w*th.
MATHEMATICAL
AND
PHYSICAL PAPERS.
CAMBRIDGE WAREHOUSE,
17, PATERNOSTER ROW.
Cambrtogt: DE1GHTON, BELL, AND CO.
lUtpjts: F. A. BROCKHAUS.
MATHEMATICAL
AND
PHYSICAL PAPERS
GEORGE GABRIEL STOKES, M.A., D.C.L, LL.D., F.R.S.,
FELLOW OF PEMBROKE COLLEGE AND LUCASIAN PROFESSOR OF MATHEMATICS
IN THE UNIVERSITY OF CAMBRIDGE.
Reprinted from the Original Journals and Transactions,
with Additional Notes by the Author.
VOL. I.
CambriUg? :
AT THE UNIVERSITY PRESS.
1880
[The rights of translation and reproduction are reserved.]
v/, I
WJftth.
CTantfcrfogc :
PRINTED BY C. J. CLAY, M.A.
AT THE UNIVERSITY PRESS.
PREFACE.
IT is now some years since I was requested by the Syndics
of the University Press to allow my papers on mathematical and
physical subjects, which are scattered over various Transactions and
scientific Journals, to be reprinted in a collected form. Many of
these were written a long time ago, and science has in the mean
time progressed, and it seemed to me doubtful whether it was
worth while now to reprint a series of papers the interest of which
may in good measure be regarded as having passed away. How
ever, several of my scientific friends, and among them those to
whose opinions I naturally pay the greatest deference, strongly
urged me to have the papers reprinted, and I have accordingly
acceded to the request of the Syndics. I regret that in con
sequence of the pressure of other engagements the preparation
of the first volume has been so long in hand.
The arrangement of the papers and the mode of treating them
in other respects were left entirely to myself, but both the Syndics
and my friends advised me to make the reprint full, leaning rather
to the inclusion than exclusion of a paper in doubtful cases. I
have acted on this advice, and in the first volume, now presented
to the public, I have omitted nothing but a few papers which
were merely controversial.
As to the arrangement of the papers, it seemed to me that the
chronological order was the simplest and in many respects the
814004
VI PREFACE.
best. Had an arrangement by subjects been attempted, not only
would it have been difficult in some cases to say under what head
a particular paper should come, but also a later paper on some one
subject would in many cases have depended on a paper on some
different subject which would come perhaps in some later volume,
whereas in the chronological arrangement each paper reaches up
to the level of the author s knowledge at the time, so that forward
reference is not required.
Although notes are added here and there, I have not attempted
to bring the various papers up to the level of the present time. I
have not accordingly as a rule alluded to later researches on the
same subject, unless for some special reason. The notes introduced
in the reprint are enclosed in square brackets in order to distin
guish them from notes belonging to the original papers. To the
extent of these notes therefore, which were specially written for
the reprint, the chronological arrangement is departed from. The
same is the case as regards the last paper in the first volume,
which suggested itself during the preparation for press of the
paper to which it relates. In reprinting the papers, any errors
of inadvertence which may have been discovered are of course
corrected. Mere corrections of this kind are not specified, but
any substantial change or omission is noticed in a footnote or
otherwise.
After full consideration, I determined to introduce an innova
tion in notation which was proposed a great many years ago, for
at least partial use, by the late Professor De Morgan, in his article
on the Calculus of Functions in the Encyclopaedia Metropolitan^
though the proposal seems never to have been taken up. Mathe
maticians have been too little in the habit of considering the
mechanical difficulty of setting up in type the expressions which
they so freely write with the pen ; and where the setting up can
be facilitated with only a trifling departure from existing usage as
regards the appearance of the expression, it seems advisable to
make the change.
Now it seems to me preposterous that a compositor should be
called on to go through the troublesome process of what printers
call justification, merely because an author has occasion to name
PREFACE. Vll
some simple fraction or differential coefficient in the text, in which
term I do not include the formal equations which are usually
printed in the middle of the page. The difficulty may be avoided
by using, in lieu of the bar between the numerator and denomi
nator, some symbol which may be printed on a line with the type.
