to the term j^''*',
in J'(x', y, as') we immediately deduce the complete value of V by giving
to i and t the various values of which these numbers are susceptible,
and taking the sum of all the parts corresponding to the different terms
hi the expansion of the function fix', y', &').
Athough in the present Article we have hitherto supposed J" to be
the characteristic of a rational and entire function, the same process will
evidently be applicable, provided y"(a;', y, z') can be expanded in an
infinite series of the entire powers of x', y, z' and the various products
of these powers. In the latter case we have as before, the development
fix', y, z') =/'<»> +/'<•> + /'® +/'<^) + &c.
of which any term, as for example f'^''> may be farther expanded as
follows,
/'« =/;«r" + /'«r"+^ +/'«/•"+*+ &c.
and as we have already determined F"/*' or the part of V corresponding
toyt'''V'+^'', we immediately deduce as before the required value of V,
the only difference is, that the numbers i and t, instead of being as
in the former case confined within certain limits, may here become in-
definitely great.
D 2
28 Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS.
In the foregoing expression (11) /3 may be taken at will, but if we
qq ^
assign to it such a value that -~ — may be a whole number, the
series contained therein will terminate of itself, and consequently the
value of Vt^^ will be exhibited in a finite form, capable by what has
been shown at the beginning of the present Article of being converted
into a rational and entire function of x, y, %, the rectangular co-ordinates
of p. It is moreover evident, that the complete value of V being com-
posed of a finite number of terms of the form Vt-'^ will possess the same
property, provided the function fix, y , %) is rational and entire, which
agrees with what has been already proved in the second Article, by a
very different method.
(5) We have before remarked, (Art. 2.) that in the particular case
where /3 = — — — , the arbitrary constants contained in y(a;', y' , %) are just
sufficient in number to enable us to determine this function, so as to
make the resulting value of V equal to any given rational and entire
function of x, y, z, the rectangular co-ordinates of p, and have proved
that the corresponding functions V and J" will be of the same degree.
But when this degree is considerable, the method there proposed becomes
impracticable, seeing that it requires the resolution of a system of
^ + 1 .^ + 2,.s + 3
1.2.3
linear equations containing as many unknown quantities ; s being the
degree of the functions in question. But by the aid of what has been
shown in the preceding Article, it will be very easy to determine for
this particular value of /3 the function J'{x', y, %') and consequently the
density p when F' is given, and we shall thus be able to exhibit the
complete solution of the inverse problem by means of very simple
formulae.
For this purpose, let us suppose agreeably to the preceding remarks,
that p the density of the fluid in the element dv is of the form
p = {l-r^)-^/{x',y',z);
Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 29
f being the characteristic of a rational and entire function of the same
degree as V, and which we will here endeavour so to determine, that
the value of V thence resulting, may be equal to any given rational
and entire function of x, y, % of the degree s.
Then by Laplace's simple method {Mec. Cel. Liv. iii. No. 16.) we
may always expand F" in a series of the form
r= r<«> + r(» + r® + &c + r«.
In like manner as has before been remarked, we shall have the
analogous expansion
f{x',y', ,')=/''«' +/'<'>+/'^=>+/'<'>+ &c +/'«,
of which any termy*'' for example, may be farther developed as follows,
/'« =^'('V' +y;'''V"+'^ +//»/'+* + &c. = r" (/'« +y;'('V'^ +/'»;.'^ + &c.)
y", yj'<'>, j^''*^, &c. being quantities independent of / and all of the form
K'"' {Mec. Cel. Liv. iii.) Moreover F/" the part of F' due to the general
term Jl'^'^r''+^* of the last series, will be obtained by writing for (i
in the equation (11), and afterwards substituting for
(n — 2\ _ f4i-n
r(!t^)r(l^) us value-
n-2
sin
In this way we get
27r;/;'V ±-„,e-n 2t-2f + 2-n
' . fn-2 \ 2.4 2t~2t'
sm (-^.j
n-2.n w + 2#'-4 w — 1 . w + 1 n + 2i + 2f — 3 ^
^ 2.4 2? ^ 3 . 5 ...... 2i + 2t' + l '
yj<'^ being what J]''-^ becomes by changing 6', -ar' into 6, sr, and the finite
integral being taken from t' = to t'=<x .
Let us now for a moment assume
^(0=
n-2.n w + 2#' — 4 «-l.« + l n + 2i + 2t'-3
X
2.4 2t' 3.5 2i + 2t' + l '
30 Mr green, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS.
then the expression immediately preceding may be written
■^J7\r 4<-n.6-n Q,t-2t' + 2
(n-2 \ 2.4 2t-2t'
dn [-^ .)
sin
and by giving to t the various values 0, 1, 2, 3, &c. of which it is sus-
ceptible, and taking the sum of all the resulting values of F/'' the quantity
thus obtained will be equal to V^^ or that part of V which is of the
form Y^\ Thus we get
27r^ r'
■fr(i),
sm
.«/)(0)./„«
+ &c &c &c.
since aU the terms in the preceding value of Vi-^ in which t'>t vanish
of themselves in consequence of the factor
/ 2^-2^ +4-w \
= (when t > t).
2 . 4 2i?-2^ „,. ., . _ „ /4-w
(-'■«) rC-?)
But F"^'^ as deduced from the given value of V may be expanded in
a series of the form
r«=?''. {r„®-i- r;(v^+ r,»./^+ v^'>t^+kc.\
and if in order to simplify the remaining operations, we make generally
__„ 27r^ n-2.n M+2/-4 n-\.n + l n + 2i+2t-3 ^„.,
' . [n-2 \ 2.4 2t 3.5 2« + 2^+l '
sm [—.]
Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 31
27r'
X<J>{t).W,
. /n-2 \
sm (— .)
the equation immediately preceding will become
^(0= ^I-:£ X {(j) (0) . f7o"' + 0(l) . t7;(W(^ (2) C7,». r' + &cc.\
[n — 2 \
which compared with the foregoing value of F^'\ will give by suppressing
the factor '— , common to both, and equating separately the
sm(-^.)
coefficients of the different powers of the indeterminate quantity r the
following system of equations
&c=...&c &c &c.
for determining the unknown functions fo'-^, /<*', f/\ &c. by means of the
known ones f7"o*'\ Ui'-'\ ZJg*", &c. In fact the last equation of the system
gives U^^=fP, and then by ascending successively from the bottom to
the top equation, we shall get the values of fs^\ /,%, f}%, &c. with
very little trouble. It will however be simpler still to remark, that the
general type of all our equations is
where the symbols of operation have been separated from those of
quantity and e employed in its usual acceptation, so that
32 Mr green, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS.
But it is evident we may satisfy the last equation by making
/«=(l-e)''^C7-«.
Expanding now and replacing e?7„'''; e^UJ^^, &c. by these values UJIi,
U^%, &c. we get
from which we may immediately deduce ^'® and thence successively
/'« = r" (/„'« +/'« r'^ +/;« r" +//« r'» + &c.)
fW, !/, 85') =/'<"' +/'«+/'<^' + &c +/'«
and > = (1 - X'"- - y" - z'"-y^.f(x', y , %),
Application of the general Methods exposed in the preceding Articles
to Spherical conducting Sodies.
(6) In order to explain the phenomena which electrified bodies
present. Philosophers have found it advantageous either to adopt the
hypothesis of two fluids, the vitreous and resinous of Dufay for
example, or to suppose with jEpinus and others, that the particles of
matter when deprived of their natural quantity of electric fluid, possess
a mutual repulsive force. It is easy to perceive that the mathematical
laws of equilibrium deducible from these two hypotheses, ought not to
differ when the quantity of fluid or fluids (according to the hypothesis
we choose to adopt) which bodies in their natural state are supposed
to contain, is so great, that a complete decomposition shall never be
effected by any forces to which they may be exposed, but that in
every part of them a farther decomposition shall always be possible by
the application of still greater forces. In fact the mathematical theory
of electricity merely consists in determining p* the analytical value of
* It may not be Improper to remark that p is always supposed to represent the density
of the free fluid, or that which manifests itself by its repulsive force; and therefore, when
the hypothesis of two fluids is employed, the measure of the excess of the quantity of either
fluid
Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 33
the fluid's density, so that the whole of the electrical actions exerted
upon any point p, situated at will in the interior of the conducting
bodies may exactly destroy each other, and consequently p have no
tendency to move in any direction. For the electric fluid itself, the
exponent n is equal to 2, and the resulting value of p is always such
as not to require that a complete decomposition should take place in
the body under consideration, but there are certain values of n for which
the resulting values of p will render fpdv greater than any assignable
quantity ; for some portions of the body it is therefore evident that
how great soever the quantity of the fluid or fluids may be, which
in a natural state this body is supposed to possess, it will then become
impossible strictly to realize the analytical value of p, and therefore some
modification at least will be rendered necessary, by the limit fixed to
the quantity of fluid or fluids originally contained in the body, and
as Dufay's hypothesis appears the more natural of the two, we shall
keep this principally in view, when in what follows it may become
requisite to introduce either.
7. The foregoing general observations being premised, we will proceed
in the present article to determine mathematically the law of the density
p, when the equilibrium has established itself in the interior of a con-
ducting sphere A, supposing it free from the actions of exterior bodies,
and that the particles of fluid contained therein repel each other with
forces which vary inversely as the w"" power of the distance. For this
purpose it may be remarked, that the formula (1), Art. 1, immediately
gives the values of the forces acting on any particle p, in virtue of
the repulsion exerted by the whole of the fluid contained in A. In
this way we get
1 dV
- _ .-jr = the force directed parallel to the axis X,
1 dV
- ■ _ . y— = the force directed parallel to the axis Y,
fluid which we choose to consider as positive over that of the fluid of opposite name in any
element dv of the volume of the body is expressed by pdv, whereas on the other hypothesis
pdv serves to measure the excess of the quantity of fluid in the element dv over what it
would possess in a natural state.
Vol. V. Paet I. E
34 Mr green, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS.
1 dV
.-^- = the force directed parallel to the axis Z.
1 — » d%
But since, in consequence of the equilibrium, each of these forces is
equal to zero, we shall have
„ dV J . dV , , dV . .-,
= -5— dx + -7— dy + -J- d% = dV\
dx dy d%
and therefore, by integration,
F = const.
Having thus the value of V at the point p, whose co-ordinates are
X, y, %, we immediately deduce, by the method explained in the fifth
article,
/w-2 \
sm
P =
2'
^.(l-r'*)
seeing that in the present case the general expansion of K there given
reduces itself to
If moreover Q serve to designate the total quantity of free fluid in
the sphere, we shall have, by substituting for
sin f TT j its value
rl^)r[^y
\ 2 / rrz-ij >i.Ji/i '2\""S~ ^"^
sm
Q = /pe/«; = ^5 i F/liW^dril-r")
See Legendre. Exer. de Cal. Int. Quatrieme Partie.
In the preceding values, as in the article cited, the radius of the
sphere is taken for the unit of space ; but the same formula may
readily be adapted to any other unit by writing — in the place of r',
and recollecting that the quantities p, V, and Q, are of the dimensions
0, 4 — «, and 3 respectively, with regard to space; a being the number
Mi GREEN. ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 35
which represents the radius of the sphere when we employ the new
unit. In this way we obtain
P = — V4 — - r{a'-r")~, and Q = lii — ; . V.
Hence, when Q, the quantity of redundant flviid originally introduced
into the sphere is given, the values of V and of the density p are like-
wise given. In fact, by writing in the preceding equation for
ry, and sin(^7r),
their values, we thence immediately deduce
and F= ^ ' ■ ' a'-.Q.
\/7r
The foregoing formulae present no difficulties where « > 2, but when
H < 2, the value of p, if extended to the surface of the sphere Jl, would
require an infinite quantity of fluid of one name to have been origi-
nally introduced into its interior, and therefore, agreeably to a preceding
observation, could not be strictly realized. In order then to determine
the modification which in this case ought to be introduced, let us in
the first place make n>2, and conceive an inner sphere S, whose
radius is a — Sa, in which the density of the fluid is still defined by
the first of the equations (12); then, supposing afterwards the rest of
the fluid in the exterior shell to be considered on ^'s surface, the portion
so condensed, if we neglect quantities of the order Sa, compared with those
retained, will be
-- r f^±i)
2* V 2 / ^,
' V2/
E 2
(1)
QJa
2
36 Mr green, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS.
and since, in the transfer of the fluid to ^'s surface, its particles move
over spaces of the order ^a only, the alteration which will thence be
produced in V will evidently be of the order
n — 2 n
and consequently the value of V will become
k being a quantity which remains finite when ^a vanishes.
In establishing the preceding results, ti has been supposed greater
than 2, but p the density of the fluid within S and the quantity of it
condensed on ^'s surface being still determined by the same formulae,
the foregoing value of V ought to hold good in virtue of the generality
of analysis whatever n may be, and therefore when w is a positive quantity
and hi is exceedingly small, we shall have without sensible errors
v;^m^m«'--* '
Conceiving now P' to represent the density of the fluid condensed
on A's surface, 47ra^P' will be the total quantity thereon contained, which
being equated to the value before given, there results
-y/TT
(I)
and hence we immediately deduce
fn + V
n — 4 -p
2~
m
Moreover as Q represents the total quantity of redundant fluid in the
entire sphere A, the quantity contained in B is
Mh green, on the laws of the equilibrium of fluids. 37
-)„=.
If now when w is supposed less than 2, we adopt an hypothesis
similar to Dufay's, and conceive that the quantities of fluid of opposite
denominations in the interior of A are exceedingly great when this
body is in a natural state, then after having introduced the quantity Q
of redundant fluid, we may always by means of the expression just
given, determine the value of Sa, so that the whole of the fluid of
contrary name to Q, may be contained in the inner sphere S, the
density in every part of it being determined by the first of the equa-
tions (12). If afterwards the whole of the fluid of the same name as
Q is condensed upon A's svirface, the value of V in the interior of S
as before determined will evidently be constant, provided we neglect
n
indefinitely small quantities of the order ht'\ Hence all the fluid con-
tained in J3 will be in equilibrium, and as the shell included between
the two concentric spheres, A and S is entirely void of fluid, it follows
that the whole system must be in equilibrium.
From what has preceded, we see that the first of the formulae (12)
which served to give the density p within the sphere A when n is
greater than 2, is still sensibly correct when n represents any positive
quantity less than 2, provided we do not extend it to the immediate
vicinity of A's surface. But as the foregoing solution is only approxi-
mative, and supposes the quantities of the two fluids which originally
neutralized each other to be exceedingly great, we shall in the follow-
ing article endeavour to exhibit a rigorous solution of the problem,
in case w < 2, which will be totally independent of this supposition.
8. Let us here in the first place conceive a spherical surface whose
radius is a, covered with fluid of the uniform density P', and suppose
it is required to determine the value of the density p in the interior
of a concentric conducting sphere, the radius of which is taken for
the unit of space, so that the fluid therein contained, may be in equi-
librium in virtue of the joint action of that contained in the sphere
itself, and on the exterior spherical surface. "•
38 Mr green, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS.
If now V represents the value of V due to the exterior surface,
it is clear from what Laplace has shown, {Mec. Cel. Liv. ii. No. 12.) that
^ = !y^ = (3^:^ K«+^r"-(«-'-M;
rfo- being an element of this surface, and g being the distance of this
element from the point p to which V is supposed to belong.
If afterwards we conceive that the function V is due to the fluid
within the sphere itself, it is easy to prove as in the last article, that
in consequence of the equilibrium we must have
V +V= const.
But V and consequently V is of the form F^"', therefore by employ-
ing the method before explained, (Art. 4.) we get
/(ar', y', %) =/'(»' =/„("> +/(">. r'' +//>. r'' + &c. = B, + B,r'' + B, r" + &c. ;
where, as in the present case, ^''°>, yi'<°', ^''% &c. are all constant
quantities, they have for the sake of simplicity been replaced by
J?o, jBi, B.^, &c.
Hitherto the exponent /3 has remained quite arbitrary, but by making
/3= — -— ^ the formula (11) Art. 4. will become when « = 0,
ir(o)_o 7? < 2y V 2 ; ^ ,,, 4-».6-w 2t-9.t' + ^-n
^' -2'^^'- YW) ' 4 . 6 2^-2^+2
n-2.n-l « + 2^- 3
"^ 2.3.4 2^ + 1
{/i-2)Tr'Bt ^ ,„ 4-W.6-M 2^-2/' + 2-« w-2.«-l « + 2/-3
2.?^ . ^ -r^ ^w . n ^
. (71-2 \ • 4.6 2^-2^ + 2 2 . 3 2^' + l '
sm(— .)
Giving now to / the successive values 0, 1, 2, 3, &c. and taking
the sum of the functions thence resulting, there arises
r= F<°' = Fo<"> + rr + T^-P + ^3<°* + &c. = s. r/">
(«-2)7r'^ ^P^ ,,, 4-W.6-W 2t-2f + 2-n n-2.n-l «+2^'-3
"""T^ri"^ ' 4.6.8 2^-2^' + 2 "" 2 . 3 2/' + l '
sm (^.)
where the sign S is referred to the variable t and 2 to ^.
Mn GREEN, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 39
Again, by substituting for V and V their values in the equation
V^ + V= const, and expanding the function V we obtain
,/^*' ti-2.n-l.n n + 2f-3
const. =47rP'«'-".2
«^"" 2 . 3 .4 2^ + 1
(w-2)7r' „ „ ,, /4-».6-?? 2^-2^'+2-w n-2.n-l ti+2t'+3
"^ (n-2 X '^^^''' 4 . 6 2f-2f + 2 ^ 2.3.4 2^ + 1
sm
..)
2
which by equating separately the coefficients of the various powers of
the indeterminate quantity r, and reducing, gives generally
(n-2
2P'«'-"-"'. sin /•
TT
2 I „2 — «.4 — « 2s — n
-^ 2 . 4> 2s ^''^'-'
Then by assigning to t' its successive values 1, 2, 3, &c. there results for
the determhiation of the quantities B^,Bi, JB^, &c. the following system
of equations,
2P' „ . (n-2 \ p ^ 2-n „ 2-n.4<-n „ , »
2P' . fn-2 \ „ „ 2 — « „ 2 — ;«.4 — ra ^ .
=^ «'-".sin — r-TT .«-^ = P,-f — r- P. + — ^— r— P3 + &C.
TT V 2 / 2 2.4
2P' , . (n-2 \ ... „ 2-M„,2-ra.4-/<
a'-", sm ( -^— t] • '^ = P.' + —^ -^^ + ~~;> 4 — P4 + &C.
&c &c &c &c
But it is evident from the form of these equations, that we. may satisfy
the whole system by making
B, = B^.a\ B, = B.a-\ B, = B,,(r\ B, = B^.a-M>ic.
provided we determine Po by
2P' . /n-2 \ „ ,, 2-n , 2 — n.4>-n ... . ,
= P„(l-a-)'
„ 2P . (n-2 \ . , ,,^
n-2
,-2\ 2
n — 2
Hence as in the present case, ^ = —^ , we immediately deduce the
successive values
2P . fn-2
40 Mr GREEN, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS.
/(or', y\ «')=/'(") = ^„ + P,r'^ + ^.r'^ + &c. = if„ (l - ^']",
and p = (1 - O ^ ./(x', y', %') = ^ sin {—^ ^) . («= - 1)"...
(«^ - /'')-Ml - 0~-
In the value of p just exhibited, the radius of the sphere is taken
as the unit of space, but the same formula may easily be adapted to
any other unit by writing j and y- in the place of a and / respectively,
and recollecting at the same time that in consequence of the equation
•dv.p . rdaP'
const,
= r^r^j!^ + ji
s -^ g
before given, ^ , is a quantity of the dimension — 1 with regard to
space: h being the number which represents the radius of the sphere
when we employ the new unit. Hence we obtain for a sphere whose
radius is bg, acted upon by an exterior concentric spherical surface
of which the radius is a,
2P'a.sin {—-■"] 2-n ^
(/3) p = -^ ff-b') ' {a'-r")-' {b'-r") ^ ;
7r
P' being the density of the fluid on the exterior surface.
If now we conceive a conducting sphere A whose radius is a, and
determine P' so that all the fluid of one kind, viz. that which is re-
dundant in this sphere, may be condensed on its surface, and afterwards
find b the radius of the interior sphere S from the condition that it
shall just contain all the fluid of the opposite kind, it is evident that
each of the fluids will be in equilibrium within A, and therefore the
problem originally proposed is thus accurately solved. The reason for
supposing all the fluid of one name to be completely abstracted from
S, is that our formulas may represent the state of permanent equilibrium,
for the tendency of the forces acting within the void shell included
between the surfaces A and B, is to abstract continually the fluid of
the same name as that on ^'s surface from the sphere S.
Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 41
To prove the truth of what has just been asserted, we will begin
with determining the repulsion exerted by the inner sphere itself, on
any point p exterior to it, and situate at the distance r from its centre
O. But by what Laplace has shown {Mec. Cel. Liv. ii. No. 12.) the
repulsion on an exterior point p, arising from a spherical shell of which
the radius is r', thickness dr and center is at O will be measured by
I'Kr'dr'p d_ (r + r)^-" - (r-/)^-"
1 — ti.S — n'dr' r '
the general term of which when expanded in an ascending series of
r'
the powers of — is,
+ ^"- 2.3.4.5 27Tl ^-.r-prfr,
and the part of the required repulsion due thereto will, by substituting
for p its value before found, become
%F . (n-2 \ ,o ,,,^ -2 + wx«.» + l M+2*-3xw + l*-l
Si' . fn-2 \ , , ,„,:
2.3 . 4 2« + l
,'2\ -1 »-2
X
It now remains to find the value of the definite integral herein con-
tained. But when 11 -\ is expanded, and the integrations are
effected, by known formulae, we obtain
(14) M 1 - ^ I (*' - O ' r''^^'dr'=/o' 2^ -jj {b'-r") ' .r"'+'dr'
^^ "a'' r^, ,3 X "^ 'f n 3\
, 2^ + 3 y 2^ + 3.2^ + 5 ¥
* 25 + 3 + w a' 2* + 3 + ».2« + 5 + wa' "*■ ^'^
^,^, ^ (2) ^ r "^ 2) {^s+l-^n){l-xy'' r^
_, , ^, ,.„ . _ „ , , „ 'dx
, 1J2S + 1
2° • / « 3>
r (.+ - + -) (1-.^)-
Vol. V. Pakt I.
42 Mr green, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS.
r f^^l r (-\ — '
\2J \2) 1.3.5 2s + lb''+'+" (l-af)' fx'^^'dx
1+w ^ ^•'^+'+" ^o'
2
J {\-x)
Avhere after the integrations have been effected, x ought to be made
equal to - .
The value of the integral last found being substituted in the expres-
sion immediately preceding, and the finite integral taken relative to s
from * = to * = X gives for the repulsion of the inner sphere.
a ^ ll+n\ f2 — n
¥~J V'~2
)
-2.7i.n + 2 n + 2s-4> (by"(l-x')'' raf'^'dx
''^" 2.4.6 2s [r) ic^'+'-^" J„
(1-x^y
.2\2
-47rv/,rP^a^/-" » n-2.n.n + 2 w + 2^-4 (a^ rj..^„ /, _^.^\
„fl+n\ l2-n<:^' 2.4.6 2s (rj ■'"'^^'^ ' ^* ^' '
i ) [ 2 J
since F (^) =\/ir, sin ( tt] =
^'D^C-?)'
and as was before observed, a; = - .
a
But we have evidently by means of the binomial theorem,
/ _ ff-j;-\i^ _ . n-2.n. w4-2 w + 2^-4 /«^y,
I rW ' "" ° 2.4.6 2* \r] '
and therefore the preceding quantity becomes
(15) 7^^ L dxaf 1 ' (l-x^)" .
Mb green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 43
T X*
If HOW we make x — — , the same quantity may be written
(16) , . , -; tx'dxiX-x^)' 1 —\'-
Having thus the vahie of the repulsion due to the inner sphere B
on an exterior point p, it remains to determine that due to the fluid
on ^'s surface. But this last is represented by
, 2 7raP^ (L_ {a-\-rf-''-{a-rf-
' l — 7i.S — ndr r
{Mec. Cel. Liv. ii. No. 12.) Now by expanding this function there re-
sults
1~ ■ r "*" 4.5 ^ "*" 4.5.6.7
^TrP'a'-'r. ^^ ■ U + "T l^ 2'- + """,T^":" ' " .g^+^cl
. r./ i» 2-w .^, ra.w + l.?« + 2 /i + 2*-l, ^.r"
= 4.PV-V.-g-.2„ ,.5.6 ,,^3 (^-^1)^-
The last of these expressions may readily be exhibited vinder a finite
form, by remarking that
flx"dx{l-a^) ^ (l - -^) ^ =/lx''dx(l-x') " S ^^ g ^^^ .-^
/ 2^ + w + l \ / 4-/A
«.?f + 2.;? + 4 w + 2jf-2 y^' V 2 / V 2 /
~ " 2. 4 .6 2s 'a^'' ^^l2s + 5\
2 r
r {lzl\ r fl+^^
V 2 / \ 2 I 2-n ^^ n.n + l.n + 2 n + 2s-l , ,.r^'
©
3 ' " 4. 5 . 6 2* + 3 ' 'a
Hence, since r (^) = v^tt, the value of the repulsion arising from ^'s
surface becomes
F 2
44 Mr green, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS.
Now by adding the repulsion due to the inner sphere which is given
by the formula (16), we obtain, (since it is evidently indifferent what
variable enters into a definite integral, provided each of its limits re-
main unchanged)
f'afdxCl -x") '' .{1 , ,
1 + w\ / 2-w \ •'^ ^ ' V (f )
\ 2 j '
for the value of the total repulsion upon a particle p of positive fluid
situate within the sphere A and exterior to S. We thus see that
when P' is positive the particle p is always impelled by a force whicli
is equal to zero at JS's surface, and which continually increases as p
recedes farther from it. Hence, if any particle of positive fluid is
separated ever so little from 2?'s surface, it has no tendency to return
there, but on the contrary, it is continually impelled therefrom by a
regularly increasing force ; and consequently, as was before observed,
the equilibrium can not be permanent until all the positive fluid has
been gradually abstracted from B and carried to the surface of A,
Avhere it is retained by the non-conducting medium with which the
sphere A is conceived to be surrounded.
Let now q represent the total quantity of fluid in the inner sphere,
then the repulsion exerted on p by this will evidently be
qr-',
when r is supposed infinite. Making therefore r infinite in the expression
(15), and equating the value thus obtained to the one just given, there
arises
q= — :: tclx-afil-x')'.
2 J \ 2
When the equilibrium has become permanent, q is equal to the total
quantity of that kind of fluid, which we choose to consider negative,
originally introduced into the sphere A ; and if now qi represent the
Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 45
total quantity of fluid of opposite name contained within A, we shall
have, for the determination of the two unknown quantities P' and b,
the equations
5', = 4nra'.P',
and ^ = — , "^""^ X" dxx" (1 - af)^,
and hence we are enabled to assign accurately the manner in which the
two fluids will distribute themselves in the interior of A; q and «/, , the
quantities of the fluids of opposite names originally introduced into A
being supposed given.
9. In the two foregoing articles we have determined the manner
in which our hypothetical fluids wiU distribute themselves in the interior
of a conducting sphere A when in equilibrium and free from all exterior
actions, but the method employed in the former is equally applicable
when the sphere is under the influence of any exterior forces. In fact,
if we conceive them all resolved into three JT, Y, Z, in the direction
of the co-ordinates x, y, « of a point j9, and then make, as in Art. 1,
r pdv
we shall have, in consequence of the equilibrium,
1 dr „ ^ \ dV ^ ^ \ dV „
0= -J— + X, = 5- + ^' = 7- + Z,
1 — ndx \ — ndy 1 — ndz
which, multiplied by dx, dy and d% respectively, and integrated, give
const. = =-^ V + f{Xdx + Ydy + Zdz) ;
X ^~ ft/
where Xdx + Ydy + Zd% is always an exact differential.
We thus see that when X, Y, Z are given rational and entire functions
V will be so likewise, and we may thence deduce (Art. 5.)
p = (1 _ ;j;'^ _ y'2 _ «'^)f^ ./{a;', y', x'),
where / is the characteristic of a rational and entire function of the Same
degree as V.
46 Mr green, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS.
The preceding method is directly applicable when the forces X, Y, Z
are given explicitly in functions of x, y, x. But instead of these forces,
we may conceive the density of the fluid in the exterior bodies as given,
and thence determine the state which its action will induce in the con-
ducting sphere A. For example, we may in the first place suppose
the radius of A to be taken as the unit of space, and an exterior con-
centric spherical surface, of which the radius is a, to be covered with
fluid of the density U"^'^: ZJ"'"* being a function of the two polar co-
ordinates 6" and ■zsr" of any element of the spherical surface of the same
kind as those considered by Laplace {Mec. Cel. Liv. iii.). Then it is
easy to perceive by what has been proved in the article last cited, that
the value of the induced density wiU be of the form
p = [/-'Wr'' (1 - r"'y' .f{r") ;
r', &, -ar' being the polar co-ordinates of the element dv, and £/'<'* what
Z7"<'> becomes by changing Q", -sr" into 9', tst'.
Still continuing to follow the methods before explained, (Art. 4. and 5.)
we get in the present case
f{af, y', «') = t7'<Vy(r'^) =/«,
and by expanding y(r'^), we have
/(r'^) = i?o + B,r" + B,i'' + B,r" + &c.
Hence, /'" = B,U'^\ and
' . ln-% \-"'^'''^ 2.4.6 2^-2^ ^ 2.4 2^'
sm(-^.)
n-l.n + 1 w + 2? + 2if-3
^3.5 2«-l-2«r + l ■
Then, by giving to t all the values 1, 2, 3, &c. of which it is sus-
ceptible, and taking the sum of all the resulting quantities, we shall
have, since in the present case V reduces itself to the single term V^\
sm (-^.)
n-l.n + 1 n + 2i + 2t'-3 ^
^ S . 5 2i + 2f + l '
the sign S belonging to the unaccented letter t.
Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 47
If now V represents the function analagous to V and due to the
fluid on the spherical surface, we shall obtain by what has been proved
(Art. 3.)
V = WK 27r«^ W'^ ^'7^ /-i(/M {i) (f-2artx-V a'i^',
X • ^ ,Ot« • • • t
{i) representing the same function as in the article just cited.
Moreover, it is evident from the equation (10) Art. 4, that
,. ,j ,., ,^ ^ , ,.^ ^,1.2.3 i ^n-l.n + 1 n + 2i + 2t'-S
/ild^{t){r^-2ar^ + a^)^ = 2«'-" ^ 3 ^■_^ 2 ^ ^ ^i + ^ TTT
n-2.n n + 2t'-4> IrV^^^'
"" 2.4 2^ \a) '
and consequently,
(.9) r'=cr<o.w-.. '';';''^';;;;;'-^f;^;:^7^
«-2.w ?« + 2if'-4 /r\' + ''
2.4 2^' \a)
the finite integrals extending from t' = Q to t'=<x).
Substituting now for F' and ?^' their values in the equation of equi-
librium,
(20) const. = r'+ f;
we immediately obtain
const. = i7".47rflr' ".2
3 . 5 2i + 2t' + l
n-2.?i w + 2^'- 4 //•\'+'''
2 .4 2f
Q
^ 2^" rm 97? v,.<+2,' ^-1-^^ + 1 « + 2i + 2^'-3
^ . fn-2 \ ' "3.5 2» + 2r-l
sm(-^.)
n-2.fi Ti-{-2t'-4> 4-W.6-W 2^-2^' + 2-«
"^ 2 .4 2? ^ 2 . 4 2^-2^
the constant on the left side of this equation being equal to zero, except
when i = 0.
48 Mr green, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS.
By equating separately the coefficients of the various powers of the
indeterminate quantity r, we get the following system of equations :
^ . (n-2 \
2 sm — — — TT . AC
2 . 4
^ • fn-2 \
, I o -n Tk 4 — w „ 4i — n.6 — n .
TT 2 2.4
^ . fn-2 \
2sm(-^.)
TT 2 2.4
&c &c &c.
But it is evident from the form of these equations, that if we make
generally ^,+i = a'^Bt, they will all be satisfied provided the first is, and
as by this means the first equation becomes
2 sm — -— TT , A AC
Tr»/, 4 — « „ 4 — M.b — « . . \
«-»- = J?„ (l + -g-«- + 3 ^ a-^ + &c.)
= J5o(l-«-'^) ^ = .Bo«*-"(a'-l) ^ .
there arises
„ . /M-2
2sm
(n-2 \
^^ La-'-'{a'-l)' , B, = B,.a-\
Bo = -— -a-'-'ia'-l)' , B, = B,.a-\ B, = B,.a-\ !>ic.
TT
Hence
f{r") = B, + By' + B,r'' + &c. = ^„ (^1 + ^ + ^ + &c.)
»8 2 sm I — — - TTJ ^_„
= ^„fl_ !l)-i = ^„a=(a^_/^)-'= L* ia'+'(a^-l)~(a^-r'^)-\
\ a I If
and the required value of p becomes
(21 ) (D = C7'< V; (1 - r'^)^/(r'0
2sm(-^.j
(a' - l)'^"a i7'» f^') '(«= - /=)-' (1 - r"y\
Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 49
But whatever the density P on the inducing spherical surface may
be, we can always expand it in a series of the form
P = C7"<°>+ Z7"<'>+ C7"<^>+ Z7"® + &c. in inf.
and the corresponding value of p by what precedes will be
„ . /n-Q,
2sm
P = -
<a{a'-l) ' .{a'-r")-H^-r")
X { t7'W+ [/'<')- + t7'<='^ + f7'<^>^ + &c. in inf.] ;
Ijm^ U'M^ jjm^ &c. being what U"^'\ U"^'\ U"^% &c. become by changing
d", w" into ff, Ts-', the polar co-ordinates of the element dv. But, since
we have generally
^d&'d-uy" sin 6)"PQ« = fdff'd^" sin 6" C7"<"Q» = ^^ C7<",
{Mec. Cel. Liv. iii.) the preceding expression becomes
-sm(-^.)
p = _> a{a'-\) ' K-r'^)-'(l-0 ' jd&'d-sr" sin &'.
2:(2e + l)PQ«^;
a*
the integrals being taken from 0" = O to 0" = 7r, and from ■bt" to ■sr" = 27r.
In order to find the value of the finite integral entering into the
preceding formula, let R represent the distance between the two ele-
ments dff, dv ; then by expanding -^ in an ascending series of the powers
r'
of — we shall obtain
a
— = ^ _ 2°°Q<*>.— -,
B Va^ - 2ar' [cos 0' cos 0" + sine' sine" cos (-ar' -•23-")+/* ° *«"
Mec. Cel. Liv. iii.). Hence we immediately deduce
^ = .r«»e^, and .^4,^^ = K(^.>1)«?»^.
Vol. V. Part I. G
50 Mr green, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS.
If now we substitute this in the value of p before given, and after-
// o ft^ __ >«'2
wards write — and „3 in the place of their equivalents,
dd"dnr" smO", andvV'^'^,
clr R
we shall obtain
. (n-2
p- i7^ — («^-i)^ (i-O^ /-^;
the integral relative to da being extended over the whole spherical sur-
face.
Lastly, if p^ represents the density of the reducing fluid disseminated
over the space exterior to A, it is clear that we shall get the corres-
ponding value of p by changing P into pida in the preceding expression,
and then integrating the whole relative to a. Thus,
, = - !iy4 (i-..)=i^/a-«.)*-?/**£i.
But dada = dvx\ dvi being an element of the volume of the exterior
space, and therefore we ultimately get
fn — 2_
4— n
. /n — 2\
(22) p= y5 -i^-r")'^ .fp^dv, ^ ,
where the last integral is supposed to extend over all the space exterior
to the sphere and R, to represent the distance between the two elements
dv and dv^.
It is easy to perceive from what has before been shown (Art. 7.), that
Ave may add to any of the preceding values of p, a term of the form
h being an arbitrary constant quantity : for it is clear from the article
just cited, that the only alteration which such an addition could produce
would be to change the value of the constant on the left side of the
Mr GREEN, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 51
general equation of equilibrium ; and as this constant is arbitrary, it is
evident that the equilibrium will not be at all affected by the change
in question. Moreover, it may be observed, that in general the additive
term is necessary to enable us to assign the proper value of p, when
Q, the quantity of redundant fluid originally introduced into the sphere,
is given.
In the foregoing expressions the radius of the sphere has been taken
as the unit of space, but it is very easy thence to deduce formula^
adapted to any other unit, by recollecting that —, -p, j^ and y^^,
are quantities of the dimensions 0, — 1, — 1 and S — n respectively with
regard to space: for if h represents the sphere's radius, when we employ
any other unit we shall only have to write, t> j, -j- > -jr- and j- in the
place of r, r, R, dvi and a, and afterwards to multiply the resulting
expressions by such powers of h, as will reduce each of them to their
proper dimensions.
If we here take the formula (22) of the present article as an example,
there will result,
• / W-Q ^ 4-n
(23).... p= 1-|_-I(i"-/^)^ fp,dv^-^-^,
for the value of the density which would be induced in a sphere A,
whose radius is b, by the action of any exterior bodies whatever.
When w > 2, the value of p or of the density of the free fluid here
given offers no difficulties, but if » < 2, we shaU not be able strictly to
realize it, for reasons before assigned (Art. 6. and 7.) If however n
is positive, and we adopt the hypothesis of two fluids, supposing that
the quantities of each contained by bodies in a natural state are ex-
ceedingly great, we shall easily perceive by proceeding as in the last
of the articles here cited, that the density given by the formula (23)
will be sensibly correct except in the immediate vicinity of A's surface,
provided we extend it to the surface of a sphere whose radius is
h—^b only, and afterwards conceive the exterior shell entirely deprived
of fluid: the surface of the conducting sphere itself having such a
G 2
52 Mr green, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS.
quantity condensed upon it, that its density may every where be repre-
sented by
ftl 2 \ „-4 „_2 4-n
Application of the general Methods to circular conducting Planes, &f:.
10. Methods in every way similar to those which have been used
for a sphere, are equally applicable to a circular plane as we shall im-
mediately proceed to show, by endeavouring in the first place to determine
the value of V when the density of the fluid on such a plane is of
the form
p = {\-ry.f{x',y'):
f being the characteristic of a rational and entire function of the degree * ;
x\ y' the rectangular co-ordinates of any element dcr of the plane's
surface, and r', & the corresponding polar co-ordinates.
Then we shall readily obtain the formula
r= ff^ = r rrdr'd9'{l-ry.f{x',y') ^ .
'' g"'' ''■^ {f^-Zrr' cos {9-9') + r"f^'
where r, 9 are the polar co-ordinates of p, and the integrals are to be
taken from 9' = to 0' = 27r, and from r' = to /•' = !; the radius of
the circular plane being for greater simplicity considered as the unit
of distance.
Since the function /{x', y') is rational and entire of the degree j,
we may always reduce it to the form
(24) f{x', y') = A^°^ + A^'^ cos 9' + A^'^ cos 20' + ^*'' cos 39' +
+ ^« sin 9' + B'-'^ sin 29' + B^'^ sin 30' +
the coefficients A'-''\ A^'\ A^'\ &c. B^'\ B^% B^\ &c. being functions
of r' only of a degree not exceeding *, and such that
^('•'=«'o°' + «<V^ + 4"V'* + &c.; ^« = («?> + alV + 4'V'V)/;
^(» = ( jw -I- J(/)r'^ + i(»r'^ + &c.) r' ; B'^ = {bf> + hfr" + &c.) r'\
Ma GREEN, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS. Sfi
We will now consider more particularly the part of V due to any
of the terms in f as -<4^'^ cos i& for example. The value of this part
will evidently be
r r /dr'dff{\ - r"fA^^ cos iff
{r'-^rr' cos (Q - ff) + r'')~^ '
the limits of the integrals being the same as before. But if we make
6' = 9 + (p, there will result dff = d<p, and cos i9' = cos id cos e0 — sin iO sin «0,
and hence the double integral here given by observing that the term
multiplied sin i<p vanishes when the integration relative to (p is effected,
becomes
cos ie/lA^'^r'dr' (1 - ry f^ ^"^ ^"^ ''I' —^ ;
° {r'' — 2rr' cos (p + r"^)~^
If now we write F"/*^ for that portion of V which is due to the term
«/*^r"+^* in the coefficient A^'^ we shall have
r,» = «/'> . cos ieflr^^'^^'dt" (1 -ry /" "^"^ ^"^ "^ ^ .
" {r^ — 2rr' cos (p + r"^)~^
But by well known methods we readily get
•^'^ d(p cos i(})
L
{r' — ^rr' cos (f> + r'^) ^
i ^i-.-iv» ^"' n-l.n + 1 n+2t'-S n-l.n + 1 n+2i+2t'-3
-2irr.r i„^„,. 2 _ ^ 2^ "" 2 . 4 2i + 2t '
when r'>r, and when /<r, the same expression will still be correct,
provided we change r into r' and reciprocally.
This value being substituted in that of Fj*'' we shaU readily have by
following the processes before explained, (Art. 1. and 2.)
F,w = 27ra/'V* cos 10 2o r ' — r 7 -—p
P^^ . ,.„ ,S + 2t-2f~n\
»-l.» + l « + 2« + 2#'-3 "^f^ ' I \ 2
X — ^:: -. 7;r- — :;^-r, X
(/3 + i)r[^
2 . 4 2i + 2/' ( 2li + 5 + 2t-2 f)
^^[ 2 J
54 Mr green, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS.
= TTtt/'V COS i6 . —5 — .
, „P n-1 .n + 1 n + 2f — 3 7i-l . ji + 1 n + 2i + 2t'-3
" 2.4 2t' ^ 2 . i 2i + 2f
3-n.5-n 1 + 2^-2^'-?? ^
^ 2(i + 5-n 2fi + 3 + 2t+2t'-n'
the sign of integration 2 belonging to the variable f.
Having thus the part of V due to the term a,''' cos i9' in the expansion
of J'iaf, if) it is clear that we may thence deduce the part due to the
analogous term J/'^ sin i& by simply changing «/" cos iQ into J/'' sin iO, and
consequently we shall have the total value of V itself, by taking the
sum of the various parts due to all the different terms which enter
into the complete expansion of y(a;', y').
11 3
If now we make iS = — - — and recollect that
2
sin
the foregoing expression will undergo simplifications analogous to those
before noticed (Art. 5.) Thus we shall obtain
TT^a/" , .^^„„M-1.« + 1 n-k2t-3
r/" = "-:^ r' cos iQ . 2 r'''
' . (n
sin I -
2
n-1 \ 2.4 2t'
sin — - — IT
(-
n-1 .n + 1 n + 2i + 2f-3 3-n. 5-n 1 + 2^-2^- w
'^ 2 . 4 2i + 2t' ^2.4 2if-2^
or by writing for abridgment
,. ^ n-1. n + 1 n + 2t'-3 n-l.n + 1 n + 2i + 2f-3
"P^^'*^'- 2 . 4 2f "" 2 . 4 2i + 2t' '
there will result this particular value of /8
,^(. ^«/'> . .^ „ ,„ 3-n. 5-n l + 2t-'2f-n ^,. „^
^' = . (n-1 . ^^osze.^r-^. ^^^ ^^_^^, .<p{t;t'),
Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 55
and afterwards by making
ro = ro» + r/'> + r.» + r « + r« + &c.
we shall have
TT*
V^'^= , r* cos i6 into x
/« — 1
sin
sin (-^vr)
«<,'>.1.0(«;O)
+ a<'>.?^.<^(e;0) + «?>.!. 0(«; 1) . r=
+^'- 2.4.6 •<^(^;o) + «^"-2r-^— 0(^;i)-^
+ «^^^.0(«; 2).r' + af.l.cj>{i;3).f^
+ &C +&C +&C +&C
Conceiving in the next place that F is a given rational and entire
function of x, y, the rectangular co-ordinates of p, we shall have since
X = r cos 0, y = r sin 0.
{25) r= C<") + C('> cos 6 + C-'^ cos 20 + C('> cos 3 + &c.
+ ^« sin + ^('' sin 29 + E^'^ sin 30 + &c.
of which expansion any coefficient as C^'> for example, may be still
farther developed in the form
C« = ""'-^ {di\(p{i', 0) + c>{K(p{i; l).r'+4^.(p{i; 2).r' + ke.}.
sm (-^ .J
Now it is clear that the term C> cos iO in the developement (25)
corresponds to that part of F which we have designated by F''', and
hence by equating these two forms of the same quantity, we get
F» = Cw cos ie,
56 Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS.
which by substituting for F"*'' and C^ their values before exhibited, and
comparing like powers of the indeterminate quantity r gives
/> , 1-^ 3 — n,. 3 — n.5 — n,i. 3 — n.5 — n.7 — n,., ,
2
&iC.— &c &c
of which system the general type is
C<'> = (1 - e)~ . «« ;
the symbols of operation being here separated from those of quantity,
and e being used in its ordinary acceptation with reference to the lower
index u, so that we shall have generally
f.m „(i) _ ^ (0
The general equation between «!'' and cll^ being resolved, evidently gives
by expanding the binomial and writing in the place of eci'', e''&i\ ^c'i\ &iC.
their values c„*j\ , cj-i\, Cu%, &;c.
(26) ««=(l-e)^c« = c<;>+^c„« + ''~^-''~^
2 — ' ■ 2
... « — 3 . ra — 1 . w + 1 (i) , s
%+ 2.4.6 "-- + ^"-
Having thus the value of af we thence immediately deduce the value
of ^<'' and this quantity being known, the first line of the expansion
(25) evidently becomes known.
In like manner when we suppose that the quantity J5^'> is expanded
in a series of the form
j5:« = — ^TTT ^^»"- *^ (^' ' 0) + ^'* "^ (^' ; 1) • ^' + ^^'* <^ («■ ; 2) . f^ + &c. ^
sin
sin(^.)
Mr green, on THE LAWS OF THE EQUHJBRIUM OF FLUIDS. 57
we shall readily deduce
A«= (1 - ef^e^^^ + ^e.%+ ""'^'""l^ e.% + &c.,
and ii,^ being thus given, B'-'^ and consequently the second line of the
expansion (25) are also given.
From what has preceded, it is clear that when V is given equal to
any rational and entire function whatever of x and y, the value of
f{x', y') entering into the expression
p={l-r'-^)-^.f{x',y'),
will immediately be determined by means of the most simple formulas.
The preceding results being quite independent of the degree s of
the function f(x', y) will be equally applicable when s is infinite, or
wherever this function can be expanded in a series of the entire powers
of x, y', and the various products of these powers.
We will now endeavour to determine the manner in which one fluid
will distribute itself on the circular conducting plane A when acted
upon by fluid distributed in any way in its own plane.
For this purpose, let us in the first place conceive a quantity q of
fluid concentrated in a point P, where /• = « and 6 = 0, to act upon a
conducting plate whose radius is unity. Then the value of V due to this
fluid will evidently be
g V'
((^ — 9,ar cos Q + r^)~^
and consequently the equation of equilibrium analogous to the one marked
(20) Art. 10., will be
(27) const. = ^ ^+ F;
(«'-2«rcos e + r^)~
V being due to the fluid on the conducting plate only.
If now we expand the value of V deduced from this equation, and
Vol. V. Part I. H
58 Mr green, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS.
then compare it with the forrnulag (25) of the present article, we
shall have generally E^^ = 0, and
C"'=-2ga-^.l^(r;O)+0(e;l)^+«^(/; 2) ^ +,^ («; 3) J + &c.^
except when i = 0, in which case we must take only half the quantity
furnished by this expression in order to have the correct value of C*"'.
Hence whatever u may be,
2 sin I ^-Q— T I
^ - 0, and cf = ^^ qa} -"-*-^" ;
TT
the particular value f=0 being excepted, for in this case we have agreeably
to the preceding remark
sin
(¥')
-i
a}-n-2u^
■w'
and then the only remaining exception is that due to the constant
quantity on the left side of the equation (27). But it will be more
simple to avoid considering this last exception here, and to afterwards add
to the final result the term which arises from the constant quantity thus
neglected.
The equation (26) of the present article gives by substituting for
d" its value just found.
^ . (n-l \
2 sm I—- — tt)
«»= l^f L qa'"-'-'". {1 + '^.a-'
n — 3.n-l ^, n — 3.n-l.n-l ., . ,
2.4 2.4.6 '
^ • /«-!
2 sm
(n-1 \
l^i i qa'-'-'-"- (1 - a-')—
2 sin I -—^
TT^
Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 59
and consequently,
sin (^.)
. (n-\
2 sin I —7: /r I 3_^ ,2 ,4
;ii^^«--(«=-ir.''(i-5)"'
•* sin 1 ^ "I B-B „M
q{a'-V)—{a'-r'y\-,
the particular value -4'°^ being one half only of what would result from
making i = in this general formulje.
But 4'' = evidently gives £^''^=0, and therefore the expansion of
f{a!, y') before given becomes
fix', y') = J^'^ + A^'^ cos ff + A-'^ cos 20' + ^''^ cos 30' + &c.
= 1-^ -g(«^-l) ^ («=-r'0-'.{| + - cos0'+ -cos20' + &c.|
or by summing the series included between the braces,
. (n — 1 \ 3-»
JKx,y)- -^ ^a^-2«r'cos0' + r"'
sin (^ .)
Q
iJ being the distance between P, the point in which the quantity of
fluid q is concentrated, and that to which the density p is supposed to
belong.
Having thus the value of /(a;', y') we thence deduce
(n — 1 'N 3-,
p = (1 - /»)-/(x', y') = - — i-| i (1 - /') ^ ? — ^.
sin 1 — ::: — TT I „_3 (a^ — Yy^
H 2
60 Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS.
The value of p here given being expressed in quantities perfectly
independent of the situation of the axis from which the angle 6' is
measured, is evidently applicable when the point P is not situated upon
this axis, and in order to have the complete value oi p, it will now
only be requisite to add the term due to the arbitrary constant quantity
on the left side of the equation (26), and as it is clear from what has pre-
ceded, that the term in question is of the form
n-3
const. X (1 - /') 2 ,
we shall therefore have generally, wherever P may be placed.
P = (l-r-)
1-3
- [n-l \ 3.„
const. -
^- 7?-' 1
The transition from this particular case to the more general one,
originally proposed is almost immediate : for if p represents the density
of the inducing fluid on any element dai of the plane coinciding with
that of the plate, p^da-i will be the quantity of fluid contained in this
element, and the density induced thereby will be had from the last
formula, by changing q into pidai. If then we integrate the expression
thus obtained, and extend the integral over all the fluid acting on the
plate, we shall have for the required value of p
p=(l-0^ .jconst. \f ^ fp^da ^" J^ };
B being the distance of the element dai from the point to which p belongs,
and a the distance between da^ and the center of the conducting plate.
Hitherto the radius of the circular plate has been taken as the unit
of distance, but if we employ any other unit, and suppose that b is
the measure of the same radius, in this case we shall only have to
.^ a r' d(Tx , R . ,, , „ , ,
write ^ > ^ ' -^ and -g- m the place of a, r, da, and R respectively,
recollecting that -^ is a quantity of the dimension with regard to space,
by so doing the resulting value of jo is
Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 61
(n-1
sin I -— — TT
(28).
.p = {]r-r')' .|const. Jp.dcr,- — ^^]
By supposing w = 2, the preceding investigation will be applicable
to the electric fluid, and the value of the density induced upon an
infinitely thin conducting plate by the action of a quantity of this
fluid, distributed in any way at will in the plane of the plate itself
will be immediately given. In fact, when n = 2, the foregoing value of
p becomes
1 7 , y/a'-b']
^ = 7ltptHst--^/^'^<^'
B'
If we suppose the plate free from all extraneous action, we shall
simply have to make pi = in the preceding formula; and thus
,„^, const.
(29) p =
Vb'-r"'
Biot (Traite cle Physique, Tom. ii. p. 277.)> has related the results of
some experiments made by Coulomb on the distribution of the electric fluid
when in equilibrium upon a plate of copper 10 inches in diameter, but
of which the thickness is not specified. If we conceive this thickness
to be very small compared with the diameter of the plate, which was
imdoubtedly the case, the formula just found ought to be applicable
to it, provided we except those parts of the plate which are in the
immediate vicinity of its exterior edge. As the comparison of any
results mathematically deduced from the received theory of electricity
with those of the experiments of so accurate an observer as Coulomb
must always be interesting, we will here give a table of the values of
the density at different points on the surface of the plate, calculated
by means of the formula (29), together with the corresponding values
found from experiment.
62 Mr green, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS.
Distances from the
Plate's edge.
Observed
densities.
Calculated
densities.
5 in
4
3
1,
1,001
1,005
1,17
1,52
2,07
2,90
1,
1,020
1,090
1,250
1,667
2,294
infinite.
2
1
,5
We thus see that the differences between the calculated and observed
densities are trifling; and moreover, that the observed are all something
smaller than the calculated ones, which it is evident ought to be the
case, since the latter have been determined by considering the thickness
of the plate as infinitely small, and consequently they will be somewhat
greater than when this thickness is a finite quantity, as it necessarily
was in Coulomb's experiments.
It has already been remarked that the method given in the second
article is applicable to any ellipsoid whatever, whose axes are a, h, c.
In fact, if we suppose that x, y, % are the co-ordinates of a point p
within it, and x', y', z' those of any element dv of its volume, and
afterwards make
X = a. cos 9, y — i.sin 6 cos w, a = c.sin 6 sin ■sr,
x'= a. cos ff, y' = J. sin 9' cos w', z'= c.sin 9' sin w',
we shall readily obtain by substitution.
l-n
2 .
■ V=abcf p. r'^dr'd&d-ar' si-n 9'. {Xr"- 2 nrr'-¥vr'^)
the limits of the integrals being the same as before (Art. 2.), and
\ = «^ cos 9^ + If sin 9^ cos ts^ -\- & sin 0- sin Tsr^,
ft. = a^ cos 9 cos 9' + U sin 9 sin 9' cos tb- cos in-' ■\-<? sin 9 sin 9' sin •ar sin w',
V = «' COS 9" + ¥ sin 9'^ cos sr'^ + e sin 9'^ sin ■ar'^
Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 63
Under the present form it is clear the determination of V can offer
no difficulties after what has been shown (Art. 2.). I shall not there-
fore insist upon it here more particularly, as it is my intention in a
future paper to give a general and purely analytical method of finding
the value of V, whether p is situated within the ellipsoid or not. I
shall therefore only observe, that for the particular value
(30) , = ^\^-ii^-%-i]' = ^^(^ -'") ' '
the series Uo + U2' + U/ + &c. (Art. 2.) will reduce itself to the single
term Uo, and we shall ultimately get
2sin("— .)
which is evidently a constant quantity. Hence it follows that the ex-
pression (30) gives the value of p when the fluid is in equilibrium
within the ellipsoid, and free from all extraneous action. Moreover,
this value is subject, when n < 2, to modifications similar to those of
the analagous value for the sphere (Art. 7.)-
G. GREEN.
11. On Elimination between an Indefinite Number of Unknown Quantities.
By the Rev. R. Muephy, M. A. Fellow of Cuius College, a?id of
the Cambridge Philosophical Society.
[Read Nov. 26, 1832.]
SECTION I.
INTRODUCTION.
Fourier, in his treatise, * Theorie de la Chaleur,' * has given an
example of the determination of an indefinite number of unknown
quantities, subject to the same immber of conditions. If n be the
number of those quantities, in order to discover their law by this
method, it will be necessary to eliminate successively the first (ni- 1)
and the last (« — »^) unknown quantities, thus determining the »^'^ by
a final equation containing that quantity only.
This process is obviously too laborious, and the results too compli-
cated, to be practically useful, in most cases.
The same objection applies to the elegant method of Laplace, which
makes the determination of one of the unknown quantities, depend
on the discovery of all the (w — 1) arbitrary multipliers introduced in
the process. It has besides the disadvantage of not seizing, in many
cases, the facilities offered by the peculiar forms of the proposed equa-
tions.
• Vid. Fourier, p. 1 69 to 174-
Vol. V. Paet I. I
G6 mk murphy, on elimination between an indefinite
In the physical investigations, which conduct to an indefinite num-
ber of equations, it is of great importance to discover the law of those
quantities, corresponding to the law by which the given equations are
connected. The method which I here propose for this object is founded
on the two following principles.
First, if we make the right-hand member of the a;*'' equation dis-
appear by transposition, the left-hand member is then a function of x,
which vanishes when x is any number of the series 1, 2, 3, w; and
therefore it must be of the form
P.(.r- 1) {x-2) (x-S) (x-n).
Secondly, if an identity exist between two formulas which are
partly integer, partly proper algebraic fractions (of which the numerators
are of lower dimensions than the denominators) the integer and fractional
parts are separately equal.
To demonstrate this principle, let
represent such an identity, where each symbol denotes an entire function
of X, and the dimensions of P, P' are respectively lower than those of
Q, Q'; then we have
(N-N)QQ = PQ- PQ'.
If therefore N—jV' be not identically nothing, we shall have the
entire function, represented by the left-hand member, identical with one
of lower dimensions ; but this is impossible, because in integer formulae
we may equate like powers of x, hence we must have iV=iV' and,
therefore also,
Z! - ^
Q- Q'
By means of this principle, we shall be able to expand a given
entire function P, in terms of other given functions, whenever such an
expansion is possible.
NUMBER OF UNKNOWN QUANTITIES. 67
SECTION II.
Application of the First Principle.
The first principle alone is sufficient, in a great number of instances,
to resolve the proposed equations ; we shall illustrate its application by
selecting three distinct classes of equations to be resolved.
First, when the terms which compose the general or a;**" equation are
proper fractions.
Example :
To find the values of the n unknown quantities ssi, %i, sss, s,, sub-
ject to the n equations following,
»1 «2 ^ »„ ^ _ 1
3 4 "*■ 5 "^ » + 2 2'
«i , ^ ^ , g« =_ 1
4 "*" 5 6 "^ « + 3 3*
» + l « + 2 ra + 3 2w »■
The general, or a;**" equation, when its right-hand member is trans-
posed, becomes
- + — Y — -^ \- H — =0.
a; x+1 x + ^ x + n
N
Suppose these fractions are actually added, and let -^ represent the
sum; where D = x{x-\-\)(x-\-9l) {x + n) and A'' is some function of x
of n dimensions. . ,- • .•
i2
68 Mr murphy, ON ELIMINATION BETWEEN AN INDEFINITE
Hence we have 7^ =0, and therefore iV=0, provided a; is any num-
ber of the series 1, 2, 3,....ti and consequently iV (which is of ?i dimensions)
has a factor (^ — 1) U — 2) (x — ti); and can therefore admit of no
other factor, but a constant c.
Hence we have in general,
^^ X x + 1 x + 2 x+n x(x+l){x + 2) {x + 3)...{x+n)'
Multiply this equation by x, and then put x = 0, hence c = ( — 1)\
Multiply the same by x + 1, and then put x = —I;
, n n + 1
hence ssi = - - . — - — .
Similarly, multiply by x + 2, and put x= —2,
_ n.{tt—l) {n + !)(« + 2)
• • '^ ~ 1 . 2 • 1 . 2 '
and generally, if we multiply equation (a) by x + m, and then put
x= -m, we get
'"'^ ' 'I. 2.3 m ' 1.2 m '
It is clear from this example, that if the general or x^^ equation were
a+bx^ a' + h'x ^ d'\¥x ^ «<"' + i<"'x ~ "'
we should find the sum of the fractions composing the left-hand member
to be
c .{x—\){x — 2) {x — n)
(a + hx) [a + h'x) («" + h"x) (a" + i^a;) '
then multiplying by n + bx and putting x= — j, we should find e,
-I
multiplying by a'+b'x and putting .r=-,,, we should find s,,
&c &c &c
NUMBER OF UNKNOWN QUANTITIES. 60
In the example above taken, we have supposed that the number
of equations and unknown quantities were the same, but if we supposed
that following the same law as in that example, the number of equa-
tions were n + m, then the numerator N which was shown to be of
n dimensions, ought to vanish when x is any number of the series
1, 2, 3 n + m; that is, the equation A^=0 has more roots than it has
dimensions, which is impossible ; it is therefore equally impossible to
satisfy all the given equations.
On the other hand, if the number of the given equations was
only n — m, then n would by the preceding reasoning have a factor
{x — l){x — 2) {x-7i-{tn),
and since it is of n dimensions, it must have another factor of m dimen-
sions, as C {x - a^) {x — a-i) (x — a„).
Hence - -\ — ^ H "* ^ ^-
X x + 1 x + 2 x + n
_ C(x—'l){x—2) .{x~ tl + m){x — ai) jx — a-^ { x~a,„) ^
■~ ''"'xT{x + l){x + 2) [x + n) '
following now the same steps as before, we find
^^ cj-iy.a.az g. . g^( ly "■('»-^) jn-m + i)
«.(« — 1) {n-m + 1)' '' ' ' ai.a-i a,„
c(-l)".(l+ai)(l+a.> (Ifg.) ^ (l+aiXl+aa) (!+«„,) w n-m + l
'~ (w-1)(m — 2) {n — m + 2) o, . a^ a,„ '1' 1
^. ., , (2 + a,)(2+a,) {2+a^) n.{n + l) (ti-m+l) {n-m+ 2)
Similarly, %■^= - ~ — • ., ^ • i ^ •
The quantities a„ a.^ a„ are evidently arbitrary, and each of the
required quantities », z-., he. x„_„, are here determined in such a manner,
as to contain the m arbitrary constants. This is therefore the most,
complete solution of the problem.
70 Mb murphy, ON ELIMINATION BETWEEN AN INDEFINITE
Another useful observation may be made in this place ; if the function
which represents the a;"' eqvxation were discontiiuious, i. e. if any of the
equations, for instance the second, were
3 + 5 ' 7 ~ '
2 2 2
and consequently an exception to the general law expressed by the x^^
equation, we should have then N—0 when x=\,S, 4 w, also when
A' = ^, but not when a; = 2, hence in this case,
iV=c. (ar-i)(x-l) {x-S) (x-4) {x-n);
after this the remainder of the process would be the same as before.
We have been thus particular about the preceding example, as being
well calculated to shew the spirit and advantages of the present method.
The next class of equations, which may be solved by the first principle
alone, consists of those in which the terms composing the a;*'' equation
contain common factors ; for if we then assign to x such values as may
successively cause such factors to vanish, the unknown quantities will
be determined.
Example :
To find the values of asj, %2, %^ a, subject to the n equations
following; viz.
a:, + 1 ,2.S!2 + 1.2.3.S53+ + 1.2.3 «s!„= -1,
2s!, + 2.3.a!2 + 2.3.4.X3+ + 2 . 3 . 4...(w + l)x„= -1,
3a, + 3.4.&, + 3.4.5.«3 + + 3 . 4. 5...(w+2)a;„= -1,
n8Sl + ?i(w + l)8:2+M(w + l)(w + 2)S83+ + W (w + 1) (« + 2)...2W2!„= -1.
If we transpose the right-hand member of the above equations, the
.r"" or general equation becomes
\ + x%,+ X {x + l)%.,^ X {x-\-\){x + ^) .%; + +a;(d;+l)(a:+2)...2x.!£„ = 0.
NUMBER OF UNKNOWN QUANTITIES. 71,
This equation is evidently of w dimensions with respect to x, and its
roots by the first principle are 1, 2, 3 n; the left-hand member
must therefore be identical with the product
c{x-l\{x -2){x-S) (x-ti),
whatever value may be assigned to x.
i — lY
Put therefore x = 0. Hence c = - — ^ ' ,
1.2.3 «
X ^^ X.»..>........V[ ^^ -^ fif
^-~" 2a— ,g ^ ,
n . (w — 1)
1\¥
_ _ «(w-l)(w-2)
* ^ a:,-- 1222.32
&c &c
and generally, «,„ = ^^ ^ . 3 ^)^~ •(-!)•
We may verify this result by observing, that if we substitute this
quantity for 25,„ in the general or x^^ equation, then its left-hand member
becomes
n . ( w-1) X . ( x + l) w.(w-l)(w-2) ^(£+_l)_(a; + 2) , '
^"■"'^'*" 1.2 •~T:2 1.2 . 3 • 1.2 7 3 ^*'''-
This quantity is evidently the part which does not contain h in the
product,
f, , x{x + l) ,„ x{x + l)ix + 2) ,3,o„l f, n , n{n- l) 1 1
or in (l-//)-'.(l-|)".
it is therefore the coefficient of //" in the expansion of
But this coefficient is manifestly when x is any positive integer, which
evidently agrees with the proposed conditions.
72 Mr murphy, on elimination BETWEEN AN INDEFINITE
Another class of equations which may easily be resolved by the
first principle, occurs when the x^^ equation is of n dimensions, and
arranged according to the powers of some function of .r ; it is then
merely necessary to expand
c.a:{x — l)(x-Q) (x — n)
according to the powers of that function ; and equate the coefficients
of like powers in both cases.
Example:
Ki + SSs + «3 + + SS, = - 1,
2a!, + 2-S2 + S'xs + + 2»ss„ = - 1,
Sz, + 3-S5, + 3^X3 + + 3'%„ = - 1,
w^i + w'asa + ?r%3 + + n''z^ = — 1,
to find «„ %2
The general or «"" equation in this case, is
1 + x»i + x"%i + + afz^ = 0,
the roots of which equation are x = l, 2, 3 ;/.
Hence, the left-hand member is identical with the product
c.{x-l){x-2){x-3) (x-tt),
or c(-iy{S„-xS„., + x'S,_,- (-l)".x"|,
where S„ denotes the sum of the quantities 1, 2, 3 n when taken
in products m and m together.
Hence, by equating, we get
c(-lY S =1- • r = \^y ..
- c(-1)".aS'„_,= «,; .-. x,= -aS'_,;
c( — 1)" .«>„_2= »2; .•. SS2=— 0,2;
and generally £., = S,,,
where S.„ denotes the sum of the reciprocals of the quantities of which
a9„ represents the sum.
NUMBER OF, UNKNOWN QUANTITIES, 73
SECTION III.
Amplication of the Second Principle.
To expand a given function of x as P, in terms of other given
functions
Qo, Q., Q. Qn,
all being supposed of n dimensions in x.
Let P=aoQo + «iQi + «2Q2+ +«nQ»,
where a^, ffi, Oa «» are constants to be determined.
Divide all the functions by Qo, and let the corresponding quotients
be respectively
P', Qo, Q'l, Q.....Qn,
and the remainders
p', g^o, q\, q'i q\-
Then by attending to the second principle, we have
P' = «oQ'o + «lQ'l + (kQ2+ +«„Q'n,
p' =aoq'o + aig-'i + «25''2 + +a„q'n,
when we obviously have Q'o=l and §''0=0.
Dividing the last equation by q'l and using a similar notation, we
get in like manner
P'=«.Q". + «2Q"2 + ««Q"„,
p"= aiq"i + (hq"2 + anq\,
where Q"i = l and q'\ = 0.
Divide the equation last obtained by q"i, and we obtain
P"' = a,Q"',+ +a„Q\,
p"'==a»q"', + +«„^"„,
in the latter of which equations the first term = and in the former
it equals unity.
Vol. V. Pakt I. K
74 Mr murphy, ON ELIMINATION BETWEEN AN INDEFINITE
The systems of the first equations thus obtained may be written in
an inverse order thus,
&c. = &;c
whence «„, «„_i, a„_2, &c. are successively known.
We have supposed all the functions to be of n dimensions, for
these necessarily comprise all of lower degrees.
Example :
To expand unity in terms of the functions
af, {x-^hy, (a; + 2A)", (a; + «A)°.
Put l=«o*" + «i(^ + ^0° + '''^(^ + 2^)" + + «„(a;+wA)"; dividing by
a;", we get
= ao + ffi +«2 + + «„,
1= ai?'i + Oag-'s + + «»<?'»,
where we have o'„ = A. {waf"*.»i-l — , ^ ' hm^af-^+ \.
^ 1.2
Divide now by g'l and we obtain
= ai + 2a2 + + wa»,
1= «2g'"2+ + a„g^'„,
where in general g'"„ = A^ |— ^ — ^— ^ a;""* {m^ — m) + &c. > .
This process is easily continued, and we obtain successively the
equations
= 1.2a2 + 2.3a3 + (»— 1). w«„,
0= 1.2. 3 03+ {n- 2) (71-1) nan.
and lastly, ^= 1.2.3 «a„
NUMBER OF UNKNOWN QUANTITIES. 75
From these equations taken in the inverse order, we get
^ 1
"' ~ 1.2 .3 nh"'
«„-!= - na,
n>
„ _ n(n-\)
1 . 2
&C.= &C
Hence the required expansion is
To apply this principle to equations, we may observe that when
the general or a;"' equation is cleared of fractions and its right-hand
member transposed, it is of the form
-P+ XiXi +^2X2 + +i8„X„ = 0,
where ssi, sss a, are the unknown quantities, and P, Xx, X^....
known functions of x.
The left-hand member must, by the reasoning of the preceding
Section, be divisible by (« — l)(x— 2) {x—n).
Let Xi, Xi, &c. when divided by this quantity leave the re-
mainders Q'l, Q'2, &c. and P, the remainder P', hence
where all the functions are necessarily of less than n dimensions, the
application of the process above described, would then determine the
quantities ssi, asj, »„.
R. MURPHY.
Caics College,
March 5, 1833.
K«
III. On the General Equation of Surfaces of the Second Degree.
By Augustus De Morgan, of Trinity College.
[Read Nov. 12, 1832.]
The present investigations are a continuation of those upon lines
of the second degree, published in Vol. IV. Part I. of these Transactions.
I have omitted various algebraical developments, as unnecessary, and
tending to swell this communication to a length more than proportional
to its importance.
As the theory of the reduction of oblique to rectangular co-ordinates
is a very necessary part of what follows, I proceed first to give the
equations which will be required under this head. Let x, y, %, be
oblique, and x', if, a' rectangular co-ordinates to the same point, with
a common origin. Let the angles made by the first system be
A A A ^
y% = ?, %x = t), xy = ^,
and let the rectangular and oblique co-ordinates be so related that
AAA
COS xsd = a, COS yx' = /3, cos xyf = a', &c. ;
whence the following equations:
a/ = ax + fiy + yx,
y' = a'x +, /3'y + y'z (1),
S8' =a"x + fi"y + 7"i8;
l = a'+a" + a'", COS ? = /37 + (i'y' + fi"y'\
l=l3f>+ fi'^+ fi'% COS t, =ya + y'a! + y"a" (2),
1 = y + y^ + y% cos ^ = a/3 + a'/3' + a")8".
78
Mr DE morgan ON THE GENERAL EQUATION OF
Make the following abbreviations, to which, for facility of reference,
are annexed those which will afterwards appear in treating the general
equation of the surface,
aaf + bif-ir cz" + 2ai/z + 2bzx + 2cxy + 2aa? + 2% + 2cs! +/ = (3),
the co-ordinates of the center of which call X, Y, and Z. Throughout
this paper, all subscript indices indicate the dimension of the quantity
signified, in terms of the coefficients of (3) :
p =/3'7"-/3"7',
/ = /3"7-/37",
p"-=^y' -d'y.
y'a"-y"a'.
^1 ff If
q = y a—ya ,
q"= 7«' -7'«'
r =a'/3"-a"/3',
t" = a"(i-a(i"..
r"=«/3' -a' 13.
(4),
a^^= be — a',
b^, = ca- b%
c„ — ab — (?.
tto = sin' I,
b^= sin'*;.,
Co = sin^ ^.
.(5),
a^= 6 + c — 2acos^,
b,= c + a — 2b cos rj,
Cf = a + b — 2c cos ^,
l^^ —bc-aa, I, = &cos^+ccos»7-« — acos^, \ = cos v cos ^— cos ^,
m, = ca — bb, mj=: ccos^ + acos^—b -bcosrj, 7»o= cos^cosf — cos ^...(6),
91^1 =■ ab— cc, n,=a cos j? + 6 cos ^—c~c cos ^, «„ = cos ? cos tj — cos ^.
=»?„«o-ao 4-^ cos A
= «o lo-boMo-T- COSri\...(7),
.= /o««o-Co«o-T-cosg
(8),
(9),
Fo=l+2cos^cos.jcos^-cos^^-cos'»j-cos'^'
V, = aai + bb^-\-cCa + 2ala + 2bma + 2cnf,
Vi = a„ + b^, + c„ + 2/,, cos '^■\-2m„ cos r\-\-<iLn„ cos ^
= */;C//-/,; ^«1
.(10),
Vz~abc^2(ibc — a<i — b¥ — c& \ =.c„a„—m,J-^b\
r; = a,,a' + J,,6^ + c,,c^ + 2/,,6c + 2»?,,ca + 2w,,«F (11),
= m„n„-a„l„ -=ra
■.nj„ -b„m„^b
= l„m„-c„n„ ^c
W=-^ +/= aX+ 6 F+cZ+/
.(12).
SURFACES OF THE SECOND DEGREE. 79
From (4), we find by inspection that the following six quantities
are severally equal:
pa + qft + ry, pa + p'a' + p" a'\
p'a' + g^/3' + r'y', qfi + ^/3' + q"fi" (13),
p"a" + q"(i" + r"7", ry + r'y' + r"y",
and moreover, that any symmetrical interchanges of accents in the first
three, or of letters in the second, give results severally equal to nothing.
Such are joa' + g'iS' + ry, p li + p (i' + p" fi" , &c. Let the common value
of the first six be T. We have then
pa -{■ q& + ry = T,
pa +ql3'+ry'=0 (14),
pa" + qli" + ry" = 0.
From which, by obvious multiplications and additions, looking at equa-
tions (2), we have
p +3' cos (^+r cos ri=Ta,
pcos ^+q +r COS ^=T(i (15),
p cos t] +q cos^+r = Ty.
From either of which sets we deduce
1^ ■¥<f -Vi^-^^qr cos + 2rp cos n-\-^pq cos^= T^ (16),
and similar relations may be deduced between jo', g', r', and jt>", ^", r' ;
T being the same throughout.
Again, form the several quantities
flo, /o, &c. or 1 - cos^ f, cos n cos ^- cos f, &c.
from the second set of equations in (2), and make the results homo-
geneous and symmetrical from the first set; for example, write for
Oa and /o
(7« + V«' + 7"«") («/3 + a')3' + a"/3") - \c?^oi^^cl'''\ (fiy+fi'y'+(i"y"),
80 Mr DE morgan ON THE GENERAL EQUATION OF
in which the factors equal to unity, and introduced for symmetry,
have the brackets []. Develope these expressions, from which we
obtain the following equations:
fl„=p^+y^+jo"% l,= qr-^qr' + q"r",
h^=q' + q'' + q"\ m,= rp^-t'p' + r"p" (17),
Co = r^ + /" + r"*, n„=pq +p'q' + jo'Y'.
These, added together, the three last having been respectively multi-
plied by 2 cos I, 2 cos rj, 2 cos ^, give from (16)
«o + *o+Co + 24 cos f + 2»?o cos »? + 2«„cos ^=3T\
The first side of which, developed from (5) and (6) gives 3 V^* whence
T=y/Vo (18).
If the process by which (17) w^as obtained from (2) be repeated
upon (17), that is, if at,ha-lo, Wana—a^la, &c. be formed, we shall have
equations of a similar form, substituting instead of p, p' &;c. such functions
of them, as they themselves are of a, y3, &c., the first sides of the equations
being from (7), ^o ^^ *^^ ^^^^ three, and V^ cos f, F^ cos ri, Vg cos ^, in
the last three. These equations are such as would arise from sub-
stituting in (2),
^ ^ ,^ — instead of a y~ — and y-^ ^ for a and a", &c...(19),
which are therefore the values of a, a', &c. in terms of p, q, &c.