The symbol " : " is frequently used in expressing ratios ; but for
employment in the text it has the fatal objection that it is appro
priated to mean a colon. The symbol " r " is certainly distinctive,
but it is inconveniently long, and dy r dx for a differential coef
ficient would hardly be tolerated. Now simple fractions are fre
quently written with a slant line instead of the horizontal bar
separating the numerator from the denominator, merely for the
sake of rapidity of writing. If we simply consent to allow the
same to appear in print, the difficulty will be got over, and a
differential coefficient which we have occasion to name in the text
may be printed as dyjdx. The type for the slant line already
exists, being called a solidus.
On mentioning to some of my friends my intention to use
the "solidus" notation, it met with a good deal of approval, and
some of them expressed their readiness to join me in the use of it,
amongst whom I may name Sir William Thomson and the late
Professor Clerk Maxwell.
In the formal equations I have mostly preserved the ordinary
notation. There is however one exception. It frequently happens
that we have to deal with fractions of which the numerator and
denominator involve exponentials the indices of which are fractions
themselves. Such expressions are extremely troublesome to set
up in type in the ordinary notation. But by merely using the
solidus for the fractions which form the indices, the setting up
of the expression is made comparatively easy, while yet there
is not much departure from the appearance of the expressions
according to the ordinary notation. Such exponential expressions
are commonly associated with circular functions; and though it
would not otherwise have been necessary, it seemed desirable
to employ the solidus notation for the fraction under the symbol
"sin" or "cos," in order to preserve the similarity of appearance
between the exponential and circular functions.
Vlll PREFACE.
In the use of the solidus it seems convenient to enact that
it shall as far as possible take the place of the horizontal bar
for which it stands, and accordingly that what stands immediately
on the two sides of it shall be regarded as welded into one. Thus
sin mrx/a means sin (mrx f a), and not (sin mrx} + a. This welding
action may be arrested when necessary by a stop : thus sin nO . /r n
means (sin nd) f r n and not sin (n9 r r n ).
The only objection that I have heard suggested against the
solidus notation on the ground of its being already appropriated
to something else, relates to a condensed notation sometimes
employed for factorials, according to which x (x + a) . . . to n
factors is expressed by x nla or by x nja . I do not think the ob
jection is a serious one. There is no risk of the solidus notation,
as I have employed it, being mistaken for the expression of
factorials; of the two factorial notations just given, that with
the separating line vertical seems to be the more common, and
might be adhered to when factorials are intended ; and if a
greater distinction were desired, a factorial might be printed
in the condensed notation as x n ^ a , where the " ( " would serve
to recall the parentheses in the expression written at length.
G. G. STOKES.
CAMBKIDGE,
August 16, 1880.
CONTENTS.
PAGE
On the Steady Motion of Incompressible Fluids 1
On some cases of Fluid Motion 17
On the Motion of a Piston and of the Air in a Cylinder 69
On the Theories of the Internal Friction of Fluids in Motion, and of the
Equilibrium and Motion of Elastic Solids 75
SECTION I. Explanation of the Theory of Fluid Motion proposed. Form
ation of the Differential Equations. Application of these Equations
to a few simple cases 78
SECTION II. Objections to Lagrange s proof of the theorem that if
udx+vdy + wdz is an exact differential at any one instant it is always
so, the pressure being supposed equal in all directions. Principles of
M. Cauchy s proof. A new proof of the theorem. A physical inter
pretation of the circumstance of the above expression being an exact
differential 10G
SECTION III. Application of a method analogous to that of Section I. to
the determination of the equations of equilibrium and motion of
elastic solids 113
SECTION IV. Principles of Poisson s theory of elastic solids, and of the
oblique pressures existing in fluids in motion. Objections to one of
his hypotheses. Keflections on the constitution, and equations of
motion of the luminiferous ether in vacuum 110
On the Proof of the Proposition that (Mx + Ny)~ l is an Integrating Factor of
the Homogeneous Differential Equation M+Ndyf dx = . . . 130
On the Aberration of Light 134
On Fresnel s Theory of the Aberration of Light 141
On a Formula for determining the Optical Constants of Doubly Refracting
Crystals 143
On the Constitution of the Luminiferous Ether, viewed with reference to
the Aberration of Light 153
X CONTENTS.
PAftT!
v Report on Recent Researches on Hydrodynamics 157
* I. General theorems connected with the ordinary equations of Fluid
Motion 158
II. Theory of waves, including tides 161
III. The discharge of gases through small orifices 176
IV. Theory of sound 178
V. Simultaneous oscillations of fluids and solids 179
VI. Formation of the equations of motion when the pressure is not sup
posed equal in all directions 182
* Supplement to a Memoir on some cases of Fluid Motion 188
^ On the Theory of Oscillatory Waves 197
On the Resistance of a Fluid to two Oscillating Spheres 230
On the Critical Values of the Sums of Periodic Series 237
SECTION I. Mode of ascertaining the nature of the discontinuity of a
function which is expanded in a series of sines or cosines, and of
obtaining the developments of the derived functions .... 239
SECTION II. Mode of ascertaining the nature of the discontinuity of
the integrals which are analogous to the series considered in Section
I., and of obtaining the developments of the derivatives of the
expanded functions 271
SECTION III. On the discontinuity of the sums of infinite series, and of
the values of integrals taken between infinite limits .... 279
SECTION IV. Examples of the application of the formulas proved in the
preceding sections 286
j Supplement to a paper on the Theory of Oscillatory Waves .... 314
Index , 327
ERRATA.
P. 103, 1. 14, for their read there.
P. 193, 1. 3, for p*~* read p y ^.
MATHEMATICAL AND PHYSICAL PAPEES.
{From the Transactions of the Cambridge Philosophical Society,
Vol. vii. p. 439.]
ON THE STEADY MOTION OF INCOMPRESSIBLE FLUIDS.
[Bead April 25, 1842.]
IN this paper I shall consider chiefly the steady motion of
fluids in two dimensions. As however in the more general case
of motion in three dimensions, as well as in this, the calculation
is simplified when udx + vdy + wdz is an exact differential, I
shall first consider a class of cases where this is true. I need
not explain the notation, except where it may be new, or liable
to be mistaken.
To prove that udx + vdy + wdz is an exact differential, in
the case of steady motion, when the lines of motion are open
curves, and when the fluid in motion has come from an expanse
of fluid of indefinite extent, and where, at an indefinite distance,
ential. JNow from the way in which this equation is obtained,
lf S. 1
CONTENTS.
v Eeport on Eecent Eesearches on Hydrodynamics ...... 157
* I. General theorems connected with the ordinary equations of Fluid
Motion ............ 158
II. Theory of waves, including tides ....... 161
III. The discharge of gases through small orifices ..... 176
IV. Theory of sound .......... 178
V. Simultaneous oscillations of fluids and solids ..... 179
VI. Formation of the equations of motion when the pressure is not sup
posed equal in all directions ........ 182
* Supplement to a Memoir on some cases of Fluid Motion ..... 188
y On the Theory of Oscillatory Waves ........ 197
On the Eesistance of a Fluid to two Oscillating Spheres ..... 230
On the Critical Values of the Sums of Periodic Series ..... 237
SECTION I. Mode of ascertaining the nature of the discontinuity of a
function which is expanded in a series of sines or cosines, and of
obtaining the developments of the derived functions .... 239
SECTION II. Mode of ascertaining the nature of the discontinuity of
the integrals which are analogous to the series considered in Section
I., and of obtaining the developments of the derivatives of the
expanded functions .......... 271
SECTION III. On the discontinuity of the sums of infinite series, and of
the values of integrals taken between infinite limits .... 279
SECTION IV. Examples of the application of the formulae proved in the
preceding sections .......... 286
j Supplement to a paper on the Theory of Oscillatory Waves .... 314
Index , 327
EEEATA.
P. 103, 1. 14, for their read there.
P. 193, 1. 3, for "* read p^.
EEEATUM.
P. 318, Equations (17) and (18). For  read + before the terms multiplied
by sin 30 and cos 30.
MATHEMATICAL AND PHYSICAL PAPEES.
{From the Transactions of the Cambridge Philosophical Society,
Vol. vir. p. 439.]
ON THE STEADY MOTION OF INCOMPRESSIBLE FLUIDS.
[Bead April 25, 1842.]
IN this paper I shall consider chiefly the steady motion of
fluids in two dimensions. As however in the more general case
of motion in three dimensions, as well as in this, the calculation
is simplified when udx + vdy + wdz is an exact differential, I
shall first consider a class of cases where this is true. I need
not explain the notation, except where it may be new, or liable
to be mistaken.