From (1), by means of (14) and (18), can be deduced the following :
^/YgX=px'+p'y'-!t-p"^,
VT,y = qaf + q'y'+q"fi (20),
-v/Fo as = r x' + / 2^' + r"%',
and the equations of the axis of x', referred to the oblique axes
X, y, and k, are any two of the three,
qx-py — 0, ry — q% = 0, p%-rx=Q (21),
SURFACES OF THE SECOND DEGREE. 81
The equations of the center, central line, or central plane, as the
case may be, of the surface expressed by (3) are
aX+'cT+bZ+a = 0,
cX+bY+aZ+% = (22),
bX+aY+cZ+c = 0,
and in the two following sets of quantities, it will be found that the
sum of the products made by taking a term from each in the same
horizontal line is = F^ ; while if the terms be taken from different horizontal
lines, it will be = 0.
a
c
b.
««
n„
^..^
c
b
a.
«„
K
K,
b
a
c,
m,,
K
c„.
Thus
aa„-\-cn„-\-bm„—Vz, an,, + cb^^+bl,, = 0, &c.
Hence, if the three equations in (22) be independent of one another,
the co-ordinates of the center are
j^^ _ a,,a + n,J + m,,c ^ y^ _ n„a^bj)^l„c ^^ _ m^^a^-l^-\-c,fi
r-
The equation of the surface, referred to this center, and to axes
parallel to the primitive axes, becomes, calling ^ {x, y, %) the first side
of equation (3),
aa? ^-bf +cz^ + 2ay% + 2bzx ■{■^'cxy+^{X, Y, Z) = ...;:.. (24),
and by multiplying the three equations in (22) by X, Y, and Z respec-
tively, and adding, we get
0(X, F, Z) = aX+bY+7z+/=JV (25).
When only two of the equations (22) are independent, there is a
central line. The conditions of this case are, that the numerators and
Vol. V. Paut I. L
82 Mr DE morgan ON THE GENERAL EQUATION OF
denominators in (23) must be severally equal to nothing; but if
f^3 = 0, the equations in (10) shew that it is sufficient that one of
the numerators should be equal to nothing; or that the conditions
may be stated thus,
r, = 0, a/o^, « + \/T,* + 'v/cIc = (26).
When F'i = 0, F't is a perfect square, (10) and (11), its root being
the second expression in (26). Hence W appears in the form - . From
two of equations (22), substitute in (25) values of any two co-ordinates
of the center in terms of the third; it will be found that the co-
efficient of the third disappears under the conditions in (26), and that
the resulting value of W, which we denote by W, may be expressed
in either of the following ways:
„^, b(^ — 2cab + a¥ , „ cb^ — 2acb + b<f ^
ab — & bc — tt
^ _ ad'-^bac + ca' .^^.
ac — V
When no two of the equations (22) are independent, there is a
central plane. The conditions of this case are, as appears from the
equations, that a„, 6,,, c^,, /„, «»,,, «,,, must be severally = ; of which how-
ever it is sufficient that any three should exist. We have moreover
a
a : c '. b (28).
From all which it appears that W is now in the form -. From
one of the equations (22) substitute in (25) the value of one of the
co-ordinates in terms of the other two; the coefficients of the last two
will disappear, as before, and the different forms of the value of W,
which we call W", will be
W"^ - I +/= - J +/= - 7 +/• (29).
By substituting W or W", when necessary, for W or <p {X, Y, Z)
in (24) the equation of the surface will be obtained, referred to any
point in its central line or plane.
SURFACES OF THE SECOND DEGREE. 83
Let the equation of the surface, referred to the principal axes, be
Aa;" + A'y" + A"z''+W=0 (30),
which must be identical with (24) when the values of x', y', «', found in
(1) are substituted. We must then have
a==Aa' +A'a" +A"a"\
h = Ali' +A'(i"' +A"(i"\
c = Ay^ +A'y" +A"y'\
(31),
a = Al3y +A'fi'y' + A"li"y",
h=-Aya +A'y'a -{■A"y"a',
'^ = Aafi+A'a'^ +A"a"li",
which equations are reduced to those in (2) by substituting unity for
A, A', A", a, h, and c; and cos f, cos n, and cos X, for a, h, and c. Thus,
whatever equation is deduced from these, we immediately find another,
containing a, /3, &c. in the same way, by the last mentioned substitu-
tion. Multiplying the first of these by p, the last by q, and the last
but one by r; and adding, we obtain by the use of (14),
pa + qc +rb =Aa\/Vo
p +qcoS(^ + rcosr]= ay/V^
from which, and similar processes, we obtain
(32),
p{A — a) + q{A cos ^—c) + r {A cos t} — b) = 0,
p{A cos^-c) + q(A-h) + r(^cosf-a) = (33),
p{Acc^ri — b) + q{Acosl^—a) + r{A — c) =0;
which agree in form with (22), if a, J, and c be struck out, and A — a
substituted for a, ^cos^ — a for a, &c. But 1^3 = is the result of (22),
with the last terms erased ; that is, if in V^ the substitutions just men-
tioned be made for a, a, &;c. the result developed and equated to zero
wiU give the equation for determining A, A', and A". That equation is
r,A'- r^A'+r,A-v,=o (34).
L 8
84 Mb DE morgan ON THE GENERAL EQUATION OF
We also find from (33), for substitution in (21),
-:-:-:: l^ — l^A + loA' : m^^ — m^A + m^A^ : n^, — n,A + n^A^ (35).
The equation (34) must have all its roots possible. For from (31)
it appears that A' and A" cannot be of the forms X + m V-l and X— m \/- 1,
unless a' and a", /3' and /3", 7' and 7" are of the same form ; from which,
since
{K + xV'^){o-(p\/'^) - («-x\/^)(0 + 0\/^T)
is of the form k\/ — 1, it will follow that p, q, and r (4) must be of
this form : which is inconsistent with (32), if we suppose V^ positive ;
since it may be seen from (31), and will presently appear otherwise,
that a is possible when A is possible.
We might find equations of the third degree to determine jh q, &c.
but it will be more convenient to express them in terms of A, &c.,
supposed to be found from (34). To do this, let a,,, a^, I,,, I,, &c. (5)
and (6), be found in terms of A, a, he. by substituting the values of
a, h, a, h, he. from (31). The results, after reduction, are
a,,=A'A"jf +A"Ap" +AA'p"', a,=U'+'^")f +{A"+A)p" +{A+A')p"%
h,^A'A"(f +A"Aq" +AA'q"\ b={A'+A")q' +{A"+A)q" +{A+A')q"%
c„=A'A"f^ +A"Ar" +AA'r"', c=U'+^"V +U"+A)r" +{A+A')r"\
..(36),
l„=A'A"qr+A"Aqr'+AA'q"r", 1,={A'+A")qr+{A"+A)q'r'+{A+A')q"r",
m„=A'A"rp+A"Ar'p+AAy'p", m={A'+A")rp+{A"+A)rp'+{A+A')r"p",
n,=A'A"pq+A"Ap'q'+AA'p"q", n=U'+^")Pq+i^"+-^)p'q'+i^+^')P'Y'
which equations, with those marked (17), give the following values
of p'', qr, he.
a,-a^A + a,A' _ l„ - l,A + kA'
^~ {A-A'){A-A")' ^ {A-A'){A-A"y
" — ^i i~^t ^ +hoA ^ _ m,,-m,A + irigA^ .
^'~ {A-A%A-A"y ''P~ {A-A'){A-A") ^^^'
•i _ C// — g, ^ +CoA ^ _ n,, - n,A + n^A'
^ ~ XA^^t^ - ^"') ' ^^~ {A- A) {A - A") ■
SURFACES OF THE SECOND DEGREE. 85
In which equations the letter p, q. A, &c. may be accented throughout
singly or doubly, striking off three accents from any A which thus
obtains three or more.
By squaring the equations (15), writing V^ for 7", substituting
the values just obtained for p^, qr, &c. and then multiplying the same
equations together two and two, and making
Li = *„Co + b^c,, - 2 IX, Z/2 = in,,n, + m^n,, - aj,, - a„k,
Ni = «„*„ + «o*// - 2 n„n^, Ni = l,,m, + lotn^^ - c^n„ — c„«o,
we get
^_ F,-L,-{F,-aK)A + F,A' r,-M,-{F-br,)A+ F.A^
**" V,{A-A'){A-A") ' ^~ r,{A-A'){A-A")
2 F-N,-{r,-cr:)A+V,A' ,„.
^- K{A-A'){A-A") ^^^^'
„ _ FiCos^ — Lz — jVi cos^—aF'o)A+ FgCosBA^
^'y~ F{A-A'){A-A")
_ Fcos tj — Mi- (Fj cos tj -bFp) A + FqCos t/A^
'y"- F,{A-A'){A-A")
^ F, cos t-'^2-{F, cos ^-c F„) A + F„ cos ^A'
"^~ F^{A-A'){A-A")
in which the letters may be singly or doubly accented as before, and
from which the determination of the position of the principal diameters
is made to depend directly upon the solution of (34).
Let the surface whose equation is (3) be referred to another origin
and other axes, and let the quantities corresponding . to those already
given or deduced, which belong to the new origin or axes, be denoted
by the same letters and accents enclosed in brackets [ ]. Thus the
angles made by the new axes are [^1 [>;], and [^] ; the coefficients of
86 Mr DE morgan ON THE GENERAL EQUATION OF
the new equation are [a], [«], &c.; the functions of these coefficients
already noticed are [«J, [/„], [F,], &c. Since the principal diameters
of the surface are the same, from whatever equation they are derived,
w r w'l
that is, since — 'T ~ ~ rlT ' ^^' *^^ roots of (34) bear to those of [34]
the proportion of W^ to [ W^ ; whence, \ being an indeterminate quan-
tity, since one coefficient in (3) is indeterminate.
.(39),
LjO'^'k' M-^'
These equations* correspond to the general relations (6), (7), and (9),
given in my former paper, and from them may be deduced the pro-
perties of systems of conjugate diameters, and the remarkable property
of the reciprocal squares of three semi-diameters at right angles to one
another.
Let wT', V, and Z', be the co-ordinates of the second origin referred
to the first, so that if the co-ordinates be changed, [y] and (p{JC', Y', Z')
will be corresponding terms of two equations, the terms of which should
be respectively proportional. Assume X, the indeterminate quantity
above-mentioned, so that
[/] = \4>{X', Y', Z') (40).
and multiply together the first and last of (39), recollecting that
W
= -r.-^f^ [^i = -[Fj^t/].
* These relations have been given by M. Cacchv, for the case of rectangular co-
ordinates, in his " Lcfons sur les applications du Calcnl Infinitesimal d la Geometrie," Vol. i.
p. 2441. The equation (34) of this paper, in as general a form, has also been given, since
this was written, by Mr Lubbock, in the Philosophical Magazine.
SURFACES OF THE SECOND DEGREE.
87
and we obtain
[^J-X^(r-,F-.Z')[^J=X.(^-/^;).
Substitute from the last of (39) for [jfl, and develope <f>{X', V, Z'),
removing the term which contains it to the left hand side; which
gives
ca=
r, + F, jaX" + br" + cZ'' + &c. &c.)
■(41),
answering to (8) in my former paper.
We shall afterwards proceed to some applications of these general
formulas, and now enquire into the several varieties of the equation (3),
and the criteria for distinguishing between them. The following table,
immediately to be explained, gives a synoptical view of the various
caseSi inhcrfHt i->9.R :i;f<>1 '"\'r^hr' 5r .-^.v:
When the Equations of the
Center denote
positive,
negative.
W changes its sign.
^ negative,
positive.
A point
A Right Line. W
substituted for W.
A Plane. W" sub-
stituted for W.
Impossible.
Single Hyper-
boloid.
Impossible.
Hyperbolic
Cylinder.
Impossible.
(W=o) Point.
(W= oc) Elliptic Paraboloid.
(W=0) Cone.
(W=oc) Hyperbolic Para-
boloid.
(W'=0) Right line.
(W"=oc) Parabolic Cylinder.
(W'=0) Intersecting Planes.
(W'=oc) Parabolic Cylinder.
fW"=0 \ ^.
^, Plane.
Ellipsoid.
Double Hyperboloid.
Elliptic Cylinder.
Hyperbolic Cylinder.
Parallel Planes.
88 Mr DE morgan ON THE GENERAL EQUATION OF
Taking the first line of this table, and the signs of W, V^, V^, and
V^, (on which, as will presently be shewn, the variety of the equation
depends,) being such as to denote that the equation is impossible, a
change of sign in W only will indicate the ellipsoid, the elliptic cylinder,
or parallel planes, according as the centre is a point, a line, or a plane.
When the sign changes, if W be then = 0, the variety of the equation
belongs to a point, a right line, or a plane ; while if W be infinite,
we have an elliptic paraboloid, a parabolic cylinder, or a plane. In
using W, we mean its real value, W or W", when the primitive form
of W becomes - .
The following table, from which the preceding may be deduced, and
which I proceed to establish, gives the signs of W, &c., and also of V^,
&c., for the different cases. When p alone, or p and n occur on the
same line, p may signify either sign, provided n stands for the other.
Also when a sign is enclosed in brackets, it is a necessary consequence
of what precedes it, and not an independent assumption. The num-
bers over the headings are references to the equations.
The last part of the table, including all the varieties under W= - ,
forms a similar synoptical table for the curves of the second degree.
The following are the values of W, W", V^ and Fi, expressed in the
notation of my former paper, the equation of the curve being
ay* + hxy + ca^ + dy->rex +f= ;
and the angle made by the axes being Q,
.^, _ cd^ + ae^ — hde „
™,„ _ _ dr-^a£_ &-^^cf
id ~~ 4c '
V, =- (b'-iac),
Vi = a + c — h cos Q.
SURFACES OF THE SECOND DEGREE.
8d
0)
•, C
0)
•5
1
§ 1
B
d
II
i
i
.5
1
■4->
s
II
.O" 4)
CO i^C
6
il
a
CQ
d
11^
o
11^
o
II
g
CO
a;
1
»
^ go
11
- E
s
S
3, and the next
y be = 0,
B
O
5, and all which
bstituted for 7,
s
•J3
u
2
CO
<»
-.' S
te
X
■s s
-^ S
e'
«
s
s
#>
e"
«
^ u-
w
e
O
e
TS
,
t3
o
u
8
o
•■6 'o
^
^
QJ
o
1
1
p-(
2-S
1^'
Si
a
1
en
s
s"
c8 o
*^ 'o
i^
} Straight lin
Intersecting
c8
1 =
1
1 S
2 CL(
a
02
1
o
a
o
(J
1
•1 .J
1
'o
d
1
PU
s^
a, a
8
a,
a.
a
a, a
a, a
«
a,
o 8
^— , '
/— \
+ +
+ 1
+ 1
+ I
+
+ 1
+
1
©
+ 1
o
O ^
■ • —
^ •
- ■ y •
'
£^
a,
8
o
0|0
.''^S
" —
O '^
Rh a
a,
a
O
a.
§
a.
a,
8
o ■
0|C
Vol. V. Paet I.
M
90 Mr DE morgan ON THE GENERAL EQUATION OF
First, with regard to the coefficients K^, Vi, V^, V3 in equation
(34) it appears from spherical trigonometry, that V^ is always positive
when '(;, J/, and ^ are the sides of a spherical triangle; while from the
possibility of the roots, as well as from the quantities themselves, we
infer that if V3 is finite, Fj and Vi can never vanish at the same
time, while if ^i = 0, and ^ = 0, Fj, must be negative.
If we suppose TV finite, and the order of signs in (34) to be
H (-- or + + + +, in which case all its roots are of one sign ;
that is, if K2 be positive, and Vi and V3 of the same sign, the equa-
tion (30) shews that the surface is impossible or an ellipsoid, according
as W and F'a have the same or different signs. From (36) it appears
that in this case, a^^, b,^, and c„ must be positive, whence a, h, and c have
the same sign ; which conditions, together with that of V^ having the
same sign as a, are equivalent to those given in the Table for the
impossible case or the ellipsoid. If we examine independently into
the conditions under which the aggregate of the first six terms of
(24) always has the same sign, we shall find them to be that a^, b„,
and c„ must be positive, and V3 must have the common sign of a, h,
and c. And it is evident that the first three terms of (30) are the first
six terms of (24) in a different form. It may be worth noticing, that
these conditions are equivalent to supposing ,-- , -- — , —7=5= to be
's/ he \/ca y/ab
the cosines of the sides of a spherical triangle. When any other order
of signs except the two already noticed, is found in (34), we shall have
one positive root only, or one negative root only, according as V3 is
positive or negative ; that is to say, one possible axis, or a double
hyperboloid, when V^ and W have contrary signs ; and one impossible
axis or a single hyperboloid, when they have the same signs.
When W—0, V^ being finite, equation (30) represents a point, or
a cone; the first when all the roots of (34) have the same sign, the
second in any other case. When V3 = 0, Vi being finite, or W infinite,
the center is at an infinite distance, and the equation belongs to an
elliptic or hyperbolic paraboloid, according as V^ is positive or negative.
Since when V3 = 0, «,,, 5„, and c,, have the same sign, (10), which is
SURFACES OF THE SECOND DEGREE. 91
also the sign of V^, a,^ may be substituted for Vi. In this case, (10)
and (9), V2 has the form
P+ Q + 2? + 2\/QK" cos ?+2\/;BP cos .7+2 ^/PQ cos ^,
which, when P, Q, and R have the same sign, is always of that sign;
and therefore can only be = when P, Q, and B are severally = 0.
When ^"3=0, and F'i = 0, in which case W appears in the form -,
and its real value is W (27), the simplest criteria of which are ex-
pressed in (26) the equations (30) and (34) assume the forms
Aaf' + A'y"+Jr'=0 (42),
KA'- r,A + r,=o (43),
the first of which, if V^ be positive, and F", and W of the same sign,
is impossible, and belongs to an elliptic cylinder if V^ be positive,
and Fi and W of different signs. As before, we may substitute a„
for Vi. If V2 or a^i be negative, (42) belongs to an hyperbolic cylinder :
and if V2 — O, in which case a^^ = 0, h,i = 0, and c^^ = and W is infinite,
we have a parabolic cylinder. It appears therefore, that any surface of
the second order, which has three parabolic sections, not having a
common line of intersection, is a parabolic cylinder. The central line
of this surface is at an infinite distance. When W' = and V is
positive, equation (42), considered as of two dimensions, represents
only the origin, and therefore belongs to a straight line, the axis of
iB'. When Fa is negative, W being =0, (42) is the equation of two
planes intersecting at an angle whose tangent is
2^/-AA' 2V-F,r,
A + A' ' °^ r,
When the equations of the center belong to a plane, and W as
well as W appears in the form -, the real value of W is W", given
in (29) and the simplest conditions are, as in (28),
«// = *// = C// = 0,
a : b : c '.: a : c : b.
M 2
92 Mr DE morgan ON THE GENERAL EQUATION OF
The equations (42) and (43) take the forms
Ax"+jr" = (44),
F,A - F, =0 (45).
The first of which is impossible if W and Fl have the same sign,
that is, if W" and a have the same sign ; for when a„ = b,^ = c„ = 0,
T^i takes the same form with respect to a, b, and c which Vs took with
respect to a„, b^,, and c„ in the last case. When a and W" have
different signs (44) belongs to two parallel planes, which coincide in
one where W" = 0. That is (29) the surface is impossible, two parallel
planes, or one plane, according as af—c^ is positive, negative, or nothing.
When W becomes infinite, or a = 0, in which case b, c, a, b, and c are
severally = 0, the proposed equation (3) is in fact of the first degree.
Though oblique co-ordinates have hitherto been used, yet they
might have been dispensed with so far as the criteria of distinction
between the different classes of surfaces are concerned. It would take
some space, and complicated algebraical operations, to prove this in
all the individual cases, but the following general consideration is equally
conclusive. So long as we only consider those distinctions which are
implied in calling the surface bounded or unbounded, of one sheet or
of two sheets, &c. in which no numerical relations of lines, &c. appear,
it is evident that any equation will preserve the same character, how-
ever the axes on which its results are measured are inclined to one
another. That is, when the sign of a quantity is alleged to be a cri-
terion of distinction, it cannot stand as such, if by any alteration of
I, >/, or ^, consistent with V^ remaining positive, the sign of that quantity
can be changed. Again, if the signs of two out of the three, a, b, and c
be changed, as well as that of the third letter in a, b, and c, (those of
a, b, and c, for example) it is evident that the surface remains the same
in form and magnitude, those parts which were below one of the co-
ordinate planes being transferred above it, and vice versa. That is,
it is impossible that any aggregate of terms of an odd degree, with respect
to a, b, and c, b, c, and a, or c, a, and b, can affect the sign of any
SURFACES OF THE SECOND DEGREE. 93
of the criteria. If we look at F'l, V^, Fs, and F^, we find that those
terms, and those terms only, which are multiplied by cosines of f, &c.,
are of the first or third degree, with respect to any of the three sets
just mentioned.
The case is very much altered when we consider any numerical
relation, however simple. For example, I give the condition which
expresses a surface of revolution, or a surface two of whose axes are
equal. If A and A' belong to the equal axes, a, a, &c. become in-
determinate; hence the numerators of the six equations (38), will, when
equated to zero, have a common root. Eliminate F - FA + FoA" from
the values of a* and (3y, &c. in (38), which gives
.-_ XaCOsf-Z^ MiCO^ri-D iVgCOS^-iV^ ,^
AFa = r"=- = -7 T~ - y - V4b),
a cos ^— a cos t] — o ccos^— c
which does not admit of any material simplification. There are evidently
other ways of obtaining corresponding conditions from (38). I have
chosen this because the corresponding formulae have been given in the
case of rectangular co-ordinates. In this case,
cos ^ = cos >; = cos ^ = 0, and Li = - l„, &c.
whence,
^ _ ^/ _ ^/
a b c
(See Mr Hamilton's Analytical Geometry, p. 323.)
To apply the formulae (39) and (41), let there be two planes whose
equations, separately considered, are
\'x+ fi'y+ i/'a + l =0 J
but which together must be one of the varieties of equation (3). Let
new and rectangular axes be taken, the intersection of the planes
being that of x. Their equation will then be
[c] z" + 2[a]yss = 0,
94
Mr DE morgan ON THE GENERAL EQUATION, &c.
g
■»■>
>
a
o
ICIIC
o
<1>
Cm
O
c
be
I
i.ei w
-i.>
§
^
-« lCk°
fcc.S
C8 O
S M
« -a
o
CO
r
aj to
fc: ^-i
bi3 +^
•l.s-a
g - ^
•^ 5 S
£ ^ c
g " §
'S -^ ■=«
o .in
bO
o Si
e9 .iH H-i
Iv^
a.
I
"a.
+
o
o
>
I
s-
ta
O
u
o
u
a.
+
"a.
o
u
o
u
+
+
O
u
!!•
CO
O
O
~a.
+
>V3
'en
"a.
a.
+
"a
CO
J3
<u
^
&I
CO
o
-§
0)
2
§ I
Sh '^
0; o
^ s
«4-i 5s
o
II
'-J II w
O F=f
I — I
II W
I I
+
I — 1
I I
+
I — I
I I
I — I
1-©
I I
+
r— 1
+
lie
^
©I
+
I
s
I
+
I
+
0)
g
o
o
p
H
CO
a
K
IV. On a Monstrosity of the Common Mignionette. By Rev, J. S.
Henslow, M.A. Professor of JBotany in the University of Cambridge,
and Secretary to the Cambridge Philosophical Society.
[Read May 21, 1832.]
Having met with a very interesting monstrosity of the common
Mignionette {Reseda odorata,) in the course of last summer (1831),
I made several drawings of the peculiarities which it exhibited. I beg
to present the Society with a selection from these, as I think they may
both serve to throw considerable light upon the true structure of the
flowers of this genus, which is at present a matter of dispute among
our most eminent Botanists, and also tend to illustrate the manner
in which the reproductive organs of plants generally, may be con-
sidered as resulting from a modification of the leaf.
It is well known to every Botanist, that Professor Lindley has
proposed a new and highly ingenious theory, in which he considers
the flowers of a Reseda to be compounded of an aggregate of florets,
very analogous to the inflorescence of a Euphorbia. Mr Brown, on
the other hand, maintains the ordinary opinion of each flower being
simple, and possessed of calyx, corolla, stamens, and pistil. I shall
not here enter upon any examination of the arguments by which
these gentlemen have supported their respective views, but will refer
those who are desirous of seeing them to the " Introduction to the
Natural System of Botany, by Prof. Lindley," and to the "Appendix
to Major Denham's Narrative, by Mr Brown," My present object will
be little more than to describe the several appearances figured in plates
1 and 2.
96 PROFESSOR HENSLOW, ON A MONSTROSITY
Fig. 1. is one of the slightest deviations that was noticed from the
ordinary state of the flower. It consists in an elongation of the pistil (a),
and a general enlargement of its parts, indicating a tendency in them to
pass into leaves. This is accompanied by a slight diminution in the size
of the central disk. The number of the sepals was either six or seven.
Fig. 2. is a portion of the ovarium of the same flower opened, in
which three of the ovules are somewhat distorted.
Fig. 3. Here the three valves of the ovarium have assumed a dis-
tinctly foliaceous character (a); the same has happened to some of the
stamens {b), and to the petals (c) ; but the sepals are unaltered. The
central disk has entirely disappeared.
Fig. 4. This is a still closer approximation to the ordinary state of
a proliferous flower bud, when developed. Those parts which would
have formed the pistil, if the flower had been completed, are no longer
distinguishable, and only a few of the stamens are to be seen, disguised
in the form of foliaceous filaments crowned by distorted anthers (h).
Fig. 5. A slight deviation in one of the petals from the usual
character. The fleshy unguis is somewhat diminished, and the fimbriae
are becoming green and leaf-like. These are aggregated into three
distinct bundles, the middle one being composed of a single strap,
and the two outer ones of five straps each, blended together at the
base.
Fig. 6. The line of demarcation between the unguis and the fimbriae
has completely disappeared, and the number of the latter is considerably
reduced. The whole is more green and leaf-like than fig. 5.
Fig. 7. The fimbrige reduced to a single strap ; the position of the
lateral bundles being indicated by slight projections only. Other in-
stances occurred in which the petal appeared as a single undivided
uniform green strap.
Fig. 8. The two exterior whorls of a flower, consisting of seven
regularly formed sepals, and eight petals. The latter deviate more or
less from the forms represented in fig. 6 and 7. The whole of a green
tint, and leaf-like.
7>^nsactM>ns afthe Ceimi. I'Ail.Sor. VoLTTTf^J
JSa-n,l,.u- ,1,1 f
.TD-CSmn^Jatrfi'
Transaetians of the C'affiiJ'ful.SrK.Til.KIt.Z.
XlkKStmnfy X'"^
OF THE COMMON MIGNIONETTE. 97
Figs. 9, 10. These are parts of one and the same flower dissected
to shew the several whorls more distinctly. The whole has assumed
a regular appearance, and is composed of seven sepals, alternating with
seven green strap-shaped petals, which are succeeded by about twenty
stamens without any fleshy disk ; the pistil is somewhat metamorphosed.
This is perhaps the most remarkable deviation that was noticed from
the ordinary state of the flower, and as several examples of it occurred,
it is not likely that there is any error in this account of it. It appears
to lead us in a very decided manner to the plan on which the flowers
of the genus may be considered to be constructed, and to shew us
that they are really simple and not compound.
Fig. 11 to 15, represent the appearances assumed by some of the
stamens, indicating various degrees of deviation from the perfect state
towards a foliaceous structure.
There were other circumstances, besides the appearances in figs. 9.
and 10, which may lead us to conclude the structvire of the flowers
of the genus to be simple and not compound. A compound flower
arises from the development of several buds in the axillee of certain
foliaceous appendages more or less degenerated from the character of
leaves, and consequently these buds and the florets which they develop
are always seated nearer to the axis than the foliaceous appendages
themselves. If we suppose a raceme of the mignionette to degene-
rate into the condition of a compound flower, we must allow for the
abortion of the stem on which the several flowers are seated, so that
these may become condensed into a capitulum, each floret of which
will be accompanied by a bractea, more or less developed, at its base.
Let us compare this supposition with the diagrams represented in
figs. 16, 17, 18.
Fig. 16. is an imaginary section of the flower in its ordinary state,
(a) the pistil, (b) the stamens on the fleshy disk, (c) the petals, {d) the
sepals alternating with them.
Fig. 17. represents the position of the several buds (e) which com-
pose the florets of the flower on the supposition of its being com-
pound. Here it will be noticed that these buds alternate with the
Vol. V. Part I. N
98 PROFESSOR HENSLOW, ON A MONSTROSITY
sepals instead of being placed in their axils where we might rather
expect to find them.
Fig. 18. represents a fact which was observed in the present case,
where some of the latent buds in the axils of the altered petals were
partially developed. This development might perhaps be considered as
indicating the construction of a compound flower, and those buds which
in ordinary cases compose the outer and abortive florets, it might be
said, are here manifesting themselves. But the axes of these buds lie
nearer to the axis of the whole flower than the petals in whose axils
they are developed; whereas it appears by fig. 17, that they ought to
be further from it, since the centres of the five outer circles marked (e)
would represent the axes of the several buds, whose partial develop-
ment must be supposed to be on the side next the axis, if we allow
any weight to the analogy between the position of the abortive
stamens on the supposed calyx, and the fertile stamens on the central
disk.
These figures are all that I have thought it necessary to give for
the purpose of illustrating the structure of the flower; but as there
were several interesting appearances noticed upon dissecting the pistil,
I have selected some of them for the second plate, as they may
possibly serve to throw some light upon the relationship which the
several parts of the ovarium bear to the leaf, and to support the
theory of their being all of them merely modifications of that im-
portant organ.
Fig. 19. is a pistil in which the three ovules have become foliaceous,
and the central, or terminal bud of the flower-stalk is developing in
the proliferous form represented in fig. 4.
Fig. 20. The central bud is not developing ; but the three axillary
buds in the bases of the transformed valves of the pistil are here
assuming the form of branches on which one or two pair of leaves are
expanded.
Fig. 21. 22. unite the appearances in fig. 19 and 20, with the
addition of a glandular body seated between the leaves at their
OF THE COMMON MIGNIONETTE. 99
junction. This apparently originates in the union of the two glandular
stipules seated at the base of the leaves of this genus, and which
may also be seen to accompany the scale-like leaves on the central
bud within.
Figs. 23. to 25. Interior views of metamorphosed pistils, in which
the ovules are seen transformed to leaves, and the glandular stipules
are all that remain of the leaves which should compose the central
bud, their limbs having entirely disappeared.
Fig. 26. The appearance of these stipules on a leaf-bud, develop-
ing under ordinary circumstances.
Fig. 27. One of them more highly magnified.
Figs. 28. 29. Their appearance on the small scale-like leaves of the
central buds in fig. 21, 22.
Fig. 30. Similar to fig. 23, but without any appearance of the
transformed ovules; the glandular stipules are seen in the bottom of
the ovarium.
These glandular bodies assume a very prominent character in the
anatomy of the metamorphosed pistils, and I was for some time
puzzled to account for them, thinking that they might represent an
altered condition of the ovules. I believe however that I have rightly
considered them as the only representatives of the various leaves which
would have made their appearance on the branch if the bud had
developed in the ordinary way. They do not appear to diminish in
size though the limb of the leaf has disappeared.
Fig. 31. Four pedicillated semitransformed ovules, seated on a pla-
centa of a pistil metamorphosed similarly to that in fig. 9-
Figs. 32. to 35. Other appearances of a similar kind, all representing
various approaches of the ovules to a foliaceous character. The little
theca-shaped appendages are hollow, with a perforation at their apex,
representing the foramen.
100 PROFESSOR HENSLOW, ON A MONSTROSITY, &c.
Fig. 36. One of these dissected, exhibiting a free clavate cellular
body within, resembling the columella in the theca of a moss, and
"here probably representing the nucleus of the ovule.
Fig. 37. In this case the theca-shaped body was partially open
exposing the included nucleus.
Fig. 38. This nucleus more highly magnified.
These appearances surely indicate a development of the investing
coats of the nucleus into leaves ; but how far these developments
might be extended, and whether the nucleus itself is capable of being
further separated into a series of investing coats does not appear from
these specimens.
J. S. HENSLOW.
TRANS A CTIONS
CAMBRIDGE
PHILOSOPHICAL SOCIETY.
Vol. V. Part II.
CAMBRIDGE:
PRINTED BY JOHN SMITH, PRINTER TO THE UNIVERSITY :
AND SOLD BY
JOHN WILLIAM PARKER, WEST STRAND, LONDON;
J. & J. J. DEIGHTON, AND T. STEVENSON,
CAMBRIDGE.
M.DCCC.XXXIV.
On the Calculation of Newton's Experiments on Diffraction. By
George Biddell Airy, M.A. late Fellow of Trinity College,
and Plumian Professor of Astronomy and Experimental Philosophy
in the University of Cambridge.
[Read May 7, 1833.]
Since the publication of Fresnel's experiments on Diffraction, it has
been usual to employ as the source of light, in all experiments of this
class, the image of the Sun formed by a lens of short focal length. On
the undulatory theory, the effect of light thus produced is precisely
the same as if the minute image of the Sun were the real origin of
the light diverging with equal intensity through a solid angle whose
diameter is many degrees. The spherical or chromatic aberration of
the lens produces no sensible effect in any of the common experiments,
in all which the angle, made by rays which afterwards interfere, is small.
In calculating experiments thus conducted we proceed therefore with
full confidence that no consideration is left out of sight, the omission
of which could cause sensible error.
Newton's experiments however were conducted in a different way.
His origin of light was a hole, from Jg^ to ^ of an inch in diameter,
through which the Sun's light was made to pass. The effect of this
light, on the undulatory theory, is not the same as if the bright hole
were the origin of light. It becomes then a matter of some interest
to examine mathematically what is the effect produced by transmitting
the sun-beams directly through a hole of sensible size ; and whether this
effect, in practice, will differ much from the effect produced by forming
an image of the Sun with a lens of short focal length.
The integrals which occur in this investigation are of such a kind
that their values cannot be exhibited even in tables of numbers (except
Vol. V. Part II. O
102 PROFESSOR AIRY ON THE CALCULATION OF
of course in any particular case, when by very tedious summation nume-
rical results might be obtained). The only thing which can be attempted
is, to shew that the integrals are precisely the same as those that occur
in a very different instance where Fresnel's method of experimenting
is adopted. Even thus far however I have not succeeded except in one
case, namely, where the hole is a rectangular parallelogram of any length,
and where the diffracting aperture is also a rectangular parallelogram
in a similar position ; including in this general case the particular instance
in which one or both parallelograms have no boundary on one side.
To consider, in the first place, a case similar to Newton's. A plane
wave is supposed to enter an external parallelogram and then to pass
through a slit with sides parallel to those of the parallelogram ; and the
intensity of the light which falls upon a screen at a given distance is to
be found. First, it is to be observed, that in estimating the comparative
intensity of light in a direction parallel to one side of the parallelograms
(suppose for instance the shorter) there is no necessity to take into ac-
count the length of the parallelograms in the other direction ; as it will
easily be seen, upon attempting an integration, that the intensity of light
is expressed by the product of two quantities, of which one depends only
on the lengths of the parallelograms and the position of the point of
the screen in one dimension, and the other depends only on the breadth
of the parallelograms and the position of the point of the screen in the
other dimension. The intensity of light along a given line parallel
to one side of the parallelogram will therefore, so far as it depends on
the other side, be affected only with a constant multiplier. Neglecting
therefore the lengths (by which term I designate that dimension of the
parallelograms which is perpendicular to the line on which the comparative
brightness is to be ascertained), suppose a normal to the front of the
wave to be di-awn, and suppose the limits of the breadth of the external
aperture measured from this line to be a, fi, (the distance of any point
of the aperture being v), and suppose the limits of the breadth of the
slit to be 7, 5, (the distance of any point of the slit being w)'. and
suppose the distance of the point on the screen, whose illumination we
wish to ascertain, to be x. Let the distance of the external aperture
from the slit be a, and the distance of the slit from the screen h. Suppose
NEWTON'S EXPERIMENTS ON DIFFRACTION. 103
the front of the wave where it enters the external aperture to be divided
into a great number of small parts ^v ; and suppose each of these to
be the origin of a small wave which diverges from it as a center. The
distance from the point v of the aperture to the point w of the slit is
^{a' + {v-wY]=a+ —(v-wy;
and the disturbance produced at w by the small wave spreading from the
space Sv at v will therefore be proportional to
^tj.sin. — {vt- A — a— —-(v — wY].
Integrating this with respect to v, the coefficient of sin — {vt — A- a)
will be
L cos ~{v-wy,
and the coefficient of cos —- (\t— A — a) will be
A
-Xsin^(«-M;)^
The first of these integrals = X cos ^ iv \/ —r — w V -r-J ''
and putting ^(s) for f. cos f- »M, this integral between the limits v = a,
v = l3, will be proportional to
<h\^\/ -- —W\/ -—] — d>\a \/ -— - W\/ ^\.
(TV \ TT
- xM , the integral - /„ sin -— (v — ivy
between the same limits will be proportional to
- ^ l*^ ^ - ■" ^^) + H° ^Fx - =" ^i)'
o 2
104
PROFESSOR AIRY ON THE CALCULATION OF
The whole displacement at the point w will therefore be
sm —
{yi-A-a)x{ (p(iB\/^-w\/^]-<p(a\/%-w\/-^]]
' I ^V aX aXJ ^\ aX aXl j
2ir f /
+ cos —— (yt — A — a) X < - \l/ [j3
aX
— w
aXJ ^ \ aX aXl ]
Suppose now this displacement to be the origin of a small wave
which diverges from it as a center. The distance of the point w of
the slit from the point x of the screen is
^{¥ + {w-xY]=h^^{w-x)\
and this distance must be added to ^ + a in the expressions
sin -^{yt-A-a) and cos ~ {vt— A-a),
X X
in order to find an expression proportional to the displacement produced
by it on the screen at the point x. The expression must also be mul-
tiplied by Sw, the breadth of the small space from which the wave
proceeds. Thus we find for the whole displacement at the point x of
the screen.
sin
-.)..{ ^(/5V^-«,v'|;)-?.(«\/|;-».V|;)}'
COS J— {w
■{vt-A-a-b)y. { — —
+COS -:^{vt- A -a-b)x j ■■
si„i(„-.)..{-*(^ v^-. V„4).«(<.v/|;-. V|)}j
.co.i (.-.)'x{-V.(/3 V|;-» Vi) .+(« V|;-«- X/|) Y
NEWTON'S EXPERIMENTS ON DIFFRACTION,' iO&
where the integrals are to be taken between the limits w = y, w = S. The
brightness at the point x of the screen will then be proportional to the
sum of the squares of the coefficients of sin---(yi—A — a — h) an(f
cos — (yt— A — a — b).
A
To consider in the second place a case in which the illumination is
produced in Fresnel's method. Let the distance from the origin of light
to the aperture be a', and from the aperture to the screen V. Let a
line be drawn from the origin of light perpendicular to the screen, and
let the limits of the aperture measured from this line, in the same
direction as the breadths of the parallelograms in Newton's case, be e and ^
(the general letter for the distance of any point in this direction being p),
and let the limits in the direction perpendicular to this be rj + np, 6 + nj),
where m is constant. (It is readily seen that this implies the figure to
be rhomboidal, with two sides parallel to the length of the parallelograms
in Newton's case.) Let q be the general letter for distance in this second
direction : also let of and y' be the distances, in the directions of p and q,
of a point on the screen from the same line. The distance from the
origin of light to the point p, q, in the aperture is
and the displacement there will therefore be proportional to
The distance from the point p, q, in the aperture to the point or', y', on
the screen, is
and this must be added to
A + a' + ^ + ^,,
in the expression for the displacement, in order to find the displacement
produced at the point x',y', of the screen by the wave diverging from
106 PROFESSOR AIRY ON THE CALCULATION OF
the point p, q, of the aperture. For the effect of the wave spreading
from the small rectangle whose sides are ip, ^q, we must multiply by
^p, Sq. Hence we find that the quantity to be integrated is
where, after integrating with respect to q, the limits of q must be ex-
pressed in terms of p before the next integration.
Puttmg A' + a' + b'+ ^ ^ = B', this expression becomes
The first integral is
27r
sm
-COS - {..-2? - ^— (p _ _^) J /^ sm {2 .-^r^ (? - ^) }
/tt 2(«'+ft')/ «y^'l , TT / ./2{a'+b') ,^/ 2m:' y
which between the limits q = r}-irnp, q — Q-^np, is proportional to
^1 ^ «7yx ^ ^ i' («' + *') x^ «'*'x J
The quantity proportional to .4 sin j^.i^lt*} L _ 4^) | will be ex-
pressed in the same manner, putting >// in the place of 0.
The whole displacement of ether at the point x', y', will therefore
be found to be
NEWTON'S EXPERIMENTS ON DIFFRACTION.
107
cos
\ a'h' V a' + b'J I ^\ ^ a'b'X ^ ^ b'la' + b')\
sm~(yt-B')x f ^
jp
+ sin;
27r, .
f-COS— (v^
A
-B')x\ <
Jp
. TT a'+b' f a'x'Y r f . . /2 (a' + b') ,./ 2a' i
+ COS;
TT a'+b' 1^ a'x' Y r , L./2(«' + 6') ,./ 2^^
where the integrals are to be taken between the limits p = €, p = ^.
The brightness at the point x', i/, of the screen will then be propor-
tional to the sum of the squares of the coefficients of sin — (v#— ^')
A
and cos -— {v t - B').
A
We have now to shew that, for a constant value of y', and a vari-
able value of x', these expressions may be made similar to those ob-
tained in the first case. For this purpose it will be necessary, first, to
make the coefficients of the expressions under the integral sign equal:
secondly, to make the limits of integration the same.
108 PROFESSOR AIRY ON THE CALCULATION OF
rr,! /. , •! i- • •^ IT a' + b' a'x'
1 he first consideration gives vis j— = - . — rrr ' x = -; — rt ;
^ h\ \ a'b' af + b"
"^aX ''^ «'6'X y ^ b'{a' + b')\' ^ a\ ^ a'b'X '
and the second consideration gives 7 = e; S = ^; whence 5 — 7 = ^-c. The
first set of equations, reduced, are
6 = ^' +
1 1 *,OaA ^./l 'aA
a\/ - = tj'S/ J- — y' \/ jn\ whence (/3 — a) v - = — >?; and w= — V-
The purport of these equations, in common language, may be stated
thus :
If in Newton's method light pass through a rectangular hole whose
horizontal breadth is /3 — a, and through a slit whose horizontal breadth
is 5-7, at the distance a from the former, and fall finally on a screen
at the distance b from the slit:
And if in Fresnel's method light pass through a rhomboidal hole,
with two vertical sides, at the distance a' from the Sun's image; and
fall on a screen or eyepiece at the distance V from the hole, so that
1 1__ 1
a'^ b'~ b'
And if the length of the vertical sides of the rhomboid be \/- x
til
the horizontal breadth of the external hole in the first case (or /3 — a);
and the horizontal breadth of the rhomboid be equal to the horizontal
breadth of the slit in the first case (or 5 -7); and the tangent of the
angle made by the sides of the rhomboid be \/ j, (the acute angle of
the rhomboid being on the side where x is negative and y positive).
NEWTON'S EXPERIMENTS ON DIFFRACTION, 109
Then the proportion of the intensities of light along the horizontal
line in the first case will be the same as the proportion of the inten-
sities of light along a horizontal line in the second case: the distance
x' = x y. -r in the second case corresponding to the distance x in the first
case.
If in the first case the center of the hole is opposite to the center
of the slit, the horizontal line in the second case must be drawn over
the middle of the illumination on the screen. But if in the first case
the center of the hole is not opposite to the center of the slit, but
deviates in the direction which makes x positive, then the horizontal
line in the second case must not be drawn over the middle of the
illumination, but on that side on which y' is negative. In general,
or when one side of either aperture in the first case is wanting, the
equations
may be used.
When the inequality of the sides of the rhomboid is considerable,
the form of the illumination is not very different from the illumination
when the hole is parallelogrammic. The coloured bars will be a little
inclined, so that those which for a parallelogram would be perpendi-
cular to its longest sides, will approach towards the direction perpendi-
cular to the longer diagonal of the rhomboid. Besides these, there is
a faint brush of light projecting from each part which corresponds to
an obtuse angle, and nearly in the direction of a line bisecting that
angle produced. These general notions will assist the reader in judging
what ought, theoretically, to be expected in the different circumstances
of Newton's experiments.
In Newton's experiments the external hole was in fact circular.
What would be the effect of this form it is impossible (theoretically)
to say: but judging from the insignificance of the effect produced by a
Vol. V. Tart II. P
110 PROFESSOR AIRY ON THE CALCULATION OF
rectangular hole, I am inclined to think that, when the apertures are
centrally opposite, the same investigation will apply well to it.
I may now without impropriety mention the circumstances which
induced me to make this investigation.
In Newton's Optics, Book iii. Observation 6, Newton describes in
very striking language the effect of narrowing a slit on which the
sun-light fell after having passed through a hole a quarter of an inch
in diameter. He states that when the breadth of the slit was about
—t\\ of an inch, the illumination on the screen was interrupted by
a black shadow in the middle. It is certain, theoretically and prac-
tically, that if the experiment had been made in Fresnel's method the
center would be the brightest part. It seemed therefore worth while
to ascertain, by the best kind of investigation that svich an un-
manageable case admits of, whether the size of the external hole
could account for the dark shadow. From consideration of the form
of the illumination in the second case above, it appears certain that
it could not. The only resource (which the dullness of the weather
at that time denied me) was to repeat the experiment. This I have
now done three separate times in the presence of as many different
persons : I have used both parallelogrammic and circular holes of dif-
ferent sizes (the largest circular hole being ^inch in diameter) and
have sometimes diminished the aperture to as little as j^ inch (by
estimation). The distances have been 30 inches each, which appear
to have been the distances in Newton's experiments. In every in-
stance the center has been bright. I can account for this inaccuracy
in Newton's observation only by supposing that his eye was in such
a state as not to recover from the sudden impression which is pro-
duced by rapidly diminishing the central light on the screen (which
makes it for an instant appear black), and by referring to his candid
avowal in the Advertisement, that " the third book and the last pro-
" position of the second were put together out of scattered papers,"
and that " The subject of the third book I have also left imperfect,
" not having tried all the experiments which I intended when I was
NEWTON'S EXPERIMENTS ON DIFFRACTION. HI
"about these matters, nor repeated some of those which I did try
"until I had satisfied myself about all their circumstances." I may
add that Newton's measures of the distances at which the first dark
bar was formed are so irreconcileable with those of his admirer Biot
that, referring to the avowal above-cited, I think no reliance ought to
be placed on the accuracy of his observations of diffraction.
Since writing the above, I find that Biot has repeated the experi-
ment with the same result which I have obtained {Traite cle Physique,
Tom. IV. p. 749). He has not commented on or even mentioned
Newton's observation.
G. B. AIRY.
Observatory,
May 6, 1833.
p 2
VI. Second Memoir on the Inverse Method of Definite Integrals.
By the Rev. R. Muuphy, M.A. Fellow of Cuius College, and of
the Cambridge Philosophical Society.
[Read Nov. 11, 1833-3
INTRODUCTION.
The object of my former Memoir on the present subject, pub-
lished in the Fourth Volume of the Society's Transactions, was to
investigate the principles by which we might revert from a function
outside the sign of definite integration, to the function under that
sign, whenever the latter belonged to any of those classes usually
received in analysis. In that case the function outside the sign of
integration possessed the characteristic property of converging to zero
when a variable quantity x was made to increase indefinitely ; in the
present Memoir I have endeavoured to complete this theory, by the
research of the forms and properties of the functions under the sign
of integration, when the characteristic above mentioned is not pos-
sessed by the function resulting from integration : and as the subject
increased in difficulty, those methods of analysis which possessed greater
simplicity and uniformity have been most adhered to, in the follow-
ing investigations.
The fourth Section is devoted to the research of the nature and
properties of the function under the sign of integration, when the
integral always vanishes between the limits (0 and 1) of the indepen-
dent variable which have been uniformly adopted in this as in the
first Memoir. The class of functions thus investigated possess the re-
markable property of vanishing an indefinitely great number of times
114 Mb MURPHY'S SECOND MEMOIR ON THE
in a finite extent; such functions correspond to an extended and
curious class of pheenomena in nature, when any principles of action
which have been observed, under peculiar circumstances cease to produce
the observed effects, as when equal charges of opposite electricities
are communicated to a body, or when a body electrised by influence
is removed from the vicinity of the influencing system ; or lastly, as
when heat in its thermometric effects disappears in the chemical
changes which bodies undergo.
The properties of this class of functions are of great use and
importance in analysis, as they conduct directly to the theory of
reciprocal functions. This term I have here employed to denote such
functions, two of which being multiplied together the integral of the
product vanishes, except in one particular case. That function which
is in this sense reciprocal to another, is also in general different in its
nature. There are however many functions which are reciprocal to
functions of their own nature, and to this class belong the only two
species of reciprocal functions hitherto introduced into analysis ; namely,
the sines or cosines of the multiples of an angle, the integral of the
product of which always vanishes (when taken between proper limits)
except in the particular case of equimultiples; and secondly, such
functions as satisfy the well-known partial differential equation in the
third book of the Mecanique Celeste; where the integral of the product
also vanishes except in the particular case where the functions are of
the same order. It is this exception which renders reciprocal func-
tions particularly useful, as is evident from the application of the
trigonometrical functions in the theory of heat, and of Laplace's functions
in investigations relative to the distribution of electricity. In the same
Section I have shewn generally the means of discovering all species
of reciprocal functions, and given several examples : as an instance of
one of the most simple species possessing properties very analogous to
those of Laplace's functions, but giving a simpler integral in the case
where that integral does not vanish, it is proved in the succeeding
h
Section that if T„ be the coefficient of h" in - — j , then when n and in
are vmequal ftT„T„ = 0, but when n = vu ftT„T„ = l.
INVERSE METHOD OF DEFINITE INTEGRALS. 115
The theory of reciprocal functions is applied in the fifth Section
to the complete solution of the question, which was the object of this
and the preceding JMemoir, namely, to revert from any function what-
ever to that under the sign of definite integration, those reciprocal
functions being employed which are most convenient in each particular
instance.
The last application in this Memoir of the theory of reciprocal
functions, is to the development of given functions of x in descending
powers or other forms which vanish when x is infinitely great; the
results of which may be applied to the valuation of functions of
very great numbers, and to a great variety of physical problems.
These series have also the peculiarity, generally, to terminate for the
functions of integer numbers.
116 Mb MURPHY'S SECOND MEMOIR ON THE
SECTION IV.
Inverse Method for Definite Integrals which vanish; and Theory of
Reciprocal Functions.
1. When the equation fif{t).t' = (p{x) is supposed to be restricted
to particular values of x, then whatever may be the form of (p {x),
J'{t) may always be determined ; the values to which x is restricted
we shall suppose to be the natural numbers 0, 1, 2, 3 (w — 1), and
the method here pursued will also apply if the values of n should be
different from those mentioned.
2. * First, let f,f{t).t' = 0, the limits of t being always and 1,
and let us seek for f{t) a rational function of t of the lowest possible
dimensions, which shall satisfy this equation when x is any integer from
to n — 1 inclusive.
Any value of f{t) which answers the proposed conditions may be
divided by the absolute term, and the quotient, it is evident, will
equally fulfil those conditions; we may therefore take the first or
absolute term in f{f) to be unity, and as the conditions to be satisfied
are w in number, we must have n coefficients in f{t), which will hence
be a rational function of the form
1 + Alt + A,f + + Ant";
and therefore (j>{x) = + — -^ + — ^ + + "—-^,
p
or = T^r by actual addition,
putting Q for (a; + !)(« + 2), (x + w + 1), and P representing a function
oi X oi n dimensions.
Hence P=0, provided x be any number of the series 0, 1, 2....(/i — 1);
these are therefore all the roots of that equation, P being of n dimensions ;
hence we must have
P = c.x.{x-l){x-2) {x-n + \)\
c representing a constant quantity.
* I have resolved this question in a different manner in the " Treatise on Electricity."
INVERSE METHOD OF DEFINITE INTEGRALS. 117
We have thus
1 Ai A2 An c,x.{x-\) ....(ar-w + l)
« + l "^ x + ^ "^ x + ^6 "^ x + n + 1 ~ {x ■\-\) .{x + ^)....{x -{-n + 1)'
Multiply by x + 1, and then put x= —\\ hence c = ( — 1)",
by ar + a, a:= — 2; -(4i= — - . — — ,
by . + 3, .= -3; 4=-"-^-^.^^^±ii^>;
&c &c.
1- j^/js -. « w + 1 ^ «.(«-l) (?i + l).(w + 2) .„ J
hence /(0 = 1- j •-]-• ^+ ^T^-^-^ H^ ^./^-&c.
dt" 1.2.3....W
3. Denoting by P„ the value of f{t) which has been investigated
in the preceding article, it possesses the remarkable property ; that
ftP„P„ = 0, except when m — n, and then
r p jj __ ^ .
•'' '" "~2w + l'
the limits being always and 1.
For when m and n are unequal, one of them as n is the greater,
P„ contains then only powers of t inferior to n, the integral of each
of which vanishes by the natvire of P„.
When m = n, the last term of P„, namely
(w + l)(w + 2)....2w ,
1 . 2 ....n ^ ^''
is the only term of which, when multiplied by P„, the integral does
• This value of _/(<) lias been shewn in the " Treatise on Electricity " to be the coefficient
of /j' in {1-2//. (1-2/)+/*^}-^.
Vol. V. Part II. Q
118 Mr MURPHY'S SECOND MEMOIR ON THE
not vanish ; and since in general
/■p.. _/_ix. ^-(^-1) ....{x-n + 1)
■'' " " ^ '^^ •{x+l){x + 2)....{x + n + l)'
it is evident that in this case ftPj
2n + l '
4. To illustrate the observation in Art. 1, with respect to the
generality of this method, let it now be required, to find a rational function
of t, as f{t), of the lowest possible dimensions, to satisfy the equation
fif{t).t' = 0, when x is any number of the series
p, p + 1, p + 2, .p + n-l.
Putting as before /(^) = \ + A,t + A-J'- + + Aj\ we have
/• f(f\ ft _ 1 , ^' , -^2 , -^-
■"•^^^' x + 1 x-^2 x + S x + n + 1'
the sum of all which fractions must by the reasoning of Art. 2, be
c . jx—p) {x—p—l)....{x — p — ti + 1) _
{x + l){x + 2){x + 3)....{x+p + x) '
and determining c, Ai, Ai, &c. in the same manner as in the Article
referred to, we have
1.2.3....W
c = (-l)».
(j9 + l).(jO + 2)....(jO + W)'
. _ n n+p+1
"^'~~1- p^\ '
. _ n.{n-\) (»+/? + l).(«+jP + 2)
'~ 1.2 • (^ + l).(;j + 2) '
&c &c.
and therefore
'^ ' 1 ^ + 1 1.2 (jo + l).(jo + 2)
t^ d^ j /t _ ^ , n.{n-l) ^ „ 1
~ip + l).{p + 2)....(p + n)-dt''-y -V 1-^+ 1.2 .^-&c.|;
INVERSE METHOD OF DEFINITE INTEGRALS. 119
or, putting l — t=^t', we obtain
J^' (jo + l).(jo + 2)....(jp + «)" r/^"
5. From this result it follows that if we put
then shall ftj'{t).t'' = 0, provided x is any number of the series
0, 1, 2 (n-l);
Op representing any constant quantity.
Now OpfP may be taken for the general term of an arbitrary function J^;
hence the most general function which satisfies the equation ftf{t)-t'' — 0,
is expressed by
.,,. _ d^jtH'T)
In fact we have (supposing the integrals to commence from ^ = 0,)
f,f{t) . r = t^f, it) - xt^-'f, {f)\X.{x- 1) ./s it), &C.
representing by fn {t) the ri^ successive integral oi fit), and putting for x
0, 1, 2....(w — 1) successively, it follows that
Mt) = 0, f,{t) = .f„{t) = 0, when ^=1;
that is,Jn{t) and its n differential coefficients vanish when t = and when
t=l; therefore y^ (/) contains a factor of the form ^".(1 — ^)", and con-
sequently the most general form of f{t) is
d"(t"t'''r)
dt" ■
6. Hence we deduce the following general property: '' If /{t) he
any function which satisfies the equation [tf{t) . t* = 0, a; being any integer
from to in — V) inclusive, then the equation f{f) = will always have n real
roots lying between and 1."
For the equation r.^'°F=0 has n roots t = and n roots ^=1; and
therefore f{t) which is the n^^ derived equation must have n roots be-
tween and 1.
q2
120 Mr MURPHY'S SECOND MEMOIR ON THE
Hence, if we suppose the equation J,f{t) .^ = to hold true for an
indefinite number of entire values of x, the equation f{t) = will also
have an indefinitely great number of roots all lying between and 1,
and the curve, of which the ordinate is f{t), and the abscissa t, would
intersect that portion of the axis of x, of which the length is unity
measured from the origin in an indefinitely great number of points;
thus we have a property characteristic of this class of functions.*
7. We have supposed J'{t) to consist of terms involving the
powers of t, but as we may proceed in like manner for any other
assumed form, we take the following as an example, because it leads
to some remarkable results.
To find a rational function of h. 1. (f) as y(h. 1. t) of the lowest
possible dimensions, which may satisfy the equation ftf(h.\.t).t' = 0,
X being any integer from to n—1 inclusive.
Put /(h. \.t) = \ + A, h.\.t+ A^ (h. 1. ff + + A„ (h. 1. t)',
and observing that J,{h.\.{t)]"'.f = {-\f. •^^■^•:;\ ,
we get f,f{\,.l.t).t' = ^^-j^^^,+~^^^^- ± -(^^nyr.T-.
and actually adding the fractions in the right-hand member of this
equation, the numerator which is a function of n dimensions, ought
to vanish when x is any number of the series 0, 1, 2...(w-l); that is,
{x + 1)» -A,{x + !)"-■ + 1 . 2^2 (a; + 1)""' - 1 . 2 . 3 ^3 (a; + 1)""'
= C.x.{x-\){x-^) {x-n->r\).
Let Si represent the sum of the natural numbers 1, 2, 3.,..(«-l), n,
Si the sum of their products two by two,
^^3 the sum of their products three by three, &c.
* Vide Art. (4) in my first Memoir on the Inverse Method of Definite Integrals.
/
INVERSE METHOD OF DEFINITE INTEGRALS. 121
Then by the theory of equations, the right-hand member of this
equation is equivalent to
c {(x + 1)' - s,{x+iy-' + SA^+iy-' - SA^+i)"-', &c.|
whence c = l, ^i = aS',, ^2=, „, ^3= J -^, &c. hence the required
function is
8. It has been proved, that the function thus obtained (which we
shall denote by L„) in common with all others which possess the
property that ftj'(t) .f = 0, when x is any integer from to n — 1 in-
clusive, is of the form
d\ {ft'" V)
dt" '
to verify this in the present case, we must sum the preceding series
which is represented by Z/„.
First, by the nature of multiplication, we have
hr + SJi^-' + S.h"-"- + +S„ = {h + \){h + ^) {h + n),
and the development of an exponential gives
i+7.h.l.(^+-A^ + + i.a.3..,:, +&c.=/-,
the coefficient of h" in the product of both the latter series is iden-
tical with that by which Z/„ is expressed.
But since that product =^(A + 1) (/« + 2) (A + w)
df
= ^{r(l+Ah.l.^4-^^l^^&C.)|,
it follows that the coefficient of h" is also expressed by
d" [f {h.\. ty\
1.2.3 ndf'
122 Mr MURPHY'S SECOND MEMOIR ON THE
this quantity is therefore the sum of the series which we proposed to
find.
Now the equation h. 1. {t) = is satisfied by ^ = 1 ; hence h. 1. t is
i' t'^
of the form t'. Q, {where Q= — (1 + - + — + &;c.)}, and therefore if we
Q"
put — ^ = J^, we get the value of L„ to be
d".{t''t''''F)
df
which was the formula we had required to verify.
We may also observe that since in the equation L„ = 0, / must have
n values lying between and 1, therefore h.l. {t), according to the powers
of which L„ is arranged, must have n real negative roots, which we
see confirmed by the positive signs of all the terms which compose L,,.
9. If we form the equation
u (1 — h h. 1. u) =t,
we have by Lagrange's theorem
. J..UW.N, ^' d{t\iA.tf ^ ¥ d'{t\\.\.tf ,
« = . + ;i.h.l.(0+^.-^^^— ^ -f-^-^3.-A^^+&c.
from whence it appears that Li„ is the coefficient of h" in the value
of -^. Similarly if in Article (12) we form the equation
u \\ -h. (1 - u)] =/,
du
we have P„ = the coefficient of h" in -rj .
10. If Q„ i<? the coefficient of h" in -j-, supposing u to he deter-
mined by the equation u{l — hU) = t, U bei)ig a function of u which
vanishes when u = l, and T the same function oft, then shall
j,Q„f ^ x.{x-\){x-2) {x-n + 1) ,
j/F'T 1.2.3 n '^ ''
INVERSE METHOD OF DEFINITE INTEGRALS. 123
For if we put ti = in the equation u{l — hU) = t we get 1 = 0,
and putting u = 1 we have by supposition U = and therefore t = I,
hence the limits of u are the same as the limits of f.
But j;Q„f = the coefficient of h" in f^^.f,
^i U/t
and l^^.if = JJ^ = !„u''{l-hUr
expanding the part under the sign of integration, and taking the co-
efficient of h" we obtain
hHnt - 1.2.3 n -(-l)^^ ■^-
11. If U he a rational and entire Junction of u which vanishes
when u = \, and if Q„ be the term independent of u in the product
U"- \\— —\ , then shall Q„ be itself a rational and entire function of
t possessing the property of ftQj'^ = 0, x being any integer from to
n—\ inclusive.
For it has been proved in my former Memoir on the Resolution
of Equations*, that the root of the rational equation <f){x) = is the
coefficient of - in — h. 1. ^— , hence the value of u in the equation
u(\ -hU)=t, is the coefficient of - in -h.l. j(l--] -hul, and
differentiating, it follows that the value of -tt is the term independant
of u in
u)
* Camb. Trans. Vol. iv. p. 131,
124 Mr MURPHY'S SECOND MEMOIR ON THE
because the u under the logarithmic sign is the same as if we had
placed there, a or any arbitrary symbol, and is therefore treated as a
(jLu
constant in the differentiation; hence the coefficient of h" in ~t- is the
term independant of m in
(> - -:)
n + i '
that is, its value is Q„, and therefore by the preceding Article /Q.r
vanishes between the limits of x, and n — \, its general value being
T being the same function of / that U is of u.
By this theorem, every possible variety of rational and entire func-
tions which possess the above-mentioned property may be found, as in
the following
Example:
To find a rational function of t, in which the powers of the variable
are in arithmetical progression, such that jiQ,nt'=0 when x is any number
of the series 0, 1, 2 {n — 1).
In this instance put U = 1 — u"", m being any positive integer.
Hence Q„ = term independent of u in
/ t\ "*""*"'*
(i-^o».(i--)
^ « (w + l)(w + 2)...(w+m) w. (w-l) (M+l)(w+2)...(?i+2OT) ^,„_.
1' 1.2...m ■ 1.2 ■ 1.2, ..2m
in which if we take in particular m =1, we get the value of P„ before
found in Art. (2).
This formula for Q„ may be written in another form by which it
will comprise the case where /w is a fraction, thus
n (m+l)(m+2)...( m+n) ,„, w(«+l) (2»^+l)(2w^+2)...(2w^-^w) „ „
^-=^-i- r^:::^ -^ ^"ttt- t:2::ji -^ "*'''•
INVERSE METHOD OF DEFINITE INTEGRALS. . 123
and it is, moreover, evident that either of those values are identical with
1.2...nde' '
which is included in the general form given in Art. 5. viz.
d\ {ft'" V)
dj" ■
12. 2'o find a rational and entire function of f of h dimensions,
which if multiplied hy a rational and entire function of t' of less than n
dimensions, the integral of the product may vanish between the limits t =
and t=l.
Let the required function be represented by (p, q),„ so that
{p,q\^l + A,t^ + A,f-f + A,,f^,
and by the proposed conditions we must have
lAp, qXt-" = 0,
ni being any integer from to « — 1 inclusive, put t^ = T, the limits
of 7' are the same as those of t.
Hence J^ip, ?)» T~^~' ■ T''= 0.
1-1
Now ij), q)„ T" , is a function of T of which the indices are in
arithmetical progression, - being the common difference, and T' the
first term ; and as the nature of the question affords m independant
equations for the determination of the n coefficients Au A-,...A„, it
follows that there is only one function of the kind, which will satisfy
the proposed conditions, and by Art. 5, it is evident that the function
5 2'"''' (1
jAyi
1 1 ,A/ 1
n+ 1 w+- -
\ q l\ q
..)...i'
' 1
answers those conditions, and is manifestly of the required form, it
Vol. V. Paet II. R
126 Mr MURPHY'S SECOND MEMOIR ON THE
l_i
follows that if we divide this function by T"^ , and then substitute f
for T, we shall obtain the value of {p, §-)„ ; we have thus,
_ ip + \){p + l+q){p^-\ + 2q)....\p + 1+{n-l).q] n
ia+g){l + 2q)....{l + (n-l).g\ l'
, (2p + l)(2p + l+q)....{2p-i-l + (n-l).q} n.( ?i-l)
l.{l+q)....{l+{n-l).q} ' 1.2 ^ .'^^•
13. The functions {p, q\ and (5-, jo)„ may be termed reciprocal func-
tions, and possess the remarkable property, that if n and «' are any
different integers, then shall
ft(p,q)n.{q,p)n' = 0.
For if n>n' then {q, p)„' is a rational and entire function of t^ of
less than n dimensions, and therefore by the preceding Article the
integral of the product must vanish ; again if n' > n, then {p, q)„ is a
function of f^ of less than n' dimensions, and therefore when multiplied
by (q,p)„' the integral ought to vanish.
To determine the value of the same integral when 71 = n', it is
evident by the nature of the function {p, q)„ that we need only attend
to the last term in the expansion of {q,p)n, namely
. {nq^l){nq + l+p)....{nq^-l + {n-\).p}
^ "->•'' 1.0.+p)....{l^{n-l).p}
INVERSE METHOD OF DEFINITE INTEGRALS. 127
Now if we put for {p, q)^ the series assumed in Art. (12) aiid multi-
plying then by f, integrate from ^=0 to ^ = 1, we have
ar + 1 a;+j9 + l a; + 2/> + l "*" x-\rnp + l
and actually adding these fractions, the denominator of the sum is
{x + '\){x +p + l)(;r + 2jo + 1) {x + np + 1);
and since the numerator is of n dimensions in x, and vanishes when
x = 0, q, 2q....{n- I) . q,
it follows that the sum is of the form
c .X . (x — q) (x—2q)....{x — {n — l).q]
{x + l).{x+p + l)....{x + np + l)
Multiply by ^ + 1 and then put x= —1; hence
^^ c.{-l)\l.(q + l)(2q + l)....{{n-l).q + l} ^
p . 2p . 3p....np
whence deducing the value of c, and substituting in the above integral,
we obtain
^'^^'^^''•^^^~P^'''l.{q + l){2q + l)....{{n-l).q+l\
^^ x.(x-q)(x-2q)....{x-{n-l).q}
{x+ l){x +p + l)....{x + np + l) '
hence y;(^, g)„ .^"^ = (-^y . ^ ^^^ ^| ; ^^^^^ ^^ ^y
nq (nq - q) [nq — 2y) .... \nq - {n — 1) . q]
{nq + l){tiq+p + l)....(nq + np + l)
from whence we obtain finally
n" 1 . 2 . 3 . . . .?{
f^ ip, q)n {q, p)., = „(^ + ^) + i • i(^q^l)„..{{n-.l).q+l\
nq(n q — q) {nq — 2q)....{nq—{n — l). q}
''~'Up + l){2p + l)....{{n-l).p + l} ■
R 2
128 Mr MURPHY'S SECOND MEMOIR ON THE
that is, it
_ {pqY V .^' . 3'
n{p+q) + l'l.{p + l)(q+l){2p + l){2q + l)....{{n-l).p+l\{{n-l).q + l\
, 2 _ y f 1.2.3....W 1^
COR. j,(p,p}n- 2n + l-\l.{p + l).{2p + l)....{{n-J).p-i.l}f-
14. To find the reciprocal function to that denoted by L„ in Art. 8,
, d" {f (h.\. ty\
^' 1.2....W df '
namci
L„ consists of the powers of h. 1. /, and possesses the property of
ftL„t' = when x<n; suppose now that we investigate a rational function
X„ which shall possess the property JtK {h.l. t)' = when x<n; then it
is evident that j^X„i„/ = when n and n are unequal; and therefore they
are reciprocal functions.
Put K=l + AJ + A,f +....AJ'',
Put Ar = 2" + 'B„ A, = 3'' + 'B, A„ = {n + lY^\B,r,
hence we must have when x<n,
1'-' + a"-'^, + 3"-"^2 + (« + i)"-^jB, = 0.
Now the left-hand member of the equation is the same as
putting t = after the differentiations.
Hence the differential coefficients from the 1" to the w* inclusive
of the function between the brackets vanishes when ^=0; that function
of e' ought therefore to contain no power of t inferior to the (w + 1)"',
and conversely, a function of e* which does not contain such a power
of t, will fulfil the required conditions.
INVERSE METHOD OF DEFINITE INTEGRALS. 129
Now this is the case with (1 — e')''''"^ which is also when expanded
of the same form as the part between the brackets; hence equating like
terms, we have
Hence A,= -\.T, A,= '^^^^ .S" ^„ + , = (- 1)". (« + !)";
and therefore
X„ = l-p2"^+'^^.3'7^- (-1)". (« + !)«. r.
Cor. 1. When ?^ and n' are unequal, then ftL„'\,„ = 0.
But when Ti'=fi, we need only take the last term of L,„ namely, (h. 1. /)";
hence
j;x„z>„ = j;(h.i.^)"{i-^.2"^+'i^j^^.3"^^-&c.|
= (-l)..,...S....„{.-f.l.^^).l-.e.}
_ (-l)''.1.2.3....w
~ ft + l
Cor. 2. j;x„(h.l.^)^'
= i - ly .1 . 2 . 3....X ll'-^-' - n Q.""-' + ^~^ .3"-"-' - kc.\
= ( - 1)"- M .2.3 ...x A" . (A"-*-'),
h being put = 1 after the operation of taking the «* finite diiFerenc<»
on the supposition that the increment of k is unity ; from whence it
is easy to deduce
^^■^' = <-')-'^--^-
Cor. 3. All the roots of the equation X„ = are real, and lie between
and 1.
For if we put h. 1. (/) = it, and X„e"= U,
then ;x„ (h. 1. ty = f. Uu^ = ti^f^ U- xw-^f.: U+ ^4^^ // U, &c.
130 Mr MURPHY'S SECOND MEMOIR ON THE
and putting x = 0, 1, 2, &c. successively, it follows that fu"U and its (« — 1)
successive differential coefficient vanish when u = and w = - oo , Hence
U=0 has n real negative roots; and therefore X„ = has n real positive
and fractional roots.
15. In general let U,„ V„ be any functions of the variable t and the
integer n, and let A-^.-.A,,, ai...a„ represent constant quantities; or de-
pending on n only.
Put T„ = C/„ + A,U, + A,U, + .... + A^U„,
and T:= K + «i^> + «-.F, + .... + a,r„.
Then the n equations
j;r„r„=o, f,T„r,=o, j,t„v,=q ;r„r;_,=o,
Avill serve to determine the constants A^, A.,....A„.
In like manner let the corresponding constants «i, a2....a„ be de-
termined from the n equations
the functions T„ and T„' which are thus determined, are reciprocal func-
tions, and possess the general property ft T„ TJ = 0, except when n - n',
and then
ft 2\ T: = aJtT^K = A,, ft T: C7„ ;
this is the general principle of reciprocal functions.
Cor. Let f{t) be any function of t represented by the series
f{t) = c,T, + c. 2\ + c, T, .... &c.
where Co, c,, Cg, &c. are constant coefficients to be determined, then
multiply by T^, T(, T~U &c. and integrate the successive products,
and we get
c,ftT,Tl = ff{t)T^,
c^ftT.TI^ ftf(f).Tl,
c.fT,T^ = f,f{t).T.I,
&c &c.
by means of which equations the required coefficients are given.
INVERSE METHOD OF DEFINITE INTEGRALS. 131
16. Let «„, h„, c„, &c. be any functions of t, the reciprocal functions
to which for simple integration are «„', J„', c'„', &c.
Let a„, &c. be any function of another variable T, and let a/, &c.
represent the corresponding reciprocal function.
Put S„ = a„a^ + Kai + C^a^ +
and S,! = an'uo + i/a/ + c„'a.2 +
then S„, Sn are general forms for reciprocal functions with respect to
the double integration relative both to t and T.
For if we put m for n in the latter series, and multiply the series
for S„ and S,„' together, the integral of the products of any two terms
which do not hold the same place in either series when taken relative
to T must vanish, since a„, a„' are reciprocal functions.
Hence frSnSJ = a„a„' fj.aoaa + b„b,„' fraiai + c^cj frO^a./ +
Integrate now with respect to t, observing that when m and n are un-
equal, then
_^ «„«,„' = 0, ftKbJ = 0, ftC„c„' = 0, &c.
Hence /_4*S'„«S',„' = 0, when m is not equal to n,
and ftfrSuSn = ftfr {a„a^aoa^ + Kb„'aiai' + CnC^'a-^a^ +...].
Cor. 1. In the same manner reciprocal functions of any number
of independent variables may be formed.
Cor. 2. The equation S„ = has n real roots or values of t lying
between and 1, whatever value be assigned to 7', when a„, b„, c„, kc.
are functions possessing the property ftaj' = 0, &c, x being any integer
from to w - 1 inclusive ; for then «„ must be of the form — —jj-„ — - >
by Art. 5, and similarly
, d\{t'-t"'V') _ d\{t^t"'F")
"~ dt" ' ^"~ df '
and therefore
Hence »S',=0 must have n real roots between and 1. (Art. 6.)
132 Mr MURPHYs SECOND MEMOIR ON THE
17. If it is necessary that the terms which compose the reciprocal
functions S,„ S,! should follow a simple law, it will be most convenient
to get first two reciprocal functions of t, as R,,, R,', which may contain
an arbitrary constant r, and to put for «„, J„, c„ &c. the values acquired
by R„ when r = 0, 1, 2, &c. ; and similarly for «„', i„', c,,', he. the cor-
responding values of R\.
Example :
Thus, put R^^iiff'"^-^, and RJ = {ttf'" '^, P„ being the
d" (tt'Y
function so denominated in Art. 3, namely, — ^,^ ; then, integrating
by parts, we have
the part outside the sign of integration vanishes between the limits of
/, and repeating the same operation any number of times, the part out-
side the sign of integration is evidently of the form
dt'-" ' dt"-' \ dt'
the latter differential coefficient will vanish between limits when k is
any number from to r inclusive, because it will always contain the
factor {tt'Y~'"^^ ; also when n and m are unequal we may suppose w to
d'P„
be the greater, and since ft'' — j-^ is of in + ;• dimensions, it follows
that if k> n + r, then k -1> in + 7-; and consequently the latter dif-
ferential coefficient will be identically zero.
^*-i / d' P
The only instance in which the factor , .._, iff' , .'" j does not
Aanish between limits is, therefore, where k lies between r + 1 and r-\-n
inclusive, but then the first factor is changed to ft'''P„; and since k — r
is now some immber from 1 to « inclusive, this factor vanishes between
limits (vid. Art. 5.), and therefore the part outside the sign of integration
vanishes in all cases, and we thus obtain ,
f,R.R.„ -(-1; j^-^^,-^.-^[tt -^jr)'
INVERSE METHOD OF DEFINITE INTEGRALS. 1S8
Put now h = r, the first factor under the sign of integration becomes
simply P„, and the second factor is then of m dimensions; and there-
fore, by the nature of P„, the integral vanishes; and therefore, when
n>m, ftIl„Bm' = 0: and the same reasoning applies when m>n, only sub-
stituting RJ instead of R^ throughout the process, hence R„ and R,„' are '
reciprocal functions.
When m = n, then in the general expression
j;R«.' = (-irj;p.^(«-^);
we need only take the term involving the highest power of t in
dr K^ dr )'
namely,
/ ,.... (« + !)•(« + 2)...(2w) d^ (..rd-.n
^ ' 1.2... n dr \ dt' 1
. ,, , (« + l) . (« + 2)...2« , . , ,^ , ,x ,
and observing that /JP„#" = ( — 1)" . -, .,, , — ''" /_ — -^. ;
it follows that ftR„R,!=- . {n + r) {n + r-1) {n + r-2)...{n-r).
The reciprocal functions a„, a„' may be obtained by putting r =
in R„ and RJ ', similarly, if we put r = l, we get b„, b„', &c., and thence
we obtain the reciprocal functions relative to double integration, namely,
dP d^P d^P
S,'=:ao'{tt'Y ^n + «.'(«')^^'^ + «^'(«T^"^-f" + «3'(«')^-"^", &c.
In the same manner if we vary the constant a while r remains constant,
we obtain the reciprocal functions
Vol. V. Part II. S
184 Mh MURPHY'S SECOND MEMOIR ON THE
Cor. 1. The simplest form for a„ is the sine or cosine of the w'"
multiple of an arc of which the limits are and 2w7r, as
A„ sin (2 nicT) + B„ cos (2 wtt T),
where A„, B„ are arbitrary constants, then we have (putting for sim-
plicity a = 0),
S„ = A,P„ + {A, sin ^TTT + B, cos ^-n-r) -^
+ {Ai sin 4 TTT + Bi cos 4 ttt)
dt
■i >
this is the most general form for all the reciprocal functions which occur
in the Mecanique Celeste. (Vid. Prop, xi. Treatise on Electricity.)
CoK. 2. If T„, T,' are arbitrary functions of t, which do not become
infinite when ^=0 or 1, then, putting
Rn = {tt'f Tr*^, and R,: = {tt'f T; .^ ,
the same reasoning as that used in the preceding example will show
that R^, R„' are reciprocal functions, and thus we get for a^^, aS",,' the
very general forms
S„ = «„ T,P„ + «. y. ^ («')* + «^ T^ -^ m + «3 T, ^ {tt'f + &c.
S: = a„' 2;'P„ + a/ T; "^ {tt'f + a.: T^ ^ {tt') + ai Ti ^{tt'f + &C.
Cor. 3. If f{t, t) is any function of the variables /, t, which is ex-
panded under the form
f{t, t) = a,S^ + a,Si + a^S; +
then, to determine the coefficients a^, Ui, a-i, &c., multiply successively
by So, Si, SJ.... and integrate from t=0 to t=l, and from t = to
T = 1 : we thus get
do ft fr So So = ftfTSo'J'{t, t),
aiftfrSiSi' = ftf,Si/{t, t),
aJJ^S.,S.; = f,f^S./f{t,T);
from whence the required coefficients are known.
INVERSE METHOD OF DEFINITE INTEGRALS. 185
SECTION V.
Inverse Method for Definite Integrals which are expressed in positive
powers of x, or under any form.
18. Let <^{x) represent any function of x, such that Stf(Jt) .f = (p{x)
when X is any integer from to n — 1 inclusive, then excluding the
case of (p {x) = 0, which has been considered in the preceding Section,
it is evident that by putting
f{t) = A, + A,t + A,f + +An-,tf-\
the conditions of the question give n equations, which suffice to de-
termine the coefficients A^, Ai, A^, A„.^\ if we represent the
particular value of f{t) thus deduced by T„^i, and seek its most
general value, we have
;/(0 .t^ = <p {x),
.-. f,{f(t)-T„.,}.t^ = 0.
Hence by the preceding Section, the most general value of f{t)— Tn-i is
dt- '
and therefore the most general value oi f(t) is found by adding this
appendage to its prime value T„_i.
19. When <p{x) is a rational and entire function of x, of m di-
mensions, we have by the proposed conditions
'P^'^'' x+l^ x + 2^ x + 3^ x + n'
and actually adding the terms which compose the right-hand member
of this equation, the common denominator is
(x + l){x + 2) (x + n),
s2
136 Mr MURPHY'S SECOND MEMOIR ON THE
and tlie numerator will be a function of ft — \ dimensions, represented
by v„, so that
v„
<p{x)
{x+ l)(ar + 2) (a; + M)'
when X is any integer from to (^^ - 1) inclusive; and if we multiply
by a;+ 1 and put x= — 1, and again by a; + 2 and put a-= — 2, &c. as in
the preceding Section, we get
A,
V -
1
1
.2.3..
..{n-
•1)'
A,
= - •
n-\
1 *
1.2.
V.
3...
-2
-1)'
4
_(^
-1)(«
-2)
«_S
'~ 1.2 ■ 1.2.3....(« - J)'
&c.= &c.
Now the equation
^{x) . {x + 1) (a; + 2) {x + n)- v„ = 0,
is of m + n dimensions, and is by hypothesis satisfied, when
^• = 0, 1, 2, («-l);
therefore if u^ represent some function oi x of m dimensions, we must
have the identity
(p{x) .{x + V) (ar + 2) (ar + «)-», = M,.ar . (^-l)(ar- 2) (x-n + \),
hence if we divide
<f>{x){x + \){x^2) (x + w) by x{x-'\){x-^) {x-n^l\
and retain only the part of the quotient which is an entire function of x,
u, will be completely determined.
Put now —1, —%,...— n successively for x in the preceding identity,
and we get
t;., = (-l)»+M .2.3....».«_,,
INVERSE METHOD OF DEFINITE INTEGRALS. 137
«_, = (- 1)"+'. 1.2.3....n/-~-.u.2,
&c. = &c.
from whence the values of A^, Ai, A-z, &c. are known, and being sub-
stituted, give
J .M_2^
n.(«.H)(.-f2) («-l)(.-2) I
^ 1.2 1.2 ' J
Example :
Let 0(a;) = 1, then «, = 1, and therefore
«.(« + l)(w + 2) (w-l)(/?-2)
"*" 1.2 ■ 1.2
.f-&e.|
20. The function Tn-\ possesses a property analogous to the charac-
teristic property of those in the former Section, that is, the equation
2\_^ = admits of n — m-l roofs between and 1, and consequently
vanishes an indefinitely great number of times between the limits / =
and t=\ when n is taken indefinitely great.
For since r„., = (- 1)-' |«M., - ^^^^jtil . ^V «., ^
n{n+l){n+2) {n-l)(n-2) ]
■^ TTa • 1.2 •«-3^&c.j
_ (-1)- ^\t^(u -VlzI u 1 1 (»-i)-(^^-a) „ t. .,„^l
= i.2.3....(.*-i)-rfrr \ ' 1 ^ ^'^^ i:% -"-a^-^c.JI
= -r 2.3..U-i) -£^^'^"'"-^^"'^-
138 Mr MURPHY'S SECOND MEMOIR ON THE
tlie operation A being performed on the supposition that the finite
increment of x is unity, and x being put =1 after the operation A""'
has been performed.
Put i=l — f, and therefore,
A"-'(M_,r-') = A"-'M_,-^A'-'M_.(a;-l)+— — A"-'M_,(a;- 1) (a--2)-&c.
and since m_^ is of m dimensions, the first term of this series which does
not vanish is
ftn-m-\
- 1 ■ 2.-(«-/»- 1) •^""'"-^^'^~ ^^ (^-^)--^^ -n + m + l),
and therefore the whole expansion is of the form
t'"-"-' r, 1.2.3 (w-l),
which being substituted gives
_ d'{t''t"'-'"-T}
and since the equation t''t''""'-'^F'=0 has at least 2n-m—l real roots,
viz. ti of them =0, and n — m—l of them = 1, it follows that the w""
derived equation T„ = has n — m—l real roots lying between and ] .
COK. Since r„_. = ,.,.3.1(,_^) • ^ {r^-^u.J-^},
if we actually differentiate we get
^-^= 1.2.3.!..(«-l) -^""'^^-^-^+^>—^^ + "~^^"-^"'^-
21. Let now <l>{x) be any function whatever, and let it be required,
in general, to find J'(i), so that ftj'it) ■ f = ^{x), provided x be any
integer from to w — 1 inclusive.
It has been shewn in Art. 18, that a function T„.i of w— 1 di-
mensions may always be found to satisfy the imposed conditions, and
for the most general value oi f{t) we shall then have
INVERSE METHOD OF DEFINITE INTEGRALS. 1^9
*
Now 7'„-i contains only n constants, being of » — 1 dimensions, and
therefore if we denote by P„ the same quantity as in the preceding
Section, namely the coefficient of h" in
{1- 2h{l-2t) + h:'}-i,
we may put
T„.i = ttoPo + a^P, + (hP2 + + a„_,P„_i,
the right-hand member being of the same dimensions with the left, and
containing the same number of constants.
Now by the properties of P„ we have j;P„P„ = 0, when m and « are
unequal, and
2« + l
Hence we have fiP^T„_.,= «„
Hi
Jl'* 2 -* n-1 — "^ •
But by the conditions of the question,
jc being any integer less than n.
Hence
j;P„7;-i = ^r„_i = 0(O) = (f>{h) when h is put =0,
iP 7'„_,=j;2;_, (1 - 2o=0(o)-20 (1)= - A ^^y^ .<^ (A),
140 Mr MURPHY'S SECOND MEMOIR ON THE
and generally
i;p.r..,=;r._,{i-f.^.*.'-ti^.<?!^<|±?l.f-&c.)
= (-l)'"A"'.^^ '\ ^ ' ^^ '-.d){h). When h is put =0.
' 1 . 2 m ^ ' ^
and by comparing the former integrals with the latter, the values of
ffo, «i, a-i, &c. are known, and being substituted give
T._, = P,0(/^)-3P,A^.0(A) + 5P.A^^^±^^^±^.0(A)
J. • <« • t7
// being put =0, after the operations are performed.
It should be observed here that the terms of this expansion are
perfectly independant of «, which only fixes the number of the terms;
hence this series may be continued to any number of terms, and we
shall always have ftT„.it^ = (p{x) provided x is any integer less than that
number, and consequently if the series be continued ad infinitum, the
equation will be true for all integer and positive values of x.
Cor. Multiply both sides by if and integrate from t = to /f=l,
hence «^ (^) = ^ • <^ ('') + ^ . ^^^j;;^^^^^ A ^ . A
+ ^-(x + l)(ar + 2)(x + 3)^ 1.2 '?>^ + *'C-
when /* is put =0.
This series may be used, not only for the integer and positive
values of x, but for any values which will not render it divergent.
(Vid. First Memoir, 'On the Inverse method of Definite Integrals,'
Art. 2.)
INVERSE METHOD OF DEFINITE INTEGRALS. 141
22. When 0(a;) is given we may obtain f{t) in an infinite variety
of forms by means of the theory of reciprocal functions given in the
preceding Section. For instance, if we denote by S^ the sum of the
products of the natural numbers 1, 2, 3. n when taken m and m
together, and put
i.=i+«.h.i,^.j«^.(h.M.+ ^.(h.M-+....+ r^.ch.ur
,5!lM!!iM. (Art. 8. Section IV.)
and \„ = l-?.2"/+^4^^.3"f- ±{n+l)''t
= (-l)"A"{(A + l)''^*}, when h is put =0,
then L„ and \„ are reciprocal functions. (Sect. iv. Art. 14.)
Put therefore y*(^)=aoZ/o + «iZ/i + a2Z/2+a3i3 + &c.
and observing that
1.2.3....W
ftKL„ = {-lY.-
w + 1
we have «„ = ( - 1)" . ^ ^"^ ^ . ftf{t) . X„.
But jl/{t) . \ = ftf{f) . ( - 1)'. A" . (A + 1)" . t. When h is put = 0,
=(-i)"A"(a + i)»j;/(o.^
= (-l)».A".(A + l)«.0(A), since ft/{t).f = (p{x).
Hence «„= ^ ^^ — - . A» . (A + 1)" . (k),
and therefore
f(f) = Lo(p{h) + 2Li — ^ ^' r ^.gjr,^ — i 2 +^-^^- — i 2 3 '
Vol. V. Part II. T
I
142 Mr MURPHY'S SECOND MEMOIR ON THE
which series when convergent will satisfy the equation jtf{t) . f' = ^ («)
for all values of x\ but even if not convergent, it will satisfy that
equation for all the integer values of x from to n — \ inclusive,
provided it be continued for at least n terms.
If we multiply by f and integrate as before, we get
which series when convergent may be used for any value of x, but
only positive and integer values when divergent.
23. In Art. 21. when ftf(t).t'^(p(x) a given function of x, we have
found y(0 in a series expressed by functions of t of the same nature
as P„, now P„ is only a particular value of the general function (jo, q)„
investigated in the former Section, Art. 12., namely, when p = q = l; we
shall now express /{t) according to this more general class of functions,
that is, under the form
fit) = «o ip, q)o + «i (p, q)i + «2 {p, q)2 + &c.
Now in Art. 12. above referred to, we have found
, . _ {p + l){p + l+q)....{p^l + (m-l).q] m
Kp,qh-i l(l^q)....{i^{m-\).q} l"^
(2p + l)(2p + l+g)....{2jo + l + (?»-l).g} m.jm-l)
■^ i.(\+q)....{\ + {m.-l).q} ' 1.2 '^ " *''•
To simplify this expression, put
77 = (/>^ + l)(M + l+9)--{p^ + l+(^-l)-g}
'■'' l{l+q)....{l+{m-l).q}
Let yj^ express the operation of changing h into h + 1 (Vid. former
Memoir, Note B. 2.), >//^ the repetition of this operation a second
time, &c. ; the preceding series will then become
INVERSE METHOD OF DEFINITE INTEGRALS. 143
{p, 9),„ = H,r - f . ^H,,f" + ^^^ >\^^H,.r
on the supposition that we put h = after the operations above indicated,
are performed.
Separate in this expression the symbols of operation and of quantity,
and we shall obtain the equation
(p,q),„ = (l-fr.H,J'':
But \U — 1 or \^ T- x//° indicates that we must subtract the original
value of Hp,q, from the value it receives when h + 1 is put for h,
that is, it is the same as performing the operation A of finite differences ;
this consideration transforms the preceding equation, to this
(p, q)m = (-iy" A" . Hf.qt"", when h is put =0.
In like manner if we put
„ ^ i,qh-\-\) {qh + l+p) {qh + 1 -\-{m-l) .p}
"■' 1(1+^) {\ + {m-\).p}
we have (g-, jo)„ = (-l)"' A'" Jf^.pi?"', when A = 0.
Now observing that by the nature of reciprocal functions we have
S* ip, q)m (q, p)n = 0, except when m = n,
and by Art. 13., fi{p, q\{q,p)^
_ ip, q)'" 1.1.2.2.3. 3 .m . m
~ 1 + mip+q) '1.1. (l+ju)(l + 9)(l + 2^)(l+29)...{l + (»w-l) .p} {l + {m-l).q} '
then since f{t) = «„ (p, q)o + «i (p, q)i + a, {p, q)^ + &c.
we have ftf{t) . (q, p)„
_ (pq)'" 1.1.2.2. 3 m . m
"**"'• l + m(p+q)'l.l{l+p){l+q) {l + im-l).p} {l+{m-l).q} '
t2
144 Mr MURPHY'S SECOND MEMOIR ON THE
But if we put for (q, p)„ the value above found, and observe that the
operations A and fi are with respect to different variables h and t, and
therefore their order is transmutable, we have also,
= {-iy A"^ H,,p<p{qh), by hypothesis.
Comparing this value of the integral with that already found, we get
'" ^ ' {pqT I'l" 2 ■ 2 • 3 • 3 ■■■
l + {m-l).p \ +{m-\) .q
"mm
X A" JZ", p {qh), when h = 0,
from whence we have finally
At) = {p, q). (qh) - ip, q), . ^+f/^ • T ' T " ^ ^V. <l> W
, , 1 + 2(0 + 0) 1 1 1+p 1+q .,„„ , ,
_(« «N l+^Ci>+g) 1 1 l+£ l+i 1±2£ 1+22 A3 W^'" ri.r«M
^^'^'°- (pqf ri- 2 •^^- 3 •—^■^■^'>f't>'^W
+ &c &c.
h being put =0, after the operation, and H', H", H', &c. being the
values of Hp,, when m = \, 2, 3, &;c. successively.
Cor. 1. Multiply by t\ and then integrate from ^ = to ^=1; for
Itf{t).t' put its value <p{x), and for ft{p,q)mt'' its value
/ ,v„„,„ 1-2. 3. .-^^ xix-q)...{x-{m-l).q\
^ 'P ■ 1 (1 + g)(l + 2y)...{l -!-(»«- 1| .^)*(a;+l)(a;+jt) + l)...(a; + »w^+l)'
INVERSE METHOD OF DEFINITE INTEGRALS. 145
by Art. 12 ; and lastly, put for H,,^ its value
(gA + 1) {qh + 1. +i)). . ..\ g^ + l+ { m-\).p\ .
1.(1 +;?).. .|i+(»w-iy:jo"i
we thus obtain
+ »(^- g> L+a(i>±2) ^. (^ ^ (^ (^
(a: + l)(a;+jo + l)(a?+2^ + l) Sg'' '^ '^^ ^ /rv"/ /
x(a; — 5')(a; — 2^)
■*" {x + l)(a:+jt> + l)(ar + 2/> + l)(2 + 3ja + l)
^ ^ 1^.2^^^ ^'^^^' + ^^^^^ +-^ + ^^^^^ + aja + 1) {qh)
+ &c. when A is put = 0,
and where /> and q are perfectly arbitrary.
Cor. 2. Put ^ = ^ = 0, and make
where ^(0), 0'(O), 0"(O), and the values of ^{x) and its successive
differential coefficients when a; = 0, and the above expansion will become
</>(x) = ^„.^ + ^,.^^-., + ^..^3 + &c.
If, moreover, we put
rr, , « , , , W.(»-l) (h. 1. O'' „
which is the same as A„ when we put f for ^ (a;), then it is easily
seen by the principles of the first Memoir, that jj Tj' = -, r-r—r , and
/ r r , ji n (a; + l)"+''
since we have also fij'it) . /^ = {x), it follows that
146 Mr MURPHY'S SECOND MEMOIR ON THE
24. The functions which have been all along designated by {p, q)„ and
{q, p)„, have been already shewn to be reciprocal one to the other; putting
p = q, the resulting function {p, p)„ must be reciprocal to itself; that is,
ft{p, p)„{p, p),„ = when m and n are unequal positive integers; when
p = l the function {p, j)),, is then identical with that denoted by P„ , which
has been before shewn to be reciprocal to itself; again, the function T„ or
n n.(n~\) {hA.ty ^ n.{n-l) .{n-^) (h. 1. If
is reciprocal to itself, for if we mviltiply by (h. 1. ty, and integrate, we get
j;r„(h.l.0"' = 1.2.3...«.(-ir{l-f^-^!^ (^±i)^_&e.}.
The expression between the brackets is the term independent of h in the
product (1+/^)"(1 + t) , or the coefficient of //-('"+'> in (1+A)"-'"-';
it is therefore zero when n>m, but when n = m its value is ( — I)'",
and when n<m, its value is
, _ (?w + l -n){m + 2-n)...m
^~ ' ' 1 .2 ...n '■
Hence fi T„ 7'„ = 0, when m and n are unequal, and
1.2. ..n
25. Put h. 1. (^) = T, and substituting in T„, we have
1.2...W J'„e'^ = e"|l.2...M + w.2.3...Wx+^^^^^\3.4...WT^ + &C.|
(dw c?"-'t" n.(n-l) d"-W „ 1
_ d"{e'^r'')
„ _, 6-^d" (e-T")
Hence 7; = -^—p^ t-^ .
1.2. ..war"
INVERSE METHOD OF DEFINITE INTEGRALS. 147
From , this formulae it appears that the equation T„ = has n real
values of t all negative; and therefore n corresponding values of t,
which are all included between and 1.
Moreover, if we form the equation
u = T + hu, or u
1-h'
it follows by the theorem of Lagrange, that T„ is the coefficient of h"
de" e'~*
in ^'^•-j-> that is, in - — y, and putting t for e% T„ is clearly the co-
h
efficient of h" in the expansion of the function y .
Conversely, we may now prove that the coefficient of h" in the ex-
h
pansion of - — - is a reciprocal function; for when h = 0, this function
A ""■ ft
is reduced to unity, we may therefore put generally
= ro+T,^+T,A^ + &c. where T, = \.
A
fX-k
1-h
Let h' represent any other arbitrary quantity, and we have
1-h
j= T,+ T,h'+T,h" + &ic.
Multiply both series term by term and integrate, the result in the
left-hand members is
{i-h){i-h')^' ~ i-hh"
/which expanded becomes 1 + hh' + h^h'^ + kc.; which being identical with
the integral of the product of the right-hand members, will necessarily
require that the integrals of those terms which are not in corresponding
places in both series must vanish, and the integrals of the products of
the corresponding coefficients to be unity, which are the same properties
that have been demonstrated in Art. 24.
148 Mr MURPHY'S SECOND MEMOIR, &c.
Cor. Put .; — r = ^» and the series
\—n
t~^ = (\~.h){ T„ +T,h+ TJi^ + &c. \ becomes
^^=^+ ^■•r4T^+ ^-7;:ttv3 + *'^-
The principles which have been used in this Section to obtain ex-
pansions such as the preceding by means of reciprocal functions relative
to simple integration, will apply with equal simplicity to reciprocal
functions relative to any number of integrations.
R. MURPHY.
Caius College,
Bee. 18, 1833.
VII. On the Nature of the Truth of the Laws of Motion. By the
Rev. W. Whewell, M.A. Fellow and Tutor of Trinity College.
[Bead Feb. 17, 1834.]
1. The long continuance of the disputes and oppositions of opinion
which have occurred among theoretical writers concerning the elementary
principles of Mechanics, may have made such discussions appear to some
persons wearisome and unprofitable. I might, however, not unreasonably
plead this very circumstance as an apology for offering a new view of
the subject; since the extent to which these discussions have already
gone shews that some men at least take a great interest in them ;
and it may be stated, I think, without fear of contradiction, that
these controversies have not terminated in the general and undisputed
establishment of any one of the antagonist opinions.
The question to which my remarks at present refer is this: "What
is the kind and degree of cogency of the best proofs of the laws of
motion, or of the fundamental principles of mechanics, exprest in any
other way?" Are these laws, philosophically considered, necessary, and
capable of demonstration by means of self-evident axioms, like the
truths of geometry ; or are they empirical, and only known to be true
by trial and observation, like such general rules as we obtain in natural
history ?
It certainly appears, at first sight, very difficult to answer the argu-
ments for either side of this alternative. On the one hand it is said,
the laws of motion cannot be necessarily true, for if they were so, the
denial of them would involve a contradiction. But this it does not,
for we can readily conceive them to be other than they are. We can
conceive that a body in motion should have a natural tendency to
move slower and slower. And we know that, historically speaking,
Vol. V. Paet II. U
150 Mr WHEWELL, ON THE NATURE OF THE TRUTH
men did at first suppose the laws of motion to be different from
what they are now proved to be. This would have been impossible
if the negation of these laws had involved a contradiction of self-evi-
dent principles, and consequently had been not only false but incon-
ceivable. These laws, therefore, cannot be necessary ; and can be duly
established in no other way than by a reference to experience.
On the other hand, those who deduce their mechanical principles
without any express reference to experiment, may urge, on their side,
that, by the confession even of their adversaries, the laws of motion
are proved to be true beyond the limits of experience ; — that they are
assumed to be true of any new kind of motion when first detected, as
well as of those already examined; — and that it is inexplicable how
such truths should be established empirically. They may add that the
consequences of these laws are allowed to hold with the most complete
and absolute universality; for instance, the proposition that "the quan-
tity of motion in the world in a given direction cannot be either
increased or diminished," is conceived to be rigorously exact; and to
have a degree and kind of certainty beyond and above all mere facts
of experience ; what other kind of truth than necessary truth this
can be, it is difficult to say. And if the conclusions be necessarily
true, the principles must be so too.
This apparent contradiction therefore, that a law should be neces-
sarily true and yet the contrary of it conceivable, is what I have now
to endeavour to explain ; and this I must do by pointing out what
appear to me the true grounds of the laws of motion.
2. The science of Mechanics is concerned about motions as deter-
mined by their causes, namely, forces ; the nature and extent of the
truth of the first principles of this science must therefore depend upon
the way in which we can and do reason concerning causes. In what
manner we obtain the conception of cause, is a question for the meta-
physician, and has been the subject of much discussion. But the general
principle which governs our mode of viewing occurrences with reference
to this conception, so far as our present subject is concerned, does not
appear to be disturbed by any of the arguments which have been
OF THE LAWS OF MOTION. 151
adduced in this controversy. This principle I shall state in the form
of an axiom, as follows.
Axiom I. Every change is produced by a cause.
It will probably be allowed that this axiom expresses a universal
and constant conviction of the human mind ; and that in looking at
a series of occurrences, whether for theoretical or practical purposes,
we inevitably and unconsciously assume the truth of this axiom. If a
body at rest moves, or a body in motion stops, or turns to the right
or the left, we cannot conceive otherwise than that there is some
cause for this change. And so far as we can found our mechanical
principles on this axiom, they will rest upon as broad and deep a
basis as any truths which can come within the circle of our know-
ledge.
I shall not attempt to analyse this axiom further. Different per-
sons may, according to their different views of such subjects, call it a
law of our nature that we should think thus, or a part of the con-
stitution of the human mind, or a result of our power of seeing the
true relations of things. Such variety of opinion or expression would
not affect the fundamental and universal character of the conviction
which the axiom expresses; and would therefore not interfere with our
future reasonings.
3. There is another axiom connected with this, which is also a
governing and universal principle in all our reasoning concerning
causes. It may be thus stated.
Axiom II. Causes are measured by their effects.
Every effect, that is, every change in external objects, implies a
cause, as we have already said : and the existence of the cause is known
only by the effects it produces. Hence the intensity or magnitude of
the cause cannot be known in any other manner than by these effects:
and, therefore, when we have to assign a measure of the cause, we
must take it from the effects produced.
In what manner the effects are to be taken into account, so as
to measure the cause for any particular purpose, will have to be
u2
153 mk whewell, on the nature of the truth
further considered ; but the axiom, as now stated, is absolutely and
universally true, and is acted upon in all parts of our knowledge in
which causes are measured.
4. But something further is requisite. We not only consider that
all changes of motion in a body have a cause, but that this cause may
reside in other bodies. Bodies are conceived to act upon one another,
and thus to influence each other's motions, as when one billiard ball
strikes another. But when this happens, it is also supposed that the
body struck influences the motion of the striking body. This is inclu-
ded in our notion of body or matter. If one ball could strike and
affect the motions of any number of others without having its own
motion in any degree affected, the struck balls would be considered,
not as bodies, but as mere shapes or appearances. Some reciprocal in-
fluence, some resistance, in short some reaction, is necessarily involved
in our conception of action among bodies. All mechanical action upon
matter implies a corresponding reaction; and we might describe matter
as that which resists or reacts when acted on by force. Not only
must there be a reaction in such cases, but this reaction is defined
and determined by the action which produces it, and is of the same
kind as the action itself The action which one body exerts upon
another is a blow, or a pressure; but it cannot press or strike with-
out receiving a pressure or a blow in return. And the reciprocal
pressure or blow depends upon the direct, and is determined altogether
and solely by that. But this action being mutual, and of the same
kind on each body, the effect on each body will be determined by the
effect on the other, according to the same rule ; each effect in turn
being considered as action and the other as reaction. But this cannot be
otherwise than by the equality and opposite direction of the action and
reaction. And since this reasoning applies in all cases in which bodies
influence each others motions, we have the following axiom which is
universally true, and is a fundamental principle with regard to all me-
chanical relations.
Axiom III. Action is always accompanied by an equal and opposite
Reaction.
OF THE LAWS OF MOTION. 153
5. I now proceed to shew in what manner the Laws of Motion
depend upon these three axioms.
Bodies move in lines straight or curved, they move more or less
rapidly, and their motions are variously affected by other bodies. This
succession of occurrences suggests the conceptions of certain properties
or attributes of the motions of bodies, as their direction and velocity,
by means of which the laws of such occurrences may be exprest.
And these properties or attributes are conceived as belonging to the
body at each j)^^^^ of its motion, and as changing from one point to
another. Thus the body, at each point of its path, moves in a
certain direction, and with a certain velocity.
These properties, direction and velocity for instance, are subject
to the rule stated in the first axiom : they cannot change without
some cause ; and when any changes in the motions of a body are
seen to depend on its position relative to another body or to any part
of space, such other body, or such other part of space, is said to
exert a Jbrce upon the moving body. Also the force exerted upon
the moving body is considered to be of a certain value at each
point of the body's motion ; and though it may change from one point
to another, its changes must depend upon the position of the points
only, and not upon the velocity and direction of the moving body.
For the force which acts upon the body is conceived as a property of
the bodies, or points, or lines, or surfaces among which the moving body
is placed; the force at all points therefore depends upon the position
with regard to the bodies and spaces of which the force is a property ;
but remains the same, whatever be the circumstances of the body
moved. The circumstances of the body moved cannot be a cause
which shall change the force acting at any point of space, although
they may alter the effect which that force produces upon the body.
Thus, gravity is the same force at the same point of space, whether it
have to act upon a body at rest or in motion ; although it still remains
to be seen whether it will produce the same effect in the two cases.
6. This being established, we can now see of what nature the
laws of motion must be, and can state in a few words the proofs
J 54 Mr WHEWELL, ON THE NATURE OF THE TRUTH
of them. We shall have a law of motion corresponding to each
of the above three axioms ; the first law will assert that when no force
acts, the properties of the motion will be constant; the second law
will assert that when a force acts, its quantity is measured by the
effect produced ; the third law will assert that, when one body acts
upon another, there will be a reaction, equal and opposite to the
action. And so far as the laws are announced in this form, they will
be of absolute and universal truth, and independent of any particular
experiment or observation whatever.
But though these laws of motion are necessarily and infallibly
true, they are, in the form in which we have stated them, entirely
useless and inapplicable. It is impossible to deduce from them any
definite and positive conclusions, without some additional knowledge or
assumption. This will be clear by stating, as we can now do in a
very small compass, the proofs of the laws of motion in the form
in which they are employed in mechanical reasonings.
7. First, of the first Law ; — that a body not acted upon by any force
will go on in a straight line with an invariable velocity.
The body will go on in a straight line : for, at any point of its
motion, it has a certain direction, which direction will, by Axiom I,
continue unchanged, except some cause make it deviate to one side or
other of its former position. But any cause which should make the
direction deviate towards any part of space would be a force, and the
body is not acted upon by any force. Therefore, the direction cannot
change, and the body will go on in the same straight line from the
first.
The body will move with an invariable velocity. For the velocity
at any point will, by Axiom I, continue unchanged, except some
cause make it increase or decrease. And since, by supposition, the
body is not acted upon by any force, there can be no such cause
depending upon position, that is, upon relations of space; for any
cause of change of motion which has a reference to space is force.
Therefore there can be no cause of change of motion, except
there be one depending upon time, such, for instance, as would exist
OF THE LAWS OF MOTION. 155
if bodies had a natural tendency to move slower and slower, according
to a rate depending on the time elapsed.
But if such cause existed, its effects ought to be considered sepa-
rately ; and it would still be requisite to assume the permanence of
the same velocity, as the first law of motion ; and to obtain, in addi-
tion to this, the laws of the retardation depending on the time.
Whether there is any. such cause of retardation in the actual
motions of bodies, can be known only by a reference to experience;
and by such reference it appears that there is no such cause of the
diminution of velocity depending on time alone; and therefore that
the first law of motion may, in all cases in which bodies are exempt
from the action of external forces, be applied without any addition or
correction depending upon the time elapsed.
It is not here necessary to explain at any length in what manner
we obtain from experience the knowledge of the truth just stated, that
there is not in the mere lapse of time any cause of the retardation of
moving bodies. The proposition is established by shewing that in all
the cases in which such a cause appears to exist, the cause of retar-
dation resides in surrounding bodies and not in time alone, and is
therefore an external force. And as this can be shewn in every in-
stance, there remains only the negation of all grovind for the assump-
tion of such a cause of retardation. We therefore reject it altogether.
Thus it appears that in proving the first law of motion, we obtain
from our conception of cause the conviction that velocity will be
uniform except some cause produce a change in it ; but that we are
compelled to have recourse to experience in order to learn that time
alone is not a cause of change of velocity.
8. I now proceed to the second Law : — that when a force acts
upon a body in motion, the effect is the same as that which the same
force produces upon a body at rest.
This law requires some explanation. How is the effect produced
upon a moving body to be measured, so that we may compare it with
156 Mk WHEWELL, on the NATURE OF THE TRUTH
the effect upon a body at rest? The answer to this is, that we here
take for the measure of the effect of the force, that motion which
must be compounded with the motion existing before the change, in or-
der to produce the motion which exists after the change: the rules for
the composition of motion being established on independent grounds
by the aid of definition alone. Thus if gravity act upon a body
which is falling vertically, the effect of gravity upon the body is
measured by the velocity added to that which the body already has :
if gravity act upon a body which is moving horizontally, its effect
is measured by the distance to which the body falls below the hori-
zontal line.
The effect of the force which we consider in the second Law of
motion, is its effect upon velocity only : and it is proper to mark
this restriction by an appropriate term : we shall call this the accele-
rative effect of force; and the cause, as measured by this effect, may
be termed the accelerathe quantity of the force.*
A law of motion which necessarily results from our second Axiom
is, that the accelerative quantity of a force is measured by the acce-
lerative effect. But whether the accelerative effect depends upon the
velocity and direction of the moving body, cannot be known indepen-
dently of experience. It is very conceivable, for instance, that the
force of gravity being every where the same, shall yet produce, upon
falling bodies, a smaller accelerative effect in proportion to the velocity
which they already have in a downward direction. Indeed if gravity
resembled in its operation the effect of any other mode of mechanical
agency, the result would be so. If a body moved downwards in
* The accelerative quantity of a force (the quantitas acceleratrix vis cujusvis of Newton)
is often called the accelerating forces and we may thus have to speak of the accelerating
force of a certain force, which is at any rate an awkward phraseology. It would perhaps
have been fortunate if Newton, or some other writer of authority, at the time when the
principles of mechanics were first clearly developed, had invented an abstract term for
this quantity : it might for instance have been called acceleralivity. And the second law
of motion would then have been, that the acceleralivity of the same force is the same,
whatever be the motion of the body acted on.
OF THE LAWS OF MOTION. 157
consequence of the action of a hand pushing it with a constant effort,
or of a spring, or of a stream of fluid rushing in the same direction,
the accelerative effect of such agents would be smaller and smaller
as the velocity of the body propelled was larger and larger. We can
learn from experience alone that the effects of the action of gravity
do not follow the same rule.
We assert that the accelerative quantity of the same force of gra-
vity is the same whatever be the motion of the body acted on. It
may be asked how we know that the force of gravity is the same
in cases so compared ; for instance, when it acts on a body at rest
and in motion ? The answer to this question we have given already.
By the very process of considering gravity as a force, we consider
it as an attribute of something independent of the body acted on.
The amount of the force may depend upon place, and even time, for
any thing we know a priori ; but we do not find that the weight of
bodies depends on these circumstances, and therefore, having no evi-
dence of a difference in the force of gravity, we suppose it the same
at different times and places. And as to the rest, since the force is a
force which acts on the body, it is considered as the same force,
whatever be the circumstances of the passive body, although the ejects
may vary with these circumstances. If the effects are liable to such
change, this change must be considered separately, and its laws investi-
gated ; but it cannot be allowed to unsettle our assumption of the
permanence of the force itself. It is precisely this assumption of a
constant cause, which gives us a fixed term, as a means of estimating
and expressing by what conditions the effects are regulated.
It appears by observation and experiment, that the accelerative
quantity of the same force is not affected by the velocity or direction
of the body acted on : for instance, a body falling vertically receives,
in any second of time, an accession of velocity as great as that which
it received in the first second, notwithstanding the velocity with which
it is already moving. The proof of this and similar assertions from
experiment produced, historically speaking, the establishment of the
second law of motion in the sense in which we now assert it. And
here, as in the case of the first law, we may observe that an important
Vol. V. Part II. X
158 Mr WHEWELL, ON THE NATURE OF THE TRUTH
portion of the process of proof consisted in shewing that in those cases
in which the accelerative effect of a force appeared to be changed by
the circumstances of the motion of the body acted on, the change was,
in fact, due to other external forces ; so that all evidence of a cause
of change residing in those circumstances was entirely negatived; and
thus the law, that the accelerative effect of the same force is the
same, appeared to be absolutely and rigorously true.
9. When the motions of bodies are not affected merely by forces
like gravity, which are only perceived by their effects, but are acted
upon by other bodies, the case requires other considerations.
It is in such cases that we originally form the conception of force;
we ourselves pull and push, thrust and throw bodies, with a view, it
may be, either to put them in motion, or to prevent their moving,
or to alter their figure. Such operations, and the terms by which
they are described, are all included in the term force, and in other
terms of cognate import. And in using this term, we necessarily
assume and imply the co-existence of these various effects of force
which we have observed universally to accompany each other. Thus
the same kind of force which is the cause of motion, may also be
the cause of a body having a form different from its natural form ;
when we draw a bow, the same kind of pull is needed to move the
string, and to hold it steady when the bow is bent. And a weight
might be hung to the string, so as to produce either the one or
the other of these effects. By an infinite multiplicity of experiments
of this kind, we become imbued with the conviction that the same
pressure may be the cause of tension and of motion. Also as the
cause can be known by its effects only, each of these effects may be
taken as its measure ; and therefore, so long as one of them is the
same, since the cause is the same, the other must be the same also.
That is, so long as the pressure or force which shews itself in
tension is the same, the motion which it would produce must, under
the same circumstances, be the same also. This general fact is not
a result of any particular observations, but of the general observation
or suggestion arising unavoidably from universal experience, that both
OF THE LAWS OP MOTION. 159
tension and motion may be referred to force as their cause, and have
no other cause.
We come therefore to this principle with regard to the actions of
bodies upon each other, that so long as the tension or pressure is the
same, the force, as shewn by its effect in producing motion, must
also be the same.
10. This force or action of bodies upon one another, is that which
is meant in the Third Axiom, and we now proceed to consider the
application of this axiom in mechanics.
Pressures or forces such as I have spoken of, may be employed in
producing tension only, and not motion ; in this case, each force prevents
the motion which would be produced by the others, and the forces
are said to balance each other, or to be in equilibrium. The science
which treats of such cases is called Statics, and it depends entirely
upon the above third axiom, applied to pressures producing rest. It
follows from that axiom, that pressures, which acting in opposite di-
rections thus destroy each other's effects, must be equal, each measuring
the other. Thus if a man supports a stone in his hand, the force or
effort exerted by the man upwards is equal to the weight or force
of the stone downwards. And if a second stone, just equal to the 'first,
were supported at the same time in the same hand, the force or effort
must be twice as great ; for the two stones may be considered as
one body of twice the magnitude, and of twice the weight; and
therefore the effort which supports it must also be twice as great.
And thus we see in what manner statical forces are to be measured
in virtue of this third axiom ; and no further principle is requisite to
enable us to establish the whole doctrine of statics.
11. The third axiom, when applied to the actions of bodies in
motion, gives rise to the third law of motion, which Ave must now con-
sider. Here, as in the cases of the other axioms, we must inquire
how we are to measure the quantities to which the axiom applies. What
is the measure of the action which takes place when a body is put
in motion by pressure or force? In order to answer this question, we
X 2
160 Mr WHEWELL, ON THE NATURE OF THE TRUTH
must consider what circumstances make it requisite that the force
should be greater or less. If we have to lift a stone, the force which
we exert must be greater when the stone is greater : again, we must
exert a greater force to lift it quickly than slowly. It is clear, there-
fore, that that property of a force with which we are here concerned,
and which we may call the motive quantity of the force,* increases both
when the velocity communicated, and when the mass moved, increase, and
depends upon both these quantities, though we have not yet shewn
what is the law of this dependence.
The condition that a quantity P shall increase when each of two
others V and M does so, may be satisfied in many ways : for instance,
by supposing P proportional to the sum M+ V (all the quantities being
expressed in numbers), or to the product, MV, or to MF'-, or in many
other ways.
When, however, the quantities ^ and M are altogether hetero-
geneous, as when one is velocity, and the other weight, the first
of the above suppositions, that P varies as M + V, is inadmissible.
For the law of variation of the formula M+ V depends upon the
relation of the units by which M and V respectively are measured;
and as these units are arbitrary in each case, the result is, in like
manner, arbitrary, and therefore cannot express a law of nature.
«
12. The supposition that the motive quantity of a force varies as
M^-V, where M is the mass moved and V the velocity, being thus
inadmissible, we have to select upon due grounds, among the other
formulae MV, MV\ M'V, &c.
And in the first place I observe that the formula must be propor-
tional to M simply (excluding M.^ &;c.) for both the forces which
* The motive quantity of a force {vis cujusvis quantitas matrix of Newton) is sometimes
called moving force; we are thus led to speak of the moving force of a force, as we
have already observed concerning accelerating force. Hence, as in that case, we might
employ a single term, as motivity, to denote this property of force; and might thus speak
of it and of its measures without the awkwardness which arises from the usual phrase.
OF THE LAWS OF MOTION. l6l
produce motion and the masses in which motion is produced are capa-
ble of addition by juxtaposition, and it is easily seen by observation
that such addition does not modify the motion of each mass. If a
certain pressure upon one brick (as its own weight) cause it to fall
with a certain velocity, an equal pressure on another equal brick wiU
cause it also to fall with the same velocity ; and these two bricks
being placed in contact, may be considered as one mass, which a dou-
ble force will cause to fall with still the same velocity. And thus
all bodies, whatever be their magnitude, will fall with the same velo-
city by the action of gravity. Those who deny this (as the Aristo-
telians did) must maintain, that by establishing between two bodies
such a contact as makes them one body, we modify the motion which
a certain pressure will produce in them. And when we find experi-
mentally (as we do find) that large bodies and small ones fall with the
same velocity, excluding the effects of extraneous forces, this result
shews that there is not, in the union of small bodies into a larger one,
any cause which affects the motion produced in the bodies.
It appears, therefore, that the motive quantity of force which puts
a body in motion is, cceteris paribus, proportional to the mass of the
body ; so that for a double mass a double force is requisite, in order
that the velocity produced may be the same. Mass considered with
reference to this rule, is called Inertia.
13. The measure of mass which is used in expressing a law of
motion, must be obtained in some way independent of motion, other-
wise the law will have no meaning. Therefore, mass measured in
order to be considered as Inertia must be measured by the statical
effects of bodies, for instance, by comparison of weights. Thus two
masses are equal which each balance the same weight in the same
manner; and a mass is double of one of them which produces the
same effect as the two. And we find, by universal observations, that
the weight of a mass is not affected by the figure or the arrange-
ment of parts, so long as the matter continues the same. Hence it
appears that the mass of bodies must be compared by comparing their
weights, and Inertia is proportional to weight at the same place.
162 Mr WHEWELL, ON THE NATURE OF THE TRUTH
Since all bodies, small or large, light or heavy, fall downwards with
equal velocities, when we remove or abstract the effect of extraneous
circumstances, the motive quantity of the force of gravity on equal
bodies is as their masses ; or as their weight, by what has just been said.
14. For the measure of the motive quantity of force, or of the action
and reaction of bodies in motion, we have, therefore, now to chuse
among such expressions as MV, and MV^. And our choice must be
regulated by finding what is the measure which will enable us to
assert, in all cases of action between bodies in motion, that action and
reaction are equal and opposite.
Now the fact is, that either of the above measures may be taken,
and each has been taken by a large body of mathematicians. The former
however {MV) has obtained the designation which naturally falls to the
lot of such a measure ; and is called momentum, or sometimes simply
quantity of motion : the latter quantity {MV^) is called vis viva or liv-
ing force.
I have said that either of these measures may be taken : the former
must be the measure of action, if we are to measure it by the effect pro-
duced in a given time; the latter is the measure if we take the whole
effect produced. In either way the third law of motion would be true.
Thvis if a ball B, lying on a smooth table, be drawn along by a
weight A hanging by a thread over the edge of the table, the motion
of B is produced by the action of A, and on the other hand the
motion of A is diminished by the reaction of B; and the equality
of action and reaction here consists in this, that the momentum {MV)
which B acquires in any time is equal to that which A loses : that is,
so much is taken from the momentum which A would have had, if
it had fallen freely in the same time; so that A falls more slowly by
just so much.
But if the weight A fall through a given space from rest, as 1 foot,
and then cease to act, the eqviality of action and reaction consists in
this, that the vis viva which B acquires on the whole, is equal to the
vis viva which A loses ; that is, the vis viva of A thus acting on B is
OF THE LAWS OF MOTION. 163
smaller by so much than it would have been, if A had fallen freely
through the same space.
15. In fact, these two propositions are necessarily connected, and
one of them may be deduced from the other. The former way of
stating the third law of motion appears, however, to be the simplest mode
of treating the subject, and we may put the third law of motion in
this form.
In the direct mutual action of bodies, the momentum gained and lost
in any time are equal.
This law depends upon experiment, and is perhaps best proved by
some of its consequences. It follows from the law so stated, that the
motive quantity of a force is proportional to the momentum generated in
a given time; since the motive quantity of force is to be equivalent
to that action and reaction which is understood in the third law of
motion. Now, if the pressure arising from the weight of a body P
produce motion in a mass Q, since the momentum gained by Q and
that lost by P in any time are equal, the momentum of the whole
at any time will be the same as if P's weight had been employed
in moving P alone. Therefore, the velocity of the mass Q will be
less, in the same proportion in which the mass or inertia is greater:
and thus the accelerating quantity of the force is inversely propor-
tioned to the mass moved. This rule enables us to find the accele-
rative quantity of the force in various cases, as for instance, when bodies
oscillate, or when a smaller weight moves a large mass; and we
can hence calculate the circumstances of the motion, which are found
to agree with the consequences of the above law.
16. But the argument may be reduced to a simpler form. Our
object is to shew that, for an equal mass, the velocity produced by a
force acting for a given time is as the pressure which produces the
motion; for instance, that a double pressure will produce a double
velocity. Now a double pressure may be considered as the union of
two equal pressures, and if these two act successively, the first will
communicate to the body a certain velocity, and the second will com-
164 Mr WHEWELL, ON THE NATURE OF THE TRUTH
municate an additional velocity, equal to the first, by the second law
of motion ; so that the whole velocity thus commvinicated will be the
double of the first. Therefore, if the velocity communicated be not
also the double of the first when the two pressures act together, the
difference must arise from this, that the effect of one force is modified
by the simultaneous action of the other. And when we find by expe-
rience (as we do find) that there is no such difference, but that the
velocity communicated in a given time is as the pressure which com-
mimicates it, this result shews that there is nothing in the circumstance
of a body being already acted on by one pressure, which modifies the
effect of an additional pressure acting along with the first.
17- I have above asserted the law, of the direct action of bodies
only. But it is also true when the action is indirect, as when by
turning a winch we move a wheel, the main mass of which is farther
from the axis than the handle of the winch. In this case the pres-
sure we exert acts at a mechanical disadvantage on the main mass of
the wheel, and we may ask whether this circumstance introduces any
new law of motion. And to this we may reply, that we can conceive
pressure to produce different effects in moving bodies, according as it
is exerted directly or by the intervention of machines; but that we
find no reason to believe that such a difference exists. The relations
of the pressures in different parts of a machine are determined by con-
sidering the machine at rest. But if we suppose it to be put in
motion by such pressures, we see no reason to expect that these pres-
sures should have a different relation to the motions produced from
what they would have done if they were direct pressures. And as
we find in experiment a negation of all evidence of such a differ-
ence, we reject the supposition altogether. We assert, therefore, the
third law of motion to be true, whatever be the mechanism by
the intervention of which action and reaction are opposed to each
other.
From this consideration it is easy to deduce the following rule,
which is known by the designation of D'Alembert's principle, and
may be considered as a fourth law of motion.
OF THE LAWS OF MOTION. 165
WJien any forces produce motion in any connected system of matter,
the motive quantities of force gained and lost by the different parts
must balance each other according to the connexion of the system. ■
By the motive quantity of force gained by any body, is here
meant the quantity by which that motive force which the body's mo-
tion implies (according to the measures already established) exceeds
the quantity of motive force which acts immediately upon the body.
It is the excess of the effective above the impressed force, and of course
arises from the force transmitted from the other bodies of the system
in consequence of the connexion of the parts. The motive quantity
of force lost is in like manner the excess of the impressed above the
effective force. And these two excesses, in different parts of the sys-
tem, must balance each other according to the mechanical advantage
or disadvantage at which they act for each part.
This completes our system of mechanical principles, and authorizes
us to extend to bodies of any size and form the rules which the
second law of motion gives for the motion of bodies considered as
points. And by thus enabling us to trace what the motions of bodies
will be according to the rule asserted in the third law of motion,
(namely, that the motive quantity of forces is as the momentum pro-
duced in a given time,) it leads us to verify that supposition by experi-
ments in which bodies oscillate or revolve or move in any regular
and measurable manner, as has been done by Atwood, Smeaton, and
many others.
18. We have thus a complete view of the nature and extent of
the fundamental principles of mechanics; and we now see the reason
why the laws of motion are so many and no more, in what way they
are independent of experience, and in what way they depend upon
experiment. The form, and even the language of these laws is of
necessity what it is; but the interpretation and application of them is
not possible without reference to fact. We may imagine many rules
according to which bodies might move (for many sets of rules, dif-
ferent from the existing ones, are, so far as we can see, possible) and
we should still have to assert — that velocity could not change without
Vol. V. Pakt II. Y
166 Mb WHEWELL, ON THE NATURE OF THE TRUTH
a cause, — that change of action is proportional to the force which pro-
duces it, — and that action and reaction are equal and opposite. The
truth of these assertions is involved in those notions of causation and
matter, which the very attempt to know any thing concerning the rela-
tions of matter and motion presupposes. But, according to the facts
which we might find, in such imaginary cases as I have spoken of,
we should settle in a different way — what is a cause of change of ve-
locity, — what is the measure of the force which changes motion, — and
what is the measure of action between bodies. The law is necessary,
if there is to be a law ; the meaning of its terms is decided by what
we find, and is therefore regulated by our special experience.
19. It may further illustrate this matter to point out that this
view is confirmed by the history of mathematics. The laws of motion
were assented to as soon as propounded; but were yet each in its turn
the subject of strenuous controversy. The terms of the law, the form,
which is necessarily true, were recognised and undisputed ; but the
meaning of the terms, the substance of the law, was loudly contested;
and though men often tried to decide the disputed points by pure
reasoning, it was easily seen that this could not suffice ; and that since
it was a case where experience could decide, experience must be the
proper test: since the matter came within her jurisdiction, her authority
was single and supreme.
Thus with regard to the first law of motion, Aristotle allowed that
natural motions continue unchanged, though he asserted the motions
of terrestrial bodies to be constrained motions, and therefore, liable to
diminution. Whether this was the cause of their diminution was a
question of fact, which was, by examination of facts, decided against
Aristotle. In like manner, in the first case of the second law of
motion which came under consideration, both Galileo and his oppo-
nent agree that falling bodies are uniformly accelerated ; that is, that
the force of gravity accelerates a body uniformly whatever be the
velocity it has already ; but the question arises, what is uniform acce-
leration ? It so happened in this case, that the first conjecture of Ga-
lileo, afterwards defended by Casraeus, (that the velocity was propor-
OF THE LAWS OF MOTION. l67
tional to the space from the beginning of the motion) was not only
contradictory to fact, but involved a self-contradiction; and was,
therefore, easily disposed of. But this accident did not supersede the
necessity of Galileo and his pupils verifying their assertion by refer-
ence to experiment, since there were many suppositions which were
different from theirs, and still possible, though that of Casrasus was
not.
The mistake of Aristotle and his followers, in maintaining that
large bodies fall more quickly than small ones, in exact proportion
to their weight, arose from perceiving half of the third law of motion,
that the velocity increases with the force which produces it ; and from
overlooking the remaining half, that a greater force is required for the
same velocity, according as the mass is larger. The ancients never
attained to any conception of the force which moves and the body
which is moved, as distinct elements to be considered when we en-
quire into the subject of motion, and therefore could not even propose
to themselves in a clear manner the questions which the third law of
motion answered.
But, when, in more modern times, this distinction was brought into
view, the progress of opinion in this case was nearly the same as with
regard to the other laws.
It was allowed at once, and by all, that action and reaction are
equal ; but the controversy concerning the sense in which this law is to
be interpreted, was one of the longest and fiercest in the history of ma-
thematics, and the din of the war has hardly yet died away. The
disputes concerning the measure of the force of bodies in motion,
or the vis viva, were in fact a dispute which of two measures of action
that I have mentioned above should be taken ; the effect in a given
time, or the whole effect : in the one case the momentum {MV) in the
other the vis viva, {MV'^) was the proper measure.
20. It may be observed that the word momentum, which one party
appropriated to their views, was employed to designate the motive
quantity of force, or the action of bodies in motion, before it was
Y2
168 Mr WHEWELL, ON THE NATURE OF THE TRUTH
determined what the true measure of such action was. Thus Galileo,
in his "Discorso intorno alle cose che stanno in su I'Acqua,'" says, that
momentum "is the force, efficacy, or virtue with which the motion
moves and the body moved resists; depending not on weight only,
but on the velocity, inclination, and any other cause of such virtue."
The adoption of the phrase vis viva is another instance of the extent
to which men are tenacious of those terms which carry along with their
use a reference to the fundamental laws of our thought on such matters.
The party which used this phrase maintained that the mass multiplied
into the square of the velocity was the proper measure of the force
of bodies in motion; but finding the term moving force appropriated
by their opponents, they still took the same term force, with the
peculiar distinction of its being living force, in opposition to dead
force or pressure, which they allowed to be rightly measured by the
momentum generated in a given time. The same tendency to adopt,
in a limited and technical sense, the words of most general and fun-
damental vise in the subject, has led some writers (Newton for instance,)
to employ the term motion or quantity of motion as synonymous with
momentum, or the product of the numbers which express the mass
and the velocity. And this use being established, the quantities of
motion gained and lost are always equal and opposite; and, therefore
the quantity which exists in any given direction cannot be increased
or diminished by any mutual action of bodies. Thus we are led to the
assertion which has already been noticed, that the quantity of motion
in the world is always the same. And we now see how far the
necessary truth of this proposition can be asserted. The proposition is
necessarily true according to our notions of material causation ; but the
measure of "quantity of motion," which is a condition of its truth, is
inevitably obtained from experience.
21. It is not surprising that there should have been a good deal
of confusion and difference of opinion on these matters : for it appears
that there is, in the intellectual constitution and facvdties of man, a
source of self-delusion in svich reasonings. The actual rules of the
motion and mutual action of bodies are, and must be, obtained from
OF THE LAWS OF MOTION.
169
observation of the external world : but there is a constant wish and
propensity to express these rules in such terms as shall make them
appear self-evident, because identical with the universal and necessary
rules of causation. And this propensity is essential to the progress of
our knowledge ; and in the success of this effort consists, in a great
measure, the advance of the science to its highest point of simplicity
and generality.
22. The nature of the truth which belongs to the laws of motion
will perhaps appear still more clearly, if we state, in the following-
tabular form, the analysis of each law into the part which is necessary,
and the part which is empirical.
First
Law.
Second
Law.
Third
Law.
Necessary.
Velocity does not change
without a cause.
The accelerating quantity
of a force is measured by the
acceleration produced.
Reaction is equal and op-
posite to action.
Empirical.
The time for which a body has al-
ready been in motion is not a cause of
change of velocity.
The velocity and direction of the mo-
tion which a body already possesses are
not, either of them, causes which
change the acceleration produced.
The connexion of the parts of a body,
or of a system of bodies, and the action
to which the body or system is already
subject, are not, either of them, causes
which change the effects of any ad-
ditional action.
Of course, it will be understood that, when we assert that the con-
nexion of the parts of a system does not change the effect of any
action upon it, we mean that this connexion does not introduce any
new cause of change, but leaves the effect to be determined by the
previously established rules of equilibrium and motion. The connexion
will modify the application of such rules ; but it introduces no ad-
ditional rule: and the same observation applies to all the above stated
empirical propositions.
170 Mr WHEWELL, ON THE NATURE OF THE TRUTH
This being understood, it will be observed that the part of each law
which is here stated as empirical, consists, in each case, of a negation
of the supposition that the condition of the moving body with respect
to motion and action, is a cause of any change in the circumstances of
its motion; and from this it follows that these circumstances are de-
termined entirely by the forces extraneous to the body itself.
23. This mode of considering the question shews us in what
manner the laws of motion may be said to be proved by their sim-
plicity, which is sometimes urged as a proof. They undoubtedly have
this distinction of the greatest possible simplicity, for they consist in
the negation of all causes of change, except those which are essential
to our conception of such causation. We may conceive the motions
of bodies, and the effect of forces upon them, to be regulated by the
lapse of time, by the motion which the bodies have, by the forces
previously acting ; but though we may imagine this as possible, we do
not find that it is so in reality. If it were, we should have to con-
sider the effect of these conditions of the body acted on, and to com-
bine this effect with that of the acting forces ; and thus the motion
would be determined by more numerous conditions and more complex
rules than those which are found to be the laws of nature. The laws
which, in reality, govern motion are the fewest and simplest possible,
because all are excluded, except those which the very nature of laws
of motion necessarily implies. The prerogative of simplicity is possessed
by the actual laws of the universe, in the highest perfection which is
imaginable or possible. Instead of having to take into account all the
circumstances of the moving bodies, we find that we have only to
reject all these circumstances. Instead of having to combine empirical
with necessary laws, we learn empirically that the necessary laws are
entirely sufficient.
24. Since all that we learn from experience is, that she has no-
thing to teach us concerning the laws of motion, it is very natural
that some persons shovdd imagine that experience is not necessary to
their proof. And accordingly many writers have undertaken to esta-
blish all the fundamental principles of mechanics by reasoning alone.
OF THE LAWS OF MOTION. 171
This has been done in two ways: — sometimes by attending only to the
necessary part of each law (as the parts are stated in the last para-
graph but one) and by overlooking the necessity of the empirical
supplement and limitation to it; — at other times by asserting the part
which I have stated as empirical to be self-evident, no less than the
other part. The former way of proceeding may be found in many
English writers on the subject; the latter appears to direct the reason-
ings of many eminent French mathematicians. Some (as Laplace) have
allowed the empirical nature of two out of the three laws ; others, as
M. Poisson, have considered the first as alone empirical ; and others, as
D'Alembert, have assumed the self-evidence of all the three indepen-
dently of any reference whatever to observation.
25. The parts of the laws which I have stated as empirical,
appear to me to be clearly of a different nature, as to the cogency
of their truth, from the parts which are necessary ; and this difference
is, I think, established by the fact that these propositions were de-
nied, contested, and modified, before they were finally established. If
these truths could not be denied without a self-contradiction, it is
difficult to understand how they could be (as they were) long and
obstinately controverted by mathematicians and others fully sensible to
the cogency of necessary truth.
I will not however go so far as to assert that there may not be
some point of view in which that which I have called the empirical
part of these laws, (which, as we have seen, contains negatives only,)
may be properly said to be self-evident. But however this may be,
I think it can hardly be denied that there is a difference of a fun-
damental kind in the nature of these truths, — which we can, in our
imagination at least, contradict and replace by others, and which, his-
torically speaking, have been established by experiment; — and those
other truths, which have been assented to from the first, and by all,
and which we cannot deny without a contradiction in terms, or reject
without putting an end to all use of our reason on this subject.
26. On the other hand, if any one should be disposed to maintain
that, inasmuch as the laws are interpreted by the aid of experience
172 Mh WHEWELL, on the NATURE OF THE TRUTH, &c.
only, they must be considered as entirely empirical laws, I should not
assert this to be placing the science of mechanics on a wrong basis.
But at the same time I would observe, that the form of these laws is
not empirical, and would be the same if the results of experience
should differ from the actual results. The laws may be considered as
a formula derived from a priori reasonings, where experience assigns
the value of the terms which enter into the formula.
Finally, it may be observed, that if any one can convince himself
that matter is either necessarily and by its own nature determined to
move slower and slower, or necessarily and by its own nature deter-
mined to move uniformly, he must adopt the latter opinion, not only
of the truth, but of the necessity of the truth of the first law of
motion, since the former branch of the alternative is certainly false : and
similar assertions may be made with regard to the other laws of motion.
27. This enquiry into the nature of the laws of motion, will, I
hope, possess some interest for those who attach any importance to the
logic and philosophy of science. The discussion may be said to be
rather metaphysical than mechanical ; but the views which I have en-
deavoured to present, appear to explain the occurrence and result of
the principal controversies which the history of this science exhibits ;
and, if they are well founded,' ought to govern the way in which the
principles of the science are treated of, whether the treatise be intended
for the mathematical student or the philosopher.
173
VIII. Researches in the Theory of the Motion of Fluids. By the Rev.
James Challis, late Fellow of Trinity College, Cambridge,
and Fellow of the Cambridge Philosophical Society.
[VieaA March 3, 1834-3
1. The subjects treated of in this communication are of a miscel-
laneous character, referring to several points of the theory of fluid
motion, respecting which the author conceived he had something new
to advance. In illustration of the principles he has attempted to establish,
solutions are given of two problems of considerable interest: — the
resistance to the motion of a ball-pendulum ; and, the resistance to the
motion of a body partly immersed in water and drawn along at the
surface in the horizontal direction. The principal object in the solution
of the latter problem is to account for the rising of the body in the
vertical direction on increasing the velocity of draught, which in some
recent experiments on canal navigation has been observed to take place.
In the course of the paper I have had occasion to refer several times
to a previous communication* to this Society respecting fluid motion,
for the purpose of giving to the views there advanced some corrections
and confirmations which have been suggested by more mature considera-
tion. For the sake of distinctness the subjects of the present essay
are divided into sections.
• Camb. Phil. Trans. Vol. HI. Part in.
Vol. V. Part II. > Z
174 Mb CHALLIS's RESEARCHES IN THE THEORY
SECTION I.
On the Integral of the Equation -j-^ + -~ = Q.
2. This equation is applicable to all problems respecting the motion
of incompressible fluids, which require for their solutions the consideration
of motion in one plane only. Mathematicians have obtained integrals
of it suited to the particular questions they were discussing ; for instance,
in solving the problem of waves propagated in a canal of uniform
width, M. Poisson has given a value of (p, which, while it satisfies the
equation in question, is exclusively applicable to that problem. But
it is well known that by the common method of finding the integrals
of linear partial differential equations of the second order between
three variables, a value of cp may be found prior to any consideration
of the circumstances under which the fluid was put in motion. There-
fore any inferences respecting the nature of the motion, which may be
drawn from this integral, must be equally applicable to all problems of
this class. To obtain such inferences is the object of the following
reasoning.
3. The integral I speak of is.
To ascertain its general signification, I propose to determine the forms
of the functions F and jf, independently of any hypothesis respecting
the mode in which the fluid was put in motion. The quantity (j) is
subject to the condition {d(p) = udx-^vdy, where u and v are the
velocities at the poiiit xy in the directions of the axes of x and y
respectively. Hence ^^=«, -j^=v, and
u = F' {x + y^/'^\)+f {x-y^/'^),
v=V^lF'ix + y\/^^)-\/'^fix-yV^l).
OF THE MOTION OF FLUIDS. ' 175
First, it may be observed that u and v are not both possible for any
values of x and y, unless the functions F' andy be the same. Again,
as the form of F' we are seeking for is to be independent of all that
is arbitrary, it will remain the same whatever direction we arbitrarily
assign to the axes of co-ordinates. Let therefore the axis of y pass
through the point to which the velocities u, v, belong. Then
y = 0, u = 2F'{x), v = 0.
If now the axes be supposed to take any other position, the origin
remaining the same, u will be equal to / ^ ^ F' {^/x^ + y^).
Hence
F'{x + y^^) + F'(x-yV-l)=-^^y^^=^.F'(./^FTr),
a functional equation for determining the form of F'. Let
x + yy^ - l=m, and x — y^/— 1 =n;
then
2x = m + n, and "s/ x^ -k-y^^s/ mn.
Therefore,
c
It is easily seen that if F\y/mn) =— f=, the equation is satisfied.
Hence ^
^ = ^-7=+ ^== = A^ and^--^^. ^-^
dx x + yV^l^ a;-yV-l x'^ + y^' dy~x^ + y'-'
2C -^ -
and consequently the velocity at xy, or \/ u'^ + v^ = —-r=^ ^- ,., ,,
These results shew that the velocity is directed to or from the origin
of co-ordinates, and varies inversely as the distance from it. But we
must observe that this limitation as to the point to which the velocity
is directed, is owing to the particular forms, x+y'\/~^, x-ys/ — \,
z2
176 Mr CHALLIS's RESEARCHES IN THE THEORY
of the quantities which the function F' involves. For the equation
^j% + -T-? = 0, is also satisfied by the following,
((> = F {{a^- X cos d - y m\ Q) + {^ + xsm9 +y cosO) ^/^^}
+y {(" + a; cos 0-y sin 0) - (/3 + a; sin f y cos 9) V~^\ '■>
and this analytical circumstance has its interpretation in reference to
the motion of the fluid. By supposing the function f to be the same
as F, and giving to F' the same form as before, we shall find,
' d^ _ 2 C(.r + g cos g + /3 sin 9)
dx ~ (a + xcos9 — i/ sin 9y + (13 + x sin 9 + y cos 9)-
d^ 2C(y + /3cosg-as in 9)
dy {a -\- X cos 9 — y sin 9'f + (/3 + a; sin + y cos 9Y
/d^y /d^Y 4C^
\dx ) \dy) ~ {a + x cos9-y sin9y + (^ + x sm9 + y cos9y
Or, if a cos 9-\-fisin9= —a, and /3 cos 9 — a sin 9= —b,
d(f> 2C(x-a)
dx (x — ay + {y — bf '
^ or V- ^^(y-*)
dy ^x-ay + iy-by
Vu' + v^ =
y/ix-af + iy-by'
This shews that the velocity is directed to the point whose co-ordinates
are a, b, and varies inversely as the distance from it. And as we have
arrived at this result without considering any circumstances under which
the fluid was caused to move, the inference to be drawn is, that such
is the general character of the motion. Nothing forbids our considering
C, a, and b, functions depending on the time and the given conditions
of motion in any proposed problem. Also if at a given instant, a line
commencing at any point, be drawn continually in the direction of the
motions of the particles through which it passes, C, a, and b, may be
OF THE MOTION OF FLUIDS. 177
supposed to vary in any manner along this line. The foregoing-
reasoning only proves that in passing at a given instant from one
point to another indefinitely near along the line, these quantities may
be considered constant.
4. The nature of the integral we have been discussing will perhaps
be understood by comparing it to the general integral of a common
differential equation, which has a particular solution. The latter, we
know, is that which gives the answer to a proposed problem, and the
general integral is used (though not necessarily) to obtain this solution.
So, I conceive, the integral above is useful for arriving at the particular
functions of x, y, and t, which give the velocity and direction of the
velocity at any point and instant in any proposed question. The
integral of -—^ + -r^ = , which M.M. Poisson and Cauchy have
obtained for the solution of the problem of waves, may be called the
particular solution of the equation, for that particular problem ; and I
think it probable that the same might have been obtained by employing
what I would call the general integral, though I am not prepared to
state exactly the process.
5. The following considerations are added in confirmation of the
foregoing reasoning. In whatever manner the fluid is put in motion,
we may conceive a line, commencing at any point, to be continually
drawn in a direction perpendicular to the directions of the motions at
a given instant of the particles through which it passes. This line
may be of any arbitrary and irregular shape, not defined by a single
equation between x and y. But it must be composed of parts either
finite or indefinitely small, which obey the law of continuity. Con-
sequently the motion, being at all the points of the line in the directions
of the normals, must tend to or from the centres of curvature, and
vary, in at least elementary portions of the fluid, inversely as the
distances from those centres. An unlimited number of such lines
may be drawn through the whole extent of the fluid mass in
motion.
178 Mr CHALLIS's RESEARCHES IN THE THEORY
6. If we put (f} = (pi + (p2 + (f>3 + Sac. we shall have
d^ d^_ (d^ d^\ Id^ dy,\ id^ d-^<pA , . _ „
daf "*" df ~ \dx' ^ dfj '^ \dx' "^ df) ■*■ \dx' ^ df)'^^-~^-
Hence if there be any number of functions which severally satisfy the
given equation, the sum of these will satisfy it. But from what has
been proved above, if
d^i Ci(x — a,)
d(p, _ C:(y-/3,)
dx ~ (ar-aO' + (y-/3,)"
dy (x-a,y + (y-l3,r'
a 02 Cs {x — aj)
d(j>, Q{y-fi,)
dx ~ ix-a,r + oj-(i,y'
dy (^x-a,f + iy-fi,Y'
&c. = &c.
&c. = &c.
<p\, 02. 03, &c. will severally satisfy it; therefore 0i + 02 1- 0, + &c.
will also. And we have,
dx dx dx
v=^ +^ +^ +&,c
dy dy dy
These equations prove that the velocity at any point may be the re-
sultant of several velocities produced by different causes; and that any
given cause will have the same effect in producing velocity at a given
point, whether or not other causes may be operating to produce
velocities at the same point.
7. We may here also determine the manner in which the motion
of the fluid is affected, when the rectilinear transmission of an impulse
tending from any centre is interrupted by a plane surface. For suppose
two impulses tending from two centres to be of equal magnitude and in
every respect alike. Then if the straight line joining these centres be
bisected at right angles by a plane, there will be no motion of the par-
ticles contiguous to the plane in a direction perpendicular to it, because
the resultant of the velocities from the two causes must lie wholly in
7)U>.1I- OF THE MOTION OF FLUIDS. 179
the plane. Hence as the division of fluids* may be effected without the
application of force, nothing will be altered if we suppose the plane to
become rigid and to intercept the communication of the fluid on one side
with that on the other. The motion on each side will then be reflected,
and the angle of incidence will be equal to the angle of reflection. -
8. I propose now to adduce an application of the proposition
above demonstrated (Art. 3.) respecting the general law of fluid motion,
which may serve to shew its utility. Suppose water in a cylindrical
vessel (for instance, a glass tumbler,) to be caused to revolve with con-
siderable rapidity about the axis of the cylinder. There is no practical
difficulty in making the fluid revolve so that every particle shall de-
scribe approximately a horizontal circle about the axis. Then, the fluid
being left to itself after the disturbance, each particle may be considered
to move as it does, by reason of a centripetal force tending to the
axis in a horizontal plane. This force must be owing to the action
of the cylindrical surface on the fluid particles in contact with it,
deflecting them continually from a rectilinear course. If V be the
velocity of the particles in contact with the surface, and a the radius
V-
of the cylinder, the force tending to the axis is — , the effect of
friction being neglected. The deflections which this force is continually
producing in the directions of radii, are transmitted through the fluid,
and as they tend to a centre, will vary, according to the proposition
above proved, inversely as the distance from the centre.f Hence the
V^ a V^
centripetal force at the distance r is — x -, or — . This shews
^ a r r
that at any distance r the velocity is still V. Experience seems to
confirm this result. For if light substances be strewed on the surface
of the water, those nearer the centre always perform their revolutions
* The introduction of this consideration here is merely reverting to a principle -which
Professor Airy (very properly, I think,) has proposed to make the basis of the mathematical
treatment of fluids. Without referring to a principle of this nature, I do not see that
problems of reflection can be satisfactorily solved.
+ The total motion is compounded of these deflections and rectilinear motions along
tangents to the circles, which by Art. 6. may be considered separately. ''' ' •"'-'■'■
180 Mr CHALLIS's RESEARCHES IN THE THEORY
in less time than those more remote. This is particularly observable
in two of the floating particles which are near each other, and at nearly
equal distances from the centre. That which is less distant overtakes
the other, as it ought to do, supposing it to describe a less circle with
equal velocity. At the centre a kind of eddy is formed, the more
observable as the motion at every point of the surface is more nearly
in concentric circles. When the revolving motion takes place in a
conical tunnel from which the water is issuing, the appearance at the
axis is very remarkable, a hollow space like a sack, being formed a
considerable way down the axis. What has been here said may serve
to explain in some measure the manner in which eddies in any case
are produced.
SECTION II.
On the Integration of the Equation -^ + -r^ + -r^ = 0.
9. M. Poisson has expressed the general integral of this equation
by means of definite integrals ; {Memoires de rAcademie des Sciences,
Ann. 1818), and this, I believe, admits of a discussion similar to that
applied above (Art. 3.) to the integral of -^ + -~ = 0. But perhaps
the following reasoning, analogous to what was indicated in Art. 5.,
may be considered sufficient. In whatever manner the fluid is put in
motion, we may conceive a surface to be described, which shall be
every where perpendicular to the directions of the motions at a given
instant of the particles through which it passes. This surface may be
of an arbitrary and irregular shape, not necessarily defined by a single
equation between x, y, and %. But it must be composed of parts either
finite or indefinitely small, which are continuous, and consequently have
radii of curvature subject to the same conditions as those of regular
curve surfaces. Hence the normals to all the points of any element
of the surface will pass through two focal lines, situated at the centres
and in the planes of greatest and least curvature, and cutting the
OF THE MOTION OF FLUIDS. 181
directions of the normals at right angles. The motion, being in the
normals, will be directed to the focal lines. If we describe another
surface indefinitely near the first, and cutting in like manner the direc-
tions of the motion at right angles, all the points of any fluid element
intercepted between two opposite elements of the surfaces, will at a
given instant ultimately have their motion directed to the same focal
lines : but this cannot be said in general of more than an elementary
portion. If we suppose the form of the superficial element to be a
rectangle, the normals through all the points of its sides, will inclose
a wedge-shaped mass, the transverse section of which at any point, it
is easy to shew, will vary as the product of the distances of that point
from the focal lines. Hence the velocity in passing at a given instant
from the first to the second of the surfaces above-mentioned wiU vary
inversely as this product. Let therefore r and r + l he the distances
of the point whose velocity is V, from the focal lines to which the
C
motion is directed. Then V= . j-, in which expression C, /, and
the positions of the focal lines are constant at a given instant, when
r varies through a space which may either be finite or indefinitely small.
Let a, /3, 7, be the co-ordinates of the middle of that focal line which
is distant by r from the point in question. The velocity (m) in x will
then be V. ; the velocity {v) in y, V. ; and the velocity
{w\ in », V. ^. Hence
' r
udx + vdy + wd%= Vi — ~dx + ^ , dy H -d%\ .