To prove that udx + vdy + wdz is an exact differential, in
the case of steady motion, when the lines of motion are open
curves, and when the fluid in motion has come from an expanse
of fluid of indefinite extent, and where, at an indefinite distance,
the velocity is indefinitely small, and the pressure indefinitely
near to what it would be if there were no motion.
By integrating along a line of motion, it is well known that
we get the equation
P^Vi^ + v + w^+C (1),
where dV= Xdx\ Ydy + Zdz, which I suppose an exact differ
ential. Now from the way in which this equation is obtained,
\ \ s. 1
ON THE STEAD I MOTION OF INCOMPRESSIBLE FLUIDS.
it appears that G need only be constant for the same line of
motion, and therefore in general will be a function of the para
meter of a line of motion. I shall first shew that in the case
considered C is absolutely constant, and then that whenever it
is, udx + vdy + wdz is an exact differential *.
To determine the value of C for any particular line of motion,
it is sufficient to know the values of p, and of the whole velocity,
at any point along that line. Now if there were no motion we
should have
t ............................ (2),
P! being the pressure in that case. But considering a point in
this line at an indefinite distance in the expanse, the value of
p at that point will be indefinitely nearly equal to p^ and the
velocity will be indefinitely small. Consequently C is more nearly
equal to G t than any assignable quantity : therefore C is equal to
Cj ; and this whatever be the line of motion considered ; therefore
C is constant.
In ordinary cases of steady motion, when the fluid flows in
open curves, it does come from such an expanse of fluid. It is
conceivable that there should be only a canal of fluid in this
expanse in motion, the rest being at rest, in which case the
velocity at an infinite distance might not be indefinitely small.
But experiment shews that this is not the case, but that the
fluid flows in from all sides. Consequently at an indefinite dis
tance the velocity is indefinitely small, and it seems evident that
in that case the pressure must be indefinitely near to what it
would be if there were no motion.
Differentiating therefore (1) with respect to x, we get
1 dp ^ du dv dw
 ^r = ^u r v r w r ;
p dx dx dx ax
1 dp ^ du du du
 =  V  W
dv du\ dw du
whence
[* See note, page 3.]
ON THE STEADY MOTION OF INCOMPRESSIBLE FLUIDS. 3
. ., , (dw dv\ fdu dv\
Similarly, w ( r  r ) + u (_ =0,
\dy dzj \dy dx)
fdu dw\ (dv dw\ _
\dz dx) \dz dy)
i dv du dw dv du dw
whence* JT=J > j~ = ^j ~r ~ ~j~ >
dx dy dy dz dz dx
and therefore udx + vdy + wdz is an exact differential.
When udx f vdy + wdz is an exact differential, equation (1)
may be deduced in another wayf% from which it appears that
C is constant. .Consequently, in any case, udx { vdy + wdz is, or
is not, an exact differential, according as C is, or is not, constant.
Steady Motion in Two Dimensions.
I shall first consider the more simple case, where udx + vdy
is an exact differential. In this case u and v are given by the
equations
J* + ^ = ........................... (3),
dx dy
0 M
dy dx~~ " () >
and p is given by the equation
The differential equation to a line of motion is
dy = v_
dx u
* [This conclusion involves an oversight (see Transactions, p. 465) since the
three preceding equations are not independent, as may readily be seen. I have not
thought it necessary to rewrite this portion of the paper, since in the two classes
of steady motion to which the paper relates, namely those of motion in two dimen
sions, and of motion symmetrical about an axis, the three analogous equations are
reduced to one, and the proposition is true. None of the succeeding results are
affected by this error, excepting that the second paragraph of p. 11 must be re
stricted to the two cases above mentioned.]
t See Poisson, Traite de Mecanique.
12
4 ON THE STEADY MOTION OF INCOMPRESSIBLE FLUIDS.
Now from equation (3) it follows that udy vdx is always
the exact differential of a function of x and y. Putting then
d U = udy vdx,
U=G will be the equation to the system of lines of motion,
C being the parameter. U may have any value which allows
d U/dy and d Ujdoc to satisfy the equations which u and v satisfy.
The first equation has been already introduced ; the second leads
to the equation which U is to satisfy ; viz.