Now since r- = {x — af + {y — fif-\-{%~yY, if we make r vary with
X, y, and %, while a, )3, 7, remain constant according to what has just
been said, we shall have rdr — {x — a)dx + {y-fi)dy + {% — y)dti. Hence
tfdx + vdy + wdz=F^dr; and as F" is a function of r and /, the right
side of the equation is a complete differential of a function of
X, y, %, and t, with respect to the three first variables, t being con-
stant. Therefore also the left side is the same. Let the function be <p.
Vol. V. Part II. A a
182 Mr CHALLIS's RESEARCHES IN THE THEORY
Then
dr ' dx ' dy ' d%
We proceed to shew next that the equation
d^d) d^d) d^d> „ du dv dw
zr^ + j^ + -j^ = 0, or J- + T- + ^- = 0,
daf df d%^ dx dy dx
is satisfied by the kind of motion we have been describing.
10. Let P (Fig. 1.) be the point whose motion we are considering;
Or, Nq, the focal lines to which the motion of the element at P is
directed. Let PNO be the straight line which passes through P and
the focal lines, cutting them in N and O. Suppose O to be the
origin of a system of axes, of which ONP is the axis of x, Oy coinciding
with the focal line Or the axis of y, and 0% perpendicular to the plane
yOx, the axis of %. The co-ordinates of P referred to another system
of rectangular axes AX, AY, AZ, are X, Y, Z: p is a point
indefinitely near to P, Pp is parallel to AZ, and the co-ordinates of
p are X, Y, Z+SZ: pqr is the straight line which passes through p
and the focal lines cutting them in q and r. Now let the equations
of Pp referred to the system Ox, Oy, 0%, be x = a% + a, y — b% + fi,
and the equations of pqr, x = dz-\-a, y = b'z + li'. Then
„ l+aa' + bb'
cos ^ Ppq = „ , — .
Let ON=l, NP=r. Hence because Pp passes through P whose
co-ordinates referred to the axes Ox, Oy, Oz, are I + r, 0, 0, it follows
that l + r = a, and /3 = 0. Thus the equations of Pp become x = az + l + r,
y = bz. Again, because the line ^^gr passes through r, whose co-ordinates
are x — 0, z = 0, we have a' = ; and because it passes through q, whose
co-ordinates are y = 0, x = l, we have l=a'z, and = i's: + /3'. Hence
a: = - = - -n, and consequently ft' = r. Thus the equations of
pqr become x — a'%^ y^V% y . Also because Pp and pqr pass
OF THE MOTION OF FLUIDS. 183
through the same point p, a; = «'x = a« + /+r, and therefore ^ = -7 .
And y = hz = h'% ^i therefore z = -rm — tx • Hence -; = —rrr, — 7t>
'' a' a{b'-h) a -a a{b-b)
which gives h' = — ; j^. From p draw ps perpendicular on Ox,
and let P.? = 5. Then ^ = x-{r + l). Bnt x = a'z = t!^±Il, Therefore
' a —a
^ = — 7 '- . Hence it will be found that a' = — — » , and
a —a 6
V = — J — -. This latter quantity, if we neglect powers of S above
the first, is equal to (l H — rj A. Therefore by substitution
„ d r \ r{l + r)J
cos / Jr«o = — -. . , , ,.
a'(^r + l)+(l+d' + b'—]s
= (neglecting ^, &c.) V ^ /
Here / „ === is the cosine of the angle pPs. Hence if ^ = the
V 1 + a^ + o^
velocity at P in Ox, and w the part resolved in the direction parallel
Va
to AZ, w — — / 2~ ^i • ^^^ ^ ~ ^^ resolved portion of the velocity
at p in the same direction. Now the velocity at p is ultimately the
same as that at s, and is therefore equal to V . -, A^ \ — r- ,
according to the law of variation from P to s determined . by the
AA2
w
184 Mr CHALLIS's RESEARCHES IN THE THEORY
considerations with which we commenced this investigation. Neglecting
powers of I above the first, this quantity becomes V \\ ^ J .
Consequently
a
But S = SZcospPs = SZ X , 'g.J p' Hence
w'-w _ „ / \-d 1 ¥-a^ 1
~IZ U +«' + *'■ / + r "^ l+«2 + fr^'r
If now a, ft, 7, be the angles which the axis AZ makes with
Ox, Oy, 0%, respectively, we have
Hence passing from differences to differentials,
-7- = (COS^'V — COS'o)^ + (cOS^/3-COS*a) - (1).
d% ' ' l + r ' r '
So if d, /3', 7', be the corresponding angles for the axis of Y, and
a", /3", 7", for that of ^, v the velocity in F", and u that in ^, we shall
have by a like process,
^ = (COSV - COs'a') ,— + (coS^/3' - COS^a') ....... (2) ,
^ = (cos^y- COs'a") y^ + (cos'/3"-COS^a") ^ (3).
OF THE MOTION OF FLUIDS. 185
But as a, a, a", are the angles which Ox makes with three rectangular
axes,
cos" a + cos" a + COS" a" = 1,
so cos-/3 + cos^/3' + cos^/3" = l,
and cos'^7 + cos'^7' + cos^7" = l.
Therefore by adding the equations (1), (2), (3),
du dv dw _
7lX^ dY^dZ~
11. The general conclusion from all that precedes is, that the law
of the variation of the velocity from any point to another indefinitely
near in the direction of the motion, at a given instant, may be expressed
C
by -^ — J-, the quantities C, r, and I, being such as we have stated
C
in Art. 9- If 1=0, we have- as a particular case, V=-^. In my
former paper on the motion of fluids, I assumed, as it now appears,
C
incorrectly, that — represents the general law of the variation of the
velocity. None, however, of the results in that paper are affected by
the assumption. For instance, the expression for
as it only requires that (p should be a function of r and /, will remain
the same. This expression may also be briefly obtained thus. We
have seen that -~- = V. Now as r is ultimately in tlie direction in
which the velocity V takes place, if a line commencing at a given
point be drawn constantly in the direction of the motion at a given
instant of the points through which it passes, dr may be considered the
increment of this line. Hence if we call its length s reckoned from
the fixed point, -j^ = -^ = F. Then integrating, c}> = jVds -^/{t);
1«6 Mr CHALLIS's RESEARCHES IN THE THEORY
and differentiating under the sign /, ^ = f -r-^^ +J''(^)- Hence
substituting for -^ in the known expression for the pressure {p),
p = f(Xdx + Ydy + Zdz) - f^ds - ^ -fit).
If f^ be always the same in quantity and direction at the same point,
dr V^
-^ = : so that, p = j{Xdx + Ydy + Zd%) - -— -f{t).
This equation may thus be considered to be strictly deduced from the
general equations of fluid motion.
Considerations analogous to those applied (Arts. 6 and 7) to motion
in a plane, might here be introduced to shew that the motion at any
point, when due to several causes, is the resultant of the motions which
would be produced by the causes acting separately ; and also to determine
the same law of reflection at a plane surface.
12. The following simple instance of fluid motion may serve to
illustrate some points of the preceding theory. BCD (Fig. 2.) is a
conical vessel with its axis vertical. A mass of fluid DBhd is made
to descend so that its lower surface hd is bounded by a horizontal
plane to which any arbitrary velocity is given. The upper surface is
also supposed to be plane and to be kept horizontal by the force of
gravity. It is required to find the consequent velocity and pressure
at every point of the fluid.
It is evident that the motion will be in vertical planes passing through
the axis, and will be, the same in all such planes. Take therefore two
planes making an indefinitely small angle with each other, and let
AB, AE, be their intersections with the upper surface, ab, ae, with
the lower. Let PQSB be an element of the upper surface, P and B
being equidistant from A, as also Q and S. If now at any instant
lines commencing at the four points P, Q, B, S, be continually drawn
OF THE MOTION OF FLUIDS. 187
in the direction of the motion at the points through which they pass,
these lines must be rectiUnear, because there is no curvilinear motion
at the boundaries of the fluid, and therefore no cause to impress a
curvilinear motion on the parts interior. The straight lines commencing
at P and R will intersect ah and ae at p and r, points equidistant
from a, and those commencing at Q and S will intersect the same
lines at q and s also equidistant from a. Now from the law of
the variation of the velocity above found, at every point of the cunei-
form element Ps, the velocity will be inversely proportional to its
transverse section. Let therefore V =^ the vertical velocity with which
(lb is made to descend, and v the vertical velocity with which the
surface DB descends. Let AB = a, AQ = x, PQ = X, ah = h, aq = x',
pq = 'S.', and the angle BAE = e. Then the element PQSR^^xeX,
and pqsr = x'e\'. These elements are proportional to the transverse
sections at P and p ; and the vertical velocities V, v, are to each
other as the velocities at p and P in Pp. Hence — = -; — , = -V-, •
-' ■* V X e\ xX
F • Wence ^, - ^
because the motion is along the slant surface. Therefore in this case
X a
r-, =" T. Suppose X to be given, and let Xi be the consequent value
X o
of x'. Then — = -r, and -. = y . If now x be taken = « — X, from
X, o b-Xi o
what has been just shewn, x' will = 6 — X, . Hence 4 — = j^, and
\0 — Xi) X2 o
consequently — = t- Therefore X2 = Xi; and so on. From this it
X2 o
follows that if AB and ab be divided into the same number of
indefinitely small equal parts, the straight lines joining the corresponding
points of division will give the directions of the motion, which is
consequently every where directed to the vertex of the cone. Hence the
velocity af^ any point W whose distance Cp W from C is p, varies as — .
P'
Let CA =h, Ca = k, z AC W= 9 ; then the velocity at p=V sec 9, and
the velocity at W= V sec 9 . ^ — ; this resolved in the vertical
But — also = jj, . Hence ^j^, = 71 • If we take x = a, x' must = h.
188 Mr CHALLIS's RESEARCHES IN THE THEORY
f^k'sec'6 VTf
direction gives j— — , which = — — = velocity at Z. Hence the
vertical velocity is the same at all points of any horizontal plane, and
the fluid will consequently descend in parallel slices.* Let us now
determine the pressure at any point on the particular supposition that
V is uniform. Then if
Vk"&eee ^, , .. , „r dw Vsec'd ^,dk 2F"Asec'0
w = :: the velocity at W, -7- = ; — x 2«-7-- = y, .
p' ■'at p^ at p-
And
Idt^' ^-Jdt'^P-J 7 = -p + ^
»
Hence
„ 2r'kse&9 r'k'sec'e
p = C-g. + . __.
And as when z = h, p — 0, and p cos Q = h, it follows that
The above solution I do not consider to be of any value, except as
illustrating the process to be followed in determining mathematically
the way in which the interior of a mass of fluid is affected as to
velocity and pressure, in consequence of given conditions at the
boundaries. This part of the theory of fluid motion is very
defective.
* I obtained this result in the number of the Phil. Mag. and Annals of Philosophy
for .Jan. 1831, but omitted to shew that it is entirely dependent on the arbitrary condition
that the inferior -surface of the fluid is bounded by a horizontal plane. Qji any other
supposition the problem would be one of much greater difficulty. This omission has not
without reason caused a misapprehension as to the application of the solution, on the part of
Berzelius in a notice taken of it in his Annual Review. {Jahres-Bericht, 1833.)
OF THE MOTION OF FLUIDS. 189
SECTION III.
Application of the Principles of the foregoing Section to an instance of
the Resistance of an Incompressible Fluid to a Body hounded by a
Spherical Surface moving in it.
13. Let a solid sphere, partially immersed in water, being of less
specific gravity than the fluid, be drawn along in a horizontal direction
with a given uniform velocity ; it is required to find the height of
its centre above the horizontal surface of the water.
We shall suppose for the sake of simplicity, that the fluid is
unlimited in extent both in the vertical and horizontal directions, and
that the surface of the sphere is so smooth that it impresses no velocity
on the water in contact with it in the direction of a tangent plane.
Let CDJBJE (Fig. 3.) be the sphere, O its centre, ADE the intersection
of the horizontal surface of the fluid by a vertical plane through the
centre of the ball; OQ a line through the centre parallel to ADE.
This will be the direction of the motion of O, since the velocity is
supposed to have become uniform, and ON to be constant. Let A,
a fixed point in ADE, be the origin of co-ordinates, AN=a, NO = 'y,
at any instant. Then the velocity {V) of O = -r-. Draw OB vertical;
let P be any point of the surface immersed; through P draw the
spherical arcs PQ, PB, and let the angle QOP=6, and the angle
PQB = to. The velocity impressed by the sphere on the fluid at P
is F'cos 9, as none is impressed in the direction of a tangent plane.
This velocity is directed to the point O, because in the case of a
spherical surface / = 0. Hence if « = the radius of the sphere,
C
FcosO = — . (Art. 11.) The velocity at every point of the line OP
produced, wiU at a given instant be in the direction of this line,
Vol. V. Part II. B b
190 Mr CHALLIS's RESEARCHES IN THE THEORY
because when the fluid is of unlimited extent, there is no cause* to
produce motion at any point of the line, but the impression made at P,
which is transmitted instantaneously, varying at different distances
from O according to the law of the inverse square. Hence if ^ be a
point in OP produced, and OR = r, the velocity at R — —, = ^ — .
Let ADE be the axis of x, a vertical through A the axis of z reckoned
positive downwards, and a line through A perpendicular to the plane
of these two the axis of y. Then if the co-ordinates of R be x, y, %,
we shall have r- = (:r — a)" -I- y^ + (s + 7)' ; and cos0= . Therefore
the velocity {v) at R,
•A-A I'?
VoH^X-a)
Ka;-ay^ + y' + (» + 7)-}5"
And
Hence
dv dv da re rr :i 4. ..k
~j- = 7- • 77 > (lor ^ and 7 are constant),
_ F«^(3cos-e-l) (la
r" ' dt
rV(3cos*^-l)
/^rf,=/(o-g5:(3cos'e-i).
Therefore, gravity being the only force acting on the fluid, the pressure
ip) at R,
* This cannot be said of the parts of the fluid adjacent to the radii produced which pass
through the circle in which the surface of the water meets the surface of the sphere, because
the water outside of the conical surface formed by these radii must be put in motion by that
within by reason of the difference of pressure occasioned by the motion. On account of the
difficulty of estimating this effect, it is left out of consideration in our solution, which can
therefore be only considered approximate.
OF THE MOTION OF FLUIDS. 191
= ^« + -27^(3008-'^- 1) - -g^-cos'e -f{t).
When r is indefinitely great this equation becomes p=g^—f{t)', and
as for this value of r the velocity = 0, p must = g% ; therefore /{t) = 0.
If now we put r = a, and i8 = ss,, the co-ordinate of P, we obtain the
?^* cos 20
pressure (/>,) at P, = gz, -\ . The portion of this resolved in
the vertical direction is jo, x cos i FOB. But from the spherical
triangle PQB, cos /. FOB = cos w sin 9. Therefore the vertical pressure
is p, cos w sin 6. The element of the surface at F = ad9 x a sin ddw.
Hence the whole vertical pressure = //jOia'sin''^ cos wt/^c^w
=ga^ff%i sin^OeoswdOdw + ——— ff sin^ 9 cos29 cos wd$dw.
M
The first term is plainly the weight of water displaced, and is there-
fore equal to — -(2«' — 3«'7 + 7*), the specific gravity of the water
being 1. The integrations with respect to u> must be taken from
a,= — cos"^ — jT—r to -f cos"^ — ^-;r , and the integrations with respect
to 6 from sin"'— to the supplement of that arc. Between these limits
of w, fcoswda) = 2\/i T ; and between the limits of 9,
a'^sm''9
2fsm'9cos29d9 x/i _ , '^^ =_ J fi _:>:!') .
•^ ^ «*sin''0 2 V «V
Therefore if JV = the weight of the sphere, which is the same as the
whole vertical pressure, and w = the weight of fluid displaced,
IV =w-
4
B B 2
i^-i)-
192 Mr CHALLIS's RESEARCHES IN THE THEORY
This result shews that the weight of fluid displaced is greater than
the weight of the sphere, and consequently that the centre O is lower
than it would be in a state of rest.
Suppose a portion of the sphere to be cut off" by a horizontal
section at the distance of b from the centre ; and let 7 become 7', the
centre being still above the surface of the water. Then if we suppose
the motion to be always in the direction of the radii*, and the horizontal
bottom to have no effect in impressing motion, the equation for this
case will be.
W=w-
TrF'a'
ttTV
= w : —
The difference between W and w is here less than before on account
of both the factors —; and 1 — -yr ; for -?- is greater than - . This
a* b* b ^ a
seems to shew that curved bottoins tend to depress the vessel when it
begins to move, and consequently to increase the resistance.
As the term —-— ff sm^6eos26 cos uidOdu) is positive from 0=:sin"' —
to = 45°, and from = 135° to 6 = the supplement of sin"' — , let us
Cv
integrate for the portion of the surface corresponding to these limits,
or what amounts to the same, take the double of the integral between
the first limits, those of w remaining the same as before. In order to
abstract from the consideration of the portion of the surface not taken
into account in this integration, we may suppose the portions for
which we integrate to be connected by a cylindrical surface, the radius
of which = a sin 45°. The length of this cylindrical part may be any
we please : the vertical pressure against it will be only equal to the
* This again cannot be true in the direction of the radii which pass through the lower
circular boundary of the surface.
OF THE MOTION OF FLUIDS. 193
weight of fluid displaced. Also the shape of the floating body above
the part immersed is of no importance to the problem. The form of
the whole body may be such as is described in Fig. 4, ABCDEF
being a half cylinder of which the axis is GH, and ALC, FKD, the
extreme portions of the body, bounded by spherical surfaces which have
their centres at M and N. Now in general ^ jjsin^d co^^O co& wdQdw,
commencing at = sin~'— , and ending at any other value of 9, will
be found to be
cosefssin^e + l-^') V sin^e-^ -\(\-—i
And if we put cos 9 = — ?= , we shall have
COS0
a-
V
W = w +
As the second term is necessarily positive, the floating body will be
higher than it would be in a state of rest, and consequently the
surface against which the resistance acts becomes less by an increase
of velocity.
To obtain a numerical result respecting the rise of the body
corresponding to a given velocity, we will suppose for the sake of
simplicity of calculation that when the vessel is at rest, the centres
of the spherical ends and consequently the axis of the cylindrical part,
are in the plane of the horizontal surface of the water. This circum-
stance may be produced by loading the upper part of the body
. without altering its specific gravity. Let / = the length of the axis
of the cylindrical portion. Then the area of the horizontal section of
the vessel at the level of the water surface is ID H ■— — , its
4 2
breadth being D. Now W—w must be equal to the difference of the
19* Mr CHALLIS's RESEARCHES IN THE THEORY
quantities of fluid displaced in the states of rest and motion, and is
therefore equal to yg \ID+'^ — j , 7 being small. Therefore
neglecting powers of — above the first,
Let ^ = 3. It will then be found that F' = 696** x 7. And if 7 = one
inch, or ^, this equation gives ^=519 miles per hour; consequently
if ^=10*4 miles per hour, 7 = 4 inches.
2
In general, neglecting "—, &c.
TV-w==
r'a'
sin e cos e (2 sin'0 + ^ ) ~ |) '
also W — w = yg llD + ^---iAO - sin 9 cos6)\ nearly ;
therefore, as I> = Zasm9, it will be found that
F- sin2 0(2sin'0 + l)-0 , . ^ /
y = -r- •'-4 '--TT, — • r.n r,n y w? bemg put for -y\.
' 4!g 4!msm'9-sm26 + 29 " * D
If 9 be assumed equal to 15°, and 711 = 3, this equation gives ^"=7-35
miles per hour when 7 = 4 inches.
These results, which probably are but very rough approximations
to matters of fact, may yet suffice to shew that when vessels and boats
of the usual forms sail in the open sea, they may be expected to rise
in some degree upon an increase of their velocity, and so much the
more as they are less adapted to cleave the water. Our theory shews
that the rise is the same for bodies of the same shape and proportions,
moving with the same velocity, whatever be their absolute magnitudes;
also that this effect is equally due to the pressures on the front and
OF THE MOTION OF FLUIDS. 195
stern of the vessel. The theory, in fact, determines these pressures to
be in every respect alike, so that if we proceeded to investigate the
total pressure in the horizontal direction, we should find it to be
nothing, when the motion is uniform. This may serve to shew that,
if friction be left out of consideration, a front ill-adapted to cleave the
water, is not unfavorable to speedy motion, if the stern be of the same
shape; and that the resistance to the motion of vessels in the open
sea is principally owing to the friction of the water against their
surface. This cause operates to produce unequal actions on the front
and stern, making the directions of the motions of the particles in
contact with the surface of the former, less inclined to the horizon
than they would be in the case of no friction, and of those in contact
with the surface of the latter more inclined. To counteract this inequality
probably the stern should be less curved than the front.
SECTION IV.
General Propositions respecting the Motion of Compressible Fluids.
14. The considerations applied at the beginning of Section II. to
incompressible fluids, are equally applicable to compressible. I shall
therefore assume that in a mass of fluid in which the density varies
as the pressure, the directions of the motion at all the points of any
element pass at a given instant through two focal lines. Let p be
the density at a point distant by r and r -vl from the focal lines, and
V the velocity : p and V the same for a point indefinitely near the
former. Also let the transverse section of a cuneiform element aclk
(Fig. 5.) which is bounded by four pli.nes passing through the focal
lines kl, mn, be at the first point efgh, and at the other, abed. The
pressure and consequently the density will be the same at all points
of the section eg; as also the velocity; at least our reasoning does not
apply to cases in which this condition is not fulfilled. The same may
be said of the section ac and of all sections intermediate to ac and eg.
196 Mr CHALLIS's RESEARCHES IN THE THEORY
Let now the area of eg = m, and that of ac = m'. Then if the motion
which exists at a given instant, be supposed to be continued uniform
for the small time t, the quantity of fluid which passes the section eg
in that time, is mpF^T, and that which passes ac is m'p'Vr. Hence
the increment of matter between the two sections is — {m' p'V'T — mpVT),
whether the velocity tend from or to the focal lines, being considered
negative in the latter case. The increment of density {Ip) of the element
in the time t, is consequently — ^^ — - — r—, — —. — — ■ But — = — ^^ =^ .
^ •' m{r'-r) m r{r + l)
Hence
pT'r'ir' + D-prnr + l) _^^^^^^Sp_^
And passing from differences to differentials,
^^^^^'dt ~ dr
or
dp dV ,.dp ,^ /i 1 \
As before udx + vdy + wd% = V dx + V^^ — — dy + V d% = Vdr,
" f* T T
if a, /3, 7, be the co-ordinates of the middle point of the focal line hi.
Now as we have supposed that in passing from one point to another
of tlie element acge, the change of velocity at a given instant depends
only on the change of r, we may consider V a function of r and t,
and Vdr a differential of a fimction of r and t. Then udx ^ vdy
+ wdfi = d(l), a complete differential of a function of x, y, and as; and
-~ = V. But in this case we have the known equation,
a' Nap. log. p^fiXdx + Vdy + Zd%) - ^ - ~ (B.)
Therefore considering X, V, Z, to be independent of the time,
d'dp _ d^(p dV d'(ji d(p d'<p
pdt ~ df dt ~d¥ ~ 'dr ' drdt '
OF THE MOTION OF FLUIDS. 197
But from (A),
pdt pdr ' dr dr' dr \r r + l) '
And differentiating (B) with respect to space only,
^1^ = Xdx+ Vdy + Zdx-d.^ - VdV.
p at
If the variation be from one point to another in the direction of the
motion, dx = dr, dy = - — — dr, dz = dr. Hence,
r ^ r r
a\dp ^ X ^-° , Y y~^ + Z ^^^ _-^ d(p d'(f>
pdr ' r ' r ' r drat dr ' di^ '
Substituting this value of — ^ in the foregoing equation, and then
equating the two values of ,'] , we shall obtain,
/ d£\d^_Q^ d^ d^t . ,.d^(l , J_\
\ ~ dt^j dr' dr ' drat df "*" dr\r "^ r + l)
+ ^ (x^^ + ry^ + Z'-^) =0 (C.)
dr \ r r r I
This is an equation of general application. If, as a particular case,
I, a, /3, 7, each = 0, we shall have the equation I obtained in my
former paper (Art. 4.) by assuming ^ to be a function of v^a^ + y^ + s!^
and t in the equation {n) of the Mecanique Analytique (Part II.
Sect. XII. Art. 8.)
It may be proved as in Art. 11, that -^ = /~77 ^*' ^^ ^'^^ incom-
pressible fluids, and that the equation applicable to steady motion is,
a' h. 1. p = fiXdx + Ydy + Zd%) - ^ + fit) .
Vol. V. Part II. Cc
198 Mr CHALLIS's RESEARCHES IN THE THEORY
15. If r be indefinitely great in equation (C), the motion is in
parallel lines, and putting r = c + s, j=j- Let -^ = w, and
suppose no force to act ; the equation for this case becomes
d''(p 2w (Pep 1 d'(p _
~d? ~ '^^' ■ dsdt "^ o^^T^ ' dF~^'
This equation combined with a* N. 1. p = — -^ — — , gives as a particular
integral, u] = al:iA. p =/"{«- {a + w)t\. By varying a little the mode of
_ ^( as.
integrating, I found also w — a^A. p =/( ■ atj, {Camb. Phil.
Trans. Vol. III. Part III. p. 399), and endeavoured to shew the way
in which each integral ought to be applied. But this enquiry was
unnecessary ; for the integral may present itself under an unlimited
number of different forms. The equations
w = a^.\.p=f{.^-{a + io)t + ^{w)], or «, = «N.]. jo=/(^^^^%i^l ,
will equally satisfy the differential equations, being, in fact, only
different forms of the first-mentioned integral. The principle according
to which it now appears to me, an integral of this nature should be
employed, is to apply it immediately only to the parts of the fluid
immediately acted upon by the arbitrary disturbance, in order to
determine the law according to which the initial velocity is transmitted
to the contiguous parts ; then to determine the law of transmission
from these to the next; and so on in succession. In the present
instance by making * and t vary so that w and p remain the same,
ds
we shall find a + w for -j~ the velocity of transmission, under whatever
form the integral may appear. The second term m of this quantity
is due to the transmission of velocity through space by the motion of
the particles themselves ; the other a is the velocity of propagation
along the particles. In this example, as the velocity and density are
propagated uniformly and undiminished, it is easy to determine at any
OF THE MOTION OF FLUIDS. 199
instant the velocity and density at any given point, which result from
a given disturbance. In other cases in which the velocity of propaga-
tion is variable, the determination would be more difficult, but must
probably be arrived at by the same principle of reasoning. Variable
propagation is analogous to variable motion, as uniform propagation to
uniform motion, and would seem to require integration to determine
the time at which the effect of a given disturbance is felt at a given
place.
16. If in the equation (C), a be an indefinitely great quantity,
the terms which do not contain a^ as a factor may be neglected in com-
'parison of those which do, and the equation will become
dr^ ^ dr\r ^ r + l) '
which by integration gives -^ = — j-, the same as for incompressible
fluids. This result was to be expected, because a, as is well known,
is the velocity of propagation in the compressible fluid, and when this
becomes infinite, the propagation is instantaneous, and the fluid there-
fore incompressible.
If / be indefinitely great, it will be found in the same way that
-r~ — — , and the motion is such as was considered Art. 3.
dr r
Let now -^ be very small compared to «, and X, V, Z, and /
each = 0. The equation (C) reduces itself to
"-11?^^"^ dr df-^' '''''• dr' -~dF~'
a particular integral of which is r^=^'P{r — at). This gives
d^ ^ F\r-at) _ F{r-at)
dr ~ r r^ ^ "'
CC2
200 Mr CHALLIS's RESEARCHES IN THE THEORY
At the same time, because a^'N.l.p= — -^ nearly, we shall have
, T , F'{r-at) ,^,
«.N.l.p = ^ -(2.)
The equations (1) and (2), involving but one arbitrary function, can
apply only to a single disturbance, which takes place in a direction
tending from a centre, as I have elsewhere shewn*. It is important
to observe that when r is very small, the term of equation (1) which
involves r"- in the denominator may be much greater than that in-
volving r. In fact, if we expand the fxmctions, supposing r to be
very small.
&c.
_ F{-at) _ F'{-at) _ F"{-at)
When therefore the disturbance is made by a sphere of very small
radius r, the motion is transmitted from its surface to other parts of
the fluid, nearly as if the fluid were incompressible.
SECTION V.
Application of the Principles of the foregoing Section to determine the
Resistance of the Air to the Motion of a Sail-Pendulum.
17. For the sake of simplicity, I will suppose gravity not to act.
The ball being spherical and perfectly smooth, the direction of the
motion of the aerial particles in contact with its surface tends at every
* Camh. Phil. Trans. Vol. HI. p. 402.
OF THE MOTION OF FLUIDS. 201
instant from its centre. Therefore / = 0. Also if the radius of the
ball be supposed very small, the equation -f- = ^-t^> obtained at the
end of the preceding Article, will be approximately applicable to the
motion of the fluid in contact with the ball. Hence the velocity which
is impressed at any point of the spherical surface may be considered
to be transmitted instantaneously in the direction of the radius through
that point, and to decrease according to the law of the inverse square
of the distance. The problem, with the limitations above made is
solved in the same manner for air as for water.
Let now the origin of co-ordinates be A, (Fig. 6.), the position
of the centre of the ball when it hangs at rest. I^et its centre perform
oscillations of very small extent in nAN, which we will consider to
be rectilinear. Suppose N to be the position of the centre at the
time t reckoned from a given epoch, and call AN, a. Take P any
point of the surface, join NP and produce it to R, and let NPR make
an angle Q with ANQ, and the plane RNQ an angle /3 with the
plane SAQ. The velocity of the centre = ^; and the velocity of
da
the air at P — -rrCosO. Hence if NP=h, and NR = r, the velocity
at ^ = -„ — . -^ . Now if AN be the axis of x, AS of a, and a
r- at
line through A perpendicular to the plane SAN, the axis of y, and
the co-ordinates of R be x, y, %, then r^ = {x — aY + y^ + %^. Consequently
the velocity (^) at R=, r^ ^ — 2 • ;77- Hence differentiating V^
With respect to the time only,
dr _ d'a b^cos9 2b^cose{x-a) d^ h^ da d.cosO
dt ~ W-' r' "^ 7 • dt^ '^ r"' df dt '
^ ^ x — a d.cosO 1 da cos^6 da sin^O da
But as cos9 = , r: — = --77 +
Therefore
dt . r ' dt r ' dt r ' dt
dV_d^ ¥cos9 ^b'cos'e da' b'sin'0 d^
dt ~ df r^ ^ r' ' dt' r' ' dt^
202 Mr CHALLIS's RESEARCHES IN THE THEORY
Hence
J dt df • r 2? • df '
Substituting in equation (B),
j\-. - d^a FC0S9 b'^ ir, 2/, -am da b* COS^ 9 da „.^^
« ^-^-P = df ■ —r- + ap (2cos=0-sm=0) ^ - -^^ . ^ -M.
When /• = infinity, /o = 1 : therefore f{t^ = 0. Hence when /• = A,
„,T. , «?^a , . COS 20 <:?a^
Where p = 1, let j9 = n = a^ Hence when (O = 1 + o-, p = e' (1 4- a-) = n + aV.
But because a- is very small, «^N. l.jo = «V very nearly. Therefore,
„ d^a , - cos 20 rfa^
^ = n+^.*cos0 + -^.^.
The total pressure resolved in the direction NA is ffp¥ eos6sm9d9dfi,
from /3 = to /3 = 27r, and from = to = 7r. It will consequently
be found to be equal to — — . -^ : and if A = the ratio of the specific
gravity of the ball to that of air, the accelerative force produced by
1 d'a
this pressure is — . -7-7 . But the accelerative force of gravity in the
same direction, if SA = A, is ^ ( 1 ~ t" ) » taking account of the weight
of air displaced. Hence
_ cP_a _g^(-._}\ j_ d^
d'a ^
or
«^__^ ±^_§^(l_l] nearly
1 + K
OF THE MOTION OF FLUIDS. 203
Therefore if L be the length of the seconds pendulum in vacuum,
2s *
I in air, / = Z« ( 1 — — j
The correction of the length of the pendulum is thus determined
to be double of what it would be if the motion of the air were not
considered. It is to be observed that these calculations apply strictly
only to the case of a very small ball. The experiments of M. Bessel
give 1"956 for the coefficient of — . Those of Mr Baily, which were
made most nearly under the circumstances which the theory supposes,
give 1"864. The effects of friction and of the suspending wire, would
tend to make the coefficient rather greater than less than 2. I am
therefore unable to account for the difference between the experimental
and theoretical determinations, which it appears by Mr Baily's experi-
ments, is greater as the radius of the ball is greater, excepting perhaps
the confined space of the apparatus may have had some effect on the
experimental results.
It would not be difficult to shew from the nature of the analytical
expressions, that if the confined space in which the balls vibrate were
taken into account in the theory, the same results would be obtained
for two balls of different diameters, vibrating in different spaces, if the
linear dimensions of the spaces were in the proportion of the diameters,
their forms being alike. If this could be verified experimentally, it
would shew that the difference of the values of the numerical coefficient
which Mr Baily calls n, for balls of different diameters, as well as its
deviation from the theoretical value 2, is very probably owing to the
confined space of the vacuum apparatus. It would at any rate be de-
sirable to ascertain by experiment whether the same ball gives the same
value of n, when it oscillates in apparatus of different dimensions.
Papworth St Everard,
March S, 1834.
* This result I obtained in the London and Edinburgh Philosophical Magazine (September,
1833), by assuming the principle of the conservation of vis viva, without employing equa-
tion (B).
205
IX. Theory of Residuo-Capillarij Attraction; being an Explanation of
the Phenomena of Endosmose and Exosmose on Mechanical
Principles. By the Rev. J. Power, M.A. Fellow and Tutor
of Trinity Hall, and late Fellow of Clare Hall, Cambridge.
[Read March 17, 1834-3
1. The curious and elegant law, according to which an interchange
takes place between two fluids separated from each other by a thin
membrane, one of the fluids generally (but not universally) the lighter
of the two, being transmitted in greater abundance, was discovered a
few years ago by Dutrochet.*
His experiments tended to show that the unknown force which
operated this effect, whether measured by the fluid transmitted in a
given time, or by the pressure required to stop the process, was, for
the same substances, proportional to the difference of densities of the
mixtures on each side of the membrane.
The vast importance of this law in animal and vegetable physiology,
renders it highly desirable that its theory should be investigated on
mechanical principles, and such is the object of the present enquiry.
2. The opinion which would attribute this phenomenon to the
existence of electrical currents, is now pretty nearly abandoned, even by
Dutrochet himself, with whom it originated, and who maintained it with
great zeal, until the publication of his later researches, in which he
* L'Agent immedial du Mouvement Vital, (Paris, 1826), and Nouvelles Recherches sur
I'Endosmose et VExosmose (Paris, 1828).
Vol. V. Part II. Dd
206 Mr POWER'S THEORY OF
confesses himself compelled to resign it, though he does so with
manifest reluctance. That electricity, artificially excited, is capable of
accelerating the process, is indeed sufficiently established by the experi-
ments of Dutrochet; but it is equally certain that this agent is by no
means essential to the operation, since, in the natural process, the most
delicate galvanometer gives no indication of its existence.
3. To me it appears unquestionable, that the phenomenon results
from the corpuscular attractions, which the particles constituting the
membrane and the fluids, exert upon each other : that electricity,
by heightening or modifying these attractions, should produce a sensible
effect upon the operation, is nothing more than its ordinary chemical
agency would lead us to expect.
4. By corpuscular attractions are meant the forces which the
ultimate atoms of different materials, whether simple or compound,
exert upon each other. These forces are enormously great (though not
infinite) when the particles are in immediate contact, but diminish with
extreme rapidity, as the particles separate, becoming insensible at a
sensible distance. The effects of corpuscular attraction are different,
according as it is exerted between particle and particle, or between
mass and mass. In the former case it gives rise to the phenomena of
chemical affinity ; and in the latter, to those of cohesion, adhesion, and
capillary attraction, which may be regarded in general, as the mutual
attraction of contiguous masses, being the combined effect of the
corpuscular attractions of their integrant particles. It is under this
point of view that La Place has considered the subject of capillary
attraction, and his theory will be of the greatest use in the present
investigation.
5. Although no pores can be detected in the membranous partition
by the help of the most powerful microscope, yet the fact that the
fluids are transmitted, is a certain proof that such pores exist. They
must indeed be extremely minute, and it will be seen that it is on
this very minuteness that the energy of the sustaining force depends.
These pores must be regarded as communicating with the opposite fluids
RESIDUO-CAPILLARY ATTRACTION. 207
at their two extremities, while the fluids meet and mix in the
interior.
6. Dutrochet argues that capillary attraction cannot be the cause
of endosmose, because it can only raise a fluid to a small height in a
capillary tube, and is utterly incapable of drawing it beyond the limits
of the tube.
In stating these objections, he perhaps does not consider that the
height at which a fluid may be sustained in a capillary tube is inversely
as its diameter, and consequently in a tube of so extremely small a
diameter as those of which it is necessary to suppose the membrane to
consist, that height might be almost indefinitely great. It is true that
in the case of a single fluid, this effect would require for its production
that the tubes themselves should be coextensive with the fluid raised ;
but this is no longer necessary when the two ends of the tube are
immersed in different fluids. The reason why a homogeneous fluid
cannot be drawn beyond the limits of the tube, is, that, were it to
be so, the tube, acting equally at its two ends, would produce no
effect whatever upon the fluid. But the circumstances are very different
when the extremities communicate with different fluids. In that case the
full residual effect, consisting of the difference of effects, which the same
tube indefinitely extended, is capable of impressing separately upon the
two fluids, might be produced by an extremely small length of tube,
not exceeding a small multiple of the sphere of attraction of the par-
ticles of the tube, and there is no doubt that the thickness of the
finest membrane is a considerable multiple of this magnitude. In fact,
if we cut off" from the ends of the tube a distance greater than the
tube's sphere of sensible attraction, it is plain that the fluids which
occupy the intermediate part, in whatever way they may communicate
there, will suffer no effective attraction from the tube, since every
elementary portion will be drawn by it equally in both directions. The
only effective attractions will therefore be those exerted by an insensible
portion at each extremity ; we may therefore imagine these two por-
tions to be brought together as near as we please without any diminution
of effect.
D D 2
208 Mr POWER'S THEORY OF
7. In order to form some sort of estimate of the forces which may
be expected to result from residual attractions of this kind, let us
suppose the fluids to be water and alcohol, and the tube to be of glass.
Now Gay Lussac found by experiments of great accuracy, that in a
tvibe of glass whose diameter was 1.29441 millimetres, water would
stand at the height of 23"'.3791, and alcohol of specific gravity O.8I96
(that of water being 1) at the height of g^^'.SgSOS. This column of
alcohol would be equivalent to 7™.7176 of water; the difference of
effects would therefore be measured by a column of water of I5"'\66l5.
Suppose now the diameter of the tube to be diminished a thousand
times, or to become 0'"'.001294, the column of water which measures
the difference of effects would be 1566l™'.5: or, since the French
millimetre = .0393708 of an English inch, a glass tube of diameter
0'".0000507, or about the twenty-thousandth of an inch, would produce
a residual effect, with water and alcohol, measured by 616.6 inches or
51" 4'" of water, which is equivalent to the pressure of nearly two
atmospheres. When it is considered that a platina wire of one three-
thousandth of an inch in diameter may be seen by the naked eye, it is
probable that the magnitude we have assigned to the capillary tube
is considerably greater than the diameter of the membranous pores,
which evade the powers of the strongest microscope. From this ex-
ample I think the conclusion may be fairly drawn, that, so far at least
as the magnitude of the force is concerned, we need be under no
apprehension but that the residual capillary forces are sufficient to
account for the sustaining force of endosmose. How far they will
account for the law of its variation will be seen hereafter.
8. An attempt to explain the phenomenon by the principles of
capillary attraction has been already made by a distinguished mathema-
tician, Mons. Poisson. He first abstracts from the pressure of the
adjacent fluids, by supposing their altitudes above the membrane to be
inversely as their densities. The fluid in the tube being now equally
pressed on both sides, he supposes that that liquid, for which the tube
has the stronger attraction is drawn by this attraction to the opposite
end, thus filling the whole tube. The fluid within the tube, he now
argues, will be urged by two forces : 1st, the attraction of the liquid
RESIDUO-CAPILLARY ATTRACTION. 209
to which it belongs ; 2dly, the attraction of the opposite liquid. If then
the latter attraction be superior to the former, the fluid which fills the
tube, he says, will be drawn in an uninterrupted stream into the
opposite vessel.
Dutrochet justly objects to this theory, that it will only account for
a motion in one direction, whereas the phenomenon of exosmose requires
a corresponding motion in the opposite direction.
Professor Henslow, in a number of the Foreign Quarterly, suggests
as a modification of Poisson's theory, that whilst the fluid within the
tube is carried in the direction of the stronger attraction, the natural
tendency of the fluids to mix, may carry the other fluid (or, perhaps,
a slight infusion of it) in the opposite direction, and thus produce the
exosmose.
I perfectly agree with Professor Henslow that the natural process
of mixture is the cause of the exosmose, it being only necessary to
suppose that the rapidity with which this process extends itself witliin
the tube is somewhat greater than the velocity with which the whole
mass of fluid which fills the tube is drawn in the opposite direction.
But the theory of Poisson is further objectionable on this account,
that it makes the continuation of the process solely dependent on the
action of the fluids, whereas the experiments of Dutrochet incontestably
demonstrate that it depends mainly on the action of the membrane.
No doubt, the effect both of the fluids upon themselves, and of the
membrane upon the fluids, ought to be taken into consideration, and
this will be done in the following theory.
9. If a capillary tube be divided into two parts by a plane perpen-
dicular to its axis ; the attraction of one of these parts upon a fluid
which exactly fills the other part is \cH, c being the contour of the
inner surface of the tube, and H a certain definite integral or constant,
depending solely on the materials of which the tube and the fluid
consist. The contour of the tube may be of any shape whatever, curved
or polygonal. (See Mec. Cel. Sup. au X* Liv. pp. 14 — 21.)
210 Mb POWER'S THEORY OF
It is convenient to give a name to the quantity H \ we will call
it the capillary affinity between the two materials of which the tube
and fluid are composed.
It is easy to see that the quantity H will remain unchanged if we
conceive the tube and the fluid to exchange their materials; for, by
the equality of action and reaction, the elementary attractions, of which
cH
—— is the sum, will be equal in the two hypotheses. The tube may
be regarded either as solid or fluid, and this fluid may be either the
same as that which fills its interior or a different one.
If we conceive the density of the inner fluid to be diminished in
any ratio, all the elementary attractions, and therefore H, will be
diminished in the same ratio ; and if, further, the density of the tube
be diminished in any ratio, H will be diminished in the compound
ratio.
10. Next, let u and v be the original quantities by volume of two
vmmixed fluids. Then, if no penetration of dimensions takes place,
u + v will be their volume after mixture. If we regard the fluids after
mixture as coexisting, each with a diminished density, within the same
volume u + v, calling r, and pi these diminished or partial densities,
(r) and {p) the densities of the unmixed fluids, we shall have
,^ J and 7— ,
{r) u + v \p) u + v
whence
^ +-^ =1
{r) ^ (p)
Again,
ri + pi = r,
r being the total or ordinary density of the mixture. The two last
equations serve to express ri and pi in terms of /•, (;•) and {p).
RESIDUO-CAPILLARY ATTRACTION. 211
If then we have a second mixture of the same two original fluids,
we shall have
— + -^ = 1
and r-i + p-i = p ,
where rj and p-i are the two partial densities, and p the total density of
this second mixture. These equations serve in like manner to express
r-i and p-i in terms of p, {r) and {p).
11. Let us now endeavour to express the mutual capillary affinities
which exist between the two mixtures just mentioned, and a third
material (as that of a membrane or tube), in terms of the densities
of these mixtures and the mutual capillary affinities between this same
material and the unmixed fluids.
Let the former affinities be denoted by H, K, L, M, N, namely,
H between the tube and the first mixture,
K between the tube and the second mixture,
L between the first mixture and the second,
M between the first mixture and its like,
N between the second mixture and its like;
and let the latter affinities be denoted by {H), {K), (L), {M), (A^),
namely,
{H) between the tube and the fluid of density (r),
(K) between the tube and the fluid of density (p),
{L) between the fluids of densities (r) and {p),
{M) between the fluid of density (/•) and its like,
(iV) between the fluid of density (p) and its like.
The attraction ^cH of No. (9) will be the sum of two partial
attractions, namely, that of the tube upon two coexistent cylinders of
the opposite fluids, whose densities are those of the original unmixed
fluids diminished in the ratios r^ : (r) and p^ : (p). Hence by the latter
part of that No.,
ic^=ic(^)^ + ic(^).^;
212 Mr POWER'S THEORY OF
whence
By similar reasoning, superposing all the different attractions, each
diminished in the ratio of the densities of the attracting and attracted
materials, we shall have
.■.i^=.(/.,^.g+(M,.^H-W^..
By combining each of the last five equations with the four equations
of No. 10, and eliminating r,, ^2, /a,, p^, we shall obtain H, K, L, M,
N, in terms of the actual densities r, p, the original affinities (H),
(K), (L), (M), (N), and the original densities (r) and (p).
12. Let us now proceed to apply the principles of the three last
numbers to explain the experiments of Dutrochet. And first let us
consider those which relate to the statical force of endosmose. In
these experiments the process was allowed to continue until the fluid
raised, or rather the mercurial column which was hydrostatically sub-
stituted for it, attained its maximum altitude ; at this moment the
densities of the two liquids were experimentally determined ; and
instituting different experiments with different mixtures of the same
substances, Dutrochet found that the maximum altitudes were propor-
tional to the corresponding differences of densities.
The substances employed in his experiments were saccharine or
gummy solutions on the one hand, and water on the other, and the
RESIDUO-CAPILLARY ATTRACTION.
213
water was found to be transmitted in greater abundance. Common
treacle is a very convenient substance for experiments.
Let us suppose then that the lower part of the endosmometer is
filled with treacle, and having a thin membrane tied over its mouth,
is immersed in water ; and let us suppose that the fluid is allowed
to ascend until the operation ceases.
At this moment we may regard the capillary pore which traverses
the membrane, as communicating at its two extremities with fluid in
the same state of mixture as the fluid in the contiguous vessels,
there being a gradual transition from one end to the other.
Let C1C1C2C2, be a portion of
the membrane, AiAiAsA^ one of
its capillary pores, with its axis at
right angles to the plane of the
membrane, communicating originally
with the water at A^A^, and with
the treacle at A^A^, but when the
fluid has reached its maximum al-
titude, communicating with the ^
first mixture of No. (10) at AiA^,
and with the second mixture of
that No. at A2A2.
^,
^
c^
^.
A^
c^
\
^.
S*
'
1
-B/
B,
\
1
<i
Ay
9
Imagine the geometrical figure of the tube, (not its material) to
be produced both ways to D, and D^, and cut off" from each end
of the tube a distance AiBi, AiB^, equal to the tube's sphere of
sensible attraction.
Since A^Bi, and A^Bi, are insensible, we may regard the fluids in
AiA^BiBi, and A^A-^BiB^, as in the same state of mixture as the
fluids in the contiguous vessels.
Vol. V. Part II. E e
214 Mr POWER'S THEORY OF >!
Let us now estimate all the forces which tend to move the central
column DyD^D-iDi in direction of its axis.
It is plain that, in whatever manner the fluids may communicate
in the interior of the tube, the tube can produce no effect upon
ByBiBiBi, since every elementary portion of this part of the fluid
will be drawn in both directions as by an infinitely extended tube.
We may also neglect, as producing equal and opposite forces in
both directions, the attraction between the tube A^B^ and the fluid
AiAiBiBi; between the tube A^Bi, and the fluid A^AzBiB,; be-
tween the fluid tube dA^Di, and AiA^D^D^ ; between C2A2D,, and
AzAzDiDs, between the membrane and C^A^D^-, between the mem-
brane and CiA-iD-i.
Lastly, we may neglect all the mutual actions of the particles
composing the central column DyD^DiDi, their tendency being only
to mix the opposite fluids, and not to move the column as a
mass.
Of the remaining attractions we shall have at one end the
attraction of the tube B^Bt, upon B-^B^A^A^, ( = \ cH) + the
attraction of the tube A^B^, upon D^D^A^A^, {= ^ cH) — the
attraction of the fluid tube C^A^D^, upon A^A^B^B^, {=\cM);
c
constituting the capillary force - {2H — M). This will be opposed
by a similar force — {^K—N) exerted at the other end of the tube.
The residual sustaining force is therefore
I .{2H-2K-M+N').
It now only remains to express this force in terms of the actual
densities r and p, and the initial constants
{r), ip), {H), (K), (L), {M), {N).
RESIDUO-CAPILLARY ATTRACTION. 215
13. For this purpose let
^-=... and ^=..;
therefore by No. (10).
making these substitutions in the equations of No. 11., we have
K = s,{H) + il-s,){K).
L =s,.{l-s,){L) + s,.{l-s,){L) + s,s,{M) + {l-s,){l-s,){N').
M= 2s, (1 -*0 {L) + s,' {M) + (1 -*,)' (^)-
N = 2s, (1 - s,) {L) + si (M) + (1 - s,y (N).
Hence 2H-2K-M+ N=A (H) + BiK) + C {L) + D{M) + E{N-),
where A = 2. {s, — S2).
B = 2.{l-s,)-2.{l-s,)
= -2{si-s,).
C=-2s,.{l-s,) + 2s, (1 - s,)
= -2{s,-s,) + 2{8{'-si).
D=-{s,'~si).
E=-{l-s,Y + {l-s,Y
= 2{s,-s,)-{s,'-'Si).
EE2
216 Mr POWER'S THEORY OF
The residual force is therefore
I {s,-s,){2{H)-^{K)-2{L) + 2{N)}
2
+ l-{s,'-s.'){2{L)-{M)-{N)}.
Again, r = r^+p, = {r).^^+{p).■^
= ('•) *! + (/») (1-*.);
(p)-r
••. *,
=={r)s, + {p).{l-s,);
■• '~ip)-irr
_ p — r
ip) - {r)
„ . „_?ie)zik±rl.
i_^
{ip)-ir)r {{p)-ir)r'
The expression for the residual force is, therefore,
p^ -r
i-fvFW*'<'^>-''^>-<'^>''
RESIDUO-CAPILLARY ATTRACTION. 217
which may be put under the form
cA{p-r)-{-cB(p'-t^)*,
making
^-(^)-(r)-r^) ^^^)+(^)-(r)L^^> {p) + {r) J}'
and^=..^{(Z)-W^H
{ip)-ir)rv' 2 r
The agreement of theory with experiment, then, requires that
jM) + jN)
(^) 2
should be either nothing, or very small compared with
14. When I first began to investigate this subject, certain con-
siderations, which it would be tedious to detail, led me to imagine
that the fluids might communicate in the interior of the tube,
forming a series of interlacing cylinders one within another, and I
found the forces which tended to protrude the cylinders into the
opposite fluids, all multiplied by (L) — - — '-—^ — - . I therefore looked
upon this expression as a measure of the tendency of the fluids to
mix, and this tendency being, as experience shows, very small in the
case of treacle and water, as well as in the case of the gummy
solutions and water, afforded an explanation why the force should
be so nearly proportional to the difference of densities, as Dutrochet's
* I have elsewhere erroneously stated, that the residual force is c A(p—r) + c B(p—ry,
a mistake which I am glad to have this opportunity of correcting.
218 Mr POWER'S THEORY OF
experiments seemed to indicate. But the preceding theory being
perfectly independent of the mode in which the fluids communicate,
it is better not to have recourse to a supposition, which is in the
slightest degree precarious, especially as I am now prepared to show,
that, in whatever way the fluids may arrange themselves within the
tube, the rapidity of the mixing process will depend upon the mag-
nitude of (X)- (^);W .
15. In fact, in whatever manner the mixing process may be
effected,- we may at any moment imagine the fluid to be divided
into an indefinite number of contiguous strata, of any arbitrary or
convoluted form, the density being the same for the whole extent of
any one stratum, but varying from one to another.
If the surface which separates two contiguous strata be a perfect
plane, it is evident, by the equality of action and re-action, that this
would be a position of momentary equilibrium, (abstracting from
gravity, which I am not here considering.)
Suppose, now, that this surface becomes
undulated in an arbitrary way, and take any
point A upon it, and draw a tangent plane
BAD, including with the surface EAC, a kind
of lens BDEC, which, with La Place, we
may call a meniscus. Draw the normal FAG ;
and let Ri, and R^ be the radii of greatest and
least curvature at the point A.
Now La Place has shown that the attraction of such a meniscus
upon the column of fluid AF is ("»"+ p")--^> where H is the
capillary affinity between the material of the meniscus, and that of
the fluid in the sense already defined. (See Supp. au X* Liv.
page 14 — 17.)
RESIDUO-CAPILLARY ATTRACTION. 219
He has also shown that the attraction of the meniscus is the
same whichever way it be turned.
If the meniscus instead of consisting of the left hand fluid, (as
in the figure), consisted of the right hand fluid, the common boundary
being the plane BAD, there would be equilibrium, the column
AF being attracted by the right hand fluid, just as much as the
column AG is by the left.
Since then the meniscus consists of the left hand fluid instead of
the right, the effect of the disturbance upon the column AF, tending
to draw it in the direction FA, is the attraction of the meniscus
upon AF, regarding it as consisting of the left hand fluid, minus
the attraction of the same meniscus regarding it as consisting of the
right, that is
\R, ^ EJ \2 2
supposing the left hand fluid to be the first mixture of No. (10),
or the lower fluid of No. (12).
If then we estimate the effect in the direction AF, it is
1 1\ /L M\
(1_ J_\ (± _M
In the same way, the effect of the disturbance upon AG, in the
direction GA, is the attraction of the meniscus, regarded as consisting
of the left hand fluid, minus the attraction of the same meniscus,
regarded as consisting of the right, that is
Ui "^ BJ • \2 ^
2 j-
Hence the whole attraction in the direction GF, is
{i^k){--^)-
220 Me POWER'S THEORY OF
If we substitute for L, M, N, the expressions at the commence-
ment of No. (13), we shall find
^L-M-N = A{L) + B{M) + C{N), where
^ = 2*i.(l-«2) + 2*2.(l-*,)-2*i.(l -*i)-2 52.(l-«2)
= 2 *, - 2 *,«2 + 2 *2 - 2 «i*8 - 2 *, + 2 «i' - 2 *2 + 2 «/
= 2*,*- 4*1*2 + 2*/
= 2(*i-*,f.
= -(*!- s^f.
= -{(i-*0-(i-*.)}'
= -(«i-*2)';
.-. ^L-M-N={s,-s,)\{2 (L) - {M) - {N)}
The effect of the disturbance in the direction GF, is therefore
consequently if (i) be greater than ^ — '-^ — '- , or, if the capillary
affinity of the opposite fluids exceed an arithmetic mean between
the capillary affinities of the two fluids for fluids of their own kind,
the tendency will be to depart still farther from the position of
RESIDUO-CAPILLARY ATTRACTION. 221
equilibrium, and the tendency is the greatest where the curvature is
the greatest.
16. Hence it is easy to see that the protruding segments of each
fluid will become more and more pointed at their summits of greatest
curvature as they advance into the opposite fluids, thus forming
interlacing spiculse, shooting into the opposite fluids, and at the same
time inosculating with each other by their lateral protrusion, and
that this process cannot cease until the fluids have divided each
other into segments of a magnitude comparable with that of the
sphere of sensible attraction.
Beyond this limit the theory does not hold. It is very possible then,
that in some cases a limit may be attained where the mixing fluids
have arrived at such a state of subdivision, that the conditions for
continuing the subdivision are no longer satisfied ; in other cases it
is possible that the subdivision may proceed until the ultimate atoms
of the opposite fluids act upon each other by ones, twos, and threes,
thus effecting a chemical decomposition : nature presents numerous
instances of both kinds.
17. But though the mathematical theory is not strictly applicable
when the subdivided segments are of less magnitude than the sphere
of sensible attraction, it may be considered as an approximation to the
truth considerably beyond this limit. For, the most effective part of
the attraction of each segment being that exerted by the particles
in immediate contact with the normal column, the diminution of
the segments will only have the effect of removing the more feeble
part of the attractions which the theory takes into the account. It is
therefore probable that, even in cases where no chemical decomposition
takes place, the subdivision of the fluids may be carried to a limit far
beyond that to which the theory is strictly applicable. Besides, the
processes of nature are not interrupted of a sudden; the tendency
therefore to farther subdivision cannot be suddenly arrested, but in
cases where it is ultimately reduced to nothing, it must be so by
passing through all degrees of magnitude. This reasoning is further
Vol. V. Paet II. F f
222 Mr POWER'S THEORY OF
confirmed by those experiments which demonstrate the almost infinite
subdivision of matter by repeated dilution, experiments which are
familiar to every one. This infinite subdivision is, in fact, involved in
the mathematical conception upon which this theory is founded, namely,
that in the state of mixture the two fluids may be regarded as
coexisting within the same volume, each with a diminished density.
This conception cannot of course be a rigorous representation of nature ;
but is sufficiently so for the application of La Place's theory, or, which
comes to the same thing, for the summation of the attractions by the
principles of the Integral Calculus.
18. In cases of simple mixture, unattended with a chemical change,
the ultimate segments of the opposite fluids, though in an extreme
state of subdivision, have a separate and independent existence, which
renders it highly probable, that the volume of the mixed fluids should
equal the sum of the volumes of the unmixed fluids. This supposi-
tion has been made in the preceding theory, and I find by experiment
that in mixtures of treacle and water it is accurately true. The same,
I believe, is true in all cases of simple mixture, where no chemical
result takes place, such as the precipitation of solids, or the disengage-
ment of heat or other volatile constituents. To liquids whose union
is accompanied by such phenomena the present theory is inapplicable,
not only on account of the penetration of dimensions, with which
such phenomena are generally attended, but on account of the change
of affinities, which the escape of some of the constituents must
necessarily produce, including heat, which, regarded as a chemical
constituent, is as important as any.
19. The addition of a third fluid to one of the liquids, by altering
the chemical affinities-, must likewise alter the capillary aflfinities, which
are only a different modification of the same corpuscular attractions
which produce the former. It is not surprising then, that Dutrochet
should have discovered some substances which accelerated the process
in his experiments, and others which retarded it or stopped it
altogether.
RESIDUO-CAPILLARY ATTRACTION. 223
Water impregnated with sulphuretted hydrogen was found not only
to stop the process, but to destroy the energy of the membrane for
subsequent experiments with pure water and pure saccharine solutions.
No doubt the sulphuretted hydrogen had decomposed the surface of
the capillary pore, leaving a coating of putrid matter, which was not
possessed of such capillary properties as to supply the place of the
material of the membrane. That this is the true explanation is shown
by the fact, that when the membrane was for a long time steeped in
water and well washed, its energy was restored : in fact, the putrid
matter being washed away, the membrane presented an unvitiated
surface to the fluids. Heat and electricity may be classed amongst
these chemical agents, as they operate their effect precisely in the same
way, namely, by changing the chemical and consequently the capillary
affinities.
20. If we wish to compute the height to which the fluid will
rise in the endosmometer, let ^ be the height of the supported column
above the surface of the membrane, and z the height of the lower fluid
above the same, w the transverse section of the tube; the difference
of the pressures of the cylindrical columns w^ and wg, having the
common section w, is gpco^—grioz: this must be counterbalanced by the
sustaining force cA(p — r) + cS{p^ — r"), which denotes a pressure on the
same scale ;
" o ff \ p J oi ff \ p / p
If a column of mercury be hydrostatically substituted for the
ascending fluid, as in the experiments of Dutrochet, calling Z the
altitude of the mercury, and R its density, we must have
„ c A (p-r\ cB (p^-r^\ r
^ = -.- g • K-w) ■" -.^ ■K-R-) "■ R^'
this of course being subject to a correction when the cistern of the
mercury is not on a level with the membrane.
F F2
224 Mr POWER'S THEORY OF
21. If the pore be circular, let ^ be its diameter, then
c = 2'7r.-, and to = tt . — :
2 4t
€ 4
•'. - = -J ;.
CO
the sustaining force is therefore inversely proportional to the diameter
of the pore, as in ordinary capillary attraction.
Hence we see how the membrane^s delicacy of texture contributes
to the intensity of the sustaining force.
22. It is now easy enough to see in what manner the process is
effected. The residual force cA{p — r) + cB{p'^ — r"), which would result
if the ends of the tube communicated with fluid of the densities
r and p, being greater than the altitudinal pressure upon the section w,
would cause the fluid within the tube to move as a mass into the
endosmometer, thus bringing fluid more and more diluted to the
issuing orifice; this will continue until the residual force is weakened
to such a degree as exactly to counterbalance the altitudinal pressure.
Contemporaneously with the former motion, the mixing process will
transfer the two fluids in opposite directions, the current from the
endosmometer towards the water producing the exosmose, and the
opposite current supplying the deficiency caused by the exosmose, and
therefore not contributing to the endosmose. The diluted fluid which
was carried into the endosmometer by the residual force, will gradually
mix with the treacle within, whether that mixture be carried on near
the orifice of the tube, or whether the diluted fluid be raised by its
specific levity higher up in the endosmometer. The extremely small
portion of diluted fluid which has thus been transmitted, and the
viscosity of the treacle, render it most* probable that it would not be
* This probability amounts nearly to certainty when we consider that the denser fluid
has no access to the lower part of the transmitted fluid. It is only when a lighter body
is insulated, or partially insulated^ in a denser that it rises by its specific levity.
RESIDUO-CAPILLARY ATTRACTION. 225
carried up by its specific levity, but rather adhere to the membrane
in the way that bubbles of air adhere to the sides of vessels containing
water or mercury. But, be this as it may, the end of the tube which
communicates with the endosmometer, will soon be surrounded by a
stronger infusion of the treacle, which will again bring the residual
force into action ; thus a fresh portion of the fluid will be introduced
into the endosmometer, and the same process will be repeated as before.
For the sake of explanation, I have supposed the residual force to
produce its eflPect discontinuously, but it is easy to see that the process will
really be continuous, the united actions of the endosmose and exosmose
always keeping the orifices of the tube surrounded by fluid in such a
state of dilution that the magnitude of the residual force will be exactly
sufficient to create a supply proportioned to the demand arising from
the mixing process which is continually proceeding within the endosmo-
meter. The residual force cannot be less than this, for if it were, the
encroachment of the treacle upon the issuing orifice would immediately
increase it ; nor can it be greater, for then the accumulation of the more
diluted fluid at that same orifice would immediately diminish it,
23. The quantity transmitted in a given time must depend more
upon the rapidity with which the mixing process is carried on within
the endosmometer than on the magnitude of the residual force. This
force is certainly essential to the transmission, but its effect is no other
than that of a pump which supplies the fluid from below as fast as it is
wanted, and no faster, and that of a catch or valve to sustain it when it
is once elevated. The moving force at the summit of any protruding
spicula is by No. (14) represented by [^ + ~p) ^(p — ^)^ and is,
therefore, for spicule of given shape, as the square of the difference
of densities. It might appear then, at first sight, more probable that
the quantity of the lower fluid absorbed by the fluid in the endosmo-
meter in a given time, would be more nearly as the square of the
difference of densities, than as the simple power of this difference, which
is the law the experiments of Dutrochet tend to establish. But such
a conclusion would be very precarious, as will appear by the following
considerations.
226 Mb POWER'S THEORY OF
24. Let us imagine two different experiments, all circumstances,
as regards the materials, form and disposition of the apparatus, being
exactly similar, but the proportions in which the substances are mixed
on each side the membrane, being different in the two experiments.
Let us suppose also that the mixing process takes place in both experi^
ments after exactly the same type, only with different velocities, that
is to say, that at certain times, t and t', t + T and t' + r, # + 2t and
#' + 2t', &c., the protruding spiculae from the lighter fluid exist in
exactly the same state in both experiments, as regards their number,
shape, size and situation.
This supposition being made, the volume of the lighter fluid absorbed
by the fluid in the endosmometer in the two experiments, will be equal
in the intervals t and t' : also the summits of the spiculae will have
described the same paths in the two experiments during these same
corresponding intervals. Let t and t be indefinitely small, and let us
equate the spaces described by the summits of any two corresponding
spiculae between the epochs t and # + r, t' and t' + t', and also between
the epochs t and ^ + 2t, t' and #' + 2t'.
Let a be the sphere of sensible attraction, and imagine a small
normal column 2 a at the vertex of each spicula, being half in one
fluid and half in the other.
The two spiculse having by the hypothesis the same shape, the
moving forces upon these columns are as {p — rf and [p' — r'f, and the
masses moved are as ap + ar and ap' + ar', that is, as p + r and p' + r;
the accelerating; forces will therefore be as — and '^ , / ; let
^ p+r p +r
(p _ rY (p — r'f
them be k ^ '- and ^ k . ^ f- . Then if v and v' be the velocities
p+r p +r
of the two summits at times t and t', equating the corresponding spaces,
we shall have
and
P + r " ■ ^' p+r'
■^ p + r ^ p +r
RESIDUO-CAPILLARY ATTRACTION. 227
These equations are equivalent to the following:
VT = VT, and — = -^-—, — —. — ;
p-irr p ^■r'
whence
v' ~ T ~ p' — r' ' p + r '
Let q be the volume of fluid absorbed in the times t and t', which
we have seen to be the same in each experiment; and let Q and Q'
be the quantities absorbed during a given time T, T not being so
great but that r, p, r' and p may be considered the same during this
interval.
If then there be a law connecting the quantities absorbed in a given
time with the densities, we must regard this absorption in each experi-
ment as uniform during the time T\
.: Q : q y. T : T,
and Q '. q V. T : r'\
... Qr = qT=Q'T';
+ /
^ T p' — r' p + r
■ The supposition we have made, as to the exactitude of type in the
two mixing processes, is particular ; but if there be a general law whicli
is applicable to all cases, that, must include the case supposed, and
therefore the result of the particular case must coincide with that of
the general law. If then there be such a law, it is expressed by the
proportion
Q.Q .'.
p-r . p-r
"s/p + r ' \/p' + r' '
This being true in different experiments, must be true in different
stages of the same experiment.
228 Mr POWER'S THEORY OF
Now in the same experiment p diminishes and r increases as the
experiment proceeds, and therefore the variation of p + r is small com-
pared with that of p — r; the quantities absorbed will therefore be
pretty nearly in the ratio of the difference of densities, as Dutrochet
found them to be. Whether the proportion
Q:Q' :: -E^ : -IzL
y/p + r Vp + r'
may be a more accurate representation of nature than the law of
Dutrochet, is left to the test of experiment.
25. It may perhaps be objected to the theory of No. (12), that
the ordinary theory of capillary attraction supposes the dimensions of
the tube to be incomparably greater than the sphere of sensible attrac-
tion, whereas the fact, that these pores are so small as to elude
microscopic observation, might lead us to apprehend that their dimensions
were of a size comparable with that sphere. The example which has
been calculated in No. (7), does not seem to leave any cause for such
an apprehension. But supposing this were the case, the only difference
it would make in the theory is this : that, whereas, on the former
supposition, the quantities \cH, \c K, &c., denoted the results of
integrations extending from nothing to infinity, and not otherwise
depending on the form of the tubes than by involving the contour c
as a multiplier; on the second supposition, the limits of the integra-
tion will depend on the form of the tubes and the texture of the
membrane : but these limits being the same in the cases compared, it is
easy to see that the theory will be still true on the latter hypothesis,
provided we look upon ^c{H), ^c{K), &c., as denoting certain
unknown limited integrals depending not only upon the nature of the
materials, but also upon the form and size of the capiUary pores. The
residual force will, therefore, on this hypothesis also, be of the form
a{p-r) + h{p'-r').
26. By the application of similar reasoning to the theory of No. (15),
it is not difficult to conclude that the moving forces upon the normal
RESIDUO-CAPILLARY ATTRACTION. 229
columns at the summits of spiculae of given shape and sixe will be as
(p — r)-, even when the dimensions of the spiculse are indefinitely less
than the sphere of sensible attraction. For, the attraction of a meniscus
bounded on one side by a plane surface, upon the conterminous normal
column, will in all cases be a definite integral depending on the shape
and size of the meniscus, and the demonstration of La Place, by which
he shows that the attraction of such a meniscus is the same whichever
way it be turned, is perfectly independent of its size and the shape of
its curved surface.
Let then I be the attraction of any meniscus upon the conterminous
normal, the meniscus consisting of one mixture, and the normal of the
other; m, the attraction of the same meniscus when the meniscus and
column consist both of the first mixture; and n, the same thing when
they consist of the second mixture. Then reasoning exactly as in
No. (15), the moving force upon the column GF will be 2l—m — n;
and if (/), (m), (»), be the initial values of I, m, n, it may be shown
exactly as before, that 2l—m — n = c/C_^ >.^g • {2(/) - (m) - (n)}, the
theory of No. (11) being equally applicable in this case. Hence, how-
ever minute the spiculae may be, the moving force upon the central
column will, for spiculse of given shape, be as the square of the difference
of densities.
This consideration applied to the theory of No. (25), gives it a
generality which renders it as satisfactory as can well be desired.
J. POWER.
Trinity Hall,
Marck 29, 1834s
Vol. V. Part II. Gg
231
X. On Aerial Vibrations in Cylindrical Tubes. By William
Hopkins, M.A. Mathematical Lecturer of St Peter's CoUege,
and FeUow of the Cambridge Philosophical Society.
[Read May 20, 1833.]
The problem which has for its object the determination of the
motion of a small vibration propagated in an elastic medium along a
prismatic tube of indefinite length (the motion of every particle in
each section of the tube perpendicular to its axis being the same) was
long since solved by Euler and Lagrange. The problem, so nearly
allied to this — to determine the motion of an aerial pulsation in a tube
of definite length — has not been so satisfactorily solved, the tube being
either open at the extremity or stopped with a substance possessing
some degree of elasticity. In addition to the difficulties of the former
problem, we have in this latter one those still more formidable difficulties
which exist in the determination of the circumstances of the motion
at the confines of two elastic media in the closed tube, or at the
extremity of the open one, where the air in the tube communicates
with the circumambient air. These motions must no doubt be deter-
minable from the nature of the media, and the causes producing and
maintaining the vibrations, having nothing arbitrary in them, except
what may be so in the original disturbance ; but I am not aware
of any progress having been made in the direct solution of these
questions, which now forms one of the greatest desiderata in the appli-
cation of mathematics to physical science; and in our inability to
determine these motions at the extremity of the tube, either by theory
or direct observation, we are driven to the necessity of assumptions.
It is from a difference in these assumed conditions that we have the
GG2
232 Mb HOPKINS ON AERIAL VIBRATIONS
different solutions which mathematicians have given of the problem in
question. The principle on which we ought to proceed in making such
assumptions is obvious ; they should be subjected to no restrictions,
(not imposed on them by our theory), which are not necessary to draw
those deductions and inferences from our mathematical results^ which
admit of verification by experiment, to the test of which an assumption,
in any degree arbitrary, must necessarily be subjected before it can claim
our confidence. The physical conditions however on which the solutions
of this problem depend, (as far as it is distinct from that of the motion
of a wave along a uniform tube of indefinite length), have neither
been assumed on this principle, nor subjected, as far as I am aware,
to this experimental test. It has been principally with the view of
remedying these defects that I have prosecuted the researches, an account
of which I have now the honour of laying before the Society.
1. The physical conditions assumed by Euler, and by most of those
who have since written on the subject, are, that the particles of air at
the extremity of a closed tube are always at rest; and that no con-
densation of the air takes place at the extremity of an open one. The
first condition involves the supposition of the perfect rigidity of the
material with which the tube is stopped. This cannot be accurately
true, but probably leads to no error very appreciable to observation.
The second condition assumes an eqviality in the densities of the external
air, and of that within the tube immediately at its open extremity,
during the whole time of the vibrating motion, in the same manner as
if the air were at rest. This supposition carries with it but little
appearance of being even very approximately true; for it is difficult
to conceive how a sonorous wave could thus be produced and maintained
in the surrounding air from the open extremity of the tube, and it
appears perfectly irreconcileable with the fact of the sudden cessation
of sound after the cause producing it has ceased, M. Poisson, struck
with these objections, has assumed another physical condition as appli-
cable to any tube, whether open or stopped, viz. that there exists at the
extremity of the tube, during the whole motion, q constant relation
between the velocity of the particles of the fluid at any instant, and
its condensation, this relation depending on the nature of the substance
IN CYLINDRICAL TUBES. SSS
with which the fluid at the extremity of the tube is in immediate
contact. This condition is manifestly less restrictive than those of
Euler, since it involves no supposition of the perfect rigidity of bodies,
and leaves room for a certain degree of condensation and rarefaction
of the fluid at the extremity of the open tube, thus removing the
difficulty above-mentioned respecting the maintaining of aerial pulsations
from the open end, in the circumambient air ; while it enables us also
to account in some measure for the rapid cessation of sound with the
cessation of the cause producing the vibratory motion of the air in
the tube.
2. The two authors above-mentioned have written elaborately on
this subject of the vibrations of elastic fluids in tubes. Mr Challis
also in his paper published in the Transactions of this Society, (Vol. III.),
has been led to the consideration of the conditions which hold at the
closed or open extremity of the tube in which the air is in a state
of sonorous vibration, though the determination of this point forms
with him a collateral rather than a principal object. He assumes that
a pulse proceeding along a cylindrical tube will be reflected from the
further extremity if the tube be stopped, the intensity of the reflected
pulse being equal to that of the incident one; and that if the extremity
of the tube be open, it will pass into the circumambient air, sending
back no reflected wave within the tube. If this were the case, it
would immediately account for the apparently instantaneous cessation
of sound above-mentioned ; but there are other equally obvious
phenomena, for which this hypothesis appears to offer no adequate
solution.
3. It will be observed, that Euler has supposed either the velocity
of the particles or their condensation to have, at the extremity of the
tube, a constant value, independently of the time ; while M. Poisson
has supposed this constancy of value to belong to the quantity ex-
pressing the relation between the velocity and condensation. It does not
however appear to me probable that any such conditions, independently
of the time, should hold. All the above assumptions are equally
arbitrary, and equally require to be put to the test of experiment. In
234 Mr HOPKINS ON AERIAL VIBRATIONS
applying this test, I find that the deductions from the results, derived
from any of the three hypotheses above-mentioned, do not sufficiently
accord with the observed phenomena to be perfectly satisfactory. This
discrepancy is more particularly observable in the position of the nodes
or points of minimum vibration in the open tube. According to Euler's
hypothesis, these nodes would be places of perfect rest ; and they would
be distant from the open end by an exact odd multiple of -, where
\ = length of a whole undulation. From the hypothesis of M. Poisson,
their positions would be the same as in the above case, but they would
become points of minimum vibration, and not of perfect rest. Mr
Challis's supposition would lead to the conclusion that no nodes existed
in this case, except they should be produced by some vibration of the
tube itself, a cause the total inadequacy of which to produce any appre-
ciable effect, must be immediately recognized by every one who has
made experiments on this subject. The facts, as determined by experi-
ment, are very obvious ; and it appears that there are nodes, which
are points of minimum vibration and not of perfect rest ; that they are
equidistant, but that denoting this distance by -, the distance between
the open extremity and the nearest node is considerably less than -.
I shall not in this place proceed further with the detail of experimental
facts ; but shall first shew how the theory of this subject may be
generalized by the assumption of conditions less restrictive than those
which have been made by the writers I have mentioned. In the second
section, I shall describe the experiments which have suggested these
assumptions ; and shall conclude with some observations on the resonance
of tubes, so far, more particularly, as it is allied to the investigations
contained in this paper.
The form under which I shall consider the problem, is that under
which it presents itself, as nearly as possible, in the experiments I have
to describe.
IN CYLINDRICAL TUBES. 235
SECTION I.
4. Suppose the tube AB, (fig. I.), open at A, and stopped at B,
with some substance possessing any degree of elasticity ; and suppose
the vibrations first produced and kept up by a rigid diaphragm, vibrating
according to a given law at A, and perfectly excluding the air within
the tube from any communication with the external air. We have
the usual equations
v=f{at-x) + F{at + x)]
(A),
as=f{at-x)-F{at + x)]
V denoting the velocity of a particle at distance a; from the origin,
and s the condensation at the same point at the time t, and a being
the velocity of propagation of an aerial pulse along the tube.
One of our conditions must necessarily be, that the velocity of the
air within the tube and immediately in contact with the diaphragm,
must constantly have the same velocity as the diaphragm itself, con-
strained to move according to a given law. Let this velocity = <p{at).
Then shall we have
(j){af)=/{ai) + F{at) (1).
5. To ascertain the nature of the second condition, which must
hold at B, where the motion of the wave propagated along the tube
is interrupted, we must consider the effect which will be produced on
the stop by the action of the air within the tube. The vibratory motion
wUl produce alternations of condensation and rarefaction at the ex-
tremity B, which will tend to put the substance forming the stop in
vibration; and if it will admit of vibrations having the same period
as those of the air in the tube, this effect will be produced by the
constant reiteration of the cause above-mentioned. If the substance is
not susceptible of vibrations of this kind, no appreciable effect will be
produced upon it.
236 Mr HOPKTNS ON AERIAL VIBRATIONS
The determination of the nature of these vibrations, or of the
function expressing the velocity at any instant of the extreme section
of the stop, will necessarily depend on the material of which it is made;
and any solution of the problem in question, independently of this
consideration, cannot be regarded as complete. Still, whatever may be
the nature of the stop, we know that the period of its vibrations must
be the same as for those in the tube; and it is also manifest, that each
vibration of the stop must begin at a time later by an interval at least
nearly = -, (/= the length of the tube), than the corresponding vibration
in the diaphragm at A, whence the original disturbance is supposed to
proceed. I say that this interval is' nearly equal -, because certain
phenomena, of which I shall speak hereafter, seem inconsistent with
its being in particular cases exactly = -. I shall therefore, to give the
ct
assumption all the generality possible, consider it as generally = — f- arbi-
trary quantity, to be determined in each particular case by experiment.
Hence then, if ^ denote the form of the function of the time expressing
the velocity of the extreme section of the stop, we shall have the
velocity = v/'l «/ — (/ + c)}, c being arbitrary. This must also be the
velocity of the extreme section of the air at B, consequently we have
as a second condition
•^{at-{l-^c)}=f{at-l) + F{at^-l) (2).
We have from (1)
(t>{at + l)=f{at + l) + F{at + l);
and eliminating F(at + l),
f{at + l)-/{at-l) = <p{ai + l)-f{at-{l + c)\ ;
or, writing at + 1 (or at,
f(flt-ir^l)=f{at)-y\f{at-c) + <t>{at + ^l) (B).
IN CYLINDRICAL TUBES. 237
The substance forming the stop being known, so that we might
regard the vibrations produced in it under given circumstances de-
terminable, the relation between the functions xj^ and J" would be
known, and the function y would be the only unknown one in the
above functional equation, from which, any particular form being
assigned to (p, that of y must be determined. The arbitrary quantity
which will be involved in the solution of this equation, must be
determined by the original value of the function jf.
6. We have here supposed the tube to be stopped, but the
equation (B) will still be true for the open tube, \|/ {«/-(/ + c)}, de-
noting always the velocity of the 'extreme section at the time f.
Equation (2) gives us
F{at + l)=-f{at-l) + y\,{at-{l+c)},
and writing at + x, for at + l,
F{at + x)= -f{at-{2l-x)} + >// {at -{2l + c -x)}'.
Hence,
v = f{at-x)-f{at-{2l-x)} +^ {at-{2l + c-x)\-\
as = f{at-x)+f{at-{2l-x)}-yl^{at-{2l + c-x)}]
The form of J" being determined by equation J?, these last equations
will give the complete solution of the problem.
7. Before we proceed to consider particular cases, we will exhibit
these equations (C) under another form, which will be useful in
deducing some general inferences as to the nature of the motion in
the tube.
Let T denote a period of time, from the commencement of the
motion at A, less than that which is necessary for the pulse to
travel twice the length of the tube ; consequently at will be less
than 21.
Equation (B) gives us
/(ar + 9.1)=/ {ar) - v// («T - c) + ^ {aT + 2l),
Vol. V. Part II. Hh
•(C).
238
Mb HOPKINS ON AERIAL VIBRATIONS
and for ar, writing ar — x,
/{(aT + 2/)-^}=/(aT-ar)-x//{«T-(a; + c)}+0(«T + 2/-;r) (3).
Also putting ar + ^sl—x, for ar,
/■{(aT + 4/)-ar}=/(aT + 2/-ar)-v|/{«T + 2/-(a; + c)} +0(«t + 4/-^)
=f{aT-x)-^{aT-{x-^c)}
-x//{aT + 2/-(a; + c)}
+ 0(aT+2/-a;)
+ (i>{ar + 4!l-x).
And similarly, we have
2/^
/{»(.+ ^v^*
^{«('^ + — )-(*' + c)},
■■/{aT-x)-<
f {"(-r +—)-(« + c)},
&c.
, ( r 2{n-l).l-\ .
&c.
.^{«('^ + -|-)-^}
IN CYLINDRICAL TUBES.
239
In the same manner,
1.
=f{aT-{%l-x)\-\
+ (
■>/^{«T-(2/+c-a!)},
(2 A
&c.
./.{a(x+?i^^^)-(2/ + c-^)}.
&c.
2«/
^{«(- + ^)-(2/-^)}.
Hence we have at the time (t+ j,
V =f(aT-x)-f{aT-(2l-x)}
-{>|/[rtT-(«+c)]->|/[«T-(2/+c-a;)]}
— &c.
f, , r 2(n-l)J-\ , x> . , r 2(w-l)./n -^, ,J
+ ^|/ {« [t + — j -(21 + C-X)}
+ &c.
2w/^
+ ^{a[r + ^)-a^}-ct>{a[r+^)-i2l-a^)},
HH2
240
Mr HOPKINS ON AERIAL VIBRATIONS
or,
v=-f{aT-x)-f{aT-{^l-x)} , ^
r^n ^ it /
. . . ^nl
Similarly, we find
as = f(fiT-x)+/{aT-{2l-x)},
+ 2,., {<t> [« (r + ^) -(.r + c)] + [« (^. 4-^^) -(2/ + C- .r)]}. ^
\...(D)(1).
\.. .(D)(2).
8. The function /(ar — x), in the expression for v, represents the
velocity of any particle produced by the first wave, propagated
along the tube from the original disturbance at A, so long as t
is less than - ; and if this wave were reflected entirely from B,
a
the first line of the above expression for v, would give us the velocity
of any particle within the sphere of the reflected wave, the time t
not exceeding — .
With our supposition as to the original disturbance, the form of f
T less than — I will be immediately known from that of (p. The
IN CYLINDRICAL TUBES. 241
other terms in the general vahie of v, shew how the general waves
in which we have
If
v, = /,(af-x), and v, = Jl{at-(2l-x)},
are formed by the superposition of successive waves, as the time
increases. If the velocity becomes by this superposition so large, that
it can no longer be considered extremely small as compared with
the velocity of propagation (a), our analysis will be no longer ap-
plicable ; but if V never exceed a certain value, the motion will
become regular, and follow the law which our investigations indicate.
Let us consider in what cases we may expect these effects to be
produced.
9. We have at present imposed no restrictions on the forms of
the functions denoted by cp, f and \//, except that their greatest
values shall be small compared with a. In order however that the
undulations may be sonorous, <p, and consequently y and \f/, must
denote periodical functions, so that the values of (p {z), f (2), and ^ (ss),
will recur as often as % is increased by a certain quantity. We will
also iinpose an additional limitation upon them, to which, in all
practical cases they will probably be subject very nearly, as will
certainly be the case in the experiments to which I shall hereafter
more immediately refer. Supposing then their values to recur, when
s becomes %-Vm\, {m any whole number), we will also suppose them
to recur with different signs when z becomes x±m' -; {m! being
any odd number).
10. First suppose the greatest value of \//, small as compared with
that of y or 0, as must be the case in a closed tube. In the above
expression for v, it will be observed that the quantity represented by
% increases as we proceed from one term to the next, in a vertical
line by 2/.
Suppose then
%l = m' . -, or l = m' -
2 4
242 Mh HOPKINS ON AERIAL VIBRATIONS
In this case it is manifest that the consecutive terms taken in the
order just mentioned will destroy each other ; and there will con-
sequently be no accumulation of motion in the tube, and the
vibrations will go on uniformly. Again, let
2l = m\, or / = 2m. -.
4
In this case the values of the successive terms taken as before in
the expression for v will be equal, and with the same sign. Hence,
if we take x of any value, except such as would render
<(>{at-x) = <p {at-{9.l-x)],
f which value of x is I — m -\ , it is manifest (since the value of (p
is greater than that of \|/), that the motion will constantly increase
for such points, and will soon become greater than is consistent
with our original suppositions. Such a vibration then cannot be
maintained. .
11. Again suppose the functions (p, f, and ^, to be continuous,
and suppose
2/=m'^+2\', or / = m'^+\',
2 4
X' being any quantity less than -; the consecutive teims of 1.(f>(%),
tit
will not then destroy each other, but as the number of pairs of terms
increases, the sum will increase till ^(s; + 2r/) becomes negative, it will
then decrease, after having thus attained a maximum value. Maxima
and minima values will thus occur alternately, and the same will hold
for 2. >//(»). If these maxima values do not render v greater than our
original suppositions allow, the vibrations may be maintained.
Since these maxima values are 0, when l = m'.-, and greatest
when l=m' .-, we conclude that they will be intermediate for inter-
mediate values of I, following some continuous law. Hence we infer
IN CYLINDRICAL TUBES. 248
the possibility of maintaining sonorous vibrations of which the period
is - , in stopped tubes of which the length differs considerably from
?«' . - , particularly if the greatest value of V/ should not be very
small. If the supposition we have made respecting the continuity of
the function (p more particularly, should not be quite true, it is not
likely in those practical cases to which we can best refer, to be so
far wrong as to render the above reasoning otherwise than at least
approximately true.
12. Our supposition has been that the intensity of the distvu-bance
denoted by v//, is considerably less than that indicated by (p, the tube
being stopped with some substance having a certain degree of elas-
ticity ; if the tube be open, it seems probable from certain pheno-
mena, that the reverse of this supposition is true.
Assuming this to be the case, the expansion of the expression
for V may be put under a more convenient form.
Let
y{r {at-{2l+ c-x)} =2f{at-(2l- x)] - f, {at- (21+ c' -x)],
Then
v=f(at-x)+f{at-{2l-x)}-xj.,{at-(2l + c'-x)} (a),
and equation (3) becomes
/(aT + 2l-x)= -/(aT-x) + x|/, {aT-(x + c')} +<p(aT + 2l-x) (4).
By proceeding exactly as in the former case, we obtain
v = {-irif(aT-x)+flar-(2l-x)}}
-fAci{'r+^)-(2l + c'-x)}
+ 2,^,(-l)-{<^[a(T + ^) -X] + 0[« (t + ^y(2l-x)]}
1
ME)(1)-
.(E)(2).
244 Mr HOPKINS ON AERIAL VIBRATIONS
Similarly, we find
«*=(-l)"{/(ar-^)-/[«T-(2/-a;)]} >^
+ ^l,,{a[T +—j-{2l■^■c-x)}
+ 2,^,(-l)»-{<^[« (t + ?^) -.V] - 0[« {'r+~) - (2/ - x)\\. ^
Reasoning on the expression for v, exactly similar to that used
above, will in this case show that sonorous vibrations cannot be
maintained if / be too nearly equal to an odd multiple of - ; but
that they can be continued, if / do not differ too much from an
even multiple of - .*
13. If we examine the expressions for as in the last article, and
in Art. 7, it will appear that the condensations and rarefactions at
the surface of the vibrating plate within the tube, are such as to
produce forces opposing more strongly the motion of the plate as
the lengths of the tubes approximate respectively to those particular
lengths for which it will be impossible to maintain the vibrations in
the tube ; and when the lengths differ from the above by - , these
condensations and rarefactions are such as to promote the motion of
the plate, instead of opposing it.
14. The expanded expression for v may be put also under another
form, which it may be useful to point out for the case in which
the intensity of the disturbance denoted by \//, is considerably greater
than that denoted by <^.
* The quantity c' in these general inferences is not taken into account. Its value
however is considerable, as will be seen hereafter.
IN CYLINDRICAL TUBES. 245
This is deduced, by assuming
i,,{at-(x + c')}= (if (at -a;)+yl.'{ai-(x + c")},
or,
x/. {«/- (x + c)} = (2 - /3) /(«^ -x)+ir' {at -(x + c")}.
Then the equation (a) (Art. 12) becomes
v=f{at-x) + {l-l3)f{at-{2l-x)}-i,'{at-{2l+c"-x)} (/3).
We may observe, that since the vibration denoted by \j/, is pro-
duced by that denoted by Jl it seems a necessary consequence that
their periods must be the same. Their phases also are nearly so ;
and if in addition we assume that the Jbrm of the function ex-
pressing the one motion, does not differ very widely from that ex-
pressing the other, (however the intensity of the vibrations may differ)
it is manifest that /3 may be so taken that the intensity of the
vibrations denoted by the unknown function \j^' shall be small com-
pared with that indicated by <p.
Equation (4) becomes
f{ar + 2l-x)=-Cl-ft)f(aT-x) + i.'{aT~(x + c")}+(j>(aT + 2l-x) (5),
= -hf{ar -x)^-^' {a-r - (^ + c")} + («t + 2/- x),
if 1-/3 = *.
This gives us
And the equation (/3) becomes, (when t=T-\- j,
v^{-hY {f{aT-x)+hflar-{2l-x)]}
+ S,,,(-&)-|>/.'{«[t+ ^^^^^)-^ ]-(x+0}+&^^1«[t+ ^^''^^^-/ ]-(2/+c"-;»^)}I
-^'{a(r + ^)-{2l + c"-x)].
r-n ^ W/ \ (t )
Vol. V. Part II. 1 1
246 Mr HOPKINS ON AERIAL VIBRATIONS
Since b is less than unity, and n soon becomes a very high
number, after an extremely short time the first line in this expression
may be neglected, as may also all the terms in the other lines in-
volving high powers oi h.
Whence it follows that the original disturbance (on which the
form of the function f will depend), will cease in an extremely short
space of time to have any effect on the form of the existing vi-
bration, supposing the vibrations maintained by some cause distinct
from that producing the original disturbance.
Also, if the cause maintaining the vibrations cease, the vibrations
themselves may cease in an extremely small space of time.
The inferences we have drawn from the former developement (E)
of the expression for v, may be drawn from this and perhaps with
still greater facility.
15. If we suppose >|/' (ss) always = 0, the expression for v will
reduce itself to the same as that given by M. Poisson. But in this
case it will be observed that all the functions involving the quantity c"
disappear, which renders it impossible to account on this theory for the
position of the modes or points of minimum vibration as determined
by experiment*. For the purpose of determining the positions' of
these points theoretically we will recur to the equations (C), the first
of which is
~ v = f{at-x)-f{at-{^l- X)} +^ {at -{2l + c - X)} (6).
If we neglect ^{at-{2l+c — x)}, (or suppose the substance with
which the tube is stopped perfectly rigid) we shall have » = 0, when-
ever
{at — x) - {at — {^l- x)}=Q, or mX,
{m being any whole number), or when
{l-x) = m.-.
* See Art. 36, Sec. II.
IN CYLINDRICAL TUBES. 247
This condition is independent of t, and consequently at all points
distant from the stopped end, any multiple of -, the motion will be
the same as at that extremity, i.e. it will always equal 0, and there
will be perfect nodes at those points.
16. We may take the general case, and let
f\at-{il-x)\-^ {a/-(2/ + c-ar)} =j(; {at-{<il->rc, -x)},
and :.v=f{flt — x) — x\a't—{^l^-Cx — x)\,
^ being still small. The forms of J" and x// being known, that of ^
will be determined ; its period will also be the same as that of J"
and ■^. It expresses the velocity of each particle produced by the
whole wave actually reflected from B. The nodes will in this case
be points of minimum vibration, and not of perfect rest.
For the sake of clearness we will assume that y(x), and >//(x), are
such that
and therefore
x(-»)=-x(x),
that y(»), and ^(z), {and therefore x(*)} admit of only one maximum
value between x = 0, and 8;=-; and that the ratio which y(s!) bears
to ylr (%) is always considerable, as by hypothesis it is when those
functions have their maximum values. There can be little doubt but
that these assumptions are at least approximately true in all practical
cases ; and appear as simple as any we can make (and some must
be made), in order to give distinctness to our inferences as to the
positions of these points of minimum vibration.
17. For the determination of c, in terms of c, let the origin of
t and X be so taken that y(0) = 0, then making at- {2l — x) — 0,
we have
-^(-c) = x(-c,);
or =\l/{ — c).
112
248 Mb HOPKINS ON AERIAL VIBRATIONS
By our hypotheses, x (*) must be always greater than \// (%) ; and
if we suppose c and c^ less than the least value of z, which gives
to ^ (%), or X (^) its maximum value, it is manifest that from this
last equation, c, must be considerably smaller than c, and must be
c
affected with a different sign. Suppose c^ = j^, where k is consider-
ably greater than unity. It follows then that if the phase of the
vibration of the extreme section of a stopped tube be retarded by a
certain quantity c, the phase of the actually reflected wave will be
c
accelerated by a quantity t.
18. Giving then the proper sign to c„ we have
v=f(at-x)-x{at-(2l-^-a;)} (7),
and to determine the points of minimum vibration, we may observe
that this expression is exactly the same, as if the wave for which
v, = x{at-{2l-^-x)},
were reflected immediately from a section B' whose distance from A = l — —x.
Suppose a rigid diaphragm at this section constrained to move
exactly as the fluid does there ; we may then suppose the actual
stop B removed, and the points of minimum vibration will remain
the same.
Now to determine them in this case, we observe that whenever
at — x = at—{2l — T — x) + m\.
the value of v will be the same as when
c
at—x = at—{2l— T - x).
In the latter case
IN CYLINDRICAL TUBES. 249
and in the former
or l-^ = m\ + ^^',
consequently, at any point in the tube whose distance from B" = m .-^,
the velocity will be the same as at B'. These then will be points of
minimum vibration in this hypothetical case, and therefore also, from
what precedes, in the actual case.
Making c = 0, we have l—x = m.-, which will give the positions
of the nodes when there is no retardation.
Hence we have this general conclusion with respect to the stopped
tube — that if there be a retardation in the phase of the vibration of
the extreme section, the positions of the points of minimum vibration
will all be further from the stopped end by —j, than if there were
no such retardation, the distances between these points respectively
remaining unaltered.
19. We will now consider the case of the open tube, in which
we suppose >|/(a!) to be always considerably larger than J'{%). Assume,
as in Art. (12),
yl,{at-{2l + c-x)}-y{at-{^l-x)}~^, [at - {21 + c' - x)} (8),
v=f{at-x)-¥f{at-{2l-x)}-^,{at-{2l + c'-x)}.
First neglecting the function >|/, , v will = whenever
f{at-x)^-/\at-{2l-x)}; ^
i. e. whenever
at—x = at—{2l—x) + m'.- {m' an odd number),
or I— x = m .-,
250 Mr HOPKINS ON AERIAL VIBRATIONS
a condition independent of /. Consequently, at every point whose
distance from the open end is an odd multiple of -, there would be
a perfect node.
20. Put
f{at-{2l-x)\-y},,{at-{2l + c'-a;)] =x {at- (2l+c,-x)\ (9).
Then
v=/{at-x) + x{at-{2l + c,-a;)} (10).
To find the relation between d and c, we have from equation (8),
(proceeding as in Art. 7, and with the same assumptions),
^(_c)=-x/„(_c'),
or >|,i(c')= -x|/(c);
and since >//(») is much larger than >/'i(i8), we shall have c'. considerably
larger than c, and affected with a different sign. We may therefore put
ki being greater than unity.
Again from equation (9),
-^.(-0=x'(-c.), •
or x'(<=-^)=-Uc').
If we suppose x'(«) nearly equal to v//^,(i8), (which probably is not
far from the truth), we shall have
C2= —c' nearly,
Hence in this case if the phase of the vibration of the extreme section
be retarded by a quantity c, that of the actually reflected wave will
be retarded by kic; and it will appear by the same reasoning as in the
case of the closed tube, that the distance of the points of minimum
vibration from the open end will be m' -r 1-, {m' being any odd
number).
IN CYLINDRICAL TUBES. 251
21. If e and e' be the distances through which the nodes are moved
by a supposed given retardation of phase, the same for each, at the
extremities of the open and closed tubes respectively,
e = — kki e ;
6 will consequently be much larger than e'.
The quantities m' i- in the open tube, and m- + -^ in the
4 2 ^ 4 2«
closed one, must be determined by experiment.
22. I will recapitulate the principal inferences from this theory.
I. In the tube AB, open at the extremity B opposite to that at
which the vibrations are produced, there will be a series of nodes
equidistant from each other by -, or half a whole undulation, the
distance of the nearest node from the open extremity being considerably
less than -, the whole system of nodes being thus brought nearer to
the open end than the position assigned to it by the investigations of
Euler or of M. Poisson. The distance of each node from the open
end will be independent of the length of the tube. (Art. 20.)
II. If the tube be closed at B, the nodes will still be equidistant as
X
2
before by - . The distance from B of the node nearest that extremity
will be - , or a quantity rather greater than that, if we suppose a cause
of displacement of the whole system of nodes to exist in this case of
the closed tube, similar to that which exists in the open one ; the dis-
placement however being necessarily much smaller in the former than
in the latter case, and in the opposite direction. (Art. 18.)
III. These nodes are not places in which the air is perfectly at
rest, but points of minimum vibration. (See Art. 16.)
252 Mb HOPKINS ON AERIAL VIBRATIONS
IV. Sonorous vibrations, whatever be their period, may be main-
tained in a tube of any length, except that of which the length does
not approximate too nearly to something less than an even multiple
of J in the closed tube, or to an odd multiple of - in the open one.
(Arts. 11, 12.)
V. The intensity of the general vibrations in the tube varies with
the length of the tube, being greatest for the lengths just mentioned,
and least in the closed tube when its length is rather greater than an
odd multiple of -; and in the open one, when it is something less than an
even multiple of -r . (Art. 10.)
VI. In these latter cases also of both tubes, the opposition afforded
by the vibratory motion of the air within the tube, to the vibrating
of the plate, is least; and greatest for the lengths which approximate
to those mentioned in (IV.), as those with which the vibrations cannot
be maintained. (Art 13.)
VII. When the cause producing the vibrations in a tube ceases,
the vibrations themselves may cease, not instantaneously, but in a period
of time not exceeding the small fraction of a second, supposing the
tube not to exceed a few feet in length. (Art. 14.)
VIII. If we suppose the original disturbance to produce an un-
dulation different in any respect to those produced by the cause which
afterwards maintains the vibratory motion of the aerial column, this
original disturbance will cease to affect the form of subsequent undula-
tions in a period of time not exceeding the small fraction of a second,
depending on the length of the tube*. (Art. 14.)
* Similar inferences to the above may be drawn equally from M. Poisson's investigations,
except that the phenomena according to his solution would take place for lengths of the open
tube materially different from those above-mentioned.
IN CYLINDRICAL TUBES. 253
SECTION II.
23. I WILL now proceed to describe the experiments which have
been made with a view of putting the different theories on this subject
to an experimental test. Sonorous vibrations are usually excited in a
tube, either by directing a stream of air across the open end, as in
blowing across the embouchure of the flute; by means of a vibrating
tongue, as in all reed instruments ; or by placing an open end of the
tube close to the surface of a vibrating body. In the two first cases it
seems impossible to conceive that the same disturbance can be com-
municated to each part of the extreme section of the air in the tube
where the original motion is produced, a condition which is always
assumed to hold at least approximately in all our mathematical investi-
gations of the subject. This irregularity of the motion will no doubt
extend to some distance within the tube, and it is impossible to say
how it will affect the phenomena even in those parts of the tube in
which the motion may become more uniform. In the second case too
in particular, a stream of air must constantly be passing through the
tube, a circumstance not contemplated in our analysis of the problem.
This may or may not influence materially the observed phenomena,
but at all events the danger of derangement from any such cause
must be avoided, if we would render our experiments decisive tests
of the truth of any theory professing to account for phenomena of so
delicate a nature as those which are now the objects of our investigation.
The third method, however, above-mentioned, is entirely free from the
latter objection, and may be made almost entirely so from the former,
and is, therefore, that which I have adopted.
24. The apparatus is very simple. Figure I. represents it. A
plate of common window glass is held firmly in a horizontal position
by a pair of pincers at its middle point. AB is a gltiss tube, having
a short brass tube closely sliding within it at the upper end B, so
that the whole tube AB can be lengthened or shortened at pleasure.
Within the tube a small* brass frame M, having a delicate membrane
* Fig. (2) represents this frame with the membrane ab, which may be tuned, or rendered
sensitive in different degrees, to the vibrations produced by any proposed note, either by
Vol. V. Paet II. K k
254 Mr HOPKINS ON AERIAL VIBRATIONS
stretched across it, is suspended by a fine wire or thread from the upper
extremity of the tube, in such a manner that it can be heightened or
lowered at pleasure. The other parts of the apparatus are merely such
as are adapted for facility and -accuracy of arrangement of the tube
and plate.
25. The air in the tube is put in a state of sonorous vibration
by means of the plate, which is made to vibrate by drawing the bow
of a violin equably across its edge in a direction perpendicular to its
plane ; the vibratory motion of the air is communicated to the membrane
suspended in the tube, and the degree of motion is indicated by the
agitation of a small quantity of light dry sand sprinkled upon it*.
Suppose the tube to be open at the upper end B, and let the membrane
be drawn up near that extremity. Tf the sand indicate a considerable
motion when the plate is vibrating, let the membrane be gradually
lowered ; a position will thus be found in which the sand has little
or no apparent motion, thus indicating the existence of a node. On
lowering the membrane still further, the sand will again become strongly
agitated, and will then come to another place of rest, (or at least of
minimum vibration), and so on till it reach the lower end of the tube.
These alternations of points of rest and motion can of course only take
place when the tube is sufficiently long in comparison with the length
of an undulation produced by the vibrating plate, to admit of them.
These nodal points are thus found to be at equal distances from each
other, the distance of the upper one from the top of the tube being less
than half that between the nodes. This is independent of the length of
the tube. These results are accordant with our theory, (Art. 22, I.), from
which it appears that this constant distance between two consecutive
nodes must be -.
2
If we call the distance of the upper node from B, -— C, C denotes
what I have termed the displacement of the nodes.
altering the tension by means of the small cylinder round which the end b of the membrane
passes, or by moving the small bridge cd, and thus altering the length of the vibrating part.
* This was the method adopted by Savart in such a variety of caseSj in which he wished to
ascertain the intensity of sonorous vibrations in air.
IN CYLINDRICAL TUBES. 255
26. If the membrane be rendered very sensitive by being exactly
tuned to the note produced by the vibrating plate, it will not indicate
perfect rest at the nodal points, shewing them in fact to be points of
minimum vibration, which agrees with our theory, (Art. 22, III.).
With such a membrane it will be difficult to determine the position of
these points with accuracy, and its sensibility should be diminished,
till the sand appears perfectly at rest when it is placed exactly at the
node. If the membrane be rendered still less sensitive, it will appear
at rest for a space on each side of the node, the position of which will
in such case, be determined by observing those points immediately
above, and below the node at which the motion of the sand is just
sensible. The middle point between them will of course be the
node.
27. Now suppose the length of the tube to be any odd multiple
of -, and the membrane to have such a degree of sensibility, as just
to remain at rest only when placed in a node or within a very small
distance of it. After it has been placed in this position, let the brass
tube sliding within the upper part of the glass one be raised through a
space less than - . While the whole tube is thus lengthened, let the
distance of the membrane from the upper end B remain the same;
the membrane will consequently be still in a node. The plate being
now put in vibration, the membrane will remain perfectly at rest, not
only in this position, but also when moved to one considerably above
or below the node, the new length of the tube remaining the same.
This indicates a less degree of motion in the tube than in the former
case, and we find that the intensity of the vibration in the open tube
is least when its length is equal to something less than an even
multiple of -T, or 2m.j — C; and becomes greater as the length
approximates to rather less than an odd multiple of -, or {2m' + 1)-—C,
m and m' being any whole numbers. (Art. 22. V.). This diminution of
motion is also very obvious when the membrane is placed in those
KK2
Si56 Mr HOPKINS ON AERIAL VIBRATIONS
parts of the tube where the motion is most sensible. In all cases,
however, the distances of the nodes from B is independent of the
length of the tube.
28. If we take a tube closed at B instead of the open one, we
observe the same alternations of points of greatest and least vibration,
and (the plate being made to vibrate in the same manner as before)
at exactly the same distances from each other as in the closed tube;
but the distance of the upper node from the closed extremity of the
X I
tube is now observed to be -, the same as the distance between the
2
nodes. Proceeding as in the former case, it is found also that the
strongest vibrations are excited when the length of the tube is about equal
to a multiple of - ; and the least vibrations when the length = an odd
multiple of - . I find also that in the open tube stronger vibrations exist
4
in the nodal points than for corresponding cases of the closed tube.
29. In performing the above experiments with reference to the
intensity of the vibrations in the tube, care must of course be taken
to prevent the influence of any other cause than that of which I have
spoken, viz. the length of the tube with respect to X. It has been as-
sumed that the vibration of the part of the plate immediately in contact
with the mouth of the tube is in all cases the same, which requires
that the tvibe should always be placed over exactly the same portion
of the plate. This portion also should be included in one and the same
ventral segment; for if a nodal line on the plate pass across the mouth
of the tube, the vibrations transmitted from opposite sides of this line
will be in exactly opposite phases, and will consequently neutralize each
other in a degree depending on the ratio which the intensity of one
of these undulations bears to the other. If the nodal line divides the
part of the plate in contact with the mouth of the tube into two
equal portions, parts of similar ventral segments, the interference
will be so complete as to destroy all sensible motion in the
IN CYLINDRICAL TUBES. 257
tube*. It is only however as regards the intensity of the vibrations
that this precaution respecting the relative position of the nodal
lines and mouth of the tube is important ; it does not affect the
positions of the nodes. The reason is obvious — it does not affect the
value of X.
30. Again, taking the tube open at B, let the extreme section
at A be made to coincide nearly with the surface of the vibrating
plate. If the plate (the bow being applied to it) vibrate freely, let
the length of the tube be gradually increased or diminished. It will
thus be found, that as the tube approximates to certain lengths, the
plate vibrates with less facility, requiring a greater pressure of the
bow, and continuing to vibrate audibly for a shorter time after its
removal; and in many cases, between certain limits in the length of
the tube, it becomes almost impossible to make the plate assume that
state of vibration which it assumes freely for other lengths ; and the
vibration, if it be produced, appears to cease almost instantaneously
on the removal of the bow, instead of being audible for several
seconds, as it would be if the tube were removed, or were of a
different length. These phenomena recur for every increase of — in
the length of the tube ; and if I be any length with which it becomes
almost impossible to make the plate vibrate in the manner proposed,
then will / + - be that length with which it vibrates with the same
facility as if the tube were removed.
* It is easy by a very simple experiment to give ocular demonstration of the fact that the
union of two intense sounds may produce perfect silence. Take a branch tube ABA' (Fig. 3.),"and
stretch over the open end B a fine membrane or a piece of common writing paper. Place the
open extremities A, A' of the equal and similar branches CA, CA' over portions of two ventral
segments of a vibratory plate in the same phase of vibration. A small quantity of sand strewed
over the membrane at B, will immediately shew it to be in a state of strong vibration. Let A, A
be then carefully placed over suiiilar portions of similar ventral segments of the plate, in opposite
phases of vibration ; the sand on the membrane will remain perfectly at rest, shewing that the
waves propagated along AC and A'C in opposite phases so completely interfere at Cas to produce
no undulation along CB. In other words, no sound would in this case be transmitted along the
tube to its mouth B.
258
Mk HOPKINS ON AERIAL VIBRATIONS
So far these phenomena are in accordance with the results of
theory, (Art. 22, VI.) ; but when we examine the length I just men-
tioned, we find it entirely at variance with them. In fact on
investigating the circumstances more narrowly, we find that the value
of / depends in a considerable degree on the small distance between
the vibrating plate, and the extreme section A of the tube, a cir-
cumstance which nothing in our theoretical deductions has led us
to anticipate. This will be seen in the results of the following ex-
periment made with an open tube.
Diameter of the tube = 1 . 35 inches.
Value of- ,.=4.82 for temperature 63°.
Position of the mouth (^A) of the
tube (Fig. I.)
Value of the length /
above mentioned.
Theoretical value of /.
As close to the plate as
possible without interfering
with its vibrations
About T^ inch from thel
lo I
vibrating plate.
.12.25 inches.
* 1 1 . 46 inches.
12. 6
31. This discrepancy however between the theoretical and ex-
perimental results is only apparent. It arises from the circumstance
of one of the conditions assumed in our mathematical investigation,
not being accurately satisfied, namely, the perfect prevention of all
communication between the external air and that within the tube at
the extremity next the plate. And this is easily proved experi-
mentally, by placing the extremity of the tube as near as possible
* In this value of Z I have taken account of the displacement of the nodes, which is .59
inches, as determined by experiment. (See Table, Art. S6.)
IN CYLINDRICAL TUBES. 259
to the surface of the plate, without interfering with its vibrating
motion, and then putting round the edge of the tube, a small
quantity of fluid which by its adherence to the tube and the plate
fills up the interstice between them, and prevents communication with
the external air. When this precaution is taken, the lengths of the
tube which correspond to the above mentioned phenomena exactly agree
with theory; that is —
The .vibration of the plate is unaffected by the presence of the open
tube, Avhen its length is equal to something less than an even multiple
of — , or 2 m. J— C, and of the closed one when its length is equal to
4 4
an odd multiple of -; but as the lengths of the tubes approximate
respectively to quantities differing by - , from the above lengths it
becomes almost impossible to make the plate assume the same vi-
bratory motion. (Art. 22, VI.)
32. It might at first appear probable that the neglect of this
precaution might have some effect on the position of the nodes, as
well as on the phenomena above mentioned. This however is not
the case; and the reason will be obvious if we recollect that the
position of the nodes depends on the periodicity of the vibrations, or
the value of X, which is unaffected by the communication with the
external air at A ; whereas the force opposing the vibration of the
plate depends on the condensations and rarefactions of the air, at the
surface of the plate within the tube, which will necessarily be much
affected by the communication just mentioned.*
33. If we take a closed tube, a similar discrepancy or accordance
in the results of theory and experiment will be found under the
same circumstances as above described.
* It does not appear so easy to account for the phenomena as above described, when the
influence of external air is not prevented. This, however, does not immediately belong to the
object I have proposed to myself in this paper, which is, to establish as accurately as possible the
identity of the results of theory and of experiment in those cases in which the conditions assumed
in our mathematical investigations are experimentally satisfied.
260 Mr HOPKINS ON AERIAL VIBRATIONS
The phenomena above mentioned, agree with those observed by
Mr Willis, and described in his paper on the Vowel sounds, pub-
lished in the Transactions of this Society, Vol. III. The manner
however in which his experiments (having a different object from
mine) were conducted, render them unfit for the verification of any
of our mathematical results in this subject.
34. From what I have above stated, respecting the difficulty of
making the plate vibrate with certain lengths of the tube, it is manifest
how we may avail ourselves of this phenomenon, in the determination
of the value of X, corresponding to any particular mode of vibration
of the plate, supposing those particular lengths of the tube can be
ascertained with sufficient accuracy. Now this can be done almost
as accurately as the position of a node can be determined by the
vibrating membrane, and consequently the value of X may thus be
found. For if A and 4 denote two observed values of /, we shall have
— = , n being a whole number easily ascertained. (See Arts. 30, 31.)
35. Though I have had frequent occasion to speak of this displace-
ment of the nodes in the open tube, from the positions assigned to them
by the common theory, I have hitherto said nothing as to the ex-
perimental determination of its magnitude. The most direct way of
accomplishing this, is to determine the actual positions of the nodal
points by means of the vibrating membrane ; but this method becomes
inconvenient when the diameter of the tube is small, as, for instance,
less than an inch. Those which I have used most commonly are
from 1.3 in. to 1.5 in. diameter. If the tube be larger than this, it
will generally be too large to admit of the extreme section of it
being placed entirely upon the same ventral segment of the plate,
as is always desirable, (see Art. 29) ; and if much smaller it becomes
necessary to make the surface of the membrane so small as to be
inconvenient, in order that it may not bear too great a ratio to the
area of the section of the tube, in which case the presence of the
membrane might be supposed to render the vibrations in the tube
materially different from what they would otherwise be.
IN CYLINDRICAL TUBES. 261
The best method therefore of determining the positions of the
nodes in tubes considerably smaller than those I have mentioned, is
that by which the value of \ is determined, as described in the
last Article.
Thus, suppose / to be the length of tube, with which it is found
most difficult to make the plate vibrate ; then (the tube being open)
we shall have
l={2m + l)^-C,
where m is a whole number, which will be known when \ is de-
termined by either of the methods pointed out above. The quantity
C evidently shews how much the distance between the open ex-
tremity, and the nearest node differs from — , or it expresses the
displacement.
From the above equation,
C={2m + l))-l,
4
and the displacement is thus determined.
36. The following table exhibits the magnitude of this displace-
ment in a tube of given diameter, as determined experimentally for
different values of - . The positions of the nodes were in these cases
carefully ascertained by means of the membrane suspended in the
tube.
Vol. V. Paet II. L i.
262
Mr HOPKINS ON AERIAL VIBRATIONS
Diameter of the tube = 1.35.*
Value of — .
at temp. 63".
Computed dist. of a Node
from B, (fig. 1.)
Observed dist. of the
same Node.
Displacement of
the Node.
2.044
3.994
4. 82
fll.24
I 7.15
9.98
7.23
10.88
6.78
9.51
6.64
.36]
} mean = .36.'i
.37]
.47
.59
The above values of - were determined by means of a membrane
2 •'
and a tube closed at the upper end, nearly 100 inches in length. The
distance of a node from the closed end being found = b, we must
Or, if bf be the observed distance, sub-
have n . - = o, or - = - ,
2 2 w
ject to an error /3, and therefore b ± (i the true distance, we have
- = - + —. The value of /3 will probably be less than ^ inch, and
t^ Tt ft
in the determination, for example, of the first of the above values
of -, « was about 45, so that that value of - may probably not be
subject to an error exceeding .001 inch. We may also remark, as an
indication of accuracy in the numbers 10.88 and 6.78, given in the third
* The measures are all given in inches.
t In the determination of the quantity b, the temperature at the time of observation must
be carefully noted, since the variation in the velocity of aerial undulations produced by a varia-
tion of temperature of even less than 1°, is sufficient to make a very sensible difference in the
value of h, this value being as much as nearly 100 inches.
Since the distance of any proposed node from the upper end of the tube will be proportional
to the velocity of the undulation, it is manifest that by observing the values of b, corresponding
to different temperatures, we may estimate directly the effect of temperature on the velocitj' of
sound. This method is capable of great accuracy.
IN CYLINDRICAL TUBES.
263
column, that 10.88-6.78 = 4.10 must =2.-, which gives us - = 2.05,
differing but .006 from the more accurate value. The error in the
two numbers above mentioned, 10.88 and 6.78, does not probably exceed
.01 or .02, and cannot, I conceive, exceed .04, and consequently, I think,
the utmost limit to the error in the corresponding numbers in the
fourth column cannot exceed .05, and is probably considerably less. The
same may be concluded respecting the numbers .47, .59, in the same
column.
The above results may, then, be considered sufficiently accurate to
determine the fact of the magnitude of the displacement increasing
with increased values of X, though not sufficiently so to determine with
certainty the law of this corresponding increase.
The displacement does not depend only on the value of \ ; it depends
also on the area of the mouth of the tube, as appears from the following
table.
Values of ^ •
Displacement.
Diameter of tube = 1.35.
Diameter of tube = .8.
2.04,4
3.99^
.23
.4
.08
.1
These values of the displacement of the nodes have been obtained
by the method mentioned in Art. 35, as that best applicable to small
tubes. The results in the second column of this table ought to be
the same as the two first in the last column of the former table; but
this method is liable, I conceive, to greater error and uncertainty than
the former, and to this, I doubt not, the discrepancy is due.' All
these latter results, however, are probably subject to an error of the same
L L2
264 Mr HOPKINS ON AERIAL VIBRATIONS
kind, and are too small both in the large and' small tube. They can
leave no doubt of the fact of the magnitude of the displacement being
dependent on the diameter of the tube.
It is important to observe, that the values of X determined in the
large tube and the small one, from the consideration that the distance
between any two nodes must equal some multiple of - , was exactly
the same, being for the first case in the table 2.05, very nearly agreeing
with the accurate value 2.044. This proves that the distance between
the nodes is independent of the diameter of the tube, provided the dis-
turbance take place uniformly throughout its extreme section.
37. I have before remarked, that there can be nothing arbitrary
or indeterminate in the vibratory motion of the air at the extremity
of the open tube when the vibrations in it are excited according to
some known law ; and consequently, if our theoretical knowledge of
the subject were complete, we should undoubtedly find in our investiga-
tions the cause of the retardation of phase, of which I have spoken,
in the reflected wave of the open tube, supposing it to be the actual
cause of that displacement of the whole system of nodes which I have
established as an experimental fact. Our knowledge at present, how-
ever, is totally inadequate to this purpose, and therefore we can only
conjecture what may be the probable cause of this retardation in the
reflected wave; but at all events, our formulse, with the modifications
1 have introduced into them, do become perfect representations of all
those phenomena which we can distinctly determine by experiment,
in the cases to which our mathematical investigations apply. The fact
too, of a retardation of phase in the reflected wave may not be very
difficult to conceive, or appear improbable, if we suppose the undulation
proceeding from the open end of the tube to advance through a certain
space before it assumes that form in diverging into free space, which
it must ultimately assume when it sends back no reflected wave from
any point of its path. Before it reaches this state, a partial wave may
be reflected in its course from each point towards the tube; and an
indefinite number of these reflected waves will form a general reflected
IN CYLINDRICAL TUBES. 265
wave, of which the period will be the same as that of each of its
component waves, but the phase of which will be retarded as compared
with that of a wave reflected immediately from the extremity of the
tube. This is equivalent to our supposing a certain space beyond the
extremity of the tube as subject to a disturbance (acting at consecutive
instants along this space) such as to produce a wave diverging in all
directions, and consequently sending a portion of this general wave
back along the tube.
To give generality to the investigations of the preceding section,
I have considered the effect on the position of the nodes which would
be produced by any retardation of the phase of the wave reflected from
the stopped end of a tube. It appears, however, that there is not in
this case any displacement of the nodes appreciable by the mode of
experimenting I have described. The only reason, in fact, for supposing
any retardation of phase in this case, is founded in the imperfect
analogy between the cases of the open tube and the tube closed with
an elastic substance. The cases are far too different, however, to admit
of any thing but vague inferences from such analogy ; and it is
manifest that no reasoning similar to that above applied to the open
tube, can be applied to the closed one. If any retardation do exist in
this case, I can only conceive it to arise from a cause similar to that
suggested by Mr Willis*, viz. that time must be necessary for the
action between the elastic stop and the air to produce its effect. This,
however, appears much less probable in this case than in that which
suggested the idea to Mr Willis, in which the action between the air
and the vibrating body (a membrane) was lateral instead of being direct,
as in the present instance. I have not been able to detect any indica-
tion of such law of force in a displacement of the nodes in the closed
tube, though I have examined the case with great care, conceiving
that any facts bearing directly upon the nature of the mutual action
of two elastic media at their common surface must necessarily be of
importance.
* Cambridge Transactions, Vol. IV. Part III. p. 346.
266 Mr HOPKINS ON AERIAL VIBRATIONS
The experimental deductions in the preceding part of this section
are based on the evidence afforded by the exploring membrane, because
it is more direct than any other evidence which the phenomena appear
to admit of, and therefore better calculated to supply those decisive
and positive tests for ascertaining the accuracy or fallacy of our theoretical
results, which it is my object to supply. We have seen the perfect
accordance of these results with the general indications of the membrane,
and also with the striking and well-defined phenomenon of the im-
possibility of making the plate vibrate in a certain manner with tubes
of certain lengths. It remains for us to consider also how far our
theory agrees with the phenomena of resonance, in those cases in
which the conditions assumed in our mathematical investigations are
satisfied, viz. where the communication between the external air and
that in the tube at the surface of the plate is prevented, and the
disturbance extends uniformly over the whole orifice. In such cases
it will appear from the following enunciation, that the intensity of
the sound is proportional to that of the aerial vibrations, as indicated
by the membrane, and by the difficulty or facility with which the
vibrations of the plate may be maintained. (See Arts. 27, 31.)
The resonance of the open tube is scarcely perceptible when the length
of it does not differ much from something less than an even multiple
of -, or 2m • j - C ; but as it approximates to something less than an odd
multiple of that quantity, or (2m'+ 1)- — C, the resonance increases, and
at length becomes of painful intensity, increasing till it is no longer possible
to maintain the same mode of vibration of the plate. Whether the length
of the tube be gradually increased or diminished in approximating to
the above-mentioned lengths, the phenomena are precisely the same.
I was the better pleased to obtain this result, inasmuch as those
which I first obtained (when the precaution of preventing communication
with the external air was not attended to*), as well as those of previous
* In such cases the resonance was always greatest (as in the case considered in the text)
when the difficulty of making the plate vibrate was greatest. The corresponding lengths of the
tube may be seen in Art. 30.
IN CYLINDRICAL TUBES. 267
experimenters, appeared either to contradict theory, or at least to be
altogether anomalous. According to our common notion on the subject,
an open tube gives the strongest resonance when its length is nearly
equal to an even multiple of 7, instead of an odd multiple, as above
stated ; and Savart* has given this as the result of his own experiments
for tubes of about the same diameter as those I have usually employedf ;
but asserting also that the length is less as the diameter is increased,
and this too whether the disturbance extend over the whole orifice of
the tube or not. My results, however, are entirely at variance with
this latter assertion, for I confidently conclude from them that if the
disturbance extend uniformly and equably over the orifice of the tube,
the phenomena will be independent of its diameter:]:, with the exception
of the effect it may have on the displacement of the nodes |. If, however,
the disturbance extend but partially over the orifice, I see no reason
to doubt the accuracy of the last-mentioned results of M. Savart ; and
this supposition will also account for the apparent discrepancy between
his results and mine as respects the length of the open tube (of which
the diameter does not much exceed an inch) producing the greatest
resonance; for it is manifest that with this partial disturbance none
of that condensation and rarefaction on the surface of the plate can
take place, which in my experiments necessarily attends, and may be
considered as causing, that powerful resonance of which I have spoken.
It is easily seen, in fact, that when the length of the tube is neany
equal to an odd multiple of -, the phase of the waves reflected from
any considerable part of the orifice not occupied by the vibrating plate,
will be directly opposite to that of the waves propagated by the plate
itself; and that thus a great part of the vibration within the tube will
be destroyed by interference.
There is no difficulty, therefore, in explaining the non-existence of
resonance in this case. If the tube, however, be lengthened or shortened
by about - , (still supposing the disturbance at its mouth partial), a
* Annates de Chimie, Tom. XXIV. p. 56. t See Art. S6.
t See Art. 36, p. 264. § Art. 36.
268 Mr HOPKINS ON AERIAL VIBRATIONS
resonance will be heard, though extremely feeble as compared with
that I have found in my experiments. This is, in fact, the kind of
resonance which has been observed by all experimenters. It does not
appear to me to admit of the same obvious explanation which the
other admits of ; that which is usually received being, as I conceive,
in itself insufficient, when subjected to those restrictions which must
be imposed upon it by the general laws which govern the communication
of motion from one particle of matter to another. At present, however,
it is not my object to enter on the discussion of this and of some
other points relative to this part of the subject. It is sufficient for
me now to have shewn that that powerful resonance which I have
observed in my experiments is exactly accordant with the results of
our mathematical investigations, when the conditions assumed in those
investigations are fully satisfied. I hope to return to the careful examina-
tion of other cases at a future period.
I have already alluded* to a paper by Mr Willis, published in the
Transactions of this Society, in which he has described some experiments
bearing on this subject, and affiarding a general corroboration of some
of the results above stated. He fixed a reed to a sliding tube, and
observed the intensity of the sound, when the reed was made to speak,
produced by different lengths of the tube, and by means of a microscope
carefully adjusted, he was able to observe the excursions of the reed
in its vibration, and to obtain micrometer admeasurements of them.
He thus found that when the length of the tube equalled about an
even multiple of - , it gave the exact note of the reed with no perceptible
resonance. As the tube was gradually lengthened, the tone was flattened,
and as the length approximated to about an odd multiple of -, the
extent of the reed's excursions was diminished, its vibrations became
irregular and convulsive, till at length it ceased to produce any musical
tone. When the tube, however, was a little lengthened beyond this
point, the reed suddenly assumed its original form of vibration,
producing a note of painful intensity, similar to that which I have
* See page 260.
IN CYLINDRICAL TUBES. 269
described in my own experiments, although the extent of excursion of
the reed was in this case less than in that in which no resonance was
produced.
One discrepancy is observable between this experiment and mine,
inasmuch as the intensity of the sound, instead of increasing as the
length of the tube approximated to the odd multiple of - , as in my
experiments, gradually decreased*. The explanation, however, of this
fact, is easily found in the diminished excursion of the reed, and still
more, I suspect, in the irregularity of its vibration, by which the
undulations produced by it are probably rendered imperfectly sonorous^.
With this explanation of this apparent discrepancy, the general results
of Mr Willis's experiments afford as strong a corroboration of those
Avhich I have obtained, as the difference between our modes of experi-
menting will allow. The flexibility of the reed, however, and its
consequent ready obedience to the vibrations of the air, as compared
with the inflexible obstinacy of a glass plate, together with the partial
disturbance produced by the reed, render it a totally unfit agent in
obtaining experimental tests for our mathematical results, though it
presents to us in its own motions many interesting points of enquiry.
Our theory will also perfectly account for one of the most striking
phenomena observable in wind instruments, viz. the rapidity with which
different states of vibration are assumed within the tube, corresponding
to different effective lengths of it, as determined by the opening or
closing of the finger holes. We have seen (Art. 22, VII. VIII.) that
* For a very clear and distinct account of these experiments, I must refer the reader to the
excellent paper from which the above is taken. It will be observed, however, that the results
mentioned in the text were not the direct objects of Mr Willis's investigations, but were such as
naturally offered themselves in the course of his experiments on the production of the vowel
sounds.
t I think it very possible that \heform of the aerial vibrations may have more to do with our
sense of the intensity of sound than has been generally supposed ; and perhaps some cases of
resonance may admit the most satisfactory explanation on this hypothesis.
Vol, V. Part XL M m
27tt Mb HOPKINS ON AERIAL VIBRATIONS IN CYLINDRICAL TUBES.
according to theory, if the cause maintaining the vibratory motion in
a tube be suddenly changed, (as in passing from one note to another),
the effect of the former mode of disturbance on the form of the
succeeding vibration will become inappreciable in an exceedingly short
period of time. Now in the most rapid musical passages, the number
of notes played in a second never probably exceeds ten or twelve,
and these usually embrace only the higher notes of the scale, for
which there must be many hundred vibrations in a second. Suppose
this number, however, not greater than about two hundred ; any undula-
tion transmitted from the reed or embouchure would still be reflected
about twenty times at the open end in the interval between two
consecutive notes in the most rapid musical passage. Now assuming
unity to represent the intensity of a wave incident at the open extremity
of the instrument*, let 1 — )3 represent that of the reflected wave,
(1 — /3)". will represent (at least sufficiently approximately) its intensity
after n reflections ; and consequently, as we have no reason to suppose /3
very small as compared with unity, it is probable that after five or six
reflections, the intensity of this wave will be quite inappreciable. Hence
the apparently instantaneous cessation of sound after the exciting cause
has ceased, and the most rapid transition from one note to another,
are perfectly accordant with theory.
M. Poisson, in the Memoir referred to in the early part of this
paper, has also investigated the vibratory motion of air in two tubes
of different diameters united together at one extremity. I hope to
examine this case also experimentally. His results must necessarily be
erroneous, as far as they depend on the physical condition he has assumed
to exist at the extremity of the open tube, and which I have shewn to
be inconsistent with observed phenomena in the uniform tube.
* See Art. 14.
W. HOPKINS.
St Peter's College,
i\r««/ 20, 1833.
271
XI. On the Latitude of Cambridge Observatory. By George Biddell
Airy, M.A. late Fellow of Trinity College, Plumian Professor
of Astronomy and Experimental Philosophy, and one of the
Flce-Presidetits of the Society.
[Read April 14, 1834.]
The accurate determination of the latitude, with an instrument
like the Mural Circle now in use at the Observatory, seems at first
sight to be an easy business. In practice, however, it is not without
difficulties. I do not here allude to the correction for refraction ;
since, though there may be a trifling uncertainty in regard to its magni-
tude, it is easy to leave , a result subject to that uncertainty, and
admitting of correction without any trouble whenever a correction of
the refraction shall be established. Nor do I allude to the uncertainty
in the corrections by which, from a star's apparent place on any day
of observation, its mean place at a fixed epoch can be inferred; since
the uncertainty about any of these is far less than the smallest quantity
for which we could pretend to answer in fixing the latitude of any
place ; and its effects being periodical, would in a comparatively short
series of observations, produce no sensible effect. The difficulties to
which I allude are instrumental: they are not periodic in time, like
the latter; nor do they admit of correction from posterior researches,
like the former of the causes of uncertainty which I have mentioned ;
they are moreover such as would scarcely be suspected to exist, until
their effects are discovered from the discordance of the results of
observations.
The Mural Circle is an instrument which gives simply the reading
of that point of the graduated limb which is opposite to an imaginary
fixed index when the telescope is pointed to the object of observation.
M M2
272 PROFESSOR AIRY ON THE LATITUDE
A single observation therefore gives us no tangible result. It is
necessary to have one other observation, or a series of observations,
by which the reading of that point of the limb can be found which
is opposite to the same index when the telescope is directed to some
point of reference; then the difference between this reading and the
former is the angular distance of the object observed from the point
of reference. It was intended originally by the maker that this point
of reference should be the celestial pole. In practice, however, it is
found necessary to descend one step nearer to terrestrial things, and to
adopt for the point of reference the zenith ; a point which, though not
marked any more than the pole by any obvious phenomena, can yet
be discovered by a process which involves less of astronomical assump-
tions, and requires a shorter time for the complete determination.
The method of determining the zenith point from observations
by reflexion at the surface of mercury, has been introduced into
observatories almost entirely by the practice of the present Astronomer
Royal at the Greenwich Observatory. The use of two similar circles
(as at Greenwich) makes the process one of little labour, though requiring
the co-operation of two observers. The same celestial objects being
repeatedly observed by direct vision with both circles, the differences
of the corresponding readings of the two circles are found ; and any
observations made with one can be referred to the other. Then when
any bright star passes the meridian, one circle is employed in observing
it by direct vision, and the other at the same time is employed in
observing it by reflexion at the surface of mercury ; the reading of the
latter circle is referred to the former circle; and then the reading
which is a mean between the reading for the direct observation and
the referred reading for the reflected observation, is the reading that
corresponds to a horizontal position of the telescope; and by adding
or subtracting a quadrant, the reading which corresponds to a zenithal
position of the telescope is obtained.
With a single circle this process cannot be adopted. In some
instances it has been imitated by observing a star directly on one
night, and observing the same star by reflexion on another night. The
OF CAMBRIDGE OBSERVATORY. 273
calculation for the zenith point then relies on our perfect acquaintance
with the variations of refraction and other corrections from one night
to another ; and thus a cause of inaccuracy is introduced, which does
not exist in the other method. In the Cambridge Observatory a different
method is regularly employed (for the idea of which I am indebted
to a suggestion of Mr Sheepshanks). When a star is to be observed
by reflexion, the circle is set approximately for the reflected observation,
and the six microscopes are read; when the star has entered the field,
and before it has reached the center, it is bisected by the micrometer
wire, (which in fact measures its distance from the fixed wire, and thus
gives a correction to be applied to the mean of the six microscopes,)
and then there is ample time to allow the circle to be turned to the
position in which the star can be observed directly, shortly after it
has passed the center of the field. Thus a direct and reflected observa-
tion are obtained at the same transit. This method is, in my opinion,
much preferable to the second that I have mentioned, and in some
respects superior to the first.
Either of the methods which applies to one circle enables us, as
will shortly be seen, to examine severely into the consistency of the
results obtained in different positions of the circle ; and this must be
considered as a most valuable property of this method of determining
the zenith point, and one which places it far above the use of a collimator
or any similar instrument.
I had hoped, on commencing observations with the Mural Circle
at the beginning of the year 1833, to be able in a very short time to
obtain a very approximate latitude. I proposed to observe some stars
every night in the manner above described, as well as circumpolar stars
(which might or might not be observed in the mercury): by the former
I should obtain a very good zenith point; and then each observation
of the latter, above and below the pole, would give me a value of the
co-latitude.
But after a few nights' observations, I found that the reading for
the zenith point, as determined by different stars, was not the same.
274 PROFESSOR AIRY ON THE LATITUDE
Had the discordance been wholly without regularity, this would have
given me no anxiety. But the first Aveek's observations enabled me to
see with certainty that one general rule could be laid down : the reading
for the zenith point as determined by northern stars was invariably
greater than that fovmd from southern stars. As the readings increase
while the telescope is turned towards the south, this discordance is of
the same kind as that which would be produced if the object end of
the telescope dropped by its own weight.
After much anxious thought and many fruitless attempts to explain
this discordance, I was obliged to give it up entirely. The method
which was adopted for approximate reduction of the observations, easily
admitting of future correction, was the following. When in one night,
or in several nights which it appeared practicable to group together,
stars had been observed by reflexion in different parts of the meridian,
1 took the three means of zenith points determined by stars far north,
by stars far south, and by stars near the zenith, as three separate results ;
and then I took the mean of these three for the zenith point. For an
approximate co-latitude I used 37°. 47'. 6",83.
At the beginning of March the telescope was moved about thirty
degrees on the circle; at the beginning of August it was again moved
thirty degrees, and on this occasion (as it appeared that the circle was
not exactly balanced) a pound of lead was attached to the eye end of
the telescope ; at the beginning of December it was again moved
about thirty degrees. It does not appear however that the fact of the
discordance has been affected, but its law seems to have been in some
degree altered.
A discordance of the same kind exists, I believe, in every circle
that has been properly examined. I am informed by Mr Henderson
(late Cape Astronomer) that he has found it in the Cape Circle. It
was recognized as existing in the Greenwich Circles : and, though the
system of observing there, which I have described, does not allow us
to trace the unmixed faults of either circle, yet from the discordance
in the places of stars as determined by the two circles, and its variation
OF CAMBRIDGE OBSERVATORY. 275
in different points of the meridian, I am inclined to think that the
defect in one circle is different from that in the other.
In vain have I endeavoured to discover the cause of this discordance.
I once thought that it might be owing to the circumstance, that for
the reflection-observation the circle is at rest for some minutes after
the microscopes are read, and possibly it might (though clamped) have
changed its position. A series of observations expressly made, showed,
however, that there was no sensible change either in a few minutes
or in many hours. I thought that the surface of the mercury might
be sensibly curved, and that from a habit of observing in one part of
the trough, an error might be produced. A set of experiments proved,
however, that there was not the least sensible difference in the results
found from observing at one or the other end of the trough. A flexure
of the wire in the field of view would not explain it, as the discordance
which that would produce is of the opposite kind. There appeared
to be no reason for supposing an error in the determination of the
coincidence of the micrometer wire with the fixed wire, in the value
of the micrometer screw, or in the observation with the micrometer
wire. The object glass, repeatedly examined by myself and once by
Mr Simms, did not appear to be loose in its cell. I am driven at last
to the supposition that the circle sensibly changes its figure ; but I
have no proof of this, nor do I see distinctly how it should produce
the discordance in question. Three sets of readings of every 10° under
all the microscopes, have not assisted me to discover such change.
My a priori opinion is, that a change in figure is hardly possible. The
telescope, it must be remembered, is attached at its ends to the limb
of the circle : the limb is in one piece (cast in several pieces and burnt
together) ; and the whole arrangement of parts seems admirably adapted
to prevent any change. If I had to fix on an astronomical instrument
which appeared less likely to change than any other, I should certainly
choose the Mural Circle.
To discover experimentally the law of discordance, I proceeded
as follows. The observations being reduced, and those of each star
being digested under the heads of D, R, SP. D., and SP. R., I
276 PROFESSOR AIRY ON THE LATITUDE
selected for the three first positions of the telescope all the un-
exceptionable corresponding observations D and R. (The stormy
weather of December made it impracticable to observe low stars by
reflexion). In each case of a double observation, the difference of
the results D and R would be double the difference between the
zenith point as found from that star, and the zenith point adopted
in the reductions. The mean of the differences of all the correspond-
ing results D and R, would therefore be double the mean of all
the differences between the zenith points found from the particular
star, and the zenith points found from all by a tolerably uniform
system : and thus it might be considered as double the difference
between the zenith point found without error of observation from
that star, and a certain imaginary well defined point. These values
for all the stars, and for each position of the telescope, were arranged
in tables (for which, as well as for some other numerical values, I
must refer to the Cambridge Observations, Vol. VI.)
The next step was, to connect these, approximately at least, by a
law. I soon found that to attempt this by calculation was almost hope-
less. Combinations of constants, sin Z.D., sin Z.D. cos^ Z.D., cos2Z.D.,
were tried in vain. I therefore adopted a graphical method similar
to that used by Sir John Herschel, in the reduction of his sweeps,
and described by him in the Phil. Trans. 1833. Taking the line of
abscissae for zenith distances, and the ordinates to represent the mean
of the differences above mentioned, I made a curve to pass among
the points so determined, as well as I could, giving to each point
an importance depending on the number of observations. From this
curve I measured off" the ordinates for every 10° of zenith distances;
half of this quantity I considered to be the correction to the ob-
served zenith distance, to be applied with different signs to the
direct and the reflected observation. The only respect in which
theoretical consideration may be said to have assisted me is the
following. Since the error in the relation between the position of the
telescope and the reading of the circle, to which the discordance
must be due, is periodical and never infinite, it may be expressed by
sines and cosines of the Z. D. and its multiples. Now it is useless
OF CAMBRIDGE OBSERVATORY. ' ' 27T
to take sines of even multiples, or cosines of odd multiples, because
when 180° — Z.D. is substituted for Z. D., the result is equal in
magnitude but opposite in sign ; and therefore when the two are
added together, (as they are in finding the zenith point from each
star), no trace of these terms would remain. Thus there may be
sensible flexure in the circle which cannot be discovered from ob-
servation by reflexion. The sines of odd multiples, and the cosines
of even ones, (all which may be expressed in finite series of powers
of sin Z.D.), will produce the same values with the same signs for
180° — Z.D. as for Z. D., and these will affect the zenith point.
Thus it appears that the terms which aff*ect the zenith point are
the same for a direct observation and for the corresponding observation
by reflexion, and it is this which justifies us in applying half the
discordance to each. It appears also that when Z. D. = 90°, the
function is maximum or minimum, and hence the curve in the
graphical process above described must there be parallel to the line of
abscissEB.
The tables of corrections being thus formed, I now considered
myself entitled to apply them to the reduced r^ults of all the
observations, whether there were corresponding observations of the
opposite kind or not. ' >»/ ^ i -
The principal steps of the succeeding process may be gathered
from the subjoined table. The first column contains the name of
the star, its position with regard to the pole, (the lower transit being
marked by S.P.), and the method of observing it (the letters D and
R being always used for direct and reflected vision). Here it is to
be observed that a star above the pole and the same star below the
pole are reduced as separate stars, which is necessary, because the
observations have been reduced with an assumed co-latitude, or an
assumed place of the pole, the error in which assumption can be
found only by comparing the separate results for the same star above
and below. The second column contains the number of observations.
The third contains its mean N.P.D. for Jan. 1, 1833, as found from
the mean of all the results in each position and mode of observation.
Vol. V. Part II. Nn
278 PROFESSOR AIRY ON THE LATITUDE
and reduced with the assumed co-latitude 37° . 47' . 6,"83 : those de-
termined from the lower transits of the star have the negative sign.
For refraction, Bessel's tables are used. The fourth column contains
the seconds only, as corrected for the errors above described ; this
has been done by taking the number of observations in each position
of the telescope on the circle, and finding the mean correction,
supposing that to each observation the correction proper to that
position was applied. The negative sign has still been retained for
the lower observations. The fifth column contains the whole number
of observations in each position of the star : and the sixth contains
the mean N.P. D. for each position, as inferred from the combina-
tion of direct and reflected observations. The seventh contains the
whole number of observations for both positions. The eighth contains
the algebraic sum of the two determinations of N.P.D., as the star
is above or below the pole. If the assumed co-latitude were correct,
this sum would = ; if the assumed co-latitude be increased by x,
this sum would be increased by ^x, and therefore to make it now
= 0, X must be taken = — i x sum in 8th column. The results, as
might be expected, are however different for different stars, though
the difference is much smaller than I could almost have hoped ; the
extreme difference in the correction of latitude being 1,"3, and this
being the difference between two results from stars nearly in the
same parallel (shewing that it does not arise from error in the cor-
rections above described), and which had been not much observed.
It now becomes necessary to determine how the relative importance
of these results shall be estimated. It would not be right to give
a value proportionate to the number of observations, because part of
the discordance may be produced by errors of division and other
causes which, in the observations of a single star, produce constant
errors. The ninth column contains the immbers by which (from my
estimation of the comparative influence of constant and variable errors)
I suppose the value of each result to be estimated. The tenth con-
tains the product of the corresponding numbers in columns 8 and 9-
The sum of the numbers in column 10 being divided by the sum of
those in column 9, gives + 2",82 for the double correction, or + 1",41
OF CAMBRIDGE OBSERVATORY. 279
for the single correction, of the co-latitude ; and the co-latitude thus
corrected is 37° . 47' . 8",24, or the latitude 52°. 12'. 51",76. This result
I conceive to be correct within a small fraction of a second. The
number of circumpolar stars used for this determination is 10, and
the whole number of observations 917.
In describing the process by which I have arrived at the above
result, it has been my wish to present to the Society not only a
determination possessing considerable local interest, but also an account
of instrumental anomalies which are of general scientific importance.
In further illustration of the latter point I will allude to the dis-
cordances in the determinations of the obliquity of the ecliptic. It
is well known that most astronomers have found the obliquity smaller
from observations at the winter solstice than from those at the
summer solstice. Now if I had used only the latitude found from
direct observations of circumpolar stars, and had applied no correction
to the observations of the Sun, I should also have found two values
for the obliquity discordant by about 5", the winter obliquity being
the smaller. With the corrections above described, (and which were
formed entirely from observations of stars, and before I had even
examined my sun observations) the two values of the obliquity
agree within 1". I might have altered the corrections so as to re-
move part of this discordance, but I prefer leaving them in
the shape in which they were given by independent considerations.
Indeed if I had confined myself to the January observations for the
winter solstice, and omitted those of December when the correction is
less certain, the discordance would wholly have disappeared. A very
small alteration of the constant of refraction (such as would not alter
the latitude much more than 0",1), or a very small alteration in the
law of refraction (which would not be sensible in the latitude) would
remove this difference. But I hardly venture to assume that obser-
vations of the Sun, near the winter solstice, can be relied on to this
degree of accuracy.
I will only add, in conclusion, that I believe the method which
I have used to be the only one of those in practice from which a
280 PROFESSOR AIRY ON THE LATITUDE OF CAMBRIDGE OBSERVATORY.
good result can be obtained. Had I determined my zenith points
by a floating collimator, the result of observations on Polaris and
^ U. Minoris would have given the latitude more than a second
wrong, and the polar distance of every southern body more than
two seconds wrong : the result of observations on the Sun would have
given nearly the same error in the latitude but with the opposite
sign. If a circle reversible round a vertical axis had been used
(as at Dublin, Palermo, &;c.) its errors would (supposing the mere
circle exactly as good,) have been just as great as if a collimator
were employed. The method adopted above appears most valuable,
not only because it gives numerical conclusions more accurate than
any other, but also because it enables us to observe discordances and
to suspect faults which, though they confused our results, might
otherwise have wholly eluded our discovery.
G. B. AIRY
Obskkvatouy,
March 23, 1834.
Tratnsajclions of theCsanb.rhil.Soc-.Vol VTl. 7.
W MOicalfi^. UefM^^CamJirui^e-.
jTvnja-ciimus ofOu. Camji. f/vu. Jodefy, Vol V Tt S.
Ft^. I
jri^ Z.
J^r^. 3.
Jkfetcalfl, Ucho^'^ Cojnbnd^e'.
281
Table exhibiting the Calculations j^r correcting the Latitude o/" Cambridge
Obseuvatory ; the Observations having been reduced with the assumed
Latitude 52M2' . 53",17.
star's Name.
No.
of
Obs.
Polaris D
Polaris R
Polaris S.P D
Polaris S.P R
8 Urste Minoris D
S Ursa; Minoris R
S Urs» Minoris S.P. ..D
2 Ursae Minoris S.P. ..R
/3 Ursae Minoris D
/3 Ursae Minoris R
/3 Ursifi Minoris S.P. ..D
/3 Ursae Minoris S.P. ..R
/3Cephei D
/3 Cephei R
ySCephei S.P D
/3 Cephei S.P R
2 Draconis D
I Draconis R
S Draconis S.P D
S Draconis S.P R
a Draconis D
a Draconis R
a Draconis S.P D
a Draconis S.P R
a Ursae Majoris D
a Ursae Majoris R
a Ursae Majoris S.P. ..D
a Ursae Majoris S.P. ..R
Uncorrected mean
N.P.D.
Corrected
for
Discordance.
« Cephei D
a Cephei R
a Cephei S.P D
a Cephei S.P R
S Ursae Majoris D
S Ursae Majoris R
Ursae Majoris S.P. ..D
g Ursae Majoris S.P. ..R
a Cassiopeiae D
a Cassiopeiae R
o Cassiopeiae S.P D
o Cassiopeiae S.P R
113
42
111
58
43
37
39
23
20
17
22
2
4
none
12
7
3
10
6
14
11
7
32
32
5
none
43
35
13
8
26
26
3
3
34
15
26
9
1 . 34 . 52,22
51,00
1 . 34 . 53,77
55,70
- 3
3.24.45,21
43,48
24 . 46,34
48,42
15. 9-41,65
42,02
■15. 9.44,58
47,54
20 . 10 . 15,91
- 20 . 10 . 16,97
17,24
22 . 37 . 54,40
53,95
- 22 . 37 . 54,99
56,31
24 . 49 . 24,80
24,51
- 24 . 49 . 26,90
28,90
27 . 20 . 55,75
55,73
- 27 • 20 . 59,30
28,
■28.
7.11,43
11,64
7 . 12,67
13,47
32. 2.18,12
18,40
■32. 2.20,38
23,23
34 . 22 . 45,38
46,36
- 34 . 22 . 47,37
47,00
No.
of
Obs.
51,38
51,78
- 54,62
- 54,83
44,44
44,24
- 47,40
- 47,28
41,25
42,40
- 45,52
-46,18
15,74
- 17,74
- 16,52
54,34
54,01
- 56,14
-55,31
24,76
24,47
-28,10
- 27,90
55,84
55,64
-59,86
11,58
11,50
■13,27
• 12,93
18,41
18,09
■21,16
- 22,45
45,73
46,03
- 47,93
- 46,54
Concluded
N.P.D.
155
169
80
62
37
24
4
19
6
16
25
10
64
5
78
21
51,49
- 54,70
44,35
- 47,35
41,78
- 45,57
15,74
- 17,29
54,17
- 55,83
24,63
• 28,04
55,74
■ 59,86
11,54
-13,13
No. Algebraic
of Sum of
Obs. Determin.
52
18,25
6
-21,80
49
45,82
35
- 47,57
324
142
61
23
22
35
99
58
84
•3,21
-3,00
■3,79
- 1,55
Weight
of
Result.
1,66
■3,41
69 - 4,12
1,59
•3,55
-1,75
Product.
- 16,05
•9,00
-7,58
-1,55
1,66
-3,41
•4,12
-3,18
-3,55
-3,50
Vol. V. Part II.
Oo
TRANSACTIONS
CAMBRIDGE
PHILOSOPHICAL SOCIETY.
Vol. V. Part III.
CAMBRIDGE:
PRINTED BY JOHN SMITH, PRINTER TO THE UNIVERSITY:
AND SOLD BY
JOHN WILLIAM PARKER, WEST STRAND, LONDON;
J. & J. J. DEIGHTON, AND T. STEVENSON,
CAMBRIDGE.
M.DCCC.XXXV.
XII. On the Diffraction of an Ohject-glass with Circular Aperture. By
George Biddell Airy, A.M. late Fellow of Trinity College,
and Plumian Professor of Astronomy and Experimental Philosophy
in the University of Cambridge.
[Read Nov. 24, 1834.]
The investigation of the form and brightness of the rings or rays
surrounding the image of a star as seen in a good telescope, when a
diaphragm bounded by a reetihnear contour is placed upon the object-
glass, though sometimes tedious is never difficult. The expressions
which it is necessary to integrate are always sines and cosines of mul-
tiples of the independent variable, and the only trouble consists in
taking properly the limits of integration. Several cases of this problem
have been completely worked out, and the result, in every instance,
has been entirely in accordance with observation. These experiments,
I need scarcely remark, have seldom been made except by those whose
immediate object was to illustrate the undulatory theory of light.
There is however a case of a somewhat different kind; which in
practice recurs perpetually, and which in theory requires for its com-
plete investigation the value of a more difficult integral ; I mean the
usual case of an object-glass with a circular aperture. The desire of
submitting to mathematical investigation every optical phaenomenon of
frequent occurrence has induced me to procure the computation of the
numerical values of the integral that presents itself in this inquiry :
and I now beg leave to lay before the Society tlie calculated table,
with a few remarks upon its application.
Let a be the radius of the aperture of the object-glass, f the focal
length, h the lateral distance of a point (in the plane which is normal
Vol. V. Part III. Pp
284 PROFESSOR AIRY, ON THE DIFFRACTION OF
to the axis of the telescope) from the focus. Then, the lens being
supposed aplanatic, and a plane wave of light being supposed incident,
the immediate effect of the lens is to give to this wave a spherical
shape, its centre being the focus of the lens. Every small portion of
the wave, as limited by the form of the object-glass, must now be
supposed to be the origin of a little wave, whose intensity is propor-
tional to the surface of that small portion ; and the phases of all these
little waves, at the time of leaving the spherical surface above alluded
to, must be the same. If then Sx x Sy be the area of a very small part
of the object-glass, q the distance of that part from the point defined
by the distance b, the displacement of the ether at that point, caused
by this small wave, will be represented by
Sx X. Sy X sin—- {vt — q — A) ;
A
and the whole displacement caused by the small waves coming from
every part of the spherical wave will be the integral of
sin — (vt—q — A)
through the whole surface of the object-glass, q being expressed in
terms of the co-ordinates of any point of the spherical surface.