The integral of this equation may be put under different forms.
By integrating according to the general method, we get
Now it will be easily seen that U must be wholly real for all
values of x and y t at least within certain limits. But ^(a) may
be put under the form F l (a) 4 \f^l F z (a), where F l (a) and F 3 (a)
are wholly real. Making this substitution in the value of U, we
get a result, which, without losing generality, may be put under
the form
U = F(^ + V^l y)+F(x  V^l y]
=l y} f(x a#)},
changing the functions.
If we develope these functions in series ascending according
to integral powers of y, by Taylor s Theorem, which can always
be done as long as the origin is arbitrary, we get a series which
I shall write for shortness,
2 cos (A y) F(x)  2 sin (^ y]f (),
the same result as if we had integrated at once by series by
Maclaurin s Theorem.
It has been proved that the general integral of (5) may be
put under the form
U=
ON THE STEADY MOTION OF INCOMPRESSIBLE FLUIDS. 5
where a 2 + {3* = 0. Consequently a and /3 must be, one real, the
other imaginary, or both partly real and partly imaginary. Putting
then a = 1 + Vla 2 , /3 = ^ + V 1 /3 2 , introducing the condition
that a 2 + /3 2 = 0, and replacing imaginary exponentials by sines and
cosines, we find that the most general value of U is of the form
U = 2Ue w ( cos Y* sin Y y +a \ cos n (sin 7 . x + cos 7 . y + 6),
where A, n, 7, a and & have any real values, the value of U being
supposed to be real.
If we take the value of U
and develope each term, such as ax n , in F (x) or f (x), in a series,
and then sum the series by the formula
cos nO + V 1 sin nO = cos n <9 (l + j V^T tan ... V
we find that the general value of U takes the form
As long as the origin of x is arbitrary, only integral powers
of x will enter into the development F (x) and f(x), and there
fore the above series will contain only integral values of n. For
particular positions of the origin however, fractional powers may
enter. The equation
d 2 U I dU 1 d*U _
dr* + r ~dr + r* dP "
which (5) becomes when transferred to polar coordinates, is satis
fied by the above value of U, whatever n be, even if it be
imaginary, in which case the value of U takes the form
U = 2Ar m e nd cos (mO  log e r n + B).
We may employ equation (5), to determine whether a proposed
system of lines can be a system in which fluid can move, the
motion being of the kind for which udx + vdy is an exact
differential.
Let / (x, y) = U^ C be the equation to the system, C being
the parameter. Then, if the motion be possible, some value of
6 ON THE STEADY MOTION OF INCOMPKESSIBLE FLUIDS.
U which satisfies (5) must be constant for all values of x and y
for which U^ is constant. Consequently this value must be a
function of U . Let it=^(C^). Then, substituting this value
in (5), and performing the differentiations, we get
\ dx \dy
Now, if the motion be possible, the second term of this equa
tion must be a function of U l ; a?, y and U^ being connected by
the equation f(x, y}= U^. Consequently, if by means of this
latter equation we eliminate x or y from the second term of (6),
the other must disappear. If it does not, the motion is impossible ;
if it does, the integration of equation (6), in which the variables
are separated, will give < (U^) under the form
A and B being the arbitrary constants. The values of u and v
will immediately be got by differentiation, and then p will be
known. Nothing will be left arbitrary but a constant multiplying
the values of u and v, and another added to the value of p.
I shall mention a few examples. Let U = ar 2 cos J^. In this
case the lines of motion are similar parabolas a,bout the same
focus. The velocity at any point varies inversely as the square
root of the distance from the focus.
Again, let U = axy. In this case the lines of motion are
rectangular hyperbolas about the same asymptotes. Also,
dU dU
u = j = ax, and v = j = ay.
dy dx
In this case therefore the velocity varies as the distance from the
centre, and the particles in a section parallel to either of the axes
remain in a section parallel to that axis.
I shall now consider the general case, where udx f vdy need
not be an exact differential.
ON THE STEADY MOTION OF INCOMPRESSIBLE FLUIDS.