Now let X be measured from the center of the lens in a direction
parallel to i; y perpendicular to x and also perpendicular to the axis
of the telescope; and % from the focus parallel to the axis of the
telescope. Then
q=.^{{x- by + y- + x} = -y/ix' +f+x'-2bx)
omitting squares and superior powers of b. But x^ + y^ + z' —f^^
since the wave is part of a sphere whose centre is the focus ; therefore,
q = VW^-^bx)=f-j,x nearly;
and the quantity to be integrated is
sm— \vt - f - A + -x).
^ J
AN OBJECT-GLASS WITH CIRCULAR APERTURE. 285
The first integration with regard to y is simple, as y does not
enter into the expression, which is therefore to be considered as con-
stant. Putting y, and y^ for the smallest and greatest values of y
corresponding to x, the first integral is
{yi-yx)y-^m-^{vt-f-A^r-x).
To this point of the investigation the expressions are general, including
every form of contour of the object-glass.
We must now substitute the values of y^ and y^ in terms of x,
before integrating with regard to x. For a circular aperture
y, - y. — ^y/a^-x"
where the sign of the radical is essentially positive. Hence the dis-
placement of the ether at the point defined .by the distance A is re-
presented by
2 f, Va' - x" . sin — {vt-f- A + ^x)
= 2sm -^{vt-/— A) f^\/a^-af .cos-— .^x
\ Ay
+ 2cos — - {vt —f— A) X a/«^ — x\sm—-.^x,
A ^ J
and the limits of integration are from x = — a to x = + a. Between
these limits it is evident that
;- /-: « . 2-ir b
f^Va' — x^ .sm—- . ^x = 0,
^ J
(as every positive value is destroyed by an equal negative value) ; and
the displacement is therefore represented by
2sin— -(«^— /— ^) ji\/«^ — ar'.cos ^ .^x,
^ ■ , ^ ./
the integral being taken between the limits x= -a, x— -^a.
p p2
286 PROFESSOR AIRY, ON THE DIFFRACTION OF
If we make - = w, — — . -2r = n, the expression becomes
2a^.sm-—{vt-f~A)J^V^-uf'Cosnw, fromw=— 1 tow=+\,
A
or 4«^ sin — - {vt-f~ A) j„\/l — tt;''. cos nw, irom w = to w = l.
A
It does not appear, so far as I am aware, that the value of this
integral can be exhibited in a finite form either for general or for
particular values of w. The definite integral
J„^/\ — vf . cos nw (from w=-0 to w = \,)
(which will be a function of 7i only) being expressed by N, it may be
shewn that N satisfies the linear differential equation
n ' dn dv? '
which may be depressed to an equation of the first order that does
not appear to yield to any known methods of solution.
If we solve the equation by assuming a series proceeding by powers
of n, or if we expand cos nw and integrate each term separately, we
arrive (by either method) at this expression for the integral
TT . rf_ n^ _ _ "" Xr \
4 "" ^ 2.4"^2.4^6 ^:^\Q'.S^^^-'
The table appended to this paper contains the values of the series
in the bracket, for every 0,2 from w=0 to w = 12. Each value has
been calculated separately, the logarithms used in the calculation have
been systematically checked, and the whole process has been carefully
examined. The calculations were carried to one place further than the
numbers here exhibited. I believe that they will seldom be found in
error more than a unit of the last place; except perhaps in some of
the last values, where the rapid divergence of the series for the first
five or six terms made it difficult to calculate them accurately by
logarithms.
AN OBJECT-GLASS WITH CIRCULAR APERTURE. . 287
In the use of tins table n must be taken = — -.-^. If instead of
using the linear distance h to define the point of the field at which
we wish to ascertain the illumination, we use the number of seconds *,
then A = /. *.sin 1", and n must be taken = — as sin 1". If \ be taken
for mean rays = 0,000022 inch, n must be taken = 1,3846 x as, a being
expressed in inches. From this expression, and from the numbers of
the table, we draw the following inferences.
1. The image of a star will not be a point but a bright circle
surrounded by a series of bright rings. The angular diameters of these
(or the value of s corresponding to a given value of n) will depend
on nothing but the aperture of the telescope, and will be inversely as
the aperture.
2. The intensity of the light being expressed (on the principles
of the undulatory theory) by the square of the coefficient of
sin-^ivt-f- A),
and the intensity at the center of the circle being taken as the standard,
it appears that the central spot has lost half its light when « = l,6l6,
I 17
or s = — — ; that there is total privation of light, or a black ring, when
2 76
n = 3,832, or * = — — ; that the brightest part of the first bright ring
a
Q WQ -I
corresponds to w = 5,12, or * = — — , and that its intensity is about — of
a Oi
5 16
that at the center; that there is a black ring when n = 7,14, or s= -- — ;
a
that the brightest part of the second bright ring corresponds to ra = 8,43,
or * = — — , and that its intensity is about — r of that of the center ;
7 32
that there is a black ring when w =10,17, or *= — — ; that the brightest
288 PROFESSOR AIRY, ON THE DIFFRACTION OF
part of the third bright ring corresponds to w = 11,63, or *= — — , and
that its intensity is about ^— - of that of the center.
The rapid decrease of light in the successive rings will sufficiently
explain the visibility of two or three rings with a very bright star
and the non-visibility of rings with a faint star. The difference of
the diameters of the central spots (or spurious disks) of different stars
(which has presented a difficulty to writers on Optics) is also fully
explained. Thus the radius of the spurious disk of a faint star, where
light of less than half the intensity of the central light makes no
1 17
impression on the eye, is determined by making /* = 1,616, or s=— — :
whereas the radius of the spurious disk of a bright star, where light
of — the intensity of the central light is sensible, is determined by
1 97
making n = 2,73, or * = — — .
The general agreement of these results with observation is very
satisfactory. It is not easy to obtain measures of the rings; since
when a is made small enough to render them very distinct as to form
and separation, the intensity of their light (which varies as a^) is so
feeble that they will not bear sufficient illumination for the use of
a micrometer. Fraunhofer however obtained measures agreeing pretty
well (as to proportion of diameters, &c.) with the results above.
For verification of the numbers it would probably be best to use
an elliptic aperture. By an investigation of exactly the same kind as
that above it will be found that the rings will then be ellipses exactly
similar to the ellipse of the aperture, but in a transverse position ; that
the major axes of the rings for the elliptic aperture will be the same
as the diameters of the rings for a circular aperture whose diameter
— minor axis of ellipse of aperture, but that the intensity will be
greater in the proportion of the squares of the axes. I have not yet
had an opportunity of examining this in practice.
AN OBJECT-GLASS WITH CIRCULAR APERTURE. 289
I shall now apply the numbers of the table to the solution of
the following problem. To find the diameters, &c. of the rings when
a circular patch, whose diameter is half the diameter of the object-
glass, is applied to its center, so as to leave an annular aperture.
The radius of the patch being -, it is easily seen that the dis-
placement (using the same notation) is
2sm-—-(vt—J'—A)fr\/a^-x'.cos—-.^a; (from a;—-a to x=+a)
- 2sin ~(vt-f-A)J\/---af. cos-^ .^x (from x= -- to x= +^.
Putting - =w, — = u, this becomes
4a^ . sin -T-{vt —f — A) /„ \/l — vf . cos — .—^w
A A /
-4.^.sm yC^^-/- ^)/«vl-M'.cos— .— .M,
the limits of integration both for w and for u being and 1. Omitting
the factor oV, the intensity will be expressed by
V(»)-i*(l)}".
where (p{n) is the number given in the table.
Upon forming the numerical values we find that the black rings
correspond to values of w=3,15, 7,18, 10,97: and that the intensities
of the bright rings (in terms of the intensity of the center) are — , — .
Thus the magnitade of the central spot is diminished, and the bright-
ness of the rings increased, by covering the central part of the object-
glass.
In like manner, if the diameter of the circular patch = a ( 1 — />), the
intensity of light would be proportional to {<p {n) — {l— pf .^{n—pn)}".
MiO PROFESSOR AIRY, ON THE DIFFRACTION OF
The quantity under the bracket, if p is very small, is equal to
X) ft
2p .<p {fi) + pn<p' (n) = - . -J— {n^<p{n)}.
In the case of a very narrow annulus therefore the diameters of the
black rings will be determined by making ?i^(p (») maximum or
minimum. It appears then that there ought to be only one black
ring corresponding to each black ring with the full aperture, and that
its diameter ought to be somewhat smaller. This conclusion does not
agree with the experiments recorded by Sir J. Herschel, in the Encyc.
Metrop. Article Light, page 488 : but it is acknowledged there that
the results are discordant with Fraunhofer's : and I am inclined there-
fore to attribvite the phasnomena observed by Sir J. Herschel to some
other cause.
The investigation of cases of diffraction similar to that discussed
here appears to me a matter of great interest to those who are
occupied with the examination of theories of light. The assumption
of transversal vibrations is not necessary here as for the explanation
of the phasnomena of polarization : and they therefore offer no argu-
ments for the support of that principle. But they require absolutely
the supposition of almost unlimited divergence of the waves coming
not merely from a small aperture, but also from every point of a large
wave : and the results to which they lead us, shew strikingly how
small foundation there was for the original objection to the undulatory
theory of light, viz. that if waves spread equally in all directions.
there could be no such thing as darkness.
Obsbrvatory, Cambridge,
November 20, 1834.
G. B. AIRY.
AN OBJECT-GLASS WITH CIRCULAR APERTURE.
291
4j r •
Table of the values of 0(w) = — f^vl — vf.co& nw from w = iow=\.
11
</>(«)
n
0(«)
0,0
+ 1,0000
6,0
- 0,0922
0,2
+ ,9950
6,2
- ,0751
0,4
+ ,9801
6,4
- ,0568
0,6
+ ,9557
6,6
- ,0379
0,8
+ ,9221
6,8
- ,0192
1,0
+ ,8801
7,0
- ,0013
1,2
+ ,8305
7,2
+ ,0151
1,4
+ ,7742
7,4
+ ,0296
1,6
+ ,7124
7,6
+ ,0419
1,8
+ ,6461
7,8
+ ,0516
2,0
+ ,5767
8,0
+ ,0587
2,2
+ ,6054
8,2
+ ,0629
2,4
+ ,4335
8,4
+ ,0645
2,6
+ ,3622
8,6
+ ,0634
2,8
+ ,2927
8,8
+ ,0600
3,0
+ ,2261
9,0
+ ,0545
3,2
+ ,1633
9,2
+ ,0473
3,4
+ ,1054
9,4
+ ,0387
3,6
+ ,0530
9,6
+ ,0291
3,8
+ ,0067
9,8
+ ,0190
4,0
- ,0330
10,0
+ ,0087
4,2
- ,0660
10,2
- ,0013
4,4
- ,0922
10,4
- ,0107
4,6
- ,1116
10,6
- ,0191
4,8
- ,1244
10,8
- ,0263
5,0
- ,1310
11,0
- ,0321
5,2
- ,1320
11,2
- ,0364
5,4
- ,1279
11,4
- ,0390
6,6
- ,1194
11,6
- ,0400
6,8
- ,1073
11,8
- ,0394
6,0
- ,0922
12,0
- ,0372
Vol. V. Paet III.
Qa
» J » 0' {
^9
• JI.JL.A 4 4- ^li
XIII. On the Equilibrium of the Arch. By the Rev. Henry Moseley,
B.A. of St John's College; Professor of Natural Philosophy and
Astronomy in King's College, London.
[Read Dec. 9, 1833.]
1. Let a mass acted upon by forces applied to any number of
points in it be imagined to be intersected by an infinite number of planes,
dividing it into exceedingly small laminse. Suppose the direction of the
resultant of the forces acting upon one of these, having for its ex-
ternal face a portion of the surface of the body, to be determined.
Combining this force with those acting upon the different points of
the next, contiguous lamina; let their common resultant be ascertained.
Proceed similarly with the next, and with each succeeding lamina.
These lines will then be the tangents to a curved line, called in
the following paper the line of pressure, whose intersection with each
lamina, marks the point where a single force might be applied so as
to produce the same effect with all those impressed upon that lamina,
this single force being impressed in the direction of a tangent to the
curve.
If any of these imaginary intersecting planes be supposed to become
real sections of the mass, so as to separate it into distinct parts, the
conditions necessary that no one of these parts may slip or turn over
on those contiguous to it, will manifestly be determined by the direc-
tion of the line of pressure in reference to the plane of the section.
In general it will be observed that forces applied to a system of
variable form are, when in equilibrium, subject to the same conditions
as though its form were invariable, together with certain other conditions,
dependant upon the nature of the variation to which the form of the
system is liable. In other words the conditions of the equilibrium of
a system of invariable form are necessary to the equilibrium of a system
of variable form ; but they are not sufficient. We shall first determine
QQ2
294 Mr MOSELEY, ON THE
the form and position of the line of pressure on the hypothesis, that
the form of the system is invariable, and then consider the modifica-
tion to which these are subjected by the opposite hypothesis.
2. Let there be conceived a mass, the connexion between the parts
of which may be any whatever, and the nature of whose surface is
determined by the equation
^ xy% = 0.
Let it be intersected by an imaginary plane whose position in reference
to a given system of rectangular co-ordinates is determined by the arbi-
trary constants A, B, C, and whose equation is
z = Ax + By + C (1).
Let Ml, Mi, Ms represent the sums of the forces acting upon one
of the parts into which the mass is divided by the intersecting plane,
resolved in directions parallel to the axes of x, y, z, respectively. Also
let JVx, N'z, ^3 be the moments of these forces about the same axes.
Then Mi, Mi, Ms; Ni, N^, iVs are given in terms of the arbitrary
constants A, B, C — of the given forces — and of the constants involved
in the given equation to the surface of the mass.
Let the position of the intersecting plane be supposed to be such,
that the forces acting upon the above mentioned portion of the mass
may have a single resultant, an hypothesis which involves the known
condition
MiN, + M,N, + M,N, = (2).
The equations to the resultant in any given position of the inter-
secting plane, are
Ml ^Ni
^'=M^-'Ms
Let the arbitrary constant C be eliminated from this equation, and
from the equation to the intersecting plane by means of equation (2) ;
and let the plane be then supposed to take up a series of positions,
the law of which is fixed by its equation, and of which, each is im-
mediately adjacent to the former.
EQUILIBRIUM OF THE ARCH.
295
Further, let it be supposed that the resultant of the forces upon
the portion of the mass, cut off by the plane, in each of its positions,
intersects with the resultant similarly taken in its immediately previovis
position — an hypothesis which introduces a new condition into the
question and establishes a second relation between the quantities
M„ M„ M,; A, B, C.
That relation is determined as follows.
Since x, y, as are to be considered as the co-ordinates of a point of
intersection of two consecutive resultants; we may differentiate the
equations (3) with respect to the arbitrary constants A and B, consi-
dering X and y as constant. From this differentiation, the following
equations are obtained:
= ss
= »
KS../(t).J,KSL/_i).,
dA
■ dAv
dB
.dB
+
dA
dA
dB
dA+-^dB{
dB
■(4),
whence, eliminating z
^M^^A^-^dB
dA
dB
dA + r^ft— aJ? I
dA
dB
dA
d
dA +
N,
M,
dB
dA
dB
dBi
^^^^clA+-^dB{
= 0...(5).
This last equation determines the relation between A and B ne-
cessary to the continual intersection of the consecutive resultants ; and
the elimination of these quantities between equations (3) and (4),
produces two equations in x, y, % which are those to the locus of
that intersection. That is, they are the equations to the line of
PRESSURE.
296 Mr MOSELEY, ON THE
3. By the elimination of A,y B and C between the equations (2), (3)
and (5), a relation is obtained between the co-ordinates of a point in
the direction of the resultant force, applicable to every position of the
intersecting plane. Being in fact, the equation to that developable
surface which is the locus of the resultants, and, which has for its
edge of regression, the line of pressure. This surface will be properly
called the surface of pressure.
It is evident that at that point where the line of pressure even-
tually cuts the surface of the mass, there must be applied a force equal
to the resultant of all the other forces impressed upon the system
and in the direction of a tangent to the line of pressure at that point,
or there must be applied to the surface of the last lamina cut off
by the intersecting plane, forces whose resultant is of that magnitude
and in that direction.
4. These conditions may be expressed as follows.
Let P' be the force — or the resultant of the forces — applied to the
last lamina, x^, y,, ss, the co-ordinates of the intersection of the line of
pressure with it, a, fi, y the inclinations of P' to the axps of x, y, %.
^ Also let
be the equations to the line of pressure.
Since the point Xx, y^, %i is a point in the surface of the mass,
.-. ^Xiy.z, = 0.
Also, since it is a point in the line of pressure,
.-. Xi = Fi%i
y, = F^%x
Since the direction of P' is that of a tangent to the line of
pressure,
tan a =
tan /3 =
d%, '
d%i
EQUILIBRIUM OF THE ARCH. 297
Also
p = VM^VMr+~m,
where M^, M.^, M-j are supposed to be taken throughout the tvhole
mass.
Thus there are six equations of condition, which together with the
equation
cos^ a + cos^ /3 + cos^ 7 = 1.
determine the seven quantities P', Xi, y^, %i\ a, /3, 7 in terms of the
forces (other than P') which compose the system, and the constants
which enter its equation. These fix the relations necessary to the equi-
librium of the mass considered as one continued geometrical solid.
Before proceeding to the discussion of the additional conditions
requisite to the equilibrium when the mass passes from the invariable
form here supposed, to a variable form, it will be well to give an
example of the application of the principles which have been already
laid down to the actual determination of the line of pressure in a par-
ticular instance.
5. Let then ABCD (fig. 1.) represent a heavy mass, bounded at its
extremities by parallel planes AB and CD, and laterally, by the planes
AC and BD inclined at any angle to one another.
Let the mass be imagined to be intersected by an infinite number
of planes parallel to AB, of which one is mn, and to be supported
by forces acting at p and p' at angles cp and <f>' with the horizon.
It is required under these circumstances to determine the form and
position of the line of pressure.
Let the line P'G bisect AB and CD. Draw P'E horizontal and
PM vertical.
Let P'M= A, CD = 2b, P'p = k,
AB = 2a, P'G = h, Gp' = k'.
Inclination of P'G to the horizon =7, -
AB = /3.
298 Vi Mil MOSELEY, ON THE
Now,
BP' - Pm _ BP - DG
PP' ~ GP' '
2a — (mn) _ 2 (a — i) _
' ■ ^sec7 ~" ^ '
/ V « 2 (« - A) .
.'. (mn) = 2a —r — - secy . A;
.-. area (BAnm) = ^ {{AB) + {mn)\ . (P'P) . sin (PP'A)
= sin {(3 +y) secy {2a A- ^-^ .secy .A"};
d\aYea(BAnm)} - /o , \ ia 2(a — b)
.'. -^ V^ ^ = 8111 (/3 + 7) sec 7 {2a ^ , ^ sec7 . A\.
- Now each element of the area has its centre of gravity in P'G ;
,-, moment of area = 2^sin (/3 + 7) sec 7 ^{a^ ^— sec7^'}
=^sin(/3 + 7)sec7{«^* ^"7 ^ sec7^H.
Now,
iVs = moment of p + moment of area (BAmn)
2 (a — b)
= PffK sin (0 + /3) + ^sin (/3 + 7) sec7 {aA^ ^ , ' sec 7 A"").
Also,
Mx — pg cos (p, Mz = 0,
M^ =pgsm(j> - g sin (/3 + 7) sec 7 {2a^ ^ sec 7 ^^},
iv, = 0, iV3 = o.
Calling therefore x and » the co-ordinates of any point in the re-
sultant of the forces applied to the area (ABmn), we have for the
equation to that resultant.
EQUILIBRIUM OF THE ARCH. 299
or, zpgcoscp - pgsivKp + xgsm{(i + 7)sec7{2a^ -r—secy.A^}
=pgK sin ((p + /3) +gsm{fi + 7)8607 {aA^ ^-— r — ^sec7 . A^}.
Differentiating which equation with regard to the arbitrar}' constant
A, we obtain
A = x,
whence by elimination and reduction,
^^l( ^- ^ ] ( seey.sm{f3 + y) \ ^
^ \ pk J \ COS(p j '
_ /_a\ [ sec 7. sin (^ + 7) ] ^
\pj \ cos(p /■
+ tan (f) . X
_^ sin((/> + /3)
COS0
The above is the equation to the line of pressure. It indicates a
point of contranj flexure corresponding to
ah
X =■ 7 cos <h.
a — h ^
The curve is concave to the axis of x, between the origin and this
point. It is afterwards continually convex.
A minimum value of % is determined by the equation
\Jta\ f / ip\^ sin>cos^7 )
^ \a-AV^ ^ ^ Ui •sinMi8 + 7)r *
It will be observed that since all the forces applied to the system
may be supposed to act in the same plane, the two conditions.
First, " That in every position of the intersecting plane, the forces
shall have a single resultant," and Secondly, "That the consecutive re-
sultants shall intersect," are necessarily satisfied.
Vol. V. Paet III. Re
300 Mr MOSELEY, ON THE
To simplify the question, let the planes AC, BD which bound the
mass laterally be supposed to be parallel, the figure ABCD assuming
the form of a rectangle. Fig. 6.
This hypothesis will introduce the following conditions :
« = *' ^ = J - 7-
Hence, by substitution the equation to the line of pressure becomes
% — . sec 7 . sec . a;^
+ tan ^ .X
cos (7 - 0)
cos
Avhich may be put under the form
, p . ^ 1 , P , ( cos (y — (j)) P sin^ cos 7 -,
^x — -~ sm cos^r = - cos 7 .cos . \k ' + -7 ^-— ; — - — *|.
* 2a ^ "a ' ^ ^ COS0 4« cos '
It is manifest therefore, that the line of pressure is in this case
a parabola — having its axis vertical and at a distance = —- sin cos 7
^ a
from the origin — having its concavity downwards — its vertex at a height
_ cos (7 — 0) p sin' cos 7
~ cos 4 « cos '
above the axis of x — and having for its parameter the quantity
• (^j . cos cos 7.
Let us now seek to determine what relation must exist between
the forces impressed upon the mass which we have hitherto considered
of invariable form, that the equilibrium, may continue under the same
circumstances when its form and dimensions are made to admit of
variation. And let us suppose
EQUILIBRIUM OF THE ARCH. SOI
First. That certain of the sections, which we have imagined, be-
come real sections of the mass, dividing it into separate and distinct
parts, each of which retains the properties of a perfect solid.
Secondly. Let us suppose every point in the system to admit of
displacement, subject, within certain limits, to the law of perfect
elasticity.
The determination of the conditions of the equilibrium in these
two cases, will constitute a complete theory of construction.
The discussion contained in the remainder of this paper will be
confined to the first case.
6. Let the mass AB (fig. 2.) have for its line of pressure the
line PP'. Now it is clear, that if this line cut the plane QQ of any
section of the mass in a point n' without the surface of the mass ;
the tendency of the opposite resultants of the forces acting upon the
two parts AQQ and SQQ', into which that section divides the mass,
will be to cause them to revolve about the nearest point Q' of its
intersection with the surface of the mass. And, this tendency being
wholly unopposed, motion will ensue. And so in the mass represented
(fig. 6.) the force p and with it the line of pressure pp' being given,
it appears that, being cut transversely as shewn in the figure, the mass
cannot be supported by any single force p if it extend beyond CD':
any such force must, to produce equilibrium, be applied at q; and
being applied there, the portion C'C"Z)"'iy will be wholly unsupported.
The line of pressure being continued cuts the planes of the sections
CD', CD', &c., without the surface of the mass.
Thus then it is a condition of the equilibrium, that the line of
pressure should intersect the plane of every section of the body within
its mass.
This condition will be satisfied if this line nowhere cut the surface
of the mass except at the points P and P. Fig. 2. Or if the equation
■^F^z, F^z, ■ « = 0,
RR Z
302 Mk MOSELEY, ON THE
found by eliminating the values of a; and y between the equation to
the surface and the equation to the line of pressure, involve only
such possible values of z as correspond to the points P and i*, where
the intersecting plane touches the surface, or to points where the line
of pressure touches it.
It is a further condition of the equilibrium that the line of pressure
should not cut any section of the mass, at an angle with the perpen-
dicular to that section greater than a certain given angle, dependant
upon the friction of the surfaces in contact, and having for its tangent
the coefficient of friction.
The resistance of surfaces is not exerted exclusively in the direction
of the normal, according to an hypothesis, which was probably in-
troduced into the theory of Statics in order to simplify the investi-
gations of those who originated that science, but which there seems
no reason for retaining any longer. It is exerted in an infinity of
different directions included within a certain angle to the normal, or
rather within the surface of a certain right cone, having the normal
for its axis and the point of resistance for its vertex. Any force,
however great, applied within this conical surface will be sustained
by the resistance of the surface of the mass — and no force however
small, without it.
Let R represent a single force on the resultant of any number of
forces applied to a fixed surface, and let R' and R" be the resolved
parts of R in the directions perpendicular and parallel to the surface.
Also let p be the inclination of R to the vertical, and f the coefficient
of friction. The friction of the surfaces in contact is therefore repre-
sented by fR, and motion will, or will not, ensue according as R" is
K"
greater or is not greater than /R'. Or, according as -p, is greater or
is not greater than f. Or, if y = tan (p, according as tan p is, or is
not, greater than tan (p, or as p is greater or is not greater than (p.
EQUILIBRIUM OF THE ARCH. 308
In the remainder of this paper the angle 0, or tan-'^ will be called
the limiting angle of resistance*.
From the above then it appears, that unless the tangent to the
line of pressure at the point where it cuts any section of the mass,
make with the perpendicular to the plane of that section an angle,
which is not greater than the limiting angle of resistance, the surfaces
there in contact will slip upon one another.
This condition may be expressed analytically as follows :
% = Ax + By + C
is the equation to the plane of any section of the mass, therefore
x-x^ = - Ai^-z), y-y,= -B {x-z),
are the equations to the perpendicular to that section. And the angles
which that perpendicular makes with the co-ordinate axes have for
their cosines
-A -B -1
VA^TW+\' VA' + B'+l' VA' + B' + l'
Also it appears from the given equations (3) to the resultant
force, or tangent to the line of pressure, that this line makes angles
with the co-ordinate axes which have for their cosines the quantities
M, M, Ms
Hence, therefore if / be the inclination of these lines to one
another,
* It is here supposed that the coefficient of friction f is constant for the saifie surfaces,
whatever be the force B! by which they are pressed together. This is usually assumed
to be the law of friction. It is only however an approximation to that law. The ex-
periments of Mr Rennie shew that f must be considered a function of R' increasing con-
tinually, but very slowly, up to the limits of abrasion.
:fs:.
304 Mr MOSELEY, ON THE
AM, + BM, + Ms
cos /= —
{{A' + B' + 1) {Mr' + Mi + Mi)}k '
in which expression M^, M^, M^, and B, are known functions of A.
Now / must not exceed the limiting angle of resistance. Therefore
cos / must not be less than the cosine of that angle.
On the whole then we have these two conditions necessary to the
equilibrium of a mass intersected by a series of planes, under the cir-
cumstances supposed.
1. That the equation
-VF,%, F^z, » = 0,
shall involve no possible roots, except such as correspond to the ex-
tremities of the line of pressure, or to points where it touches the
surface of the mass.
2. That the fraction
AM, + BM, + Ms
shall for all values of A, corresponding to real sections of the mass,
be not less than the cosine of that arc, whose tangent is the coefficient
of friction.
The first of these conditions being satisfied, the parts of the mass
cannot turn upon one another. The second being satisfied, they can-
not slip upon one another.
We have supposed the whole of the forces impressed upon the
system to be known excepting the force P', which has been deter-
mined in terms of the rest. The force P' may be supplied by the
resistance of a point in a fixed surface, in which case the amount and
direction of that resistance will be known.
EQUILIBRIUM OF THE ARCH. 805
If, however, there enter two or more resistances of surfaces among
the forces which compose the equilibrium, since the magnitudes of
these and also their directions may be any whatever, within the limits
imposed by the friction of the surfaces; the problem remains, in so
far as the known conditions of equilibrium are concerned, indeterminate,
and recourse must be had for its solution to other principles.
7. Suppose the mass AJS to be acted upon by any number of forces
among which is the force P being the resultant of certain resistances,
supplied by different points in a surface Sb, common to the inter-
sected mass and to an immoveable obstacle SC.
Now it is clear that under these circumstances we may vary the
force P', both as to its amount, direction, and point of application,
without disturbing the equilibrium, provided only the form and
direction of the line of pressure continue to satisfy the conditions im-
posed by the equilibrium of the system.
These are manifestly, that it no where cut the surface of the mass,
except at P" and within the space JSb, and that it no where cut a
section of the mass or the common surface of the mass and obstacle,
at any angle with the perpendicular greater than the limiting angle
of resistance. "
Thus, varying the force P', we may destroy the equilibrium, either,
first, by causing the line of pressure to take a direction without the
limits prescribed by the resistance of the section through which it
passes ; or, secondly, by causing the point P to fall without the surface
Bb, in which case no resistance can be opposed to the resultant force
acting in that point ; or, thirdly, the point P lying within the surface
Bb, we may destroy the equilibrium by causing the line of pressure
to cut the surface of the mass somewhere between that point and P'.
Let us suppose the limits of the variation of P' within which the
first two conditions are satisfied, to be known ; and varying it, within
those limits, let us consider what may be its least and greatest values
306 Mr MOSELEY, ON THE
so as to satisfy the third condition ; and where, and in what direction
they must be applied.
In the first place it will be observed, that by diminishing the force
P', its direction and point of application remaining the same, the line
of pressure is made continually to assume more nearly that direction
which it would have, if P' were entirely removed.
Provided then, that if P were thus removed, the line of pressure
would cut the surface, that is, provided the force P' be necessary to
the equilibrium ; it follows that by diminishing it, we may vary the
direction and curvature of the line of pressure until we at length make
it touch some point or other in the surface of the mass.
And this is the limit; for if the diminution be carried further, it
will cut the surface, and the equilibrium will be destroyed. It ap-
pears then that under the circumstances supposed, when P' acting at
a given point and in a given direction, is the least possible, the line
of pressure touches the surface of the mass.
In the same manner it may be shewn, that when it is the greatest
possible, the line of pressure touches the surface of the mass.
Now by varying the direction and point of application of P', as
well as its amount, this contact may be made to take place in infinite
variety of different points, and each such variety supplies a new value
of P', producing the required contact. Among these, therefore, it
remains to seek the absolute maximum and minimum values of that
force.
To express these conditions analytically, let Xi, y^, z.^ represent the
co-ordinates of a point where the line of pressure touches the surface
of the body.
Since the point x^, y^, & is common to the line of pressure and
to the surface of the body,
.-. -^Xty^Xi = 0, Xi = F%o, y., = F^x^.
EQUILIBRIUM OF THE ARCH. 907
Also, since it touches the surface in the point ar^ysSSj;
dz-i i d-^x-iyiZj ^^
\ dXi J
( d'^x^.yi%i \
dF,%., V dz, ) ^ ^
d%2 l d'^X2yi%i \
\ dyi I
Eliminating x.^, y^, z.^ among these Jive equations two relations are
established between the force P'*, the co-ordinates of its point of ap-
plication, and the angles which fix its direction (see Art. 4) ; by elimi-
nation between which a further relation is established between six of
these seven quantities, and, finally, by the equations of condition
COS^ a + COS^ /3 + COS* 7=1.
a relation is obtained between four of them.
Thus then we may obtain the value of P' in terms of three of
the quantities x^, y^, s, ; a, /3, 7.
Its maximum and minimum values are then at once determined by
the known conditions of the maxima and minima of functions of
three variables.
8. It is evident that the minimum value of P', being that which
just counteracts the tendency of the mass to revolve about the point
where the line of pressure touches its surface, is also precisely that
force which would be exerted there by another equal and similar mass,
acted upon by equal forces, under the same circumstances, but placed
in a contrary position, so that its line of pressure shall have, at P,
a common tangent with the line of pressure of the first mass.
* The line of pressure is here supposed to commence at P', and the force P" to enter
among the other forces which determine its equation.
Vol. V. Part III. S s
308 Mr MOSELEY, ON THE
Two masses, therefore, thus placed together would remain in equi-
librium, without the aid of any external force, and by reason only of
their mutual pressures and the resistance of their abutments.
It is also evident that since the line of pressure is similarly situated
in both, they cannot be thus placed together so that their lines of
pressure may meet and have a common tangent at the point where
they meet, unless both lines of pressure be perpendicular to the com-
mon surface at that point.
This condition throws two new equations into the system, and de-
termines the value of P' in terms of a single variable.
The value of P' is not in this case that which we have called
the absolute minimum or minimum minimorum, but simply the greatest
or least force, which applied at a given point, in a given direction will
support the system.
If however instead of a single point of contact we suppose the
masses to be in contact throughout the whole surfaces of two planes,
it is evident that the point P' * will take up for itself that position,
which we have supposed to correspond with the absolute minimum ;
a condition to which the form of the line of pressure, and the
position of its point of contact with the surface of the mass, will also
be subjected.
Hence it appears that two masses, thus in contact throughout the
surfaces of two planes, sustain a less aggregate of pressure, on their
common surface of contact, than two similar masses in contact only
by a single point, unless that point, and the position of the masses,
be such as to correspond to the minimum minimorum.
In the preceding pages we have supposed the form of the solid
to be given, together with the positions of the different sections
made through it, and we have thence deduced the form of its line
of pressure and the direction of that line through its mass.
. ?;* The point P is here the point of application of the resultant of the resistances on
the different points of either plane. .:iorte«pi sJi -^..rnaHb
..H .Ui T ■
EQUILIBRIUM OF THE ARCH. 809
It is manifest that the converse of this operation is possible.
9. Having given the form and position of the line of pressure, and
the positions of the different sections to be made through the mass, we
may, for instance, enquire what form these conditions impose upon the
surface which bounds it.
Or we may make the direction of the line of pressure and the
form of the bounding surface subject to certain conditions not abso-
lutely determining either. '"* t'^ oxi'>»^i'« inn ouJ ifion
For instance, if we suppose the form of the intrados of an arch to
be given, and the direction of the intersecting plane to be always per-
pendicular to it, and if we suppose the line of pressure to intersect this
plane always at the same given angle with the perpendicular to it,
so that the tendency of the pressure to thrust each from its place may
be the same, — we may determine what under these circumstances must
be the extrados of the arch. ,''^'^",' '"'^
If this angle equal constantly the limiting angle of resistance, the
arch is in a state bordering upon motion, each voussoir being upon
the point of slipping downwards or upwards, according as the constant
angle is measured above or below the perpendicular to the surface of
the voussoir.
The systems of voussoirs which satisfy these two conditions are the
greatest and least possible.
If the constant angle be zero, the line of pressure being every-
where perpendicular to the joints of the voussoirs, the arch would
stand even if there were no friction of their surfaces.
It is then technically said to be equilibrated. It is impossible to
conceive any arrangement of the parts of an arch by which its stabi-
lity can be more effectually secured*.
10. The theory stated above readily explains the phenomena ob-
served in the settlement and fall of the arch.
* The great arches of late years erected by Mr Rennie, in this country, have for the
most part been so loaded as very nearly to satisfy this condition.
ss 2
8M) Mr MOSELEY, ON THE
Thus let ABS" (fig. 3) represent an arch having the joints of its
voussoirs perpendicular to the intrados as they are usually made.
Let RQPQR' be the line of pressure, touching the intrados in the
points Q and Q'. It is manifest that this curve is then perpendicular
to the joints of the voussoirs at Q and Q, and inclined in respect to
those above and below these points. The inclination being downwards,
or towards the intrados, in reference to the former, and upwards, or
from the intrados, in reference to the latter.
Hence, therefore, it appears that the tendency of the pressure is
to cause all the voussoirs above the points Q and Q' to slide down-
wards, and those beneath those points, upwards.
And that these effects may be expected to follow the striking of
the centre of the arch ; the weight being then suddenly thrown upon
the voussoirs, and these admitting of a certain degree of motion in
the directions of the forces impressed upon them.
Now this is precisely what was observed at the bridge of Nogent,
of the construction of which Perronet has left a detailed account.
Three straight lines were drawn upon the face of the arch before
the striking of the centre, shewn in the figure 4, by the polygon
nmm'n', mm' being horizontal, and the other two mn and m'n' stretch-
ing from the extremities of mm' towards the springing of the arch.
After the centre had been struck, the lines were observed to have
assumed the curved forms indicated by the dotted lines MM', MN',
M'N', indicating, in accordance with the theory, a downward motion
in all the voussoirs above Q and Q', and an upward motion in those
beneath those points.
These observations have been confirmed by numerous others, and
especially by those (made also by Perronet) at the Pont de Neuilly.
The sinking of the voussoirs at the crown necessarily tends to pro-
duce a separation of their joints at the intrados in the neighbourhood
of that point, and thus to cause the actual contact of the key and
adjacent voussoirs to take place only at their superior edges.
EQUILIBRIUM OF THE ARCH. 311
If therefore the settlement be considerable, we may conclude that
the line of pressure touches the extrados at the crown, and for
some distance on either side of it. The material of the arch may
therefore be expected to yield more particularly about that point and
the points Q and Q' than any other; a great proportion of the
pressure being there thrown upon the edges of the voussoirs.
11. If by reason of such yielding, or from any other alteration in
the forces impressed upon the mass, or in the circumstances of their ap-
plication, the form of the line of pressure be altered, it may manifestly
be expected to intersect the surface of the mass first about those points;
the least possible alteration of form being there sufficient to produce
the intersection. And this being the case, the portion of the arch above
Q and Q' must separate into two portions, revolving at those points
about the lower portions of the arch (see fig. 5) and at A, upon the
extremities of one another.
Nevertheless this revolution is manifestly impossible unless the
points Q and Q yield outwards. And this can only take place by
the yielding of the material at Q and Q', by the slipping back of
the voussoirs there, or by the portions of the arch or its abutments
beneath those points revolving outwards, in consequence of the inter-
section of the extrados by the extremities QR and QR' of the line
of pressure (fig. 3).
The last is in point of fact the cause which leads, in the great
majority of cases, to the fall of the arch.
The extremity R of the line of pressure is made to cut the
extrados of the arch, or the outer surface of the pier, by the
diminution or removal of some force which acted there in opposition
to the tendency of the arch to spread itself, and which kept the
direction of the line of pressure within its mass, — the resistance of
a mass of earth for instance, or the opposite thrust of some other
arch springing from the same pier or abutment.
On the whole, then, it appears that in the commencement of its
fall the arch will divide itself into six distinct portions, of which four
312 Mr MOSELEY, ON THE
will revolve about the points S, S', Q, Q' and A, as represented in
the figure 5. Now this is what is uniformly observed to take place
in the fall of the arch.
12. Gauthey, having occasion to "destroy a bridge, caused one of its
arches to be insulated from the rest; and the adhesion of the cement
being sufficient to counteract the tendency of the pressure to rupture
the piers, he caused them to be cut across. The whole then at once
fell, the falling portion separating itself into four parts. Having con-
structed small arches of soft stone, and without cement he loaded them
until they fell. Their fall was always observed to be attended with the
same circumstances. Before the arch finally yielded the stone also was
observed to chip at the intrados about the points Q and Q', round
which the upper portions of it finally revolved.
Some experiments made by Professor Robinson with chalk models
were attended with slightly different results. Having loaded them at
the crown until they fell, he observed first, that the points where
the material began to yield were not precisely those where the rupture
finally took place.
This fact presents a remarkable confirmation of the theory expounded
in this paper.
It is manifest, that according to that theory, with any variation
in the least force P', which would support the semi-arch if applied
at its crown, there will be a corresponding change in the position of
the point Q.
Now as the load upon the crown is increased, this least force P'
is manifestly increased. The result is a corresponding variation in the
• form of the line of pressure, tending to carry its point of contact
with the intrados lower down upon the arch.
This is precisely what Professor Robinson observed. The arch
began to chip at a point about half way between the crown and the
point where the rupture finally took place.
EQUILIBRIUM OF THE ARCH. 313
The existence of the points Q and Q', about which the two upper
portions of the arch have a tendency to turn, and about which the
material is first observed to yield, has long been known to practical
men. The French engineers have named these points the points of
rupture of the arch ; and the determination of their position by a
tentative method forms an important feature in the very unsatisfactory
theory which they have applied to this important branch of Statics.
13. The theory of the equilibrium of the groin and that of the
dome are precisely analogous to the theory of the arch.
In the former case a mass springs from a small abutment spread-
ing itself out symmetrically with regard to' a vertical plane passing
through the centre of its abutment. It is in fact nothing more than
an arch, whose voussoirs vary as well in breadth as in depth. The
centres of gravity of the different elementary voussoirs of this mass
lie all in its plane of symmetry. Its line of pressure is therefore in
that plane, and its theory is embraced in that which has been already
laid down.
Four groins commonly spring from one abutment ; each opposite
pair being addossed, and each adjacent pair uniting their margins.
They thus lend one another mutual support, partake in the properties
of a dome, and form a continued covering.
The groined arch is of all arches the most stable ; and could ma-
terials be found of sufficient strength to form its abutment and the
parts about its springing, it might be safely built of any required
degree of flatness, and spaces of enormous dimensions might readily
be covered by it.
It is remarkable that modern builders, whilst they have erected the
common arch on a scale of magnitude nearly approaching perhaps the
limits to which it can be safely carried, have been remarkably timid
in the use of the groin.
H. MOSELEY.
King's College, London,
Ocl<^er 9, 1833.
XIV. Third Memoir on the Inverse Method of Definite Integrals.
By the Rev. R. Mukphy, M.A. F.R.S., Fellow of Cuius College,
and of the Cambridge Philosophical Society.
i;;Read March 2, 1835.]
INTRODUCTION.
In the two preceding Memoirs on the Inverse Method of Definite
Integrals, the limits of integration had been fixed throughout at and
1, but in the sixth Section, which is the first of the present Memoir,
the integrations terminated by arbitrary limits are fully considered; and
when performed with respect to any function of the independant vari-
able, the proper methods for discovering reciprocal functions are given,
and it is remarkable that the forms thus obtained for the trigonome-
trical functions, for Laplace's and an infinite variety of other reciprocal
functions, are all similar, differing only by a constant.
In identities obtained between the »"" differential coefficient of a
function not containing n, and its expanded value, we may, generally,
by changing the sign of n, obtain a corresponding identity between
the ra"" successive integral and its expansion, abstracting from the ap-
pendage of integration which ought to contain ?« arbitrary constants ;
this property however extends also to certain reciprocal functions which
contain n ; and this consideration leads in the same section to the com-
plete resolution of Laplace's equation for the reciprocal functions of
one variable, which are the coefficients in the developement of the reci-
procal of the distance of two points; the w*"" coefficient when multiplied
by an arbitrary constant, satisfies that equation, as is well known, but
as the equation is of the second order, another function multiplied by
■^ Vol. V. Part III. Tr
316 Mr MURPHY'S THIRD MEMOIR ON THE
an arbitrary constant must be also represented by the same equation,
this function, which is here found, is altogether different in its form and
properties from Laplace's coefficients.
The great class of reciprocal functions above alluded to possess the
remarkable property, that their integrals vanish between any of their
own maxima or minima values.
In this Section I have noticed some curious trigonometrical func-
tions of which the properties are very elegant, particularly as affording
simple means of representing by Definite Integrals the general differ-
ential coefficients of rational and integral functions ; another applica-
tion of trigonometrical functions is made, in representing the sum of
the divisors of any given number, by means of a Definite Integral.
The seventh Section is on Transient Functions. The way of forming
reciprocal functions by means of arbitrary coefficients, when the form of
the general term was given, has been shewn in the Second Memoir on
this subject. To this I have here added the method of finding the
functions which shall be reciprocal to any proposed one, and applied
the method to the cases where the given function is r, (log. t)", and
cos" {t) ; the reciprocal functions which thence resulted are transient, that
is, they have but a momentary existence between the limits of inte-
gration ; that existence is however sufficient to make their integrals
finite, and to endow them with remarkable properties. They are capa-
ble of representing the electrical state of a body when an electrical
spark is infinitely near, and about to form a part of the system ; they
are also capable of representing, under continuous forms, the state of a
body considered as composed of absolute mathematical centres of forces,
separated mutually by infinitesimal intervals.
The eighth and last Section is on the Resolution of Equations which
contain Definite Integrals; the first method for this purpose is to de-
compose the integrals into elements, and then determine the unknown
functions by elimination. This tedious process is useful in verifying
results otherwise obtained, and in giving numerical approximations in
the most difficult cases. Afterwards I have considered separately,
INVERSE METHOD OF DEFINITE INTEGRALS. 317
Equations to Definite Integrals ; first, when they contain but one Defi-
nite Integral and one parameter ; second, when they contain two or
more Definite Integrals and as many parameters; third, Simultaneous
Equations ; fourth, Definite Integral Equations of superior orders and
degrees; besides which, the nature of the appendage analogous to the
arbitrary constant of integration is discussed in the same Section.
Throughout the whole of this Memoir, a considerable number of
examples, illustrative of the corresponding theories, are dispersed.
TT2
318 Mh MURPHY'S THIRD MEMOIR ON THE
SECTION VI.
Method of' discovering Reciprocal Functions when the integrations are per-
formed with respect to any Junction of the independant variable.
(l) When the limits of integration are arbitrary.
1. The investigations of reciprocal functions contained in the Second
Memoir on the Inverse Method of Definite Integrals, are founded on the
supposition that and 1 are always the limits of the independant
variable, but it is often of importance to possess reciprocal functions in
which the limits of integration are different from those quoted. The
principle by which this is most easily accomplished, is to suppose the
integrations performed relative to a function of the independant vari-
able, which must be so chosen, that when the values and 1 are
assigned to the independant variable, the corresponding values or the
function may be the proposed limits of integration.
2. Let Q„, R„, be functions of a variable (^), the limits of which
are arbitrary, as a and h, between which limits f^Q^Rm always must
vanish, except when the integers m and n are equal.
Suppose that a function of <p, as t, is found such that when ^ = a
t = 0, and when (p = h, t=l, conditions which it is always easy to satisfy.
We may now conversely regard as a function of t, and then the
preceding integral becomes fiQ„Rm-jr, the limits being now reduced to
and 1. Suppose that -~ is separated into any two factors, X and X';
then since f,QnX x R,„\' = 0, except when 7n — n, it follows that Q„X,
R„\' are mutually reciprocal, and may therefore be found in an inde-
finite variety of modes by the principles explained in Section iv; and
dividing these functions respectively by X, X', and substituting in the
quotients the value of t expressed in terms of ^, the required functions
Q„, Rm will be obtained.
INVERSE METHOD OF DEFINITE INTEGRALS. 319
If it be desired that Q„, R^ should be functions of the same
nature, differing only in the order expressed by m and n, that is
self-reciprocal, put \ = W = \/{-~\, and having found any kind of
self-reciprocal functions in which the limits are and 1, as for ex-
ample, the functions denoted by P,„, P„ in the preceding Memoirs, we
then obtain
3. If a function V can he determined so that the quantity
d°f(ttyV} dt^
dt" ■ d0
may he of n dimensions in t, (where t' = 1 — t as in the former Memoirs),
this quantity will he a self-reciprocal function when the integrations are
performed relative to (p.
Denote this quantity by Q„, and supposing m to be an integer
less than n, it is necessary to show that f,pQmQn — 0, or that
d''{{ttyn
^'^- di" -^'
the limits of t being and 1.
Now Q„ being of m dimensions in t, let its general term be re-
presented by Oj.f, where it is evident that p cannot exceed n — 1,
since m<n; the part of the preceding integral dependant on this term is
""'^'^ dv' — •
The latter integral may by partial integration be put in the form,
the last term being
and therefore the index of differentiation never becomes negative.
320 Ma MURPHY'S THIRD MEMOIR ON THE
The first term, and 'a fortiori', all the succeeding terms of this
series vanish between the limits ^=0, and t=\, or t' = 0, for
d''-'{{tt'rV} _rr d"-'{tt'Y ,,^ ^,dV (f-^tty
dt--' ~ dt^-' ^^ ' dt dt"-'
{n-l){n~2) dT d^-^itt'f ,
"^ 1.2 dt' ■ dt"-' "^ '
the first term of this latter series contains a factor tt', the second a
factor {tt'f, &iC., and therefore the whole vanishes between limits.
The following exception to this theorem must however be attended
to; V must not he of the form {tt')".Vj, where r is equal to, or
greater than unity, for the above reasoning will not be applicable,
since then
d"-mtt'Yr\ _d'-'{{tty-^r,}
_i
dt"-^ dt'
which being expanded as above, will not vanish unless r be less than
unity.
4i. If a function V can he determined so that the quantity
d''f(ttrvi d0
at" ■ dt
may he of n dimensions in t, then the factor hy which -^ is here
multiplied, will he a self-reciprocal function when the integrations are
performed relative to cp.
Denote this coefficient by q„, then
r -r ^0 _ /■ <^" (tf'Y ^ d<t>
and as we may suppose m<n, the general term of qm-^, as a^t^
cannot be of greater dimensions than n — 1, and therefore the part of
the whole integral dependant on this term vanishes, as has been
shewn in the preceding article, hence f^qmq„ = 0, when m and n are
unequal.
INVERSE METHOD OF DEFINITE INTEGRALS. 321
We must except, as before, from the application of this theorem
the case where V is of the form {tt')-\Vi, and r greater than, or
equal to unity.
5. If (f) be any of the transcendants contained in the indefinite
integral jj (tt')", where m is between — 1 and + x exclusive, and if
^"~ 1.2.3...ndt"'^"^ '
then shall Qn be a self-reciprocal function for integrations relative to <p.
For Q„ is evidently of the form — ~rp: ■ ~TZ' ^'^d ^ is not
of the form excepted in Art. 3., since m is between —1 and + oo.
Moreover, by actual differentiation we get
1 .^.S-.-ndt"
where a, b, c, &c. are constant quantities.
Hence,
Q„ = at"' ^btt'"-' +ctH"'-^ + kc.,
which is of m dimensions in t, and therefore all the conditions re-
quired in Art. 3. are here fulfilled; therefore Q„ is a self-reciprocal
function relative to <p.
6. If (p be any of the transcendants expressed by the indefinite
integral jj (tt')"", where m is between + 1 and — oo exclusive, and if
qn =
- d° (tt')°-
1.2.3....ndt'
n» ■
then is qn a self-reciprocal function relative to (p.
d' (tfy V
For §-„ is here of the form i ' — , and V does not belong to
the excepted cases, moreover
# _ d'^.jtt'y-'"
^'' dt ~ \.2...ndt"-^^^^
is evidently of n dimensions in t, therefore all the conditions of
Art. 4. are here satisfied.
322 Mr MURPHY'S THIRD MEMOIR ON THE
7. For the purpose of convenience both in evaluating and using
reciprocal functions, the knowledge of the functions which they generate
is very useful. The generating function, for example, being the quan-
tity denoted by q^, Art. (6), the process for finding in this case the
function generated, will sufficiently exhibit the general principle, and
therefore it is now proposed tb sum the series q^ + q^h + q^k' + q^h^, &c.
Substituting for q„ its value given in the preceding article, and
representing the required sum by S we have
o /. 'V J. ditty-'" le d'itt'f-'" M dHtfy-'" , ,
But if we form the equation, u = t + ku (1 — u), and suppose y'(M) to
be the derived function from J'{u), we have generally
^^ r(«\-f'(A^h^it^^)-^A. *' d^{f'it).(ttj}
,_f^_ d?\f{t).{tty\
+ 17273 • df *'''■
which is obtained by differentiating the value of /(«) given by La-
grange's Theorem.
The preceding series coincide by supposing
f(f) = {ttf)-"' = t-" il-t)"",
and therefore /'(«) = «"" (l-w)"" = j^-L
by the assumed equation.
(u-t)-'" du
Hence 5- = -^^ . ^^ .
Now the actual solution of the assumed quadratic equation gives
u =
2h
, where R= {l-2h{l-2t) + h'}K
, B-l-\-h{l-2t) . du 1
whence u-t= ^ , and -^ = ^i
INVERSE METHOD OF DEFINITE INTEGRALS. 323
therefore S = |/2 - 1 + A (1 - 2/)} -"
R
Knowing thus the generated function S, we can conversely find q„ by
taking the coefficient of A" in the quantity S, and substituting for t
its value in terms of <^.
An exactly similar process applied to the function Q„ of Art. (5),
woxild give
as the function generated,
and observing that
R'-{\-h{\-^t)\" ^ 4^h'tt',
this quantity may be transformed to
a III
^\R^\-h{\-^t)}-";
so that Q„ is the coefficient of h" in the expansion of this function.
8. From the theorems given in Arts. (5) and (6), we can determine
reciprocal functions relative to <p, which quantity may denote any
transcendant contained in the formula Jt{tt'y, from m—-<xi to »« = + x ;
circular arcs are amongst these transcendants, namely, when m = — ^,
and since both theorems are true simultaneously, when m is between
— 1 and + 1, we shall get in this instance the two species of circular
self-reciprocal functions, namely, the sines and cosines of the multiples
of the simple arc.
I. To evaluate Q„ when «/ = — i-
For the variable with respect to which the integrations must be
performed, we have
^ = jXtty^ = l ^y^r^ -■= COS- (1-2^),
neglecting the constant which is unimportant.
Vol. V. Part III. Uu
324 Mr MURPHYs THIRD MEMOIR ON THE
By Art. (7),
2-4
Q„ = coefficient of Jf in -^ {^ + 1 - A (1-2^)}^,
in which R represents |1 — 2// (1 - 2/) + /i'}*.
Putting for t its value in terms of 0, we obtain
J? = {1-2/i cos^ + /i^}-i = (l-Ae*^^)i.(l-//e-'''^^)^^
and l-/i(l-2it) = l-/4cos0 = 1(1 -Ae*^^) + ^ (1 -//e"*^^).
Hence, ^ + 1 -/i (1 - 2/f) = |{(l-^e*^^)^ + (1 -//e-*^^)-^'';
therefore, Q„ = coefficient of /r in x . Jj " ^t-v-!!-^4' "r'^^!
= ^ coefficient of A" in (l-/<e*^^)-* + (1 — Ae-*^'^)-*
= c
c . cos n<p]
13 5 (2« — 1)
where c = ' ' '"^ ^ , the limits of ^ are and w.
2.4.6... 2ra
II. To evaluate q„ when y« = — i.
As above, we have (p = eos"' (1 — 2t),
and q„ = coefficient of h" in ^-^ . {^- 1 +/< (1 — 2/)}''.
But i? - 1 + /. (1 - 20 = i p-^;!:l'^>' - (i::.^-_:!:^^)H'^ .
I V -1 V-l j
-. q„ = ^ coefficient of /«" + ' in
\/-i V-i
c — . = c sin (1 + n) (p,
* v — 1
, 1.3.5...(2m + 1)
^^^••^ ^' = 2.4.6■.■(2;^4-2) '
INVERSE METHOD OF DEFINITE INTEGRALS. 325
9. But whatever may be the value of m, the quantities Q„, q„ may
always be simply expressed in terms of t by the theorem of T^eibnitz,
viz.
d"(uv)_ cV'v clu d'"^v 7i.{n — l) dHi^ f/"'^ v
after {yiplying which we may substitute for t its value in terms of (p.
Thus when m= — ^
1.3.5....(2«-1) n 2n-l
2.4.6....2W ^' ~1- 1 "
• ' J. n{n-l) {2n-l){2n-3) , ,„_, _ „ .
_ 1 .3.5.. ..{2n - 1) , 2n{2n-l)
-" 2.4.6... .2» ^^ 1.2 "
, 2^(2?^-l)(2>^-2)(2«-3) ,,^,„., .
"^ 1.2.3.4 ^^ "^''•^
•^ 2.4.D....2ra ^ ' ' ^
and in the same way we have
d''(tt'Y*i
^"~ rr2. 3. ...«<//"
_ 3.5.7....(2» + l) » 2;» + l
~ 2.4.6....2« ^^^ 1- 3 ^' "
w(w-l) (2w + l)(2«-l) 5, 3 „ ,
= I 3.5.7.--.(2?? + l) , ^--- . ^ , — i /.h2»+2)
2V"=l"2.4.6....(2« + 2) ^^^ +^^^) -(^ _V-1#0 \,
uu2
826 Mr MURPHY'S THIRD MEMOIR ON THE
and passing to the variable cp, since 1 — 2^=cos^; therefore ^ = sin —
id
and /' = cos-^, whence #'^ +\/ — 1 #* = cos^ + \/ — 1 sin^ by substi-
2 2 ~" 2 -^
tuting which we obtain
„ 1.3.5....(2w-l) ,
^'-^ 2.4.6....2/. •^"^^'^'
1.3.5....(2«4-J) . , ,, ^
'?''=2:476::::(2¥T2)-''"^''^'^-'^'
which values are the same with those in Art. 8.
The numerical coefficients in these formuhe may be rejected as having
no importance in self-reciprocal functions ; it is also observable that q„
contains a different multiple arc from that in Q„, the reason of which
is that Q„, <7„ are to be self-reciprocal functions for all entire values
of n from to + oo, and then f,j,q„q,n = except when 7i = »i, this ex-
ception (on which the main value of reciprocal functions depends) would
not hold universally true if q„ were of the form sin(«0), for then 5-0 = 0,
and therefore f^qo.qo=f> contrary to the principle of the exception,
but in the form above found this irregularity does not occur.
10. From the results found in Art. 9, it follows that if we put
the real functions Q„, q„ possess a common property, viz.
except when m = n, which exception does not apply to the last integral
when m = ?i = 0.
From the same results the following identities are obtained :
, ff'^*!T\. 1. ■ («')^ = cos {n cos-' (1-20}
1.3.5....(2tt-l)</^" ^ ' ' \ Ji
(» + l)2''+'rf"(«T^ • J/ , -,. w, o*\\
i-3-^i-^^^-^,, =sm U« + 1) cos ' (1 -20}.
INVERSE METHOD OF DEFINITE INTEGRALS. 327
We shall now consider whether analogous formulee hold true for negative
values of n the index of differentiation.
Generally if u and v be functions of t and fi'u denote the w"' suc-
cessive integral of ti, then
for if we take the w* differential coefficient of each term in this series,
all the terms resulting mutually destroy each other except the first
term tiv.
Putting u-=t'-"-^, v = t""-^, and rejecting the constants of integration
in the latter, we have
also — - ^^L±lt'--l, ^ - (2« + l)(2« + 3) .,_„_! »
Hence fiitf)'"-^
(-2)"(^0-- (.,-. n 2«+l ,_^_, n{n-l) (2^+l)(2« + 3) „_ „ „ ,
= i.3.5....(2«-i)^^ "i-~T~-^ ^+-r¥-- Ts ^ ^-^^-'^
«r ^327 • dt-'^ ■ ^^^ >
= cos {mcos-' (1 — 2^)^,
the appendage which contains all the arbitrary constants being
{^o + ^it + ^.f+...A„_J"-'\ . {tt')K
328 Mr MURPHY'S THIRD MEMOIR ON THE
Dividing the last equation by {tty, and integrating witt respect
to t, we get
1.3.5...(2w-l) d-'^^-'iffy-i 1 . c i/i o.M
7 — x; — ^ • TT — 1 = - Sin \ncos-Ul-2t)}.
Putting » = ???- 1 , we get
, ,, 1.3.b...(2m-S) <?-"(«') -'" + -i . <,, , ,,, ^,,
('"-1) • {-2)'"-' • dt^ "''" {(i-»i) cos-' (l-2t)},
thus are obtained the corresponding formulae for negative indices.
11. The two series of reciprocal functions arising from the theorems
in Arts. 5. and 6., differ essentially, only in reference to the inde-
pendant variable of integration, for in Art. 5., ni may be any quantity
between —1, and +x, and in Art. 6. any quantity between +1 and
— 00 ; change in the latter theorem m into — m, and the limits of w
Avill then be the same in both ; for distinctness, also let 6 be used
instead of (p in the value of §'„.
d" itt'Y*'"
Hence, Q^= i ^.s.'.ndf ■^*^'^'""' ^"'^ <t> = !^itfY,
d" (ttY'^"
^- lALndt" ^ ^"d ^ = ;(«')-'".
Now the reciprocal functions of Art. 5., give the equation
UQnQn=0, or feQM,. ^=0.
But ^ =(«')", and ^ =(«')-'" ; therefore ^ = («7'».
Hence, feQAtt'Y ^ QAtt'Y = 0.
And since QAtt'T = qn, and QAtt'y = q„; it follows that UQnQn' is
equivalent to [dq„qn; the only difference being with respect to the
variables (f> and 9 employed for integration.
INVERSE METHOD OF DEFINITE INTEGRALS. 329
If in the formulEe of Arts. 5. and 6., we assign to m all possible
values between —1 and 4-1, we obtain two series of self-reciprocal
functions, which when m = become identical with each other, and
with the functions denominated P„ in the preceding memoirs. For
every other value of m between those limits, there are two different
kinds of reciprocal functions, one of which only is a rational and entire
function of t, for instance when m= —\, we have found the functions
cos n(p and sin {7i + l)<p, the former of which only is a rational fvmc-
tion of cos <p.
12. (1.) W/ien m= -i-
To determine cp in this case, make sin 9 = ii — / ', squaring and ob-
serving that t + f' = l, we get sin^ 6 = 1 -2 {tt')K whence »
/■i + ?;'i = V^(2 - sin- 0), and 2'\tty = cose.
Differentiate the assumed equation, and we get
^os ^ = a ^**'\\ • ~7^ ' therefore — -^ . -r^ = 2 cos . -7
;i 5
2 {tty • cie' ^"-'-— ^tt')i ■ d0~ ■ti + t'i
hence, (p = 2E {e)-F(d).
The extreme values of the amplitude of these elliptic functions
being — -, and + -; the limits of ^ are 0, and 4:Ei — 2JF\, where
El and Ei denote the complete functions when the amplitude extends
from zero to a right angle.
The reciprocal functions for integrations relative to (p, are
_>3.7.11...(4?i-l)
Q,=
4.8.12 4.W
4n{^n-l) 4n{4>n-l)(in-i){4u-5) „ ,
^* sTi * ^^ 3.4.7.8 ^ (,^^-U
330 Mb MURPHYs THIRD MEMOIR ON THE
5.913...(4« + 1)
q.
4.8.12 4«
, (,0J {t-- (4>^ + l)-4^,.-.,^ (4. + l).4^».^(4«-3)(4»-4) ^,„.,^, ^^^^
(2.) When m= — 1.
In this case <?„ = 1.2.3...,,^^,
and <^ = j;(«')-'=h.l. (I).
Hence, ^ = e*, ^ = 1 + e* ;
therefore
_(« + l)_ ^^ ^+1 nin-X) {n^\){n) ^_^ ,
9» (1 + £</>)»+'= »^ 1- 2 -^ ^ 1.2 ■ 2.3 "^ '^'^■^'
where the limits of ^ are — « and + w .
13. To express the functions Q^ and qn z« terms of t alone.
By Art. 6., we have
^'~ 1.2.3...ndf
= (w-?»)(w-OT-l)...(l-OT) _ „_ 71 n-m
1 .2.3...W v ; • J j^ . ^ _^
«j^_l) (/^-m)(«->»-l) ,
^ 1.2 • (l-»»)(2-»/) ^ ^>&c.}.
INVERSE METHOD OF DEFINITE INTEGRALS. 331
Suppose 1- 1 substituted for t' in each term between the brackets,
then expanding each, the coefficient of t" in the whole will be
n{n-'l)...{n-r + l) , ^y t-. , „ »-"^ , r(r-l) {n~m){n-m-l) , ,_ ,
1.2...r "^ ^^ ^^^^l-m^ 1.2 ' {l-m){2-m) ^^^'^
«(«-!). ..(M-r+l).(-l)' (^„-„ d't'-'" d.f-'" </'->. r-"
1.2...rx{l-tn) (2-m)...{r-m) '^ dt' dp ' dt'-'
r.(r-\) d^ . t"-" d'-' . p-'"
■^ 1.2 • dt^ '~dF^' ^^•^'
when t is put equal to unity after the differentiations.
But by the theorem of Leibnitz, the part within the latter
brackets is equivalent to
fjr fr+n—2m
— -^- — =(n-2m + l) (n-2m + 2)...{n — 2m + r).t"-'"",
hence, the required coefficient of
,_,_., «■(»- l)...{n-r+l) {n-2m+ 1) {n — 2m + 2)...{n — 2m + r)
~^ '' 1.2...r ^ il-m) (2-m)...(r-m)
Henpe,
« _ (»-ffi)(n-m-l)...il-m) . . , n n-2m+l
7"- 1.2.3...n ^'~^' ^^~T' l-m '^
n(n-l) (n-2m-\- 1) (n - 2m + 2) ^ , ,
"^ 1.2 • {l-'tn)i2-m) * > ^<^'h
Again, by Art. 5.,
_ d"(ttT'"
^"- l.2...ndf-^^^'
(n + vi) (n + m—l)...(l+m) j^,„ n n + m ,„_,
1.2...n *■ 1 1+m
t'"-' t
, n.jn-X) (n + m) (n + m -I) ^„.,^, \_,
"*■ 1.2 • (l+m)(2 + m) ' ^'^^.j,
Vol. V. Part III. Xx
3SS Mr MURPHY'S THIRD MEMOIR ON THE
the reduction of which to the powers of t is effected as before, putting
— m for m, whence
(w + ot) (w + m-l)...(l +»?) ( n n + 2m + l
^"~ 1.2...ra ^ 1* 1+m ■
When j» = 0,
« o 1 ** " + i M«:il) (w + i)(w + 2)
which is the same as the value of P„, Sect. ii. Art. 2.
When m= — ^
2
and t = sin^ ^
2
{(w + ly-l^{(>^ + ly-2-} _,^.^,0
2.3.4.5
Q»=2.4 2n •il-1.2'^ ''" 2 + 1.2.3.4 -^ ''" 2 *'''-^-
14. To express the quantities Q„, q„ by means of a differential
equation.
Suppose /{t) is a function of #, subject to the condition
t(l-i) ./"it) + (»»+ 1) (1-20 ./' (0 + « • (« + 2/»+ 1) ./{t) = 0,
where /"(^ denotes the second, and /'(^) the first differential co-
efficient of f{t) relatively to t ; differentiating this equation, we get
t{l-t) .f"{t) + (»« + 2) (1 - 2^ ./"(/) + (»- 1) (« + 2»? + 2) ./' (^) = 0,
^ (1 -0 ./"" {t) + (w + 3) (1 - 20 ./'" (0 + (« - 2) (W + 2»« + 3) ./" {t) = 0,
and generally,
/(i-0/""^'no+('»+^-i)(i-20-/""'^'"'MO+(^-^+2)(«+2/»+r-i)./"'<'-»(0=o.
INVERSE METHOD OF DEFINITE INTEGRALS. 333
Put # = in all these equations successively, thence we have
{m + l)./'{0) =-n.{n + 2m + l).f{0),
(m + 2) ./" (0) =- in~l){n + 2m + 2) ,/' (0),
m + 3 ./'" (0) = - (ra - 2) . (« + 2w + 3) ./(O),
&e.
it follows from this by Maclaurin's Theorem, that the preceding equa-
tion will be satisfied, as a particular solution, by taking
^•/^v ^/«v(i ^ n + 2m+l .n.{n-\) (w + 2w^ + 1) (w + 2>» + 2) ., , .
./(0=/(0){l-i.-i-^^-.^ + ^^. (i + ^).(2 + «.) ^ ^^-&c-}»
and ,/(0) being arbitrary if we put it equal to
(1 + m) (2 + w<) (3 + ?») (n + m)
i '. 2 ; 3 TTTT^ n '
this value oi f{t) will become the same as the value found for Q„ in
the preceding article ; hence, replacing \ — thy its equal t', we get
(«') ^ + (« + 1) (1-2^) . ^ + « . (w + 2m + 1) . Q„ = 0.
But if in the value of /{() we change the sign of m, putting
... _ (l-m)(2-?w) {n-m)
'^^"^~ 1.2 n '
then y*(#) becomes equivalent to q„ {tt')" ; and if we put this for y (#)
in the first supposed equation, and divide the result by {tf)"', we get
{ti')^ + im + l){l-2t).^+{n+l){n-2m).q„ = 0.
(2) Particular inferences resulting from the preceding theory.
15. Denoting as before by <f> the" indefinite integral fi(tt')'", and
putting
xxS
834 Mr MURPHY'S THIRD MEMOIR ON THE
then assigning to m all possible values from — oc to +00, the functions
Q». qn will give an infinite series of reciprocal functions relative to all
the transcendants contained in ^ considered as the variable of integra-
tion ; and when m is between - 1 and + 1, pairs of reciprocal functions
will be obtained, except when ?» = 0, when both coincide.
In this series are included the trigonometrical functions, namely,
when m-= —\'., and Laplace's functions, when /« = 0.
In all the reciprocal functions thus arising, there exists one common
property, namely, the definite integral always vanishes between the
limits which make the functions themselves maxima and minima; this
remarkable property I have had occasion in another place to notice, in
the particular case of Laplace's functions.*
To prove this generally take the equations of the preceding article,
viz.
/#'^ + (»» + l)(l-20.-^'' + «(w + 2»« + l)Q„ = 0,
«'^ + (?» + l)(l-20•-^+(« + l)(w-2»^)9„ = O.
Multiply both equations by {tiy, and integrate reserving the con-
stants under the integral sign ; hence,
{tt'Y^^ ^ + « (« + 2m + 1) j: Q„ (tty = 0,
' (^0'""''-^+ (« + l)(«-2»^)/,^„ («')"* = 0;
and changing the independant variable by the condition 7;r =(^^')"'"» we
have
(«')""+^^ + w (« + 2/» + 1) 4 Q„ = 0,
(<0*"*' ^ + (» + 1) (»« - 2»w) /^ ^„ = 0.
Electricity, Introduction.
INVERSE METHOD OF DEFINITE INTEGRALS. 335
But when Q„, q„ are maxima and minima, -^ and -^ respectively
vanish ; therefore, between the corresponding limits of (p, we must have
U Q» = 0, f^qn = 0, which general property is easily verified when
Q„ = «cos«0 and q„ = a sin {n + 1) (p.
16. To find the complete integral of the differential equation
tt' , -^-^ + (m + 1) (1 - 2t) ^ + n (n + 2m + 1) tr =s 0,
where n is integer and m any constant.
The differential equation for Q„ (Art. 15.) is of the same form as
the above equation, and therefore u=cQ„ is a particular solution, c
being an arbitrary constant.
The form of the differential equation for q„ will become the same
as that of the given equation, if — (w + 1) be written instead of n in
the former; hence, another particular solution is c'q-^„+^y
The complete solution is therefore
u = cQ„ + c'q.^„+iy
This solution fails first when m = 0, for then the functions Q„,
5'-(«+i) in their expanded forms become both identical with Laplace's func-
tion P„, and consequently the two constants c, c' merge into only one,
viz. their sum ; but if we put generally
b , , b
c = a -\ — and c = ,
m m
then M=«Q„ + &. ^"~ ^-'"^'
m
And putting m = 0, the latter term becomes a vanishing fraction, and
therefore,
u = aP, + ^^{Qn- S'-(«4.i)} when m =0.
336 Mr MURPHY'S THIRD MEMOIR ON THE A
The term by which b is here multiplied, is the coefficient of m in
Qn-q-i,„^^), which is easily found from the expansions in Art. 13; hence,
n{n-\) (« + ])(w + 2) r 1 1 I 1 1 Ux,.
The general solution also fails when m is an integer, for then some
of the terms in the expansion of Q„ or g-.^+u will become infinite, and
the principle of vanishing fractions will simply enough in this case
also be applicable in determining the complete solution ; but if we put
for Q„, q„ their differential forms, the solution will never fail, for the
failures arise from the entrance of logarithms into the result, and these
will actually enter in the latter forms; changing our constants, the
complete solution for all cases is
it is therefore necessary to shew that the functions by which the ar-
bitrary constants are multiplied, are particular solutions.
Putting v-itty-"^, then -t- =(» + »») (1-2^) («')"+""-',
and -^ =(m + »?) {n + m - 1) (1 - 2tf («')"+'"-^- 2 (?< +«?) {tt'f *''-'.
Hence tt' .-^ -{n + m—\){l-2f) -r.-\-2{n-\-m) .v-0,
and by successive differentiations the following equations arise:
(«■').^-(« + '«-2)(l-20.^+2(2« + 2«^-l)^=0,
(«')f^-(» + ^«-3)(l-20.^+2(3« + 3»e-3)g=0.
d"v
dF
INVERSE METHOD OF DEFINITE INTEGRALS. 337
and the law of the successive formation of these equations being very
simple, we have generally
(„.,^-(»+».-*-i)(.-.o^>.{(*+i)(»+»)-*i*±L>}.g=o.
Put k = n, hence
d"v
Transpose n(n + 2m + l)-j—, and multiply by {tt')-", hence
from which it follows that M = (it')"" .d". ' satisfies the equation of
Art. 16.
dv'
Again put «j' = («')"'"*'"^^ or tt' -j- +(n + m + l)(l-2t)v=0, and by
successive integrations we obtain
tt' . v' + (n + m) (1 -2t) ftv' + 2 (n -\-m) ft'v =0,
tt' . ftv' + {n + m-l) (1 -2t) . ft' v -\-2(2n + 2m- 1) J^'v; = 0,
and generally *
tt' ft*-'v' + {n + m-k + l){l-2t) ft'v' + 2h{n + m) - ^ '^^~^^ \ . ft"-' v==0.
Put k = n, hence
tt' fr' V + (m + 1) {1 -2t) . ft" v' + 91 (n + 2m + 1) . ft"*' v' = ;
from which it appears that m = jJ" +*(»') is also a particular solution, and
therefore the complete solution of the general equation is
338 Mr MURPHY'S THIRD MEMOIR ON THE
Laplace's equation occurs when we put m = 0, and therefore
the first term alone of which is the type of Laplace's functions, the
equation is therefore more general than the functions it was used to
designate.
The term ^"+' («')-<"+"'+') gives n + 1 constants of integration which
enter as coefficients of the appendage which is a rational function of
n dimensions, but this must be rejected, since the constants must be
determined so that the rational function of n dimensions may satisfy
the given equation, and this only identifies the appendage with the
d' ift'Y^"'
other term in u, viz. aitf)"" — , / — .
17. To find explicitly the omitted part of the complete integral in
Laplace's equation.
The general equation of Art. 16. becomes in this instance
and the complete solution is
u = a^^^^+bfr'{tt')-^'-'\
the first term being Laplace's function, and the second the transcendant,
it is required to find explicitly.
Let a, /3 be any arbitrary quantities, then we have
dar\t-a' fi-a) ~ ^-a da^\t-a)^'^da\fi-a] da'-'\t-al
n{n-l) d^ / 1 X d'-^ / 1 N
^ 1.2 Ma'[(i-a) da"''[t-a)' ^'
or
INVERSE METHOD OF DEFINITE INTEGRALS. 339
hence
^-/ 1 I
+ 1 I
_ \«-a)(/3-a)| ^ 1 r 1 W +
, (n + mn + 2) 1 L_ + &cl
(« + !)(« + 2)
Commuting in this equation the quantities a and /3, we have
(» + !)(» + 2) 1 1
,., '^"{(f-/3)(a-/3)} _ 1 r . ^ ^ + 11 1
(» + l)U + 2) 1 1 1
1.2 ■ {a-(iy{t^(iy^ ^^■j
If both equations be added observing that
1 1
+
(#-a)(/3-a) ^ (/-/3)(«-/3) {t-a){^-t)'
the sum of the left-hand members
fpn £
1^2'.3^...M'«?a"C?/3"
rf".-^ ef-. ^
^_1)„^ ^-« /3-^
1.2.3...wrfa"' 1 .2.3...W6?/3"
1 1
Vol. V. Part III. Yy
340 Mr MURPHY'S THIRD MEMOIR ON THE
Hence, we get the general identity,
1 1
K^-«)(/3-^)l"^' (/3-ar'
i 1 ,^ + 11 1 (n + l){n + 2) 1 _JL_&cl
(^-ar' "^ 1 -fi-a-it-ay^ 1.2 •(/?- a)^ ' (#-«)'- '[
^_^ ^^±1 ^^ 1 (« + l)(>^ + 2) 1 _J__£,c
■^(/S-O'*""^ 1 •/3-a-(/3-0" 1.2 •(/3-af-(/3-#)'-' •]
Put now a = 0, /3 = 1, and therefore (i - t = f, hence,
>+' "^ 1 •/" "^ 1.2 ■^"-'
[+F^"*" 1 V""^ 172 •^-^ + *'c.j
in which identity n must be one of the natural numbers 0, 1, 2, 3, &c.
and the number of terms in each series must be limited to w-f 1.
Suppose the (ra + 1)* successive integral of each term of this expansion
is taken after multiplying, for convenience, by 1.2.3 n, the result
will consist,
1st, of a logarithmic part, viz. .
(-,)-.h.i.w{i-f.^.^.^^. <''";'.'r' '''-M
where the part between brackets in the upper line is equivalent to
the function P,„ and in the lower to (-1)".P„, and therefore the whole
to (-l)".P„.h.l. ^,.
* This method is applicable in every case to the decomposition of fractions, the denomi-
nators of which contain equal factors.
INVERSE METHOD OF DEFINITE INTEGRALS. 341
2d, of a rational and entire function />„ which satisfies the equation,
dP
since the term 2 ( - 1)". -^ is the result which arises if the logarithmic
term ( — 1 )" P„ . h. 1. -> be put for u in the actual equation.
3d, of an appendage containing n + 1 arbitrary constants, which as
before remarked must be rejected altogether.
Differentiating the equation for p„ above obtained, we get
(«',^- + Mi-20.^-+>-i)(«+^)# + 2(-ir.^--o,.
(«-l)(l-20.^" + 2(2«-l)^^" + 2(-l)»^^=0,
• : ■ ^" df-^ + ^^^> ~dF~^'
when these equations terminate, since j9„ is of « — 1 dimensions.
Put ^ = 0, in all these equations beginning with the last, observing
that then
^ = (-l)-.(« + l)(» + 2)...(2«),
^^ = - ( - 1)" . « (« + 1) (« + 2)...(2« - 1),
'^=^-^)"-^^i^^-^'*''^)^'' + ^) (2«-2),&c.
Y Y2
342 Mr MURPHY'S THIRD MEMOIR ON THE
Hence -^^ = - 2. (« + !)(» + 2)... (2»- 1),
^^" = (m + 1)(« + 2)...(2m-1), &c.
and the value of j9„ is the rational function
^ 1.2...{n-l) ^^ +^'^ +A,f ...+A„^,],
in which the coefficients are successively formed from the equation
{n-m-lf.A„ + {m + 2){2n-m-l).A„^i
+ 2(-ir "i^-'^)-("^ + ^) n{n-l)...(n-m-l) _
' '1.2...{n-m-l)' 2n{2n-l)...{2n-m)
and the omitted part in the integral of the proposed equation is
6|p„h.l. (I) + (-l)».^.
18. When m = —^, the general equation of Art. 16. becomes
and putting ^ = cos"^(l — 2#), we have Q„ = cosn<p, §'_,„+,) = sin ncp, the
complete solution is therefore M = a cos«^ + 6 sin «^.
Though the trigonometrical functions were the first used in analysis
as reciprocals, for the purposes of expressing functions by means of
definite integrals and of expanding them, in the former instance of
their application there remain a few remarkable cases which do not
seem to have been noticed, with which we shall conclude this Section.
19. The two functions which possess the remarkable properties al-
luded to, are
e = e^'<«» . COS {x sin &), and 6' = e^ ""^^ sin {x sin B).
INVERSE METHOD OF DEFINITE INTEGRALS. 343
The successive differential coefficients with respect to x of the func-
tions 0, 6' follow simple and elegant laws, thus
do dQ'
= 6^'=°'* cos fa: sin + 0}, ^— = e^'=»*« sin {a; sin + 0},
d'Q d^Q'
d^ = 6"^"°'' cos {ar sin + 20}, ^^= e^'^"** sin {« sin + 20},
and generally
d" d° 0'
• ^-; = e^'°'^ cos {x sin + «0}, -j— = e^ ">"* sin {x sin + «0} . ■
Again, the successive integrals relative to x, follow the same laws,
omitting the arbitrary constants of integration,
/^0 = e^cose cos {a; sin 0-0}, /,©' = e^<=°'» sin {« sin 0-0},
//e = e^cose cos 1^ sin - 20}, f,'Q' = e^'^"^* sin {x sin 0-20},
fj-Q = e^'^"'^ COS far sin0-w0}, f/O' = 6^<=<«« sin {a;sin0-«0},
for it will readily be seen by actual differentiation that
d" d"
= ^-;; {e^'="'''cos(xsin0-«0)}, 0' = T-^ i^icose sin (a; sin0- m0)}.
Again, changing the forms of the proposed fimctions, we get
= 1 {e-'^ + 6"-'^^}, 0' = -4== {e"'^^ - e"''"^'},
whence, expanding and passing from the exponential to trigonometrical
functions
a;* of
= 1 + a; cos + - — - . cos 20 + , . ^, cos 30 + &c.
1.2 1.2.3
0' = a; sin + — — . sin 20 + , ^ „ sin 30 + &c.
1.2 1.2.3
344 Mr MURPHY'S THIRD MEMOIR ON THE
jeW cos wy — -. j— g— g ^ the limits of being and ir, these formulee
\ apply for all integer values of n, except
Now e cos nO ± e' sin nO = e^^s* cos {x sin + «0|.
Hence /ee"""^ cos {arsin^- w0| =7r . — — -— ,
l^^xcose cos {arsin0 + wej =0.
The particular case where w = is included in the first of these two
equations.
20. By the results thus obtained, we are enabled to represent any
rational and integer function of a; in a form adapted to general differen-
tiation.
By applying Maclaurin's theorem, we first have
(}>(x) = Ao + A,.x +^2- j^ + ^3 - ^ g 3 + &c.;
and passing to definite integrals by the formulae of the last article,
(h{x) = - /ge^cose 1^^ cos (x sin 9) + A^ cos {x sin 6-9)
+ ^2 cos (.r sin - 20) + &c. J
also if A^u A^i, A_3, &c. represent arbitrary constants,
= - /ee^'=°^* {A-i cos(x sin + 0) + ^_2 cos (ar sin + 20)
+ ^_3 cos (a; sin + 30) + &c.}
both of which integrals must be added before <p (x) can be subjected in
a complete form to general differentiation.
We then obtain the w*'' differential coefficient by adding n9 under
each cosine in this sum, that is.
INVERSE METHOD OF DEFINITE INTEGRALS. 345
'-pp- = i /ee'-^o'S {Ao cos (x sin + n9) + A, cos [a; sin d + {n-l)e'\
(toe '^
+ ^a cos [ic sin +(» — 2)0] + &c.}
+ _|ge'^cose |^_jCos[xsine + (re + l)0] + ^_,cos[xsin0 + (w + 2)0]
+ ^_3 cos [a; sin + (w + 3) 6*] + &c4 .
I. When n «'* a positive integer, the whole of the second line
vanishes, there will then be no arbitrary constants; also, the first n
terms of the upper line disappear.
II. When n is a negative' integer, the first n terms of the second
line remain, and these contain n arbitrary constants.
III. When n is jractional, the whole of the second line remains,
giving an infinite number of constants.
21. The theory of numbers as connected with definite integrals,
afibrds another remarkable application of reciprocal functions.
Let n be any integer of which the divisors are n, Ji', n" 1; also
let m be any intger, and d an arc of which the limits are 0, tt.
Then, generally,
1 -2Acos»?0 + A'' = (l - A6'»e^^)(l- Ae-"*^^);
and hence,
h. 1. (1 — 2 A cos ra + A") = - 2 { A cos m + ^ ^' cos 2 »J + ^ A^ cos 3 /w + &c. I .
Suppose now that m is one of the numbers n, n', n" 1; this
series must contain one term involving cos»0, viz.
— A^cos w0:
n
and therefore.
Tit —
j^cosw^h.l. (1 — 2Acos»»0 + A^) = — TT. — . A"".
But when m is not a divisor of n, there will be no term in the
expansion found to contain the arc n9, and therefore,
^cos«0h.l. (1 -2ACOSJW0 + A^) = 0.
346 Mil MURPHY'S THIRD MEMOIR ON THE
Put now for m successively every integer from 1 to w inclusive, and
take the sum of all the definite integrals thus resulting, hence
/ecos»0h.l. {(l-2Acos0 + A^)(l-2Acos2e + A^)...(l-2Acos«0 + A*)}
\n , w' -^, w" -4, 1,1
= - ttX- .h + - .h" + — . A" + ...- . h").
\n n n » J
Now the quantities -, — , — , &c. are the reciprocals of all the
Tt ft Ti'
possible divisors of n, and therefore this definite integral may also be
expressed by
"^ -^{k + -,h''' + \h"" + ...-h"}.
'^ n' n n '
For 9 in the preceding equation write 20, the limits of the latter
variable will be and - .
2
Also put h = 1, and therefore,
1 -2hcosd + h' = 2{l-cos2(p) = 4!sm^(f),
1 - 2A cos 20 + ^2 = 2 (1 - cos 40) = 4 sin' 20,
&c. ;
.-. h.l. {{l-2heos9 + h') {I -2hcos2e + h'')...{l - 2hcosne + h')}
= 2w h. 1, (2) + 2 h. 1. {sin sin 20. ..sin w0}.
The integral of the constant multiplied by cos2«0 vanishes, and therefore
7^ h.l. {sin sin 20 sin 30. .. sin »0} , cos2»0 = — t|~ + — +— , +...4-ll;
and multiplying both sides by , we get this theorem.
The sum of all the divisors of a given number n, including the
number itself and unity, is expressed by the definite integral
4t7l
/^h. 1. {sin sin 20 sin 30. .. sin w0} . cos2w0.
INVERSE METHOD OF DEFINITE INTEGRALS. 347
SECTION VII.
On Transient Functions.
22. Let ^ (h, t) be such that when h has a particular value as-
signed, the whole function vanishes whatever may be the value of t,
except in one case ; (/^, t) under those circumstances, is a transient
function having only a momentary existence.
Thus the function _ , (^ —0.t\l.hH^ ' ^^*^" ^' ^^ P"^ equal to
unity is a transient function, because its value is zero in every case
except when t = 0, for then it becomes t- — j-^ when h is put equal
to 1, that is, it acquires momentarily an infinite value.
If the value of the function had been always zero, its definite inte-
gral relative to t would also be zero; but if we actually integrate from
^ = to t = \ without previously assigning a particular value to h, the
definite integral
2A \\-h \+h\~ '
thus this integral is independent of h, and therefore remains the same
when h=\, that is, for the transient function.
By the principles of the Second Memoir we can always form a
self-reciprocal function in which the general term may be of any par-
ticular kind ; thus if f{t, n) were the type of the general term, and
if we put generally,
Fit, n) = a,f{t, 0)+a,f{t, l) + a^f{t, 2)+ + a„fit, n),
lastly, if we determine the coefficients a,, «2, a„ in terms of «„ and n,
by the n equations (arising from the definite integrals) following,
f,F{i,n).fit,0)=0,
Vol. V. Part III. Zz
348 Mr MURPHY'S THIRD MEMOIR ON THE
!,F(t, n) ./{t, 1) = 0,
SF{t,n).f(t,n-l) = 0;
then the function F(t, n) will obviously be self-reciprocal.
But if f{t, n) not containing arbitrary coefficients, but being abso-
lutely given as P, (cos^% &c. is proposed as a function to which some
unknown function is reciprocal, the discovery of the latter, which is
effected in the next article, is of a more difficult nature than the pro-
cess above mentioned; and in the particular cases quoted, as well as in
many others, this required function is transient, it is therefore in this
character that transient functions are here introduced.
23. Given f (t, n) a Junction of known form with respect to the vari-
able t and the integer n, it is required to find another Junction of t and
n, as ^ (t, n), such that the definite integral jjf (t, n) ^(t, n') may always
vanish when the integers n and n' are unequal.
Begin with forming a self -reciprocal function F{t, n), the general
term of which may be of the given form J{t, n) ; thus
F{t, n) = a,f{t, 0) + a,f{t, l)+a,f{t, 2)+ +a„f{t, n),
where the coefficients are determined in the manner indicated in the
preceding article.
Suppose next that the required function {t, n) is expanded in an
infinite series of which the general term is of the form F (t, n), thus
<p{t, n):=Ao.F(t,0)+A,F{t,l) + ...+A„F{t,n) + A„^,F{t, (n + l)}, &c.
Multiply by f(t, 0), f(t, 1), f{t, 2) f(t, n - 1) successively, and
integrate the products between the given limits of t, observing that
f,F{t, 1) .fit, 0) = 0, f,F(t, 2) .fit, 0) = 0. ..J,F{t, n)f{t, 0) = 0,
by the property of the functions F {t, n) ;
INVERSE METHOD OF DEFINITE INTEGRALS. M9
and similarly,
jlFit, 2) fit, 1) = 0, f^Fit, 3)/{t, l) = 0...f,F{t, n) .f{t, 1) = 0,
&c. &c.,
we thus obtain the following equations ;
j; {t, n) .fit, 0) = A, j,f{t, 0) . F (f, 0),
^,<^{t, n) .fit, 1) = A,S,f{t, 1) . F{t, 1),
^0 (A n) .fit, w - 1) = A._,!,f{t, n) . Fit, n-l);
hence the imposed condition of reciprocity requires that the first n co-
efficients Ao, Ai...A„-i in the expansion of 0(#, w), may be each equal
to zero ; and therefore,
0(^, n)=A„F{t, n) + A„^,F{t, n + 1) +A„+,F{t, n + 2), &c. ad inf.
Multiply successively both sides by f{t, n + 1), f{t, n + 2), &;c., and
integrate; and since n + \, n + 2, &c. are each > n, the definite integrals
must vanish.
Hence,
AJ,F{t, n) .fit, n + l) + A„^J,Fit, n + 1) .fit, m + 1) = 0,
AJtFit, n) .fit, » + 2) + An^,^,Fit, w + 1) .fit, n + 2)
+ A„^2ftFit,n + 2).fit,n + 2) = 0,
&c. &c.,
from whence the coefficients An+i, ^„+2, &c. are known in terms of A^
and ti, and therefore the required function ^ it, n) is known.
24. To find the function which is reciprocal to t°.
First, we must form a self-reciprocal function, of which the general
term is of the form /"; this has been already effected in Section
IV., namely,
:,. n n + 1 ^ , nin-\) in + !)(» + 2) ^
^,,-1- j.-y- t+ ^^ . j-^ .t -&C..
z z2
350 Ma MURPHY'S THIRD MEMOIR ON THE
which has been also proved to be the coefficient of A", in the ex-
pansion of {l — 2h{l — 2f) + h^\~^, (Section IV. Art. 9), and to be equal
cl" (tt'Y
*^ 1 — oQ — ~Tf^' where t' = \ — t, (Section iv. Art. 2.)
Then representing by V„ the required function which is reciprocal
to f, we have by the preceding article
where it is obvious that when n' is less than w fiVj"' = 0, and it is
only necessary that the coefficients may be so determined, that the
same equation may remain true when n is greater than n ; and since
one of these coefficients is arbitrary, we may put ^„ = 1.
Now in general, we have by Section iv. Art. 2.
x{x~l) {x — 2)...{x-n + \)
f,Pj'' = {-iy.
(a; + 1) (a: + 2) (a; + 3)...(;r + w + 1) '
hence, i F„ #" +^ = ( - 1 )" { 7 ^^ zr~r ^^-h-r^ — v^
■' ^ ^ \(w + a; + 1) (w + a; + 2)...(2« + ar + l)
_. {n-¥x) {n+x-\)...x . {n+x) {n+x~\)...{x—l) . 1
~ "*'■ (w+ar+l)(ra+ar+2)...(2w+a;+2)^ '"'"{n+x+\){n+x+9)...{'in+x+S)~ ]
Therefore, when x is any integer from 1 to x , we must have
A ^ A X{X — Y)
2w + X + 2 "■" (2» + a; + 2) (2 w + j; + 3)
. x{x-\) (;r-2)
~ "*" (2» + x + 2) (2w + a; + 3) (2w + ar + 4) "^ '
and putting for x the successive integrals 1, 2, 3, &c.
1
= l-^„+i.
2m + 3'
« , ^ 2 . 2.1
0=1— .4„+i. X—-—: -r-4„+2
2« + 4 "+'■ (2w + 4)(2m + 5)'
^ , . 3 . 3.2 . 3.2.1
2w+5 ^"""+^- (2w+5) (2w+6) "+" (2«+5) (2«+6) (2m+7)*
&c. &c.
INVERSE METHOD OF DEFINITE INTEGRALS. 351
From whence we obtain
A„^, = 2n + 3, A„^,= ^ ^ .(2n + 5), Jn+3=~ fgg '.(2n + 7),
and to prove that this law of formation is general, we may observe
that since
/ iy+2^+i_ 2n + 2x+l 1 (2w + 2a; + 1) (2n + 2x)...{x + 1)
V~ h) ~ 1 ' h^ "^ 1.2...(2w + ^ + l)
iy+^-^' I X 1 x.jx-l) i,^\
^ \ hi '\ 2n + x + 2' h {2n + x + 2){2n + x + S)' K" ]'
Qfi 4- 2
and (l-hy^'-^'Hl + h) = l + {2n+.3) . h + . {2n + 5).k'
(2« + 2) (2w + 3) ,„ „. ,3 J
+ ^ o 9 • ^^^ + 7).h^ + &c.
Multiply both, and take the coefficient of ,^„^^^, in the products,
and we get
{2n + 2x + ]) {2n + 2x),..{x + 1) ^ x ,
\.2.3...{2n+x + \) * ~ 2n + x + 2'^ '
x{x-l) (2w + 2)(2w + 5)
■*"(2w + a; + 2) (2« + a; + 3)' 1.2 ' ^^
= coefficient of -1- in (-IV (l+^Xl-^r''
= (-1)'. coefficient of A' in (1 +^) (1-^)^'-.
Now the coefficient of h" in (1 +^) (1 -Af""', is evidently the sum
of the coefficients of h'~\ and of h\ in the expansion of (1 — A)*'-';
that is, the sum of the coefficients of the two middle terms in a
binomial raised to an odd power, and with alternate signs of + and
352 Mr MURPHY'S THIRD MEMOIR ON THE
— , hence the quantity we are considering must be zero, and there-
fore
2w + 3 X {2n + 2) (2w + 5) x(x—l)
~ i '2ra + a; + 2'*' 1.2 ' (2n + x + 2) {2n + x + 3)
_ (2» + 2) (2?^ + 3) i2n + 7) x{x-l) {x-2) „
~ 1.2.3 ' {2n + x + 2){2n + x + 3){2n + x + 4>) "'
which shews the generality of the observed law of the coefficients ^„+i,
Substituting now these values in the general formula for V,„ we
get the required function which is reciprocal to t", namely,
rr- r. /„ „v T, (2« + 2)(2w + 5) „
K = P„+{2n + 3).F„^, + ^ ^ ^ . P,+8
(2w + 2) (2w + 3) (2w + 7) „ „
"- 17273 •^"*^' *'''•
25. The Junction which is reciprocal to t" is transient.
For in general
d".(tt'Y _ d^itt'T
\.2.3...ndt'' ^ ^ ' 1.2.3. ..ndt'"'
and putting 1-t' for t, and expanding the binomial (l-t')", and
lastly actually performing the differentiations indicated, we have
^ A) .^n-A 11*^1.2 1.2
and therefore
V-l) ''n-JA ii'^+i2 1.2 *
-(2w + 3)|i T" • ~T~ ~~T72~~ • 1T2 '
(2«+2) (2W+5) ,, »+2 n+3 - (w+2)_(^+l) (w+3) (w+4) ,,2_^ >
+ 172 t^~ 11 1.2 • 1.2 ■ "^
-&c. &c.
INVERSE METHOD OF DEFINITE INTEGRALS 353
The term which is independent of t' is
but in general we have
(l-A)(l+A)-(^"+^' = l-(2« + 3).A + (^" + ^^)(|^ + 5) ^._^^
and putting h = \, we find that the term independent of if is zero.
Again, multiplying the last equation by A"+'", we get
(*"+"■ - A"*""*') (1 +^)-<'"+'* = A"+'» - (2 w + 3) . A"-^""*' + (^"+yv^"+^) . ^.+".. >^ _ &c.
Now it is easily seen that when h = 1, we have
-Tj^ (A"+'"-A"+"'+') = (w+w)(«+»?-l)...(«-/»+l)-(«+/» + l)(«+»»)...(«-w+2)
= — 2»w . (w + >w) (w + TW — l)....(w — m + 2),
-(A"+'"-A"+'"+')= - (2m-l)(«+m) («+m-l)....(«-»w+3).
&c. &e.
and therefore when A is put =1 after differentiation, we have
Jim
_^^ {(A"+'»-A"+'"+>)(l+A)-'"+'} = -2-<""+''.2>«. («+»«) (w+»«-l)...(w-M + 2)x
, , 2w+2 2OT-1 1 (2w + 2)(2w + 3) (2m - 1) (2?»-2) . .
* ^* i ■«-»^^-2"^2^■ 1.2 ■(w-w + 2)(»-»» + 3) *^"»
which series consists of only Im terms, and is equal to the infinite
series obtained by differentiating the other side of the equation, viz.
w(« — !)....(« -OT+l)x (w + l)(« + 2)....(w + »«)
- (2«+3. («+!)». ...(«-»» + 2) X (w + 2)(» + 3)....(» + »« + l)
+ (^^+^)(^" + 5) (n + 2)(w + l)...■(w-»^+3)x(w+3)(w + 4)....(w + »? + 2)
— &c. oe? infinitum.
354 Mk MURPHY'S THIRD MEMOIR ON THE
Now it is obvious by putting m = 1, 2, &c. successively, that the
finite series is always =0, and therefore the infinite series [which is
( — t'Y
the same as the coefficient of ^ ^ — ^ in the expression for ( — 1)°^„]
vanishes also, so that if V„ be arranged according to the powers of t',
it is + 0. ^' +0^'^ + &c,, nevertheless its value is in one instance infinite,
namely, when t = 0, for then P„ = P„+i = &c. = 1, and therefore
F„ = l + (2. + 8) + ("^ + f^f-^^^^^" + ^)^fV")(^^ + ^) + &c.
= (l+A)(]-A)-«''+^ when h is put =1.
= X .
And if V„ did not possess this infinite element ft Vj", from i> =
to t = 1 would vanish, whereas its actual value is the same as
26. To express the transient Junction Vn in a finite form .
Since by Art. (24.) K = P, + (2w + 3)P„+,
, (2w + 2)(2w + 5) (2« + 2)(2w + 3)(2w+7) « „
"• j~^2 ■ '^^ 2 2 3 • "..+3, «c.
therefore
1.2.3...(2w+l) r'„ = l .2. 3.. .2?? X (2« + l)P„
+ 2.3...(2w + l) X (2w + 3)P„+,./f + 3.4...(2w + 2) x (2« + 5)P,+s^\ &c.
when A is put equal to unity.
But in general,
{1 - 2A (1 - 20 + K"] -^ = P„ + P,h + P^A^ + ...P„A" + P„+, A"*' + &c. ;
c?^''^"{l-2^(l-2j?)+^'|-i^
= 1.2.3...2?«.P„ + 2.3,.,(2w4-l).P„+, A + 3.4...(2« + 2).P„+2AH&c.
INVERSE METHOD OF DEFINITE INTEGRALS. 355
Multiply by 2A"+i, and diiFerentiating once more, we get
^Thr djf^ 1
= 1.2.3...2wx(2« + l)P„A'-i + 2.3...(2w + l)x(2» + 3)P„+,A"+J+&c.
Hence, F„ = ^h-^^ . j. [jf^l ^^ j^ I "^ ' " "^> -^,f ^ .
c?A\ 1.2.3...(2« + 1).</A^° j'
when A is put = 1.
Put for abridgment the radical {1 — 2A (1 - 2#) -f A^}-J = ^, then
rf^ . {Rh") _ 2«.(2w-l)...(w + l) ^^B
rfA^ ~ 1.2...« • • </A''
. 2w(2w- l)...w ^ ^ ,c?"+'J? .
^ ' ' ' \c?A" W + 1 1 C?>&»+1
w(w-l) _A^ c?»+^B 1
(« + l)(« + 2)'1.2'rfA»*'" *''^7
Whence 2 ~ {^-i ^^^^| = 2» (2«- !)...(« + 1) {(2« + 1) A«-4 ^
2w + 3 n ,^^ d'^'R 2n + 5 h"*^ .n .{n-l) d'^^R
■^ 1 •« + ! • dh"^' "^ 1 . 2 * (» + l) (w + 2) •• d¥^ ■•■ *'*'•
c?A"+' w + 1 c?A"+- J
u 100 Tr d'R . 2w + 3 « .d^^'R
Hence 1 .2.3.. .wF^, = -jT- + r — -^. .^ „ ,,
2w + 5 w (w-1) ^^ rf»^^jB
"'' 2» + 1 ■ (» + 1) (ra + 2) ■ 1 . 2 • rfF^ "*■ *'''•
2 ^^ c?"^^Jg 2 n d'^'R
2« + 1 * t/A"+' "^ 2w + l» + l ■ </A"+^
2 n(n-l) h^ d'^^R
2w + l*(w + l)(« + 2)*1.2' <^A»^' ■•" *'^-
Vol. V. Part HI. 3 A
356 Mr MURPHY'S THIRD MEMOIR ON THE
h being put = 1, after the differentiations; this value of 1.2,..?iF^, is
expressed in two finite series, each containing only w + 1 terms.
If we actually add the terms in this formula, which contain the
same powers of A, we get
V - 1 K:? w + 2 h d"^^R (n + 3)
"~ 1 . 2...W \dh" "^ w + 1 ■ 1 ' dh"^' "^ (w + 1) («
, n
h' d'^'R
+
n + 2)' 1.2" c?A"+'
(« + 4).ra(w-l) k" d'+^E
^ SL 1
{n + l){n + 2){n + 3)'l.2.3' dh
when h is put equal to unity.
27. -Discussion of the transient function N ^.
Put « = in the general expression for ^„ in the preceding article,
dR
hence V. = R -^ 2h -y^-
dh
= U-2A(l-20+A^}-^ + 2A(l-2jf-A) {l-2A(l-2^) + A^}-i
(1-A)(1+A) , , .
This function, as has been observed in Section vii. (22), is in general
zero, except in the particular case when ^ = 0, when its value is infinite.
If we imagine a curve of which the equation is
y {l-2A(l-2a;) + ^^}4'
where h is less than unity but nearly equal to it, the limiting values of
y as A approaches unity, will give the geometrical interpretation of
the transient function V^.
INVERSE METHOD OF DEFINITE INTEGRALS.
357
Take (Fig. 1.) AB = 1, AH = 1 - h, or
BH = h both along the axis of x, and make
A the origin, then putting x = 0, we have
1 -i- h
y — jz — T7^, which is very great, and tends to be
infinite as h approaches unity, and is represented
by AC\ next putting x = l — h = AH, we get the
corresponding ordinate HE = (jZThf) ' (T+W'
which also tends to infinity ; lastly, putting x = l
we have y =
l-h
= BD, which tends to vanish
(1 + h)i
in the ultimate case representing V^.
Now varying the parameter h so as to make
it approach unity, the points C and £1 recede
indefinitely from the axis of x, and the point
7> approaches it indefinitely.
Yet the area DBACE remains constant (for
the integral between x = 0, and x = \ of
{l-2^(l-2x) + A''}J
relative to x is evidently unity).
And the altitude GN of the centre of gravity of this area is also
constant, for
ANH H
h!^f
and therefore is the same as that of the parallelogram HF, when
AF=^AB, for the distance Gg from the axis of y
x{\-h){\^h) \~h AH
{l-^h(X-2x) + ¥\l 2 2'
Hence G tends ultimately to the point g in the axis of y, which
shews that the area DBH'E' tends absolutely to vanish, HE' being
3 A 2
358
Mr MURPHY'S THIRD MEMOIR ON THE
an ordinate drawn near the origin at any small distance not varying
with the parameter Ji, and since -r- has the same sign in the interval
from B to H', H or A, it is evident that the portion of the curve
BE' tends to coincide with the axis BH', the curve therefore which
represents V„ coincides with AB, except infinitely near the origin
A, when it suddenly mounts to an infinite height.
Since the general function V„ is reciprocal to t", it follows that
fi Vaf = 0, except when w = 0, and then the definite integral is unity ;
hence if f{t) be any function containing only the positive and integer
powers of t, the transient function Vo possesses the remarkable pro-
perty expressed by the equation [tV'o ■/{f)=J^{Q).
Fig. 2. Let 2a = AB, equal the
length of the axis in a solid of revo-
lution, the surface of which is covered
with an indefinitely thin stratum of
fluid, let any abscissa ON measured
from the centre O be put equal to
a (1 - 2#), the limits of t will evidently
be O and 1 .
Let the law of density or accumulation at any point P of a section
perpendicular to the axis be expressed by the transient function \V^,
X being constant, and let the total action of the fluid on any point Q
in the axis be required, the law of force being capable of expansion
according to the positive and integer powers of t.
Put PA'' = y, then the whole quantity E of fluid is manifestly
ds
equal to ^Xtt ft T^^y -r, , s representing the arc AP.
ds
Now it is easily seen that the value of y-yr at the point A where
y vanishes is iaR, R being the radius of curvature at that point, and
by the nature of V^ this quantity is the value of the above integral,
or E = SXttuR.
INVERSE METHOD OF DEFINITE INTEGRALS. 359
Again, if we represent the distance PQ by r, and the law of force
by y(/-) and put AQ = k the initial value of r, the total action is
r rr ds „, ^ ON
2ATJ;r,y^./(r).^,
which by the property of F"o is equal to S\traRf{k), or to E .f{k).
Let us now suppose an equal quantity of fluid, but of a contrary
nature in its action, and therefore represented by — E to be collected
in a single point C in the axis produced to a small distance AC- a.
The total action of the compound system on Q will then be
E{f(,k)-f(k + a)},
which tends to vanish as C approaches A.
Lastly, suppose a unit of fluid when distributed over the surface
according to a law expressed by {t), which depends on the figure of
the solid, will exert no action on any point Q in the axis; then if
the law of distribution of the fluid be expressed by X V^ + c (p {t), the
total action on Q including that of C, will be still E {f{k) - f{k + a) ^ .
From which it follows that when an electrical spark -Eh in-
finitely near to the vertex of a conducting solid of revolution charged
with a quantity of electricity E', the distribution of the latter under
the influence of the former is expressed by the law
pi
\Va + c<i>{t) where \ — ^ ^,
otraH
and where c is determined by the equation*
Having thus given the geometrical and physical interpretations of
Vo, it will not be necessary to discuss the transient functions V^, V^,
&c., of which the properties are very analogous.
* Vide First Memoir, Art. 35, the expression there obtained for a sphere being in-
cluded in that obtained above, when the influencing point is infinitely near the sphere.
S60 Mb MURPHY'S THIRD MEMOIR ON THE
28. To find the quantity to which V„ is the generating function.
By Art. 26.
_ d ^'iRh") ^hd''*^ (Rh")
I .2.S.,.2ndh;"' 1 .2.3...(2» + l)rfA"^"
where R= {l~2h(l — 2t) +h^}~K and h is ultimately equal to 1.
Forming the equation u==h + &uK we have
A^ d^f{h)h\ k^ d^f{h).m}
hence {l-2«(l-2^) + «1-^ ■^^ = ^iZ + ^' ^^ ^ 1T2 <//^' +&c.;
dh'
du
dh
^l^^'-^^^^^' rS^F^ = '^' "°"^'^""' °^ ^"" ^" {l-2t.(l-20 + >/'M
In like manner,
«-i{l-2«*(l-2/) + «n-^.^ = 2?A-^+^'|f^
/fe^ d\{RM) ¥ d'jRh)
therefore,
_^M!!!1(^) = the coefficient of >P«- in ., , ^ f^^ .,, ;
1 .2...(2« + l)rf^''+* {1-2m(1-20 + «'H
(* + 2Am-^)^
consequently, V„ = the coefficient of ¥'^*' in v^ _ ^^(i _ 20 + «'}^ '
INVERSE METHOD OF DEFINITE INTEGRALS. 361
Now, by the assumed equation we have
ui — hu~i = k,
du
and by differentiation («J + Am"*) -jr = 2m ;
but also (m^ + hu~^) = k + 2hu~i;
hence, {k + 2hu~i)'Tr=2u;
and therefore, F„ = 2 the coefficient of ^"+' in u {l-2u(l - 2t) + u"} -^
from which it follows that if we form the two equations,
u'=h + (ku')i] . \U' =u' {l-2u' (l-2t) + u''}->^
} putting i
u" = h + (ku")i] [t7" = «"{l-2«"a-20 + «"'i^^;
then ^^j— = ^0 + J^^k + V^¥ +r,k'kc. ad inf.
supposing that in the left-hand member h is finally put equal to unity.
It may be observed that the quantities u', u" are the two roots of the
equation u'^ — {2h + k)u' + h!' = 0.
29. To expand a given ^function (pit), in terms of the transient
function \ ^ .
Let the general term of the expansion be A„V„, then by the nature
of reciprocal functions we have
= AJtPj", (Art. 24.)
M.(W-l) 1
= (-l)".^„
(w + !)(« + 2) (2« + l)'
lience,(p(t)=Kft<l>(t)-^^rj,(p(t)t-\-^^.FJt<}>(t).f-&ic.
362 Mr MURPHY'S THIRD MEMOIR ON THE
Examples :
/» - -J_ V _ ^-3 V J. g-4-5 ^ .
p_j^ 2«+3 „ (2w+4)(2«+5) (2m+5)(2« + 6)(2w+7) .^ ,
-rn—r„ J . ;'„+,+ I ^ . f'n+a J . f'n+g&C.
the latter series would also result by reverting the series for V„, in
Art. 24.
30. To find a function U„ which shall he reciprocal to (h.l.ty.
Following the steps indicated in Art. 23, we must first form a self-
reciprocal function of which the general term is a constant multiplied
by (h. 1. ty ; this has been already effected in Sect, v, namely,
and then the form of the required function will be
[/„ = t; + a 2;+ , + 6 1;^^ + c 7;+3 + &c.
Multiply by (h. 1. ty, supposing m>n, and observing that
j;r„(h.l.0" = 1.2.3...>^.(-ir. "-^'^-;)^:-f-("-'^-^^) bySect.v,
and ir7„(h. 1. /)•" =
by the nature of reciprocal functions, we get the general identity
m{m — \){m — Q)...{m — n + l) , m — n , {m-n){m -n—\)
® r.2.3...« •^^""•^TT+*- (« + i)(« + 2) — *'*'-^'
but on the same supposition that m is greater than w, we also have
= (l-l)'"-" = l-(m-w)+^^ 4p— ^-&c.;
INVERSE METHOD OF DEFINITE INTEGRALS. 363
and by comparing the corresponding terms
- ^ + 1 ^_ (« + l)(w + 2) _ {n + l)(w + 2)(w + 3) .
therefore,
rr-7--L'' + ^ T , (w +!)(« + 2) . (w + 1)(w + 2)(m + 3)
Ly„ — J „ -I J . ^ „+i + r — . -I „+2 i r — - — . X „+3, Cue.
31. To express the function Un which is reciprocal to (h. I. t)" in
a finite form, and also the function which Un generates.
l.2.3...nU„ = 1.2.3...nT„+2.3. 4...(w + l) T„+,+3 . 4 . 5...(w + 2) . T,.+,+&cc.
=^ { r„ + r,A + 7;a^+ ... T^h'+T^^.h"^^ + &,c.],
h being put equal to unity after the differentiation.
But by Section v, we have
h
J— ^ = T,+ T,h + T^h' + &c. ad inf. ;
"■■(A)
therefore, U„ = - — — — -jt when A = 1.
1.2. 3...ndh"
Now by Taylor's Theorem, this quantity is the coefficient of k" in
the expansion of - — j~r^ the latter is therefore the function which
U„ generates.
32. Properties of Un-
.1. jiUn (h. 1. ty = f,T„ (h. 1. ^)" = 1 . 2 . 3...W, by Sect. v.
II. Changing the sign of k in the quantity which U„ generates,
we get
Vol. V. Paet III. sB
364 Mr MURPHY'8 THIRD MEMOIR ON THE
pT'' = Uo- U,k + U,¥ - U^¥ + he.
— f-f^§--}
III. Since f =\ +x\i.\.t+ -^ . (h. 1. tf + j-^-j ■ (h. 1. <)' + ««:■
by means therefore of a single integral, x" may be adapted to general
differentiation.
As this result is remarkable, we may confirm it by the general
rule in the First Memoir. (Vide Sect, i.) Thus,
1
put (x) = :j T— =l-\-xk + a^J^ + &c.
*±i
/-*
then fit) = — -T =U,+ U,k+ U^k" &c. ;
for all values of k, whence JiUnf — xf as before.
33. Discussion of the Junction U^.
h
fl-h
By Art. 31. Uo = - — r when h is put eqifal to unity. Like the
transient function Vo, discussed in Art. 27-, the quantity Uo is always
zero for values of t between and 1 ; but when t = 1 its value is
infinite, and thence its integral between the limits and 1 of / is finite,
viz. unity.
INVERSE METHOD OF DEFINITE INTEGRALS.
365
To prove this property, conceive a curve Y
APC, of which the abscissa measured from
A along AB is taken equal to /, and the
corresponding ordinate y is equal to f''',
and let us suppose h very nearly equal to
unity, and at any point P draw a tangent
PT; then since
dt
h
y = t'-\
therefore, 11^ is the limiting value of the tangent of the angle PTB.
Take AB = 1 and the ordinate BC = 1, then it is evident that
A and C are constantly points of the curve when the parameter h
varies so as to approach unity.
Again, for the entire area APCB the expression is Ji^'~*, from t=0
1 — A
to t=l, that is, - — Y, which evidently tends to vanish as the para-
.« — fl
meter k approaches unity ; and as no part of the area is negative, it
follows that the curve APC tends ultimately to coincide with the two
right lines AB, BC, and therefore when T is sensibly distant from
B the tangent of the angle PTB tends to vanish, but when indefinitely
near to B it tends to infinity ; and therefore Ug, which ultimately re-
presents these tangents, is zero from A to indefinitely near to B where
t is unity, when its value becomes infinite.
In like manner the remaining functions C/i, U^, &c. may be dis-
cussed with similar results.
It may be observed that for values of t>l (which however do not
enter the definite integral), the values of t/'^ are infinite.
34. Expansion of given Junctions in terms of the functions Un .
The general formula for this purpose is
0(0 = V^kW) + UJt<p{t). hA.it)
1.2
^,<t>{f).ih.\.tf +
u.
1.2.3
.j;<^(0.(h.i. /)' + &c.
3 b2
366 Mr MURPHY'S THIRD MEMOIR ON THE
T -u - ^±1 u + (^ + i)(^+^) r/ _ &c
1 1.2
which latter series is also produced by reverting to that which expresses
C7„ in terms of 71 in Art. 30.
35. To find a function reciprocal to t" when the limits of t are 0,
and GO .
Let M„ be the required function, and put t = e"',
then ^lUj'^ = 0, from # = to ^ = x ;
«» /V- 1 N™ ^ i>_ = to T = 1,
therefore /"- (h. 1. t)-" = 0, from t =
^» "
h_
1-*
hence m„ = t C/„ = t ; — ^ttt when h = \
1 . ^...ndh''
36. Tb ^«c? « function F„ ^t>A^cA *Aa?/ ie reciprocal to cos" ^, ^A^
- awa -.
2 2
ZmeV* o/*^ Je^?^ — - and -
Following similar steps to those adopted in the preceding Articles
we shall obtain,
w + 2
in cosines F„ = cos n<p — . cos {n + 2)(f>
(« + l)(w + 4) , ,,,^ (w+l)(w + 2)(« + 6) , ,-,-, .
+ —^-12 ^•cos(« + 4)0 -^ ^ g g ^cos(w + 6)<^, &c.
INVERSE METHOD OF DEFINITE INTEGRALS. 367
in sines Fn = 2 sin (p {sin {n + l)(p — . sin {n + 3)(p
+ ^^ -^ . sm (« + 5) <^ - &c. ]
37. The Junction F„ is transient.
Either of the preceding values of F„ give F„ = Fn—F", where
F: = cos {n(t>) - ^±i . cos {n + 2).(p + ^^"^^^^^^^ • ^^^ (« + 4) <^ + &c.
F„"= cos (w + 2) <^ - ^^ . cos (« + 4) . <^ + {n + l){n + 2) ^^^ („ ^ g^ ^ ^ ^^.
passing from trigonometrical to exponential values,
1 1.2'
1 1.2"
_ ("g.^vrr ^ g-</>vrT\-"
= 2cos«0,
2F„" = £("+2)*^^ - ""'"^ . e("+4)*v^ + (" + l)(^ + 2) _ ^(„^g)^^— _ ^^
1 1.2
+ g-(n + 2),^V:rT _ ^jt2,e-(n + 4)0V^ ^ (w + 1) (W + 2) ^-(„ + 4)^vri _ j^^.^
= 2 COS «^,
hence F„=-F';-F„"=0.
368 Mr MURPHY'S THIRD MEMOIR ON THE
However, if n be even, and our limits be — ^ and ^, the function
becomes suddenly infinite at the limits, for the expansion of F„ is
then identical with that of (1 -!)-'"+'•.
38. To express infinite terms the transient function Fn.
Put
i?'„' = cos ra(^ - ^ . ^ cos (« + 2) + ^^i|^-^^ . A'' cos (« + 4) «/) - &c.
F:'= hcos (n+2)(f> - ^ . A^ cos {n+4>).<l> + <"+!) (^ + ^) ^^ cos (w + 6)0-&c.
1 1 • ^1
Then F„ is the limit of F^—F" when h approaches unity.
Put also 2 cos = a; + - ,
^ X
hence 2-F„'
x~^ X
~ {l+A(a:^+a;-^) + *'}"+'
W + 1 (W + X^ Tt W + 1
COSW0+— j— .Acos(w-2)0+^— — ^^.A^cos(«-4)0...— — -.A"cosw0+A"+^cos(«+2)0
~ fiTaFcosa^TFp^^ *
the number of terms in the numerator being w + 2.
In like manner,
2F„" X
+
h (aT-' + Aa;)"+i (x + Aa;-')"*'
_ X (a; + Aa;-')"+' + a;-' (a;"' + Aa;)"+'
INVERSE METHOD OF DEFINITE INTEGRALS. 369
eos{n+2)(l)+'^r— .h cosn(f)+- -^ .k^cosin-2)<p+ . . . —— .h" cos{n—2)(p+ h"*' cosn<p
^ fTT2rcos2^1~Fp^' '
Hence,
costKj) .{I -h"*^)+h\——cos{n-2)(p-cos{n+2)<p\+hH— — ~-cos(w-4)0 — — cosw0>
^" {l + 2Acos20 + ^^}»+' ' ^'
when h is put = 1.
Thus F = (^-^)(^ + ^^
which is evidently a transient function, as its general value for A = 1
7r
2
is zero, except ^ is an odd multiple of -, when its value becomes
infinite.
And in general F„' and F„" are equal, when h is put equal to
unity, and therefore F„ has a factor 1 — A in its numerator, which causes
TT
its general vanishing state, except when ^ = „, or an odd multiple of
^, when the denominator becomes (1-A)^"'*"^ and as the numerator is
of only n + 2 dimensions, it is evident F„ in this case is infinite, when
k= I.
In general f , ~i . r.^ 1: 2 = 2 tan"' . I^^^ . tan 0> + const.,
which taken from = to ^ = - is equal to tt, a quantity independent
of h, a result similar to those already obtained from other transient
functions.
39. When the sum of a series containing transient functions is
required, the following process, with only such modifications as may
simplify particular cases, will apply.
370 Mk MURPHY'S THIRD MEMOIR ON THE
Let S = UaVo + fli Fi . ss + (hV.i.%^ + ... + «x ^^K' + &c.
be the series proposed.
By the inverse method, put a^ = U/{t) . t' from t = to t = 1 .
Then S = /^/(r) \V, + F.tz + V.t'z' + &c.|
But V^+ V^k + VJe^, &;c. is the function which V„ generates, and may
be represented by ^ {t, k), we have then
S = X-/(t) . <p{t, T%), from T = to T = 1.
INVERSE METHOD OF DEFINITE INTEGRALS.
S7l
SECTION VIII.
On ike Resolution of Equations involvings Definite Integrals.
(l) By the Decomposition of the Integrals into Elements.
40. The utility of the method of decomposition consists principally
in the verifications it offers to results obtained by other analytical pro-
cesses, the difficulty in the eliminations which it requires.
Pm-i-^ Pm+if Pa Put/
Suppose a cylindrical shell exerts no force on any point in its axis
AB, the law of force tending to each particle of the shell being given,
but the law of density of the shell unknown, then the application of
the method of decomposition is this :
Divide the shell into « + 1 equal portions by planes perpendicular
to the axis PiQi, P2Q2, &c.
Let the density throughout each portion be supposed uniform, and
let the successive densities be pa, pi, p->....pn-
Let the total actions on the points of division Qj, Q2...Q„ be equated
to zero, which will give n equations, and another will be obtained by
considering the mass of the shell.
From these n + 1 equations, let po, p^, p^, &cc. be determined in terms
of ». ..
Finally, make n infinite.
Vol. V. Part III.
3C
S72 Mr MURPHY'8 THIRD MEMOIR ON THE
41. General Calculus for the Cylinder with any law of force.
Let «o, «i. a^.-Mn represent the total actions on the point A which
would be exerted by the successive portions P^Pi, PiPz of the shell,
if the density of each were unity; these quantities are given, since the
law of force is supposed known.
Then aopo, a^p^, a^p^, &c. represent the actual forces on A.
Again, the action of any portion as P„+4P„+5 on any point Q„ of
division in the axis, will be to the action of the similarly situated
portion P^P^ on the point A in the ratio of the corresponding densi-
ties, and in this case would be atpm+i.
By this consideration the total actions on the points Qi, Q2...Q„
are easily estimated, and equating each to zero, we get the following
system of n equations, which serve to determine the ratios — , — , &c. viz.
pa Pa
aopo- Ctopi — aipz — Chps — asPi — dn-lpn-l — (tn-iPn = 0,
aipo + aopi — aop-i — aipi — aipi —ctn-spn-i — an-ip„ = 0,
Oipo + aipi + ttopi- ttops—aipi — «n-4/'n-l — «n-3/'« = 0,
a^po + a^pi + aip2 + aopa— Oopi —ctt,-ipn-i — (i„^ip„ = 0.
a„-2po + an-3pl +«„_4p2 + «n-5/03 + «»-6P4 — «2/0«_l — fl!l/0» = 0,
ffn-lpO + fin-2Pl + «»-3p2 + «»-4j03 + «n-5/04 +«l/On-l " aoPn = 0.
Comparing the first equation with the »*'', the second with the (w-l)"",
&c. it is obvious that po is involved in the same manner as /o„, p^ as
Pn-l, &c.
Hence, p„ = po, p„_i = pi, p„.i = p^, &c.
Form now two functions in the following manner:
a known function, M = «osin0 +«isin30 + a8sin50+ .,.«„_isin(2w-l)0,
an unknown, aS'„ = jOoCOSw0+|OiCOs(w-2)0 + ^sCOs(«-4)0+
INVERSE METHOD OF DEFINITE INTEGRALS. 373
the first series may be continued to n terms or infinity indifferently,
and the last term in the second series will be ^p^ when n is even,
2
and p„_x . cos 9 when n is odd.
Suppose now that the product 9,u.Sn is decomposed into the sines of
the multiples of 9, and that all the multiples higher than the «'" are
rejected from this product, the remaining part will evidently be,
— {aopo—aopi — aip2 — a»-i/Oo} .sin(n — l)0,
— {aipo + Uopi — aopi —a„.2po}.sm{n — 3)0, •
— {(hpo + ctipi + aop2 —a„-3po}.sm{n — 5)$, &c.
the whole of which by the given equations is equal to zero.
Hence,
2S.u = A„sm{n + l)9 + B„sin(n + 3).9 + C„sm{n + 5).B, &c. ;
.-. 4 cos . S„u = A„ sin (nB) + {A„ + B„) sin in + 2)9 + {B„ + C) sin (« + 4)0, &c.
and 2Sn.iU = A„^i sin {n9)+B„.i sin {n + 2).9 + C„_i sin (« + 4) . ;
.-. 2{2cos9.Sn-j^S,.^} .u = i^A„ + B„-A„.^\.sm{n + 2) . 9, &c.
Hence it follows that if we put So=po, S^ = po cos 9,
and u = aoSin9 + a^ sin 3 + a^ sin 5 9 &c. ad inf., then.
First, Supposing S^.^ and S,n known, form a quantity \„ by dividing
the coefficient of sin(/» + l)0 in 2S,„u, by the coefficient of sin(/»0)
in S/S'm.i .u. •
Secondly, Form a quantity S^^^, by the equation
-S'„+i = 2 COS0 . /y^ - X^iS*™.! ,
by which S^, S3 a^^ may be successively formed.
Then it is obvious that the product 2S„u contains no multiple of 9
below the «'\ and therefore the coefficients in S„ must be the required
quantities po, p^, pi pn-j^ when n is odd, or p^, pi, pt ^p^ when
2 3
n is even. _
Sc2
374 M^a MURPHY'S THIRD MEMOIR ON THE
42. Applications, when the law of force is the inverse square of the
distance.
(1) Let AB be the axis of a very broad cylindrical plate, the
round side of which is covered with a fluid, attractive or repulsive,
and so distributed as to exert no action on any point in the axis.
Put AB = 1, APo = a the radius of the base.
Let ab be one of the very small annuli into which the edge is
divided, and put aPo = x.
Then it is easy to prove that the action of the annulus a 5 on the
11
point A is proportional to -^ -jr, or ultimately to the differential
1 ■ X .
coefficient of -7— with respect to x, that is, to -t-t? ryj, which quan-
Aa {o'' + arj»
tity expanded is proportional to a; — f rj + &c. ; and as b is very great
compared with x, we need only take the first term of this expansion.
In this case we may therefore put ao = l, a, = 2, ai = 3, &c.,
and therefore, M = sin0 + 2sin30 + 3sin50 + 4sin70 + &c.
The calculus of S„ as indicated in the preceding article will be as
follows :
So = pay Si = |OoCOS&,
f ^ coefficient of sin2g in ZS^u ^31
\ ' "~ coefficient of sin0 in 2SoU ~ 2)'
■ • Si = 2coseSi - x^So
= Po Jcos20-^}; ■ •
INVERSE METHOD OF DEFINITE INTEGRALS. 375
i _ coeffi cient of sin30 in 2SiU _2\
\ * ~ coefficient of sin29 in 2«S^m ~ 3 j
S:i=2COS0S,-\Si
= Po{cos39 — -cosO], ....
J _ coefficient of sin 40 in 2S3U _ 5\
\ ' ~ coefficient of sin30 in ZSaU ""6/
*S. = 2cos9S3 - Xs.S^ .
= po|cos40 — -cos 20 — ->,
f coefficient of sin 50 in 2SiU _ 91
\ '^ coefficient of sin4!0 in gAysM ~ lOj
Si = 2cos9S^ - XiSi -
= po |cos 50 - - cos 30 — -> .
{2 2 11
cos60 — ^cos40 — ^cos20 — ^>
{2 2 2 1
cos 70 — = COS 50 — - COS 30 — - cos 0> .
Generally when n is an odd integer, suppose
-^^ = cos(w-l)0-^— {cos(m-3)0 + cos(« - 5)0 + ... + cos20 + i},
and — = cos »0 {cos (w — 2) + cos (« — 4) + ... + cos 30 + cos 0}.
po n - -
The coefficient of sin w0 in 2»S'„_,« = -. p„,
n — 1 ^
of sin(M+l)0 in 2S„u = .p^;
n
therefore, x„ = <^±fi^ = 1 - ^ + ^ .
n(n + l) n n + 1
Stfi Mw MURPHY'S THIRD MEMOIR ON THE
Hence, ^=2cos0.— -X^*^
pa pa po
= cosln+l)9 -{cosin-l^e + cos(n-S)e + ...+cos29+i},
and by a repetition of the same process,
O Q . ,■> .*,{ ,-; . •-:..
— ^ = cos(»+2)0 -{cbsw0 + cbs(«- 2) . + ... + cos30 + cos^}.
pa n-TXt
Hence the laws by which S^-x and S^ are expressed are uniform,
and therefore we get for the required unknown quantities,
2 2 2 _
po = Pa, pi=--pa> P8=~^Po P—i -P<>> Pn-Po-
The positive values may be taken for the repulsive and the negative
for the attractive parts of the fluid, and if E denote the excess of the
former, we have
[n n n n n] n [ n } n
.-. po=»^-T — > which gives the complete solution of the problem.
Thus the application of a process purely algebraical, conducts in this
instance to a transient function, for if we suppose the final and equal
densities po, p^ to be finite, all the intermediate values of the densities
p^, pa pn-x become indefinitely small when n is made infinite; yet
they are not to be rejected, for if so, the total charge would be 4nra^,
it
whereas its actual value is only inra ~, an infinitesimal of the second
pa
as Its itutuiu vaiuc is fJiii-y •±'iru, —
order.
.Vo
Pm/
INVERSE METHOD OF DEFINITE INTEGRALS 377
(2) Let AB be a right line perpendicular to the bounding planes,
which terminate a solid composed of parallel strata of indefinite extent,
but uniformly dense throughout that extent ; and let the law of den-
sity of the different strata be such that there is no action on any
point Q„ within.
Let the solid be decomposed into n + 1 equal portions in which the
densities are as before represented by po, pi, p% /o„.
In this case the quantities ao, flj, 02 ci„ are all equal, and putting
them equal to unity, we have
u = sm9 + sin39 + sm56 + SiC.
So = po, Si=poCOS9, \ = 1»
S2 = 2cos9 . S^ — XiSo=poCos29, X8 = l,
Ss=-2cos9.Sz — \2Si = poCOs39, X3=l,
and generally, S„=pocosn9, and\„ = l.
Hence the solution is pi = 0, p2 = p»_i = 0, pn = po'
And if E be the whole mass and A the area of the bounding planes,
which is supposed very great, we have
E = 2iA.po.
This result is analogous to the well-known fact, that electricity can
reside only on the surfaces of bodies, and affords another instance of
a transient function.
The method of decomposition may always be applied to obtain
numerical approximations in cases which involve Definite Integrals;
for instance, in the distribution of electricity on bodies, and in esti-
mating the forces between bodies which are electrised.
(2) By means of Reciprocal Functions.
43. Equations which contain only one definite integral.
Let f(f, a) be a function involving a variable f, and an arbitrary
parameter a; F{a) a function containing a only, and (p (t) a function
378 Mr MURPHY'S THIRD MEMOIR ON THE
containing t only, the first and second of these functions being given,
it is required to find the third so as to satisfy the definite integral
equation
!,<l>{t).f{t,a) = F{a),
the limits of t being given.
Suppose (p {t) expanded according to any given class of self-reciprocal
functions as P„, that is,
^(^) = CoPo + CiPi + C2P2 + C3P3, &c. ad infinitum,
where the coefficients Co, c,, Ca, &c. are unknown.
Let J^{t, a) be expanded according to the same reciprocal functions,
f{t, a) = AoPt, + A^P^ + A2P2 + A3P3, &c. ad infinitum.
Then j?P„P„ = 0, and fiPnPn = a„ a known numerical quantity depend-
ant on n, and on the particular species of reciprocal functions which
are employed.
Multiply both series and integrate between the given limits of /,
and the proposed equation gives us
F (a) = Aoao.Co + Ai ai.c^+ A2 as . C2 + ^303 . C3, &c. od infinitum.
Now An being a known function of a and n, we can by Art. 23.
Sect. VII., find another function of a and n, as An such that fiA„A„' = 0,
when m and n are unequal integers.
Multiply the equation successively by Ao, A^', Ai, &c. and take the
definite integrals relative to a, hence
jaA(s-P\a) = CoOojaAaAt, ', .'. Co ^ C A ' A '
f.A,'F{a) = e,aJ.A,A,'; •.: c, = ^^4^,
and generally c„ = r'^'j •
Hence <b(f) = ^ /^^°'--^(«) + ^ fgA^Fja) ^ P. fa-A^'Fja) ^ ^^
^ ' ao ' faAo'Ao a,' faAi'Ai aj ■ faAa'A^
INVERSE METHOD OF DEFINITE INTEGRALS. 379
44. Examples.
In the following examples two things are to be observed. First,
that the given functions are supposed to be continuous, and therefore
the equation proposed must hold true for all values of the parameter a.
Secondly, In the final equation for determining the unknown coeffi-
cients, instead of using a reciprocal multiplier any means more simple
may be occasionally employed.
Ex. 1. Given ^^(/), cos («^) = 1 to determine <^{t\ ^he limits of t
being and tt.
Put (^{t) = Co + Ci cos/ + d cos (2/) + d cos (3/), &c. ml infinitum,
and cos {at) = Ao + A^ cos t + A2 cos (2/) + A^ cos (3/), &c.,
where to determine Ao, A^, A,., &c. we multiply successively by 1,
cos t, cos 2 A &c., and integrate from t = to t = ir, whence
, _ sin {a-n) J _ 2asin«-7r , _ 2«sin«7r
J n ^ / ,x„ 2« sin air ,
and generally A^ = ( — 1) . — ri ^ when n> 0.
7r {a — n J
Multiply both series and integrate, and we get by the proposed
equation,
[Co a.€i , «C2 aCi . \
1 = sm a-K { — =- + ~„ — -„ z — -5 + &c.>
{a a^—1 «^ — 2^ «^ — 3^ J
Put a = 0, 1, 2, 3, &c. successively, and we get
_ 1 2 3 .
C(j — , Cj — , C2 = — , oZC.
•TT TT TT
Hence ■tr(p{t) = 1 + 2cos/ + 2cos2# + 2cos3#, &c.
The value of <t)(t) is therefore the transient function - . =^^ — ^-.'^ ^ — i-^ .
^^ ^ TT I — Hh cos t + h^
{Vide Art. 38. Function Fo), when h is put equal to unity.
Vol. V. Part III. 3D
330 Mr MURPHY'S THIRD MEMOIR ON THE
Ex. 2. Given fi(p(t) .cos {at) = cos (a 0).
As before (p{t) = c„ + Cj cos t + c^ cos 2t + CaCosSt + &c.
sin flTT fl 2«cos^ 2« cos 2^ 2a cos 3^ „ ]
cos at = < J \ h &c.>
therefore cos a0 = sin(a7r)l- - -~- + f^' „ ^^j + &c.l
' [a a^ — 1 a^ — 2' a- — 3^ J
But also by reciprocal functions we get
sinaTrQ 2acos0 2acos20 2acos30 „ 1
cosae = __ |- _ _,__ + --^-^^ -,__ + &c.}
TT 1 2COS0 2cos20 2cos30 „
Hence Co = - , c, = , c, = , Cs — , &c.
TT TT TT TT
therefore 7r^(#) = 1 +2cos0 cos^ + 2 cos20 cos 2^ + 2 cos36 cos 3# + &c.
or 27r(pt= 1 +2COS {9 + t) + 2 cos 2{e + 1) + 2COS 3{9 + 1) + Sic.
+ l + 2cos{9-t) + 2cos2{9-t) + 2 cos 3{9-t) + &c.
^ (1-A)(1+^) (1-A)(1+^)
l-2h cos{9 + + A' 1 - 2A cos (0 - ^ + *'
when A is put equal to unity.
Ex. 3. Given ft <{> (t) : cos {at) = 27'(a).
jP(a) must be such (in continuous functions) as not to change when
— a is put for a, since cos (at) which is under the sign of integration
will not then alter its value.
Proceeding as in the former examples we get
ET/ ^ • \<^o ac, ac, acs „ ]
F{a) = sm«. |- _ -,_^ + -,_^ _ -^-^^ + &c.}
Put successively a = 0, 1, 2, 3, &c. hence
Co = - . 1^(0) , c. = - . F{1), c, = -. F{2), &c.
ir IT TT
hence 7r(p{t) = F(0) +2F(1). cos #+ 2F(2) . cos 2# + 2F(3) cos(30 + &c.
INVERSE METHOD OF DEFINITE INTEGRALS. 381
Ex. 4. Given ft(p{t) . {/{a + t) +f{a-t)\ = F{a),
where the forms of the functions f and F are known, and that of
required.
Put (f){t) = Co -1- Ci cos t + d cos %t + c-i cos St + kc.
f{a) = oo + a, cos« + 02 cos 2«+ "3 cos 3a + &c.
where a,„ a^, a^, &c. are known numerical quantities; hence
J'{a + t)+J^(a-t)=^2ao+2ai cosacos t+2a2 cos 2a cos 2l + 2a3 cos 3« cos 3^-r&c.
and JP(«) = 27raoCo + 7raiCi COS« + wa^d COS2« + TrogCs cos3« + &c. ;
therefore Co = — — - , c, = —^— . fa F{a) . cos « , c^ = -^— L F{a) cos 2«, &c.
J w.. 1 r t:t/ X f 1 2C0S«C0S^ 2cOS2«COS2# „ ]
and 7r(p{t) = — f„F{a) \—- + + + &c.}
Tr [Zao Oj as J
the hmits of all the integrals being and tr.
Ex. 5. /■*%=J-,.
Jta — t a — h
In this case we shall employ the functions V^ reciprocal to t".
Put <^{t) = CflFo + Ci F", At C'^V-i + &c. «c? infinitum,
1 1 !?;<'. , • ^ .
and ;: = — \ — ; ^ — : + &c. «« infinitum;
a-t a a' {^ "^
^, f 1 c„ 1 c, 2.1 6-2 3.2.1 Cs „
therefore r = ;r^ • -5 + ^ . e • ^ — ~. — ^ c <-, • -7 + &c-
a-o a 2.3 e^ 3.4.5 a^ 4.5.6.7 «'
1 * A= 6'
= -+—,+— + —4 , &c.
a a^ ci^ a
TJ 1 2.3 , 3.4.5 ,, 4.5.6.7 ,3 „
Hence Co = 1, c, = ~ . *, c, = ^ ^ . V, c^ = .^ ^ ^ . h\ &c.
and <^{t) = r,-^.br, + ^^.b^F.,-^^^^.b^r.. &c. ■
= r„ - 1. r,.(4i) + — •FAuy - 1^^. r3(4*r + &c.
3d2
382 Mb MURPHY'S THIRD MEMOIR ON THE
Put F{k) = Fo + F,A + V-it + kc. ad infinitum,
as found in Art. 28. Sect. vii.
Hence F{ - kr') = K- V.kr" + V.,kr' - &c.
theretore j^ ^^^-— ^ - 2 ^^» a ^^*+ 2.4" '^'^ 2.4.6-'^'^''^
the limits of t being and 1 ;
TT ^^ v'l-T^ »2' 2.4 2.4.6
Ex. 6. j,<l>{t).f{a-t)=f{a-h).
Denote by Pi,„ the reciprocal function P„ when ^ is the variable,
by Pi_ri when 6 is the variable.
Let/(«-0 = J,P,,, + A,Pt,, + A,P,,, + A^Pt,^ + &c.
and (j>{t) = CoPf.o + c,Pu + CaP,,^ + c^.Pt.z + &c.
.-. f{a-h) = ^oCo + g . AxCi + g . ^sC^ + \ .A^Ci + &c.
but changing t into J in the expansion of J'{a — t) we get
f{a-h) = AoP,,o + ^,Pm + A,P,,, + A,P,, + &c.
which values are identical when Cc = Pi.,o, c, = SPs,,, c^ = SPh.n, &c.
therefore (j){t) = PmP.o + SP^P., + 5P„,P,. + 7 P.^P.s + &c.
45. Ow ^A^ appendage necessary to complete the Solution of' a
Definite-integral Equation.
In the examples in which f{a, t) = cosa^ given in the last article,
the function F{a) is adapted to general differentiation relative to a,
under the definite integral ; but besides the prime value thus obtained,
there must be an appendage to represent the same operation on zero.
INVERSE METHOD OF DEFINITE INTEGRALS. 383
which contains an infinite number of constants multiplied by functions
of a, which may vanish or not, and be connected or unconnected ac-
cording both to the nature of the particular operation and the nature
of the calculus in which it is employed ; this has been already shewn
by Mr Peacock*, and in Art. 20. Sect. vi. of this Memoir. The same
remark applies to the value of (t) in the general equation
to complete it we must add ^{t) where ft^{t) .f{t, a) = 0.
To obtain \|/(/) in the equation ft<p{t) .cos (at) = F {a) above mentioned.
Let us suppose (pi {t), (p-^ {t), found by the method of Art. 44., to
satisfy the equations
Jt(px (t) . cos (at) = 1 for continuity,
ft(p2{t) . sin {at) = 1 for discontinuity,
differentiating with respect to a, the first 2n times, the second 2«— 1
times, we get
f,<pi{t).f"' cos (at) = 0,
ft (p.2 it), t"-' cos {at) ^ 0.
Hence,
^{t) = (p,{t) {At + Bf + Ct\ &c.} + 0,(0 {A'f + Bt^ + C't'^c.},
where A, B, C, &c. A', B', C, &;c. are absolute constants.
When transient functions appear in the appendage or even in the
prime solution, they must not be neglected (particularly in the mole-
cular investigations) except they are inadmissible by the nature of the
particular question, for they have a physical as well as a geometrical
meaning, as they are capable of expressing in continuous analytical
forms, the state of bodies and their mutual actions when they are com-
posed of absolute mathematical centres of forces, all separated mutually by
infinitesimal intervals.
Q/ Q; a> 04 Qf Jr
* Third Vol. Report of British Assoc, p. 212, &c.
384 Mr MURPHY'S THIRD MEMOIR ON THE
Thus let the ratio of the weight to the extent of an element P of
a straight rod AB be expressed by the transient function
(\-h){\+h) I, . ,
- — —^ /o -^N ■ J.2 ' when ^ = 1 ;
and where AP=(p, and the whole length AJB = ir, and n is very great
and integer.
Then the whole weight is finite, viz. f - — ^—^ — '—- — - — '—n = 1, vet
" J^l — 2hcos2n(p + h^ ^
this function has only an existence when = 0, -, — , — ...&c., and
therefore the rod is actually composed of disjoint particles Q,, Qa, Q3,
&c. which are separated by equal intervals, each infinitesimals, viz. -,
when n is very great, and equal to the actual number of particles ;
the action of such a system on another given one, may always be
estimated by using the transient function in its general form, and lastly,
putting h equal unity.
46. Equations which contain two or more Definite Integrals.
Given, jj cp (t) .f(t, a, b) + f,^l.{t) .F {t, a, b) = E {a, b),
the forms of the functions^ F, E being known, the forms of and
■<\f are required.
Put /(#, a, b) = ^oPo + A,P, + A^P^ + A^Ps + &c. ad inf.
where A^, Ai, A2, &c. are known functions of a and b, and Po, Pi, &c.
any self-reciprocal functions of t, such that ftPr!^ = a„, which will be a
known numerical quantity.
Similarly, F {t, a, b) = B,Po + B,Pi + B,P, + B,P^ + &c. ad inf.,
where B^, Bi, B^, &c. are known functions of a and b.
Again, let (p{t) =CoPo + c,Pi +CaP2 + C3P3, &c. ad inf.
where Co, c,, Ca, &c. are unknown numerical quantities,
and \l/{t) = eoPo + ejPj+e2P2 + e3P3,&iC. ad inf..
INVERSE METHOD OF DEFINITE INTEGRALS. 885
where eo, e^, e.,, &c. are also unknown.
The proposed equation then becomes
+ eoOoBo + eiUiBi + eia-iS^ + &c.)
Now to the function A„ there may be found a function A„ reciprocal relative to a,
and to B„ B„ b.
Let f„AaB„ = U„ a function of b only,
ftBoA,, = V„ a only.
Hence, f^AoE (a, b)-CoaofaAoAo = eoaoUo + eia^Ui + 6.^0^112 + kc. ad inf.
ft BoE {a, b) - e^a^ fiBoB^ = c^a^ K + c, a, F; + c^a^ F; + &c. ad inf.
Let t/„ be the function of b, which is reciprocal to f7„,
V„ of «, V^.
\L k {Ao U,E {a, b) - c,a,A,A^ t7"„) = e„a„ /j t7„ U^
Hence, \ \,
\fJ,{Bo KEia, b) - e„a,B,B,K) = c„«„/„F„rJ
by which equations the constants Co, e,, are immediately determined.
\fa fb (Ao U„E (a, b) - c^a^A^A^ U„) = e„ a„ /j Un UA
Also, \ >;
(/„ /, (^0 KE{a, b) - e,a,B,B, K) = c„a„ f„ V„ Vj\
and since c„, e^, have been found, the latter equations determine gene-
rally the coefficients c„, e„, and therefore the required functions <p{t),
^ {t) are known.
In like manner by employing reciprocal functions relative to double
integration, we may solve equations containing three unknown func-
tions, &c.
The problem of the distribution of electricity on bodies of which
the surfaces are not . continuous, introduces equations of this nature.
386 Mb MURPHY'S THIRD MEMOIR ON THE
47. Simultaneous Equations to Definite Integrals.
Given l-^*^^^^ ••^^^' *) +-'^^(^) • ^^^' «) = ^^^H
\!i<p{t)Mt, a) +f,i.(t) . FM a) = EAa)i '
the forms of the functions j^ F, E,fi, Fi, Ei, being known, the forms
of (p and \{^ are required.
Multiply the second equation by an arbitrary quantity \, and adding
to the first, put
f(t, a) + X/ {t, a) = A,Po + A,P, + A,P, + &c.
F{t,a) +xF^{t,a) = AoQo + A,Q, + A,Q2 + &c.
(pit) = CoPo + c,P,' + c,P^ + &c.
^^(0 = ^oQo' + e,Q, + e,Q: + &c.
where P„, P,, P2, &c.\ « .. /? ^ i
^ ^ .-k o f are functions of t only,
Qo, Qi, Q2, &C.J ^
A^, Ai, Ai,\ known functions of a, X, and self-reciprocal relative
to a,
PI, Qn reciprocal to P„, Q„ respectively, hence
(putting /,P„P„'=;),„ j;Q„Q„'=^„) E{a) + XEM
= c^poAo + c,p,Ai + C2P2A2 + &c. + eoqoAo + e^q.A, + e^q^A-^, &c. ;
.-. faAoEia) + X faAoE^ (a) = {copo + eoqo)faAo\
faA,E{a) + XlA,E,{a) = iCiP^ + e,q,)faA^
and giving to X any two values in each of these equations, the first
will produce two equations which determine Co, eo, the second will
similarly give Ci, e^, &c., and thence the functions (p{t), \l/t are known.
The same method is applicable to any number (n) of simultaneous
equations involving n unknown functions.
48. Definite-integral Equations of superior orders and degrees.
Methods similar to the preceding are applicable in most cases of
the former class thus :
INVERSE METHOD OF DEFINITE INTEGRALS. 387
Given fj,<t>{t, T)f(t, T, a) = F{a),
the forms F and J" being known to determine (p.
By Art. 16. Sect. iv. let a function Q„ be formed which shall be
self-reciprocal, relative to double integration for t and t.
Put ^(#,t) = Co Q„ + CiQi + C2Q2 + &C. 1 _rrri'i
and/(^,T,«) = ^„Q„+^,Q, + ^,Q, + &c.r'''* ^^^ a»-i.^t^»,
hence F(a) = aoCo^o + aiCi^, +0203^2 + &c.
Let ^„' be a function of a reciprocal to A„,
then faA,'F{a) = c^aJaA.A^,
faA,'F{a) = c,aJaA,A„
&c. &c.
whence Co, Ci, &c. being determined, the function (p{f,T) is known.
Equations of superior degrees must generally be converted into equa-
tions of superior orders to be easily solved, thus;
Given f,(p{t) .fit, a) x [,cp{t) . F{t, «) = >/.(«),
the forms ^ F, and -^^f being given to find the function 0.
Introduce another variable t having the same limits as t, then it is
evident that
J,<p{t) . F(t, a) = /^«^(t) . F{t, a) ;
.-. U^cp{t) .(pi-r) ./{t, a) . F{t, a) = f (a),
and since y(#, a) . F{t, a) is a given function of t, t and a, the unknown
function (p{t).(p(T) will be determined as above, and representing it by
<p^(t,T), let a be a root of the equation 0(t) = 1, then since (p{t).(p{T)
= 0i(#, t), we get the required function (}>{t) = <pi{t, a), and again putting
^ = a we get ^1 (a, a) = 1, from which equation a is known, and there-
fore <p{t) = <pi{t,a) is also known.
49. In researches on the subjects of electricity, and the phaenomena
dependent on the molecular construction of bodies, the only data which
can be furnished by experience are the total actions, and consequently
Vol. V. JPart III. SE
388 Mr MURPHY'S THIRD MEMOIR ON THE
the analytical processes of calculation require the solution of definite
integral equations: some of these have been resolved by Laplace and
others, by means of particular artifices by which the unknown functions
were subjected to differential equations ; but as no general method
existed for this purpose, the resolution of such equations has been ex-
tremely limited, and apparently simple physical problems, such as the
distribution of electricity on surfaces, (with the exception of a very
few cases) have consequently defied the powers of analysis. Besides,
an abundance of facts connected with the interior arrangement of the
molecules of bodies are of such a nature, that mathematics possessed
but little power of reducing them to analytical forms, calculated to
produce any valuable inferences ; these facts are daily increasing in
number, and the analyst is far behind the cultivator of Experimental
Physics. The Memoirs on the Inverse Method of Definite Integrals
which are now concluded, and which have been pursued when the
absence of ordinary engagements permitted, originated in the belief
that by proceeding gradually from the simplest classes of Definite
Integrals to the more complex, the general principles of an Inverse
Method would be discoverable. The formation of all possible classes
of Reciprocal Functions, and the Transient Functions included amongst
them, have at length furnished means for the resolution of equations
to Definite Integrals. The author is however well aware that there
must exist numerous imperfections in the manner in which his design
is executed, but believing also that by those endeavours, however weak,
some fresh powers have accrued to analysis, as an instrument of investi-
gation, he trusts they will deserve the approbation of the Society.
R. MURPHY.
Caius Colleob,
Dec. 24, 1834.
INVERSE METHOD OF DEFINITE INTEGRALS. 389
Analytical Table of Reference to the "Memoirs on the Inverse
Method of Definite Integrals."
FIRST MEMOIR, Vol. IV. Page 353, &c.
PAGE
Introduction 353
Section I. Principles relative to Continuous Functions.