In this case p, u and v, are given by the equations
I dp du du
 ^=Xu j  VY ................... (7),
p dx dx dy
1 dp v dv dv
T = Yuj  v j .............. ..(8)
p dy dx dy
du dv
We still have ^ =  , for the differential equation to a line of
motion, where udy vdx is still an exact differential, on account
of equation (9). Eliminating p by differentiation from (7) and
(8), and expressing the result in terms of U, we get the equation
which U is to satisfy, viz.
dU d^ (d*U <PU\ __ dU d^ (d*U d*U\ _
dy dx (da? " h dy 2 ) dx dy ( dx 2 + ~dtf) ~ 0)
or, for shortness,
d__dU _
(dy dx dx dy) (~%? + ~df) ......... (10) *
* [This equation may be applied to prove an elegant theorem due to Mr F. D.
Thomson {see the Oxford, Cambridge, and Dublin Messenger of Mathematics, Vol.
in. (I860), p. 238, and Vol. iv. p. 37}, that if a vessel bounded by a cylindrical sur
face of any kind and by two planes perpendicular to its generating lines be filled
with homogeneous liquid, and the whole be revolving uniformly about a fixed axis
parallel to its generating lines, then if the vessel be suddenly arrested the motion
of the liquid will be steady.
If w be the angular velocity, we shall have for the motion before impact
dx)  J w (ce 2 + ?/ 2 ) =  s wr 2 ,
omitting the constant as unnecessary. If u, v be the components of the change of
velocity produced by impact, it follows from the equations of impulsive motion that
udx + vdy will be a perfect differential d<p, where satisfies the partial differential
d 2 d 2
equation V0 = 0, V standing for ^ 2 + ^. If U be the 17function corresponding
to this motion and such a function exists by virtue of the equation of continuity
whether the motion be steady or not we have
, r fd<t d(f> , \ /Yd0 1 d<t> . \
U = n~ dy^ dx\= ( ~rd6 / dr } ,
J \dx b dy ) J \dr r dd J
where the quantity under the sign J is a perfect differential by virtue of the equa
8 ON THE STEADY MOTION OF INCOMPRESSIBLE FLUIDS.
In this case, since p = I (~ dx \jr dy\ , equations (7) and
(8) give
tion V0 = ; an ^ we see at once *^ at V^ =  Hence for the whole motion just
after impact
which satisfies the equation of steady motion (10); and as the condition at the
boundary, namely that the fluid shall slide along it, is satisfied, being satisfied ini
tially, it follows that the initial motion after impact will be continued as steady
motion.
To actually determine the function </> or U , and thereby the motion in any given
case, we must satisfy not only the general equation y0=0 but also the equation
of condition at the boundary, namely that there shall be no velocity in a direction
normal to the surface, which gives
()*(!?)* ................................. <>
at any point of the boundary. If f(x,y)=Q be the equation of the boundary, we
must substitute  df / dx r df I dy for dy/dx in (a), and the resulting equation will
have to be satisfied when/=0 is satisfied.
There are but few forms of boundary for which the solution of the problem can
be actually effected analytically, among which may be mentioned in particular the
case of a rectangle. But by taking particular solutions of the equation V0 = >
substituting in (a) and integrating, which gives
or what comes to the same thing taking particular solutions of the equation v^ =
and substituting in (/3), which gives the general equation of the lines of motion, we
may synthetically obtain an infinity of examples in which the conditions of the
problem are satisfied, any one of the lines of motion being taken as the boundary
of the fluid.
Thus for U = fcr*cos 30 we have for the lines of motion
iwr 2 + fcr 3 cos30=C ....................................... (7),
or Icor 2 + &{4(rcos0) 3 3r 2 .rcos0}=<7 ................ (5),
which therefore are cubic curves, recurring when is increased by 120. (5) is
satisfied by
r cos 6 = a,
giving a straight line, provided
Hence when Tc has the above value the cubic curve (7) breaks up, for the particular
value of the parameter G above written, into three straight lines forming the sides
of an equilateral triangle, and the vessel may therefore be supposed to be an equi
lateral triangular prism. The various lines of motion correspond to values of the
parameter C from to  f wa 2 . This case is given by Mr Thomson.
U = kr*cos 20 leads to the case of steady motion in similar and concentric ellipses
considered in the text a little further on, which therefore may be conceived to have
been produced from motion about a fixed axis as pointed out by Mr Thomson. In
fact, any case of steady motion in two dimensions in which yU= const, may be
conceived to have been so produced.]