Art. 1. Method of reducing the given limits of integration to and 1 in all cases 358
Arts. 2, 3, 4. In the general equation ft f{t) .t''= <p(x), x is understood to lie between
— 1 and + 00 , then cyj (x) converges to zero as x increases, when y(<) is any of the
functions usually received in analysis; consequent division of the subject 35g
Art. 5. Rule; When the known function <p{x) is rational, seek the coefficient of - in
X
<p (x) . t~', dividing it by / we obtain fit) 362
Art. 6. Examples s6S
Arts. 7, 8. Means of facilitating the Calculus oi f{t) 3Q5 .
Art. 9. and Note (A). When {x) is a logarithmic function 366, 400
Art. 10. When (pix) is expressed by an equation to finite differences 367
Art. 1 1 . When <f> (x) is a fraction, the denominator containing imaginary factors 369
Art. 12. When {x) is irrational 37O
Art. 13. Cases when equations of the form J',f(J,').(t'' ±t~')=if>x, may be resolved by
the preceding method 37]
Art. 14. Extension of the general rule to successive integration with respect to any
number of variables , 373
Section II. Principles relative to Discontinuous Functions.
Art. 15. Cases of discontinuity in Physical Problems quoted 374
Art. 16. To find a formula which shall represent the least of the two quantities a, /3. . 375
Art. 18. To find a formula which shall represent_/(a) or y(/3) according as a is < or > /3. 376
Art. 19. To find a formula which shall represent r^, or -= — j— , according as a is
a — lip p — na
< or > /3 377
Art. 20. To find a formula which shall represent ■~^^ , or ^^^, , according as a is < or > /3 378
Arts. 21, 22. Method of representing discontinuous functions of any number of breaks 380
Arts. 23, 24. Geometrical Illustrations of the theory of discontinuity 382
3E2
390 Mr MURPHY'S THIRD MEMOIR ON THE
PAGE
Section III. Application of the preceding principles to the Phaenomena of Developed
Electricity/ 386
Note (A), No. 2. On the general separation of the positive powers of the variable from
the negative 402
Note (B), No. 1 . On the apparently improper forms of (p (x) 404
No. 2. Method of valuing the results of operative functions 406
SECOND MEMOIR. Vol. V. Page 113, &c.
Introduction 113
Section IV. Inverse Method Jbr Defitiite Integrals which vanish, and theory of Reci-
procal Functions.
Arts. 1,2. X being restricted to the natural numbers 0, 1, 2, (»— 1) to &nd fit) so
tha.tf,f(t).f = 116
Art 3. P„ denoting the function y(<) above-found, when m and n are unequal /,P„P„=0,
and when equal /,P,P„ = 117
Art. 4. To find a rational function _/(<) which may satisfy the equationy^y(<)'''=0, x being
any number of the series p, ^+ 1,. . .p-{-n—l 118
Art. 5. The general form of fit), when x is from to n— 1 inclusive, is
d" (ft'" V\
At) = • ^^^„ ^ , where t' = \ - t 118
Art. 6. In this case the equation^" (/) = 0, has n real roots lying between and 1 1 19
Arts. 7, 8. To find a rational function of h. 1. /, such that /,y{h. 1. (<)}.<' = 0, when
X is from to n— 1 inclusive 120
Art. 9. Denoting this function by Z.„, the function which it generates is the value of
u in the equation u {l — hh.\. u) = t 122
du
Art. 10. If Q„ be the coefficient of *" in -r- , u being found from the equation
m(1 — A £/) = <, where J7 is a function of u vanishing when u = \, and T the same
function of t, then -^l = "("- V""^"~""^'^ 122
Art. 11. If t/ be a rational and entire function of u vanishing when m=1, and Q, be
the term independent of u in the product i/" I 1 I , then shaliy]Q„<'= 0, when
X is from to «— 1 inclusive 12S
Art. 12. To find (p, q)„ a rational and entire function of P" of n dimensions, which
multiplied by a rational and entire function of f of less than n dimensions, the integral
of the product may vanish from t — Otot=l 125
Art. 13. Reciprocal Functions; such are ip,q)„, {<l>p\'; value of the integral of the
product when n=n' 126
Art. 14. To find a function A„ reciprocal to the function L„ found in Art. 8 128
INVERSE METHOD OF DEFINITE INTEGRALS. 391
PAGE
Art. 15. General principle for finding Reciprocal Functions to simple integration 130
Art. l6. The same extended to integration for any number of variables 131
Art. 17. Examples 132
Section V. Inverse Method for Junctions which contain positive ponters of x, or are
under any other form.
Art. 18. An appendage must be annexed in all such cases 135
Arts. 19, 20. When ^j; is a rational and entire function of x ; and particular example
when ^(.r) = l 136
Art. 21. To find /(<) when7(/(O-''=0W, an<i ^ is from to n—\ inclusive 138
Arts. 22, 23. Various modes of determining y(<) in this case 141
h
Arts. 24, 25. The coefficient of h" in the expansion of , is a self-reciprocal function 146
THIRD MEMOIR. Vol. V. Page 315, &c.
Introduction 315
Section VI. Method of discovering Reciprocal Functions, when the integrations are
performed with respect to any fonction of the variable.
Arts. 1, 2. General principle for varying the limits 318
d'.^t'f'V) dt
Art. 3. If V can be found so that — "' — 'Tl "™^y "'^ ''^ " dimensions in t (where
t' = \ — t) then this quantity will be self-reciprocal relative to (p $iq
Art. 4. If V can be found so that — j^ ■ j? "^^y be of n dimensions in t, then
the factor by which -j^ is multiplied will be self-reciprocal relative to 320
Arts. 5, 6. If <p=f,(Jt(y indefinite, and m between —1 and -j-oo, and if
d^ {(tt'Y'^"'\
Qn = 1 2 xdf-^" ' ("')""" *^" *^^^ ^ ^^ self-reciprocal relative to (p 321
n — m
rhfti'\
If z= ftittfy indefinite, and m be between -1-1 and —00, and if a» = — '
^ -"^ ' > ' 1" i.2...ndt'"
then shall q, be self-reciprocal relative to 321
Art. 7. To find the functions which Q„, 5, generate 322
Arts. 8, 9. When 7« = — §, Q„, q, are the trigonometrical reciprocals 323, 325
Art 10. In the identities thus obtained, the sign of n may be changed so as to pass
from differential coefficients to integrals 326
Art. 1 1 . The two series of reciprocal functions obtained from the theorems of Arts. 5 81. 6.
differ only with respect to the variable of integration 328
Art. 12. Examples of the preceding theory 329
Art. 13. To express Q„ and q„ in terms of t alone 330
Art. 14. To express Q„ and q„ by means of differential equations 332
892 Mr MURPHY'S THIRD MEMOIR ON THE
PAGE
Art. 15. The reciprocal functions expressed by the general formulae for Q,, q, all possess
a common property, viz., their integrals vanish when taken between limits which
render the functions maxima or minima 333
Art. 16. To find the complete integral of the equation
i<'^ + (m+l)(l-2/) -^^+n(n+2m + l)u = 335
Art. 1 7. To find explicitly the omitted part of the complete integral in Laplace's equation,
for the coefficients in the expansion of the reciprocal of the distance between two
points in a plane 338
Art. 18. When m = —\ the general equation of Art. l6. represents the trigonometrical
functions 342
Art. 19. Remarkable properties of the functions
G = g^cose cos(a;sin0), 0'= £■>; ™s e sin (a; sin 0) 342
Art. 20. Application to the general differentiation of rational and integer functions oi x . . . 344
Art. 21. The sum of all the divisors of a number n, including itself and unity
= fy^- 1- {sin^ sin 2^ sinn^} . cos 2n(^ 346
TT
Section VII. On Transient Functions.
Art. 22. Nature of transient functions 347
Art. 23. To find a function reciprocal to f(t, n) any given function of the variable t
and integer n ' 348
Art. 24. To find a function V„ reciprocal to /" 349
Art. 25. The function V„ is transient 352
Art. 26. To express the transient function F„ in a finite form 354
Art. 27. Discussion of the transient function Fo ; it represents the state of a body which
an electric spark is about to enter 356
Art. 28. To find the quantity to which F„ is the generating function 360
Art. 29. To expand a given function in terms of the functions F„ 36l
Art. 30. To find a function t/„ reciprocal to (h. 1. t)' 362
Art. 31. In a finite form U„ = H — 2_ when A = l 363
1 .2...ndh"
Art. 32. Properties of C7„ &&f,Uj' = x', &c S6S
Art. 33. Discussion of the function Uq 364
Art. 34. To expand a given function in terras of the functions I/„ 365
Art 35. To find a function reciprocal to /" when the limits of t are and =0 366
Art. 36. To find a function F„ reciprocal to cos"^ between the limits ^ = and <p=-ir. 366
Art. 37. The function F„ is transient - 367
Art. 38. To express F, in a finite form 368
Art. 39. Means of summing a series expressed in transient functions 369
INVERSE METHOD OF DEFINITE INTEGRALS. 393
PAGE
Section VIII. On the Resolution of Equations which involve Definite Integrals.
Art. 40. Method of decomposition into elements 371
Art. 41. Density of a cylindric shell which exercises no action on any point in its axis
with any law of force 372
Art. 42. Examples when the law of force is the inverse square of the distance 374
Art. 43. Resolution of equations which contain but one definite integral and one parameter 377
Art. 44. Examples 379
Art. 45. On the appendage necessary to complete the solution of a Definite Integral
Equation 382
Transient functions capable of representing in a continuous form the state of a body
composed of mathematical centers of forces separated by infinitesimal intervals 383
Art. 46. Equations which contain two or more Definite integrals and as many parameters 384
Art. 47. Simultaneous equations to Definite integrals 386"
Art. 48. Definite integral equations of superior orders and degrees 386
Art. 49. Conclusion 387
ERRATA.
PAOE
First Memoir. 359, line 9. 16. 18. dele y in the sign ooy.
Vol. IV. 377, line 8. for A</3 read A-^ 1.
406, lowest line and third from bottom, for terms read times.
407, line 17. supply the word, equation.
Second Memoir. Vol. V. 134, line 6, after -j^ supply (tf)*.
d'P
line 7. after ,* supply (tf).
dt
136, line 3, 4. 18. for v, put v..
Third Memoir. Vol, V. 332, lowest line, for /'"(•-')(() read /'"<'-"(«).
333, line 5, for /(O) read f"(0).
for m+3 read (m+3).
337, line 8, put (tf)-" before ~ in the last term.
346, line 16, for intger read integer.
357. line 8,/or (1-ft*) reod (1-A)*.
XV. Oil the Determination of the Exterior and Interior Attractions of
Ellipsoids of Variable Densities. By George Green, Esq.,
Caius College.
[Read May 6, 1833.]
The determination of the attractions of ellipsoids, even on the hypo-
thesis of a uniform density, has, on account of the utility and difficulty
of the problem, engaged the attention of the greatest mathematicians.
Its solution, first attempted by Newton, has been improved by the suc-
cessive labours of Maclaurin, d'Alembert, Lagrange, Legendre, Laplace,
and Ivory. Before presenting a new solution of such a problem, it
will naturally be expected that I should explain in some degree the
nature of the method to be employed for that end, in the following
paper; and this explanation will be the more requisite, because, from
a fear of encroaching too much upon the Society's time, some very
comprehensive analytical theorems have been in the first instance given
in all their generality.
It is well known, that when the attracted point p is situated within
the ellipsoid, the solution of the problem is comparatively easy, but
that from a breach of the law of continuity in the values of the
attractions when p passes from the interior of the ellipsoid into the
exterior space, the functions by which these attractions are given in the
former case will not apply to the latter. As however this violation
of the law of continuity may always be avoided by simply adding a
positive quantity, u" for instance, to that under the radical signs in
the original integrals, it seemed probable that some advantage might
thus be obtained, and the attractions in both cases, deduced from one
common formula which would only require the auxiliary variable u to
become evanescent in the final result. The principal advantage how-
ever which arises from the introduction of the new variable u, depends
Vol. V. Part III. SF
396 Mr green, ON THE DETERMINATION OF THE
on the property which a certain function F'* then possesses of satisfy-
ing a partial differential equation, whenever the law of the attraction
is inversely as any power n of the distance. For by a proper applica-
tion of this equation we may avoid all the difficulty usually presented
by the integrations, and at the same time find the required attrac-
tions when the density p is expressed by the product of two factors,
one of which is a simple algebraic quantity, and the remaining one
any rational and entire function of the rectangular co-ordinates of the
element to which p belongs.
The original problem being thus brought completely within the pale
of analysis, is no longer confined as it were to the three dimensions of
space. In fact, p' may represent a function of any number s, of in-
dependent variables, each of which may be marked with an accent, in
order to distinguish this first system from another system of s analo-
gous and unaccented variables, to be afterwards noticed, and F' may
represent the value of a multiple integral of s dimensions, of which every
element is expressed by a fraction having for numerator the continued
product of p into the elements of all the accented variables, and for
denominator a quantity containing the whole of these, with the un-
accented ones also formed exactly on the model of the corresponding
one in the value of V belonging to the original problem. Supposing
now the auxiliary variable u is introduced, and the s integrations are
effected, then will the resulting value of ^ be a function of u and of
the s unaccented variable to be determined. But after the introduction
* This function in its original form is given by
-. /• p' dx dy dz
J {{X - xy + (/ - yf + (.' - 2)2}"-^'
where dx dy dz represents the volume of any element of the attracting body of which p'
is the density and x , y , z are the rectangular co-ordinates ; x, y, z being the co-ordinates
of the attracted point p. But when we introduce the auxiliary variable u which is to be
made equal to zero in the final result,
jr _ r p dx dy dz
J{(^a:'-xf-\.{y-yy + {z-zf + u^yr'
■ - .YOii
both integrals being supposed to extend over the whole volume of the attracting body.
ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 397
of w, the function V has the property of satisfying a partial differen-
tial equation of the second order, and by an application of the Cal-
culus of Variations it will be proved in the sequel that the required
value of V may always be obtained by merely satisfying this equation,
and certain other simple conditions when p is equal to the product
of two factors, one of which may be any rational and entire function
of the s accented variables, the remaining one being a simple algebraic
function whose form continues unchanged, whatever that of the first
factor may be.
The chief object of the present paper is to resolve the problem
in the more extended signification which we have endeavoured to ex-
plain in the preceding paragraph, and, as is by no means unusual, the
simplicity of the conclusions corresponds with the generality of the
method employed in obtaining them. For when we introduce other
variables connected with the original ones by the most simple rela-
tions, the rational and entire factor in p still remains rational and
entire of the same degree, and may vmder its altered form be ex-
panded in a series of a finite number of similar quantities, to each of
which there corresponds a term in V, expressed by the product of two
factors; the first being a rational and entire function of s of the new
variables entering into V, and the second a function of the remaining
new variable h, whose differential coefficient is an algebraic quantity.
Moreover the first is immediately deducible from the corresponding
part of p without calculation.
The solution of the problem in its extended signification being thus
completed, no difficulties can arise in applying it to particular cases.
We have therefore on the present occasion given two applications
only. In the first, which relates to the attractions of ellipsoids, both
the interior and exterior ones are comprised in a common formula
agreeably to a preceding observation, and the discontinuity before
noticed falls upon one of the independent variables, in functions of
which both these attractions are expressed ; this variable being con-
stantly equal to zero so long as the attracted point j) remains within
the ellipsoid, but becoming equal to a determinate function of the co-
3f2
398 Mr green, ON THE DETERMINATION OF THE
ordinates of p, when p is situated in the exterior space. Instead too
of seeking directly the value of V, all its differentials have first been
deduced, and thence the value of V obtained by integration. This
slight modification has been given to our method, both because it
renders the determination of V in the case considered more easy, and
may likewise be usefully employed in the more general one before
mentioned. The other application is remarkable both on account of
the simplicity of the results to which it leads, and of their analogy
with those obtained by Laplace. (Mdc. C^. Liv. iii. Chap. 2.) In fact,
it would be easy to shew that these last are only particular cases of
the more general ones contained in the article now under notice.
The general solution of the partial differential equation of the second
order, deducible from the seventh and three following articles of this
paper, and in which the principal variable 1^ is a function of # + 1
independent variables, is capable of being applied with advantage to
various interesting physico-mathematical enquiries. Indeed the law of
the distribution of heat in a body of ellipsoidal figure, and that of the
motion of a non-elastic fluid over a solid obstacle of similar form,
may be thence almost immediately deduced; but the length of our
paper entirely precludes any thing more than an allusion to these ap-
plications on the present occasion.
1. The object of the present paper will be to exhibit certain
general analytical formulae, from which may be deduced as a very
particular case the values of the attractions exerted by ellipsoids upon
any exterior or interior point, supposing their densities to be represented
by functions of great generality.
Let us therefore begin with considering p as a function of the s
independent variables
»r J , x<i , x^ ••••• o/i,
and let us afterwards form the function
dxjdx^ dxj dxl . p .^.
'{{x,-xiJ^{x,-xl)^^ ^(x.-xlJ^u'-S^
r=f-
n-1
2
ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 399
the sign / serving to indicate * integrations relative to the variables
x^, x-i, X3', x/, and similar to the double and triple ones employed
in the solution of geometrical and mechanical problems. Then it is
easy to perceive that the function V will satisfy the partial differen-
tial equation
t/vr ^, d^ ^ n-s dV
" ~ dx,^ "^ dxi "*" ^ dx^ '^ du^^ u du ^^'
seeing that in consequence of the denominator of the expression (1),
every one of its elements satisfies for V to the equation (2).
To give an example of the manner in w^hich the multiple integral
is to be taken, we may conceive it to comprise all the real values
both positive and negative of the variables ar/, x^, x,, which satisfy
the condition
the symbol / , as is the case also in what follows, not excluding equality.
2. In order to avoid the difficulties usually attendant on integra-
tions like those of the formula (1), it will here be convenient to notice
two or three very simple properties of the function F".
In the first place, then, it is clear that the denominator of the
formula (1) may always be expanded in an ascending series of the
entire powers of the increments of the variables x^, x^, x„ u, and
their various products by means of Taylor's Theorem, unless we have
simultaneously
and therefore V may always be expanded in a series of like form,
unless the s + 1 equations immediately preceding are all satisfied for
one at least of the elements of V. It is thus evident that the func-
tion V possesses the property in question, except only when the two
conditions
4!fl# Mr green, on THE DETERMINATION OF THE
% + %^%+ +% z 1 and u = .(3)
are satisfied simultaneously, considering as we shall in what follows
the limits of the multiple integral (1) to be determined by the conr
dition (a)*.
In like manner it is clear that when
Z^2+ Jl+ + 77-2>^ (4)»
a?'
the expansion of V in powers of u will contain none but the even
powers of this variable.
Again, it is quite evident from the form of the function f^ that
when any one of the * + 1 independent variables therein contained be-
comes infinite, this function will vanish of itself.
3. The three foregoing properties of F combined with the equa-
tion (2) will furnish some useful results. In fact, let us consider the
quantity
fd.,d^,...d..duu-'.[[^)\ [^)\ + (g)\ (^^)] (5)
where the multiple integral comprises all the real values whether posi-
tive or negative of x^, x^, x,, with all the real and positive values
of u which satisfy the condition
/!« 2 A< 2 « 2 /|/2
^■^^^ + -^^^^F^^ ^^^
* The necessity of this first property does not explicitly appear in what follows, but
it must be understood in order to place the application of the method of integration by
parts, in Nos. 3, 4, and 5, beyond the reach of objection. In fact, when V possesses this
property, the theorems demonstrated in these Nos. are certainly correct: but they are not
necessarily so for every form of the function V, as will be evident from what has been
shewn in the third article of my Essay on the Application of Mathematical Analysis to
the Theories of Electricity and Magnetism.
ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 401
«,, «2, a, and h being positive constant quantities; and such that
we may have generally
Ur > dr.
In this case the multiple integral (5) wiU have two extreme limits,
viz. one in which the conditions
V ^ or IT 71
-\ + -^ + + -^ + t; = 1 and u— a. positive quantity (7)
are satisfied; and another defined by
% + %+ +-, /I and « = 0.
jVIoreover, for greater distinctness, we shall mark the quantities be-
longing to the former with two accents, and those belonging to the
latter with one only.
Let us now suppose that J^" is completely given, and likewise F,'
or that portion of f^' in which the condition (3) is satisfied ; then if
we regard F/ or the rest of T^' as quite arbitrary, and afterwards endea-
vour to make the quantity (5) a minimum, we shall get in the usual
way, by applying the Calculus of Variations,
/7F''
-fdx.dx, clx,u"-^^r,'~- (8)
seeing that ^V" = and SFj' = 0, because the quantities V" and F,'
are supposed given.
The first line of the expression immediately preceding gives generally
= 2'+'— — ^Hf^ {^•\
' dxr du' u du ^ '
which is identical with the equation (2) No. 1, and the second line gives
dV
= u'"'' ~7-^(^' being evanescent) (9).
402 Mr green, ON THE DETERMINATION OF THE
From the nature pf the question de minimo just resolved, there can
be little doubt but that the equations (2') and (9) will suffice for the
complete determination of V, where V" and V-l are both given. But
as the truth of this will be of consequence in what follows, we will,
before proceeding farther, give a demonstration of it; and the more
wiUingly because it is simple and very general.
4. Now since in the expression (5) u is always positive, every one
of the elements of this expression will therefore be positive; and as
moreover V" and F"/ are given, there must necessarily exist a function
Fo which will render the quantity (5) a proper minimum. But it
follows, from the principles of the Calculus of Variations, that this
function Va, whatever it may be, must moreover satisfy the equations
(2') and (9). If then there exists any other function F", which satisfies
the last-named equations, and the given values of V" and V^, it is easy
to perceive that the function
will do so likewise, whatever the value of the arbitrary constant quan-
tity A may be. Suppose therefore that A originally equal to zero
is augmented successively by the infinitely small increments SA, then
the corresponding increment of V will be
Sr={F,-V,)SA,'
and the quantity (5) will remain constantly equal to its minimum
value, however great A may become, seeing that by what precedes
the variation of this quantity must be equal to zero whatever the
variation of V may be, provided the foregoing conditions are all satis-
fied. If then, besides F"o . there exists another function F"; satisfying
them all, we might give to the partial differentials of F", any values
however great, by augmenting the quantity A sufficiently, and thus
cause the quantity (5) to exceed any finite positive one, contrary to
what has just been proved. Hence no such value as F, exists.
We thus see that when F"" and F"/ are both given, there is one
and only one way of satisfying simultaneously the partial differential
equation (2), and the condition (9).
ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 403
5. Again, it is clear that the condition (4) is satisfied for the whole
of F"/; and it has before been observed (No. 2.) that when V is deter-
mined by the formula (1), it may always be expanded in a series of
the form
r = ^ + J?«' + Cu' + &c.
Hence the right side of the equation (9) is a quantity of the order
?/"-'+' ; and v! being evanescent, this equation will then evidently be
satisfied, provided we suppose, as we shall in what follows, that
n — s \ \ is positive.
If now we could by any means determine the values of V" and
V( belonging to the expression (1), the value of V would be had
without integration by simply satisfying (2') and (9), as is evident from
what precedes. But by supposing all the constant quantities a,, «2> «3
a, and h infinite, it is clear that we shall have
= V",
and then we have only to find V^, and thence deduce the general
value of V.
6. For this purpose let us consider the quantity
w ^ ^7 n-AdVdU dVdU , dVdU dVdU\
jdxidx.i...dx,duvr '{-r—-j— + -f— -j— + ••• + i—n— + -i t-)\ (10^
{dxidx^ dx.dxi dx.dx, du du j ' ^ '
the limits of the multiple integral being the same as those of the
expression (5), and U being a function of ;r,, x^, x, and u, satis-
fying the condition 0= U" when «,, a^, a, and h are infinite.
But the method of integration by parts reduces the quantity (10) to
— fdXidxi dx,—j — u'"-' . V
du
-/..........x..»».-.r|.,«^+^.^^} (H,
since = V"\ and as we have likewise = U", the same quantity (10)
may also be put under the form
Vol. V. Part III. SG
404 Mk green, on THE DETERMINATION OF THE
dV
— fdxidXi dxi—r—u'"-' . U'
.fdx,dx,...dx,duu''-'.u\^r'^,+^ + '^^^ (12).
Supposing therefore that U like V also satisfies the equation (2'),
each of the expressions (11) and (12) will be reduced to its upper line,
and we shall get by equating these two forms of the same quantity :
idx^ dx2...dxs-j~ u'"-' V = fdxi dXi...dxs -y- «'"* U' :
au au
the quantities bearing an accent belonging, as was before explained, to
one of the extreme limits.
Because V satisfies the condition (9), the equation immediately pre-
ceding may be written
dU' dV
fdxidx2...dxs-j — u'"~' V = fdxidxi...dx,—y^u'"-' U,'.
du du
If now we give to the general function U the particular value
u= {{x, - x,"y + {x, - x,y + + {x, - xjy + u']^-,
which is admissible, since it satisfies for V to the equation (2), and gives
U" = 0, the last formula will become
dVi
/dxidx-i dxsu'"'' —j-^
du
{{x, - x^y + {x, - x:j + + (a;, - xlj + m'^}^
_r dxydx^ c?;g,.(l-w) «'"-'+' V ,
\{x, - xlj + (ar, - xij + + {x, - x:j + u''\'^
in which expression «' must be regarded as an evanescent positive
quantity.
In order now to effect the integrations indicated in the second
member of this equation, let us make
ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 405
x^ — Xi" = u'p COS 6i ; x-i—x" = u'p sin Qx cos 0^ ; Xi—x^'—u'p sin 0, sin 02 cos 03> &c.
until we arrive at the two last, viz.,
«,_! -x[^-^ = u'p sin^i sin ^^ sin0,_2 cos0,_i,
X, — ar," = «'/o sin ^1 sin 02 sin 0,_2 sin 0,_i;
u' being, as before, a vanishing quantity.
Then by the ordinary formulas for the transformation of multiple
integrals we get
dxi dx-i dx, = u''f/~^ sin^i'"^ sin 02*"' smO^^.^dp d6i dOi...dd,.i,
and the second number of the equation (13) by substitution will become
f dp d9i de, d9,_,p'-' sm9r'' sin 9,'-' sin e,_2 . (1 - ») r'
/ »+i (1*)-
But since u' is evanescent, we shall have p infinite, whenever x^, Xi,...x,
differ sensibly from x", x^',,..x"\ and as moreover w — * + l is positive,
it is easy to perceive that we may neglect all the parts of the last
integral for which these differences are sensible. Hence V may be
replaced with the constant value VI in which we have generally
Jbf ^^ vUf •
Again, because the integrals in (14) ought to be taken from 0,_, = o
to 0r-, = 27r, and afterwards from 0,. = O to 9r = -n-, whatever whole number
less than 5—1 may be represented by r, we easily obtain by means of
the well known function Gamma:
»
/sin^i'-'' sin 02'"' sin 03'"' sin0,_2C?0,</02...c?0,., = ^ZL;
and as by the aid of the same function we readily get
r» — * + l>
f P'~'dp _ V2/ V 2 )
Wi + ,f-^ 2r(^)
3g2
406 Mr green, ON THE DETERMINATION OF THE
the integral (14) will in consequence become
and thus the equation (13) will take the form
dx.dx, dx,u"^-^^ ~^'^-^ ^~V^ )
J {{x,-x;
J + {x, - x;j +.... + (x, - x:j + m'^ } -^ r (■
w-l
In this equation V '\?, supposed to be such a function of x^, x.^ x,
and u, that the equation (2) and condition (9) are both satisfied. More-
over V'^O, and Vo is the particular value of F' for which
Let us now make, for abridgment,
dV
P = u"-' -r-, {when u = 0) (A),
and afterwards change x into x\ and x" into x in the expression im-
mediately preceding, there will then result
_- s f fi—s + V
r dx^ dx2 . ...dx,'P,' ""^^'^ I 2 / „, ,,^^
/ ^^ rr — r; f^ •••U5),
{{x,'-x,f+{x,'-x,y + ...+{'>':-^sY+u"]— r(^)
--^(^).„
P' being what P becomes by changing generally Xr into x,', the unit
attached to the foot of P' indicating, as before, that the multiple
integral comprises only the values admitted by the condition {a), and
V being what V becomes when we make u = 0.
The equation just given supposes u' evanescent; but if we were to
replace u with the general value u in the first member, and make a
corresponding change in the second by replacing F'' with the general
value F, this equation would still be correct, and we should thus have
ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 407
r dx'dxi dxlP,' \ 2 J ,^ , „
f '■ '—^ ^, = -L-jl— ir...(l6).
-^ \{x,'-x,y+{x,'-x,y+... + {x:-x.y+u'\— r(^^)
For under the present form both its members evidently satisfy the
equation (2), the condition (9), and give V" = 0. Moreover, when the
condition (3) is satisfied, the same members are equal in consequence
of (15). Hence by what has before been proved (No. 4), they are
necessarily equal in general.
By comparing the equation (16) with the formula (1), it will become
evident, that whenever we can by any means obtain a value of V satis-
fying the foregoing conditions, we shall always be able to asSgn a value
of p which substituted in (1) shall reproduce this value of V. In fact,
by omitting the unit at the foot of P", which only serves to indicate
the limits of the integral, we readily see that the required value of p is
p'= \ P' {c).
r^ 'r. fn~S+l\ '
7. The foregoing results being obtained, it will now be convenient
to introduce other independent variables in the place of the original
ones, such that .
^1 = «i?i» «a = 02^2j x, = as^„ u = hv,
Oj, ttj, flj being functions of h, one of the new independent variables,
determined by
a,' = «;* + h', a,- = (h' + h\ a/ = aj' + /^^
and V a function of the remaining new variables, f,, ^2, ^3, ^s satis-
fying the equation
1 = v' + |;^ + e/+ + U;
a,', a/, Os', 0/ being the same constant quantities as in the equation
(a), No 1. Moreover, Oi, a.^, a, will take the values belonging to
the extreme limit before marked with two accents, by simply assigning
to h an infinite value.
408 Mr green, ON THE DETERMINATION OF THE
The easiest way of transforming the equation (2) will be to remark,
that it is the general one which presents itself when we apply the
Calculus of Variations to the quantity (5), in order to render it a
minimum. We have therefore in the first place
and by the ordinary formula for the transformation of multiple integrals,
dx.dx, dx,du=^^^^ (l-2r' ^') d^,dl,...dldh.
• But since 1 - 2,'+' ^^ = v + ^»S,'+' ^,
a; ' Ur
the expression (5) after substitution will become
fd^^d^i d^sdhui tti ih a.A""'!/""'"'.
Applying now the method of integration by parts to the variation of
this quantity, by reduction, we get for the equivalent of (2) the equation
^- dh^^ V" ,^ a;) hdh^^^ ^^''^ ar'dl' ^' "" ^^^a:-dlr'
+ A^2^ X 2-^^ - A'22-Ml -^^ (2")
where the finite integrals are all supposed taken from r = l to r = * + l,
and from r' = 1 to r' = * + 1.
The last equation may be put under the abridged form,
d^ . ( ^«:^ dV
dJi
provided we have generally
o = -^+(»-s5-)^ + vr (n.
ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 409
coefficient o£^-mvV=~ {1 -^^'-2.'*' 1^ ^ "' + ^ ^"}>
coefficient of ^, i» V ^ = - ^. lU
coefficient of -j^ in vF=-^|-» + 2 ^ ^i.
Moreover, when we employ the new variables
du " y- ^ ^ . ; • Y' a? d^r dh ]'
and therefore the condition (9) in like manner will become
— -(>-^r"i^ff-^} «'»^
where the values of the variables ^1,^2, ?, must be such as satisfy
the equation i;" = 0, whatever h may be; and as n-s-\-l is positive, it
is clear that this condition will always be satisfied, provided the partial
differentials of V relative to the new variables are all finite.
8. Let us now try whether it is possible to satisfy the equation
(2'") by means of a function of the form
r^Hct> (/?);
H depending on the variable h only, and cp being a rational and entire
function of ^1, f^, f, of the degree 7, and quite independent of h.
By substituting this value of V in (2'") and making
^ d'H ( ^«:^ dH , „ ,,„^
we readily get
= v<^ - '«P (18);
where, in virtue of (17) k must necessarily be a function of h only;
and as the required value of (p, if it exist, must be independent of k,
we have, by making h = in the equation immediately preceding,
= v'0 - ko(p (19);
ko being the value k, and v'^ that of v^ when h = 0.
410 Mb green, ON THE DETERMINATION OF THE
We shall demonstrate almost immediately that every function ^ of
the form (20), No. 9, which satisfies the equation (19), and which there-
fore is independent of h, will likewise satisfy the equation (18); and
the corresponding value of k obtained from the latter being substituted
in the ordinary differential equation (17), we shall only have to integrate
this last in order to have a proper value of V.
9. To satisfy the equation (19) let us assume
<^ = ^(e.^ ?/, ?3^ ?/)?,.?„ &c (20);
F being the characteristic of a rational and entire function of the
degree 2y', and the most general of its kind, and f,, ^„ &c. designating
the variables in which are affected with odd exponents only; so that
if their number be v we shall have
7 = 27' + c,
the remaining variables having none but even exponents. Then it is
easy to perceive, that after substitution the second member of the
equation (19) will be precisely of the same form as the assumed value
of (p, and by equating separately to zero the coefficients of the various
powers and products of ^1, |s, ^,, we shall obtain just the same
number of linear algebraic equations as there are coefficients in <p, and
consequently be enabled to determine the ratios of these coeflScients
together with the constant quantity ^0.
In fact, by writing the foregoing value of (p under the form
</) = aS'^„„„, „„?.•"' ?."» ?»•" (20');
and proceeding as above described, the coefficient of ^ri ^/"t ^,'',
will give the general equation
K + 2)(m. + ])
^..i K+2)K + i) .
"r
ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 411
the double finite integral comprising all the values of r and r, except
those in which r = r , and consequently containing when completely
expanded s (s - 1) terms.
For the terms of the highest degree 7 and of which the number is
7 + 1.7 + 2 y + s — l _ ^
the last line of the expression (21) evidently vanishes, and thus we
obtain JV distinct linear equations between the coefficients of the degree
7 in <p and ko.
Moreover, from the form of these equations it is evident that we
may obtain by elimination one equation in ko of the degree JV, of
which each of the iV roots will give a distinct value of the function
(p^'y\ having one arbitrary constant for factor; the homogeneous function
^''1'' being composed of all the terms of the highest degree, 7 in (p.
But the coefficients of (p'-^'' and kg being known, we may thence easily
deduce all the remaining coefficients in (j>, by means of the formula (21).
Now, since the A'' linear equations have no terms except those of
which the coefficients of ^'^^ are factors, it follows that if ^0 were taken
at will, the resulting values of all these coefficients would be equal to
zero. If however we obtain the values of N' — 1 of the coefficients
in terms of the remaining one A from iV- 1 of the equations, by the
ordinary formulas, and substitute these in the remaining equation, we
shall get a result of the form
K.A=0,
where jRT is a function of ka of the degree iV. We shall thus have
only two cases to consider : First, that in which A = 0, and consequently
also all the other coefficients of 0*^' together with the remaining ones
in <p, as will be evident from the formulae (21). Hence, in this case
= 0:
Secondly, that in which kg is one of the iV roots of = K, as for
instance, ko in this case all the coefficients of will become multiples
of A, and we shall have
Vol. V. Part III. S«
412 Mr green, ON THE DETERMINATION OF THE
(j) = Acpr.
01 being a determinate function of ^,, ^a, E<-
We thus see that when we consider functions of the form (20)
only, the most general solution that the equation
= v'^ - *o'0 (19')
admits is
or, (p = 0; or, (p = atp;
a being a quantity independent of ^,, ^2, ^„ and (p any function
which satisfies for <p to the equation (19'). But by affecting both sides
of the equation
with the symbol v, we get
= V • v' - *o' . V ^ ;
and we shall afterwards prove the operations indicated by v and v'
to be such, that whatever may be,
V v' = V' V 0-
Hence, the last equation becomes
v' (v ^) - k„' V (p;
and as V like (p is of the form (20), it follows from what has just
been shewn, that
either = v cp, or, \7 (p = acp,
a being a quantity independent of ^i, ^2, ?«•
The first is inadmissible, since it would give ^ = 0; therefore when
(p satisfies (19'), we have
V 0' = a(p, i.e. = V — "0-
But since a is independent of ^1, ^2, Bs, this last equation is
evidently identical with (18), since the equation (18) merely requires that
K should be independent of fi, ^2, ^s-
ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 41^
Having thus proved that every function of the form (20) which
satisfies (19) will likewise satisfy (18), it will be more simple to deter-
mine the remaining coefficients of (j> from those of cp^^^ by means of
tlie last equation, than to employ the formula (21) for that purpose.
Making therefore h infinite in (18), and writing ~ in the place
of K, we get
where (22) comprises the — ^ — —!■ combinations which can be formed of
1.2
the s indices taken in pairs.
If now we substitute the value of before given (20'), and recol-
lect that for the terms of the highest degree we have 2»«r = 7, we shall
readily get
0=(7-2»«,)(7+2»?r+»-l)^™,,»,,....,+(7».+2)(»w,+l)^„^, „^+2,...„^...(22),
from which all the remaining coefficients in will readily be deduced,
when those of the part 0'^' are known.
10. It now remains, as was before observed, to integrate the ordi-
nary differential equation (17) No. 8. But, by the known theory of
linear equations, the integration of (17) will always become more simple
when we have a particular value satisfying it, and fortunately in the
present case such a value may always be obtained from by simply
changing f, into ' , . In fact if we represent the value thus ob-
tained by Ho we shall have
cih ^' </e/«v(2«:')'
and by a second differentiation
3H2
414 Mr green, ON THE DETERMINATION OF THE
(22) as before comprising all the ~ — — combinations of the * indices
taken in pairs.
Hence, the quantity on the right side of the equation (17), when
we make H = Ho, becomes
+ 2(22)^^.-^;^,+ («-2^)2^.--^ (23).
d^rd^r a,«r'2«/ V «r / «^r o, 'v/(2«;^) ^
But if we recollect that we have generally
it is easy to perceive that in consequence of the equation (18) the
quantity (23) will vanish, and therefore the foregoing value of Ha
will always satisfy the equation (17).
Having thus a particular value of H, we immediately get the
general one by assuming
H= Hfzdh.
In fact, there thence results
H = KHj „„ '^"^^ .
±l(,~ Ux, Oi, (h a,
the two arbitrary constants which the general integral ought to con-
tain being K, and that which enters implicitly into the indefinite in-
tegral. But the condition = V" requires that H should vanish when
h is infinite, and consequently the particular value adapted to the
present investigation is
n - jr rr f ^""dh
J^ Mo'a^, «2 «»
11. The values of (f> and H being known, we may readily find
the corresponding values of V and p. For we have immediately
ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 415
r=i/^ = jir^//./-gj£2^ «6),
and as the function (p is rational and entire, and the partial differen-
tial of f^ relative to h is finite, it follows that all the partial differ-
entials of F^ are finite; and consequently, by what precedes (No. 7.)
the condition (9') is satisfied by the foregoing value of F', as well as
the equation (2) and condition = F". Hence the equations {b) and
(c) No. 6 will give, since
du- "V ^' ~^) Y" ^■d^~~dh\'
and h must be supposed equal to zero in these equations
- r f^^ii)
p' = — , A. .---V^-^^ (where h = 0);
since where A = 0, a, = «/ ; and therefore,
1 - 2/^' ^^ = 1 - ^r' V = ^'.
If now we substitute for V its value (26), and recollect that « — * + 1 is
always positive, we get
-r(^) ^
27r^r
(^4^)
since it is clear from the form of Ho that this quantity may always
be expanded in a series of the entire powers of A^ In the preceding
expression, (27), H^ indicates the value of Ho when h = 0, and (p!
the corresponding value of or that which would be obtained by
simply changing the unaccented letter fi, ^2, ^, into the accented
ones ^1', f/, ?/ deduced from
(7) x; = a,'?/ ; x.^ = «; ^/ ; x/ = «/ ^;.
416 Mr green, ON THE DETERMINATION OF THE
It will now be easy to obtain the value of V corresponding to
without integrating the formula (1) No 1, where F is the character-
istic of any rational and entire function. In fact it is easy to see from
what precedes (No. 9), that we may always expand JF' in a finite series
of the form
F{xl, x-l xl) = bo^o + ii0i' + bo(p2 + 63^3' + &c.
after a;/, x-J, &c. have been replaced with their values (7). Hence, we
immediately get
p' = „"-«-' . {bo(po' + b,<p! + h(p; + &c.} (29).
By comparing the formulae (26) and (27) it is clear that any term,
as 5,0/ for instance, of the series entering into p, will have for cor-
responding term in the required value of V, the quantity
^ ^ i^„'«/< a:.b.<pM.f-j j^^'"/^ ^ (30):
''co -'^O "1 Ms (Is
Ha being a particular value of H satisfying the equation (17), and
immediately deducible from (p by the method before explained.
12. AU that now remains, is to demonstrate that
V'V0 = VV> (31),
whatever <p may be. For this purpose let us here resume the value
of A0, as immediately deduced from the equation (2") No. 7, viz.
+ A^2^-A^2lx2i^ (32),
P /w-1'
ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 417
where for simplicity the indices at the foot of the letters ^ and a have
been omitted, and their accents transferred to the letters themselves.
Moreover all the finite integrals are supposed taken from 1 to «+l.
By making A = in the last expression we immediately get v'<^,
and if for a moment, to prevent ambiguity, we write h, in the place
of the original «'r and omit the lower indices as before, we obtain
V>=(l-2a2^4p+(*-«-l)2|^ (33);
where to avoid all risk of confusion r has been changed into r" , and
the double accent of this index transferred to the letters ^ and h
themselves.
We will now conceive the expression (32) to be written in the
abridged form
the order of the terms remaining unchanged.
If then we recollect that the accents have no other office to per-
form than to keep the various finite integrations quite distinct, and
consequently that in the final results they may be permuted in any
way at will, we shall readily get
V'Va^ — VaV'0 =
(l-Sf)(4S2 ,^„,„ .-T^>-' +22-^, X 2^-^ I
+ 42^'x22-^.-,p^^, +22^x2^x2^-^
418 Mr green, ON THE DETERMINATION OF THE
^ ' [ a^b^ aa^\^^^ a^ c^V af (n) J
(1 - 2P^|-422— i^^^^— _22-L ^ 1
P g rf'0 P (/'^
2(l-2f)2JL.^ +22i:x2^^^
-2 (1-2^2^x2— ^ -2 2-x2^x2-^
all the finite integrals being taken from ?• = 1 to r = s + 1, and from
/ = 1 to r' = s + 1.
In order to obtain the required value
v'v^ - w'<p,
it is clear that we shall only have to add the first of the five preceding
quantities to the sum of the four following ones multiplied by A', and
to render this more easy, we have appended to each of the terms in the
preceding quantities a number inclosed in a small parenthesis.
Now since the accents may be permuted at will, and we have like-
wise or = b^ + U, it is easy to see that the terms marked (1), (6) and
(12) mutually destroy each other. In like manner, (2), (3), (7) and
(18) mutually destroy each other; the same may evidently be said of
(13) and (16), of (15) and (17), of (9) and (19), and of (8) and (14).
Moreover the four quantities (4), (5), (10) and (11) will do so likewise,
and consequently, we have
V'V0 - VV> = 0.
Hence the truth of the equation (31) is manifest.
ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 419
Application of the preceding General Theory to the Determination of the
Attractions of Ellipsoids.
13. Suppose it is required to determine the attractions exerted by
an ellipsoid whose semi-axes are a', b', c' whether the attracted point
p is situated within the ellipsoid or not, the law of the attraction being
inversely as the w"*" power of the distance. Then it is well known
that the required attractions may always be deduced from the function
j^ _ r p' dx' dy' dx
{{x ~x'f + {y-yj + {x-%jy^ '
p being the density of the element dx' dy' d%' of the ellipsoid, and
X, y, % being the rectangular co-ordinates of p.
We may avoid the breach of the law of continuity which takes
place in the value of V, when the point p passes from the interior of
the ellipsoid into the exterior space, by adding the positive quantity
M* to that inclosed in the braces, and may afterwards suppose u eva-
nescent in the final result. Let us therefore now consider the function.
r=/
p' dx' dy' d%'
{{X - x'y + (y- y'y + (z- zy + M^p ' '
this triple integral like the preceding including all the values of x', tf, »',
admitted by the condition
,/2 ^-^
— + — + — Z 1
If now we suppose the density /o' is of the form
f^'^i^-T^^-h-z^ ' /(^',y.«') (34). .
which will simplify / {x', y, »') when p is constant and n' = 2, and then
compare this value with the one immediately deducible from the general
expression (28) by supposing for a moment n' = n, viz.
Vol. V. Part III. 3 1
420 Mr green, ON THE DETERMINATION OF THE
we see that the function f will always be two degrees higher than F.
But since our formula become more complicated in proportion as the
degree of F is higher, it will be simpler to determine the differentials
of V, because for these differentials the degree of F and f is the same.
Let us therefore make
, _ 1 dV _ r /o' (« — x) dx dy' d %
~ 1 m' fir ~ J >rTi '
i,ia;-x'Y + iy-i/r + {z-zr + u''} —
then this quantity naturally divides itself into two parts, such that
A =xA' + A",
, ,, /- p dx dy d%
where A' — -^r J '^ ;;rr\ ,
{{x -x'Y + {y- yj + [%-%)' + u^}~
and A"=-f~
x'p dx' dy' dx
{{x-xy + {y-yy + {z-%y + u^~
By comparing these with the general formula (1), it is clear that
M — 1 = n' + 1, and consequently n = n + 2. In this way the expression
(28) gives
which coincides with (34) by supposing F=f.
The simplest case of the present theory is where y(a;', y', x') = l, and
then by No 11, we have 0o'= 1 and &„ = 1. when A is the quantity
required, and as the general series (29), No 11, then reduces itself to
its first term, we immediately obtain from the formula (30), the value
of A! following,
* A= , — -- (the \ — 7 — (35),
2
because in the present case H^, = 1, « = 3, and n = n' ^ 2.
Again, the same general theory being applied to the value of A"
given above, we get
ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 421
F («', y', %') = - x'fix, y', a') = - ar' (when / = 1),
and hence by Noll, F{x',y',%') — — a'l'. In this way the series (29)
again reduces itself to a single term, in which
and the particular value H^ corresponding thereto, by omitting the super-
fluous constant /(fj2,i,'i — tk will be (No 10),
Ho — a.
These substituted in the general formula (30) as before, immediately give
A" . ^""^^UJ ,3,, ,„ c ^'-'^dh
A = -^ ; — 7- « o C Pa / — rr — ,
and consequently by reduction since a^ = x,
A=xA' + A''=-^-l-^r^l a'b'c'x f^^ (36).
r ( "^ ] "
The value of A just given belongs to the density
' _ fi _ ^" y! _ ?!\^
Hence we immediately obtain without calculation the corresponding
values
1 dV_ ^^^^U) ,,, , f h^-'^dh
1 dV *'"'' (2) ,-, , rU-'^'dh
2^5 r
r
C = :; , -y— = ;f — r" abc% / — , ^ .
l~2~j
31 2
422 Mr green, -ON THE DETERMINATION OF THE
If now we suppose moreover
__ 1 d^ _ f f> dx dy d%
the method before explained (No 11), will immediately give
Z) = — , — T— a cu / T — ,
p/ w'+ 1 \ J^ abc
and therefore if for abridgment we make
-'^d') .,,
the total differential of V may be written
rfr=i»f{2^rf^/^-^^ + 2s.rfy/^-^^ + 2.t/./^— j-^ + 2«rf./^-^^},
which being integrated in the usual way by first supposing h constant,
and then completing the integral with a function of h, to be after-
wards determined by making every thing in F variable, we get
A being a quantity absolutely constant, which is equal to zero when
w' > 1. What has just been advanced will be quite clear if we recollect
that h may be regarded as a function of x, y, % and u, determined by
the equation
» = ?^' + 4^ + ?4^ ^- 1 - f + -' + f' -^ ' <''^>'
seeing that a' = a'*-\-h\ V = b'' + h\ and & = c" + h\
After what precedes, it seems needless to enter into an examination
of the values of V belonging to other values of the density p, since
it must be clear that the general method is equally applicable when
ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 423
where f is the characteristic of any rational and entire function.
The quantity A before determined when we make u = 0, serves to
express the attraction in the direction of the co-ordinate x of an ellipsoid
on any point p, situated at will either within or without it. But by
making « = in (37) we have
, _ a^ y" z' ^
a" + h' "*■ b'-' + li' '^ c" + h' ^ h' ^^ ^'
and it is thence easy to perceive that when p is within the ellipsoid,
h must constantly remain equal to zero, and the equation (38) will always
be satisfied by the indeterminate positive quantity — . When on the
contrary p is exterior to it, h can no longer remain equal to zero, but
must be such a function of x, y, %, as will satisfy the equation (38), of
which the last term now evidently vanishes in consequence of the
numerator o'. Thus the forms of the quantities A, B, C, D and F"
all remain unchanged, and the discontinuity in each of them falls upon
the quantity k.
To compare the value of A here found with that obtained by the
ordinary methods, we shall simply have to make n' = 2 in the expression
(36), recollecting that r(l) = 1, and r (-] =i\/7r. In this way
, .,,,,/- hdh ^ ,,, , r da
A = — Aiiraoc X \ -rrr- = — 4nrab c x / -7^—
J^ctbc J„drbc
= + ^a'h'c'x f 4?- = 4-«'*'c' J . , f " ^
But the last quantity may easily be put under the form of a definite
integral, by writing - in the place of a under the sign of integration,
and again inverting the limits. Thus there wiU result
J 47r«'J'c' /•! v"dv
^ = 'n^~ J
a •'o
a + ^«^)(i + ^-/-^^)
a' ' a-
424 Mr green, ON THE DETERMINATION OF THE
which agrees with the ordinary formula, since the mass of the ellipsoid
47r«'6'c'
IS
3
and «^ = d^ + h\
Examination of a j)articular Case of the General Theory exposed in the
former Part of this Paper.
14. There is a particular case of the general theory first considered,
which merits notice, in consequence of the simplicity of the results to
which it leads. The case in question is that where we have generally
Avhatever /• may be
a/ = a.
Then the equation (19) which serves to determine 0, becomes by
supposing kn = k . a"'
= il-'2r'^r')^r'^, + (.v-»-l)2/*'?.^-A<^ (39).
If now we employ a transformation similar to that used in obtaining
the formula (14), No 6, by making
^i = P cos 9i, ^2 = p sin 9i cos On, ^3 = p sin 0, sin 9.^ cos 63, &c.
and then conceive the equation (39) deduced from the condition that
'"^'"i- "f-d-sf =)'^ {^■*' (f )' - r^l
must be a minimum (vide No 8), we shall have
rf^,rf^2 c?f,, = p'-' sin0/-^ sina/-^ sin 9^-, dp de,d9, «?0,_„
\d^rl \dp) p" ' sin0,^sin0/ sine^,-,'
and 1 - 2^,= = 1 - p\
Proceeding now in the manner before explained, (No 8), we obtain
for the equivalent of (39) by reduction
d''(j) , ( _ _ , V cos g,. d(f>
d'<p s-l-np' dcl> 1 d9;''^^^~'' Um9rd9,. k
dp' ^ p{l-p') 'dp p' ' sm9,'sm9^' sin0^,., 1 _^2 9-V*U).
ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 425
But this equation may be satisfied by a function of the form
= Pe,e,e3 e,_,;
P being a function of p only, and afterwards generally 6, a function
of dr only. In fact, if we substitute this value of (p in (40), and then
divide the result by ^, it is clear that it will be satisfied by the system
e,,.,</e\_,
— ^<- 1
d'Q,.^ , _ cos0,_2 o?e,_2 , X._, ^ .^^^
Os-2 de^._s ' sin 0,_2 9s_2</0s_2 sin 6^,-2
+ 2 . -7— -p^ — 77 j7i h -; — —^ — = X,-
B._3d0\.3 ' sin 9^-3 Qs-3d9,_3 sin0^
&;c. &c. &c. &c.
combined with the following equation,
d'P s-l-np' dP \, k ^
P^p" ^ /" (1 -p') ■ P«?/' />' 1 -p'
where k, X,, X^, X3, &c. are constant quantities.
In order to resolve the system (41), let us here consider the general
type of the equations therein contained, viz,
- ^'Q'- , (r-i\ ^"^^-- '^^^ + ( ^'-^' X "i ft
d9\_,. ^ >sm9,_/d9,.r \sm9\., a,_,j «,.,.
Now if we reflect on the nature of the results obtained in a preceding
part of this paper, it will not be difficult to see that 6,_r is of the form
e,_. = (sine._,)*;j = (1-M^)«;>;
where j9 is a rational and entire fimction of m = cos0s_r, and / a whole
number.
By substituting this value in the general type and making
\..r^i = - i{i + r - 2) (43)
we readily obtain
= {1-M.')^: -{2i + r)^^- {X._. + i{i + r-l)}p.
426 Mr green, ON THE DETERMINATION OF THE
To satisfy this equation, let us assume
Then by substituting in the above and equating separately the coefficients
of the various powers of yu, we have in the first place from the highest
X._, = - e{e^-r—\) (44),
and afterwards generally
. e-i-9.t .e-i-M-\ .
'*' ~ ~ 2/ + 2x2e + r-2#-3 "
But the equation (43) may evidently be made to coincide with (44), by
writing «*''' for i, and t^''+'^ for e, since then both will be comprised in
\,_,+, = - e*--' {e<'-* + r-2| (45).
Hence we readily get for the general solution of the system (41),
"^ 2.4 X {2f<'-> + r-3|{2«"-' + r-5} " - &C.J ;
where w = cos 9,_r, and i*''* represents any positive integer whatever, pro-
vided ^''■' is never greater than ^*'■*".
Though we have thus the solution of every equation in the system
(41), yet that of the first may be obtained under a simpler form by
writing therein for X^.i its value — i® deduced from (45). We shall
then immediately perceive that it is satisfied by
cos [ J
In consequence of the formula (45), the equation (42) becomes
^- dp' ^ pO-p') dp \ / '^T^'i^'
which is satisfied by making ^= —\, -(«'*' + 2ft)) (e"*'' + 2a) + w — l), and
ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 427
p = „i") f„2. _ 2a>x2i'"+2^ + v?-2 o„_2
t' -(H 2 ^ 2i''» + 4w + »-3 ^
, 2«) . 2a)-2 X 2i'~' + 2ai + *-2 .2i'^' + 2w + *-4 , - „ ,
2.4 X 2«<-' + 4to + «-3,2« + 4w + w-5 ^ '
where w represents any whole positive number.
Having thus determined all the factors of (f>, it now only remains
to deduce the corresponding value of H. But Ho the particular value
satisfying the differential equation in H, will be had from by simply
making therein
since in the present case we have generally «/ = «'.
Hence, it is clear that the proper values of 0,, di, 9^, &c. to be here
employed are all constant, and consequently the factor
0102 ©3 ©s-l
entering into (f> is likewise constant. Neglecting therefore this factor
as superfluous, we get for the particular value of H,
a'
since ,0^ = ?.= + ?/ + + ?/ = ^^ = fj ,
ga a
and Pa represents what P becomes when p is clianged into — .
a' ^
Substituting this value of Ho in the equation (25), No 10, there
results since a' = a'^ + h^
H=K.P„ f ^'"""^^ , (46)
a'
K being an arbitrary constant quantity.
Thus the complete value of V for the particular case considered in
the present number is
Vol. V. Part III. 3 K
428 Mr green, ON THE DETERMINATION OF THE
v= pe.e,......e..,.irp« f "^ , (47)
a'
and the equation (27), No 11, will give for the corresponding value of p',
in - IN
-m
M-4-1
K
where P/, 9/, 62', &c. are the values which the functions P, 0i, 02, &c.
take when we change the unaccented variables fi, ^2, ^, into the cor-
responding accented ones ^/, ^/, f/, and
p «-^ + l-w — ^ + 3 n — s + 2a}-l
' ~ » + 2i + 2ft)-l .7^ + 2^ + 2(0 + 1 n + 2i + 4<w-3'
or the value of P when p = 1 ; where as well as in what follows i
is written in the place of i'''.
The differential equation which serves to determine H when we
introduce a instead of h as independent variable, may in the present
case be written under the form
. = a=(a^-«'^) Vr + «M»«'-(*- !)•«"} ^
^ ' dcf * ' ada
+ {?■(« + *- 2) a'' -(« + 2ft.)(« + 2a) + w-l)a'} H,
and the particular integral here required is that which vanishes when
h is infinite. Moreover it is easy to prove, by expanding in series, that
this particular integral is
*-l-n-2<o
provided we make the variable r to which A" refers, vanish after all
the operations have been effected.
But the constant k' may be determined by comparing the coefficient
of the highest power of a in the expansion of the last formula with
the like coefficient in that of the expression (46), and thus we have
" yfc' = Kd'^"" (-\Y « + 2^' + 2a)-l.w + 2? + 2a> + l ?^ + 2^^ + 4a,^.-3
^ ^ 2.4 . 6 2o) .
ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 429
Hence we readily get for the equivalent of (47),
rr vtc^ c^ ^ « + 2« + 2a)-l .M + 2e + 2a)4-l n + 2i + 4im-3
2.4. 6 2w
xKa'^'-'''"{-l)''a'A''a"-fdaa'-"'-'-"{a^-a"') ^
■ ■ GO
In certain cases the value of V just obtained will be found more
convenient than the foregoing one (47). Suppose for instance we repre-
sent the value of f^ when h = 0, or a = a' by V^. Then we shall hence
get
r^ i»c> o o n + 2i+2a)-l .n + 2i + 2w + l » + 2i + 4ft.-3
2.4,6 2a>
g — l~n~Suo
OD
which in consequence of the well known formula
r(,-p)r(H±f^)
/"'a-'da (a' - a'')-" = - «''-"-^? x -J^ i ,
by reduction becomes
fl+s — n\^[n + 2i + 4!w — l'^
r(l±|z^)rp±-±i^)
2r(a,+ i)r( ^ + ^'^+^" )
since in the formula (5), r ought to be made equal to zero at the end
of the process.
By conceiving the auxiliary variable u to vanish, it will become clear
from what has been advanced in the preceding number, that the values
of the function P within circular planes and spheres, are only particular
cases of the more general one, (49), which answer to * = 2 and s = 3
respectively. We have thus by combining the expressions (48) and
(49), the means of determining Vo when the density p is given, and
vice versa; and the present method of resolving these problems seems
more simple if possible than that contained in the articles (4) and (5)
of my former paper.
GEORGE GREEN.
3k2
XVI. On the Position of the Axes of Optical Elasticity in Crystals
belonging to the Ohlique-Prismatic System. By W. H. Millek,
A.M. Fellow and Tutor of St John's College, and Professor of
Mineralogy.
[Read Dec. 8, 1834.]
1. Fresnel has proved that whatever be the regular arrangement
of the medium which by its elasticity produces the optical properties
of a crystal, there are always three directions at right angles to each
other, which may be considered as axes of optical elasticity. This
being understood, it is further already established, that crystals belong-
ing to the tesseral system have three equal axes of optical elasticity ;
that rhombohedral and pyramidal crystals have two axes of elasticity
equal to each other and perpendicular to the crystallographic axis,
which therefore is the third axis of elasticity and also an optic axis;
and that crystals belonging to the remaining systems have three unequal
axes of elasticity, and consequently two optic axes (that is, axes of
optical phenomena) making with each other angles which are bisected
by the axes of greatest and least elasticity.
Sir David Brewster, who discovered the mutual dependence of the
forms and optical properties of crystals, has determined the angles be-
tween the optic axes of a great number of biaxal crystals; his obser-
vations, however, do not contain any data from which the positions of
the axes with respect to the faces of the crystals can be found.
2. In the right prismatic system the axes of elasticity coincide (as
might have been expected) with the rectangular crystallographic axes.
In the oblique prismatic system, if the three axes be XX', YY', ZZ',
the crystallographic axis {YY'), which is perpendicular to the other
two {XX', ZZ'), is always one of the axes of elasticity. This, in
Gypsum, at the ordinary temperature of the air, and in many other
crj'stals, is the mean axis, or it is perpendicular to the optic axes; in
482 PROFESSOR MILLER, ON THE POSITION OF THE AXES OF
Borax, Acetate of Soda, Felspar, Tartaric Acid and Gypsum, when heated
to about lOO^C, as was first observed by Mitscherlich, it is the greatest
or least axis of elasticity, and is therefore in the same plane with the
optic axes and makes equal angles with them.
The position of one axis of elasticity having thus an evident rela-
tion to the crystallographic form, we are naturally led to inquire if
any relation can be discovered between the other two axes of elasti-
city and the crystallographic form. The only attempts to discover any
such relation, with which I am acquainted, are those of M. Soret,
(Memoires de la Socidte de Physique de Geneve, tome I.) and Pro-
fessor Neumann of Konigsberg (Poggendorff's Annalen, B. xxvii. S.
240). Neumann shews, that in Gypsum the axes of elasticity and also
the thermal axes, or the three lines in the crystal which remain at
right angles to each other at all temperatures, constitute a system of
rectangular crystallographic axes. It appeared at first sight not im-
probable that a similar relation might be found to exist between the
form and axes of elasticity of other oblique-prismatic crystals. Though
my observations appear to disprove the law which has thus been sug-
gested, they do not establish any other in the place of it. The only
general fact which I have noticed is, that in many instances, though not
in all, one of the two axes of elasticity which are perpendicular to
W, is also the axis of one of the principal zones of the crystal.
3. To find the angle between a normal to any face {T) of a
crystal, and the apparent direction of one of the optic axes as seen in
air through any parallel faces of the crystal.
Let the crystal be attached to an index, moveable on a graduated
circle having its plane parallel to the axis of the polarizing instrument,
or a table on which the position of the index may be marked by a
line drawn along its edge with a tracing point. Let the crystal be
placed in such a position, that the apparent direction of the optic axis
in air and a normal to T may be parallel to the circle. Move the
index till the center of the coloured rings coincides with a mark in
the axis of the polarizing instrument, and observe the points in which
it meets the circle. Turn the crystal half round in the plane of T,
OPTICAL ELASTICITY IN OBLIQUE-PRISMATIC CRYSTALS. 433
taking care not to alter the inclination of T to the index, (this may
be effected by moving the crystal, the index being fixed, tiU the image
of some well defined object seen by reflexion in T appears in the
same direction after the crystal is turned as it did before.) If the
index be now turned till the center of the coloured rings coincides
with the mark, the angle it has described between the observations
will be manifestly equal to twice the angle between the apparent
direction of the optic axis in air and a normal to T. The angle
between the optic axis in air and a normal to any other known face
of the crystal being found in the same manner, the direction of the
optic axis in air wiU be completely determined.
4. To find the optic axes, their apparent directions in air being
known.
Let Qlt, Q'K (Fig. 1.) be tangents to the circular and eUiptic sec-
tions of a wave diverging from O made by a plane through the optic
axes, and therefore OQ, OQ', perpendiculars to QB, will be the optic
axes; OP the direction in which the optic axis OQ is seen in air;
OS a perpendicular to the faces through which it is seen.
The vibrations in that part of the wave which has a circular sec-
tion are perpendicular to the plane QOQ, consequently a ray polar-
ized in the plane QOQ is refracted in that plane according to the
law of sines. Let m be the ratio of the sine of incidence to the sine
of refraction for such a ray out of air into the crystal, D the mini-
mum division of the ray when refracted in the plane QOQ' through
the prism formed by two natural or artificial planes meeting at an
angle / in a line perpendicular to QOQ. Then ^ sin ^ / = sin ^ (Z) + /),
and fM sin QOS = sin POS. Whence the direction of QO is known.
0*0 being found in the same manner, the axes of elasticity O^, Oi[,
which bisect the angles qOQ, QOQ, are also known.
5. The diagram which accompanies the description of each crystal,
is the representation of a sphere, to the surface of which the faces of
the crystal are referred by means of perpendiculars drawn from the
center of the sphere. The point in which the perpendicular to any
434 PROFESSOR MILLER, ON THE POSITION OF THE AXES OF
face meets the surface of the sphere, will be called the pole of that
face. The measurements express the angles between the perpendiculars
to the faces, or the supplements to the angles between the faces them-
selves. This method of representing crystalline forms appears to have
been first employed by Neumann, in his Beitrage zur Krystallonomie,
and afterwards by Grassmann and Uhde. It has the advantage of ex-
hibiting all the faces of a crystal without confusion in one figure,
each zone being distinguished by a great circle drawn through the
poles of the faces composing it, and also of allowing all the requisite
calculations to be performed by spherical trigonometry applied to the
equations
T cos PX = T cos PY = 7 cos PZ,
h k I
or to formulse deduced therefrom, X, Y, Z being the points in which
radii parallel to the axes of the crystal meet the surface of the sphere,
and P the pole of the face {h; k\ l), which is parallel to the plane
h- + k\-^ I- = 0.
a b c
ad, /3/3', ^f, ^^' will be used to denote the extremities of diameters
drawn parallel to the optic axes, and the two axes of elasticity which
are perpendicular to YY' . In Figs. 5, 6, 7, 8 the faces are denoted
by the same letters as in the treatises of Mohs and Naumann. The in-
clinations of the faces of crystal (1) and (2) are deduced from a mean
of the best measurements of thirty or forty crystals, and are probably
within 1' of the truth.
The chemical notation and atomic weights are those employed by
Dr Turner, in the fifth edition of his Elements of Chemistry.
OPTICAL ELASTICITY IN OBLIQUE-PRISMATIC CRYSTALS. 435
EXAMINATION OF VARIOUS CRYSTALS ACCORDING TO THE METHODS
ABOVE EXPLAINED.
(1). Sulphate of Oxide of Iron and Ammonia. According to Mit-
scherlich (Jahresbericht 13), the composition of this salt, which belongs
to an extensive plesiomorphous group, is expresssed by the formula
H'^NS ■\- FeS -^ 1 H. Fig. 2. represents the poles of its faces. Their
symbols are A{1; 0; 0), C(0; 0; 1), H{0; 1 ; 1 ;) M{1', 1; 0),
P(l; 1; 1), Q(-l; 1; 1), T{2; 0; 1).
AT 42", 14'
CYQ
28»,48'
HH'
129°, 18'
MQ 42»,23'
TC 64 ,34
QYA'
44 ,54
CH
25 ,21
QC 34 ,20
CA' 73 ,12
MM'
109 ,36
QQ'
140 ,55
TP 35,14
AYP 68 ,12-1-
AM
35,12
CP
44,45
MT 52 ,46.
PYC 38 ,35i
PP'
130,37
PM
58 ,32
When yellow light is refracted through the faces TC in the plane
AC A', the minimum deviation of a ray polarized in the plane AC A',
is 41", 26'. The apparent direction of the optic axis aa in air, when
seen through the faces TT', makes an angle of 7°,10' with 2'2";
and the optic axes appear to be inclined to each other at an angle of
79" when the crystal is immersed in oil, of which the index of re-
fraction is 1,47. From these data we find Ta = 4'',47', Tfi = 71°,2',
r^ = 33'',8', A^ = 9'',6'.
Tan T^ is nearly equal to 4tan>4f. The value of A^ deduced from
the equations tan Tf = 4tan^^, T^ + ^^ = 42°, 14' is 9°, 13'i. This
would make Q = 82'',25'i. Now, 46" tan 9'^ 13'^ = tan 82'',22'| ; therefore,
if we refer the faces T, A, C, to the rectangular axes ff, YV, ^^',
neglecting the difference of 3' in the value of C^, their simplest symbols
will be (1; 0; 1), (4; 0; -1), (2 ; ; -23). The magnitude of the last
index renders the hypothesis that ^f , ^^' are crystallographic axes highly
improbable.
(2). The composition of Tartrate of Ammo7iia is expressed, according
to Dulk, (Jahrbuch fiir Chemie und Physik, 1831. B. 1.) by the for-
mula H^NT+^H. The poles of its faces are represented in Fig. 3.
Vol. V. Paet III. sL
486 PROFESSOR MILLER, ON THE POSITION OF THE AXES OF
A{i; 0; 0), C(0; 0; 1), H{0; 1; 1), 2^(1; 0; 1), L{-1', 0; 1),
3f(l; 1; 0), P(l; 1; 1), Q(-l; 1 ; 1).
Cleavage parallel to the face A.
AK 520,31' AM 55»,2' Qd gr^a?' QA' 6o»,54'i
KC 39,53 HH' 81, 4.6 QL 41 ,l6^ CP 55,34
CL 38 CH 49 ,7 ^i* 63 ,22 PM 35 ,48^
L^' 49,36 PP' 94,55 PIT 28 ,12 JWQ' 34 ,53
iIfil/'69,56 PX 42,32 /TQ 27,31i QC 53,44-i.
Z> = 25'',17', the light being refracted through CK. The apparent
angle in air between the optic axes aa and AA', is 4°,55'. In oil,
the index of refraction of which is 1,741, the apparent angle between
the optic axes =42'',20'. This gives ^0 = 8", 7', A(i = 35'',54>', A^=W,M',
i'^ = 33°,12.
In this case the positions of some of the faces A, K, C, L must be
altered half a degree before they can be referred to the rectangular
axes ff, W, ^^' with tolerably simple indices.
(3). A solution of Benzoic acid in alcohol, when suffered to eva-
porate, affords crystals of which the faces C, K, I (Fig. 4) alone are
bright. Cifr= 69",25', C/=97'',20' nearly. Z) = 64°,45', refraction taking
place through the faces CK. The apparent direction of ad in air
when seen through CC makes with CC an angle of 4°, 30'. When
immersed in oil of which the index of refraction is 1,471, the appa-
rent angle between the optic axes is 75". Hence Ca =2'',47', C/3=59'',50',
C^=28»,31', ^^=40^54'.
tan K^, tan I^, tan C^ are nearly as the numbers 3, 1, 5.
The equation 1 tan /iT^ = tan 7^ = |^ tan C^ is satisfied by making Cf=
27°,56'^, JC = 97'',17'. Hence the faces C, I, K may be referred to the
rectangular axes ^f, YY , l^ without greatly altering the observed
angles, and their symbols will be (-1 ; 0; 5), (1 ; 0; 1), (1 ; ; 3) re-
spectively.
OPTICAL ELASTICITY IN OBLIQUE-PRISMATIC CRYSTALS. 437
(4). In Felspar (Fig. 5.) the optic axes lie in the plane of the
most perfect cleavage, and make with a normal to M, angles of about
57" or 58°, (58^^ according to Sir David Brewster) which increase when
the crystal is heated. Hence, ^^' is the axis of the zone PM.
(5). The optic axes of Pyroxene (Fig. 6.) seen in air through a
slice cut perpendicular to MM are in the plane Pr, and make angles
of 16" with the axis of the zone MM. Hence, ^^' is the axis of the
zone MM', a, /3 approach ^ when the crystal is heated. At ordinary
temperatures a/3 is probably about 19"^. The best measurements of
Pyroxene shew that Pr, tr are nearly but not exactly equal, and
therefore, that its faces cannot be referred to ^^', YY', X,'C as crystallo-
graphic axes. In all the crystals of Pyroxene which I have examined,
the rings surrounding ad are brighter than the rings surrounding /3/3'.
(6). The form of Borax (Fig. 7.) closely resembles that of Py-
roxene ; its optic axes however are very differently situated. It was
observed by Sir John Herschel and also by Professor Nbrrenberg, that
the optic axes for different colours do not lie in the same plane. This
being the case, we cannot expect to find any simple connexion between
the form and the directions of the axes of elasticity.
The mean directions of the axes seen in air through the faces 7'T"
make angles of aO"^, with a normal to the faces TT', and a perpendi-
cular to them makes an angle of 55° with MM'. The rings sur-
rounding ad, /3/3' are indistinct on the sides towards M'P and MP'
respectively, the extremities a, /3 of the axes being next to the eye of
the observer. This shews that the positions of ^f', ^^' vary slightly
with the colour of the light employed.
(7). In Chromate of Oxide of Lead, as I have been informed by
Professor Norrenberg of Tiibingen, the axis of the zone MM (see the
figure in Phillips or Naumann) bisects the angle between the optic
axes, and is therefore one of the axes of elasticity. The other two
axes of elasticity are, without doubt, the lines which bisect the angles
formed by normals to MM'.
438
PROFESSOR MILLER, ON THE POSITION OF THE AXES, &c.
(8). In Epidote, (Fig. 8.) the optic axis aa seen in air through the
faces r, r, makes with r r' an angle of 8^ 50', /3/3' seen in air through
the faces M, M', makes with MM' an angle of Sl^SO'. The determina-
tion of m is rendered difficult by the complete absorption of the light
polarized in the plane MT. Assuming /x = 1,7, which is probably near
the truth, we get /•a=5'',ll', M/3=18°,5'. According to Mohs 2V=51'',41',
TM=64>'>,30', therefore, T'a=46",30', 2)3 = 46'\31'. Hence ^^' is the axis
of the zone PT. The near approximation of the values of 7'a, Tfi to
equality must be considered accidental, as the positions of the optic axes
are usually uncertain to the amount of some minutes.
The question whether any proposed lines are crystallographic axes
must be decided, as has already been intimated, by the simplicity and
symmetry of the numerical relations which the expression of the faces
requires with reference to these axes. This according to the old Hauyian
views of the structure of crystals, is equivalent to saying that the pri-
mitive form must be such that the other forms can be derived from
it by simple laws of decrement. Now, we find that by assuming the
axes of elasticity to be crystallographical axes, we have in the crystal
(1) a face (2; 0; —23), which though not very probable is not im-
possible, and in (5) a face ( — 1; 0; 5); in (2) the observed and com-
puted positions of some of the faces differ half a degree.
In (6), the optical properties are not symmetrical.
In (4), (5), (7), (8) one of the axes of elasticity f^' or ^^' is the
axis of a zone.
St John's Collegb,
Dec. 8, 18S4.
W. H. MILLER.
Tr^ff/.sojcHen.' (/^ Vu Cam^./'hjrZ. S&c. Vol.S /'i.
Ik,. VI.
Fi^. V.
fi^. IT.
AUuaiTe J.iVw9- ('■amfiritCj/e
TntAMicbjonj o/'the.Qani. Thil.Soc. VfC A fl ji>
Ft:^. I.
rr^.M. ^^
Svg JI. i
■Fi^ V
X ^/A
Ftg. iv:
■V Mff nr.
^ j'i^. im^.
M^/jy^J]k. /iJfij firm/>rT/iii^..
^^ ^rvf^'N!
B. » A c. ' ■ » k T*. ■ ■.■■■■
^Af^^
nnni ^^^^/
v.. AAAA^ ■
r^^:^'^-
'. ^ v;^
^mJ^W^
•> uM
-r^fyn
o-r^.^-
■s'^.T-^--*^*^^.
v-.^'^.,^^ V
^^
■wf^
^^"^'
A'^- . / o "> -ja ,
fyr^rs,
^-';0,^KK:JJ^eKr^rL^^
■^;:/M
.>^^^^'^.^,^ r,^^^A''Y^'
■--.' . .^ lY^^^^r:.^^-^^A^'
"^ '..r^rn.
:-v^r^">:
■f^^^M'^-^^mr^^t&r^ ^...rm
•^^ ' ^^W^C
'^■Acsr^n^Q.r
^Cf\A,^^/" Jt-^:
v^K^^^:
;>iiiS^^
mM:
Cf^.^^^^'^'
i^\M^'^^'-^'"''
5^^T^.
'^'^^>^v
■^
^^tT^nif^^^
M^
^-^^
'^^K,
"'•tA0'
m^^'
Jm
rNfA^r;t).'
Kt4^-^^¥
WJi
,/>U!^^,'
-.AKKKfs;^
iiiilii
mm
VI,