ON THE STEADY MOTION OF INCOMPRESSIBLE FLUIDS.
= V  L _^ *F_\ dx
P J\\dy dxdy dx dy* J
fdU d*U dUd*U\ }
\dx dxdy dy dx 2 ) *)
AT 1,7 f/^Y , (dU\*\ fdU d 2 U , dU d z U\ ,
Now JaujJ + j }\= j  j^ + ~j j, T\dx
\\dxj \dy J } \dx dx dy dxdy)
(dU d 2 U dU d 2 U
^\dx dxdy + dy dtf
whence,
dU d 2 U , dU d*U , dU d z U . dUd*U,
j , , dx + j j = av  = = dx  Y yg ctv
dx dy dy dx
V.AH7V) fd*U d*U\ (dU , , dU . \
I + TT" ) r ~ ^7""2" + ^j^" ^j~ "* + ~j~ dy ;
/ v dy J J \ cfo dy J \ dx dy y j
and therefore
^ 2 ^7 rf?7\ /dtr , . dU
dy
It will be observed that , 2 +j a = %(^)> ^ s a nrs ^ integral
of (10). Consequently this latter term, which is the value of C in
(1), comes out a function of the parameter of a line of motion as
it should.
We may employ equation (10), precisely as before, to enquire
whether a proposed system of lines can, under any circumstances,
be a system of lines of motion. Let f (x, y) = U^ = C, be the
equation to the system; then, putting as before, U = (f>(U l ),
we get
+
dy dx dx dy dx dy J
U d dU d\fU.
or, P<f>" ( U^} + Qf ( fTJ = 0, suppose.
10 ON THE STEADY MOTION OF INCOMPRESSIBLE FLUIDS.
Hence, as before, if we express y in terms of x and U lt from
the equation f(x, y) = U lt and substitute that value in p, the
result must not contain x. If it does, the proposed system of lines
cannot be a system of lines of motion ; if not, the integration of
the above equation will give < (Z7J, under the form
and we can immediately get the values of u, v and p, with the
same arbitrary constants as in the previous case.
One case in which the motion is possible is where the lines of
motion are a system of similar ellipses or hyperbolas about the
same centre, or a system of equal parabolas having the same axis.
In the case of the ellipse, the particles in a radius vector at any
time remain in a radius vector, and the value of p has the form
When however the ellipse becomes a circle, P and Q vanish in the
equation P</>" (tTj + Q<f> (UJ = 0. Consequently the form of <
may be any whatever. The value of U^ being x* + # 2 , we have
whence, v? + v* = 4 (</> ( U,)} 9 (a? + /) = 4 U, {< ( U,)} 9 .
Hence, the velocity may be any function of the distance from the
centre. It is evident that we may conceive cylindrical shells of
fluid > having a common axis, to be revolving about that axis with
any velocities whatever, if we do not consider friction, or whether
such a mode of motion would be stable. The result is the same if
we enquire in what way fluid can move in a system of parallel lines.
In any case where the motion in a certain system of lines is
possible, if we suppose two of these lines to be the bases of bound
ing cylindrical surfaces, and if we suppose the velocity and direc
tion of motion, at each point of a section of the entering, and also
of the issuing fluid, to be what that case requires, I have not
proved that the fluid must move in that system of lines. When
the above conditions are given there may still perhaps be different
modes of steady motion ; and of these some may be stable, and
others unstable. There may even be no stable steady mode of
ON THE STEADY MOTION OF INCOMPKESSIBLE FLUIDS. 11
motion possible, in which case the fluid would continue perpetually
eddying.
In the case of rectangular hyperbolas, the fluid appeared, on
making the experiment, to move in hyperbolas when the end
at which the fluid entered was broad and the other end narrow,
but not when the end by which the fluid entered was narrow.
This may, I think, in some measure be accounted for. Suppose
fluid to flow out of a vessel where the pressure is p { into one where
it is p z) through a small orifice. Then, the motion being steady,
we have, along the same line of motion, p/p C Jy 2 , where v is
the whole velocity. At a distance from the orifice, in the first
vessel, the pressure will be approximately p : , and the velocity
nothing. At a distance in the second vessel, the pressure will
/2 If) q} }
be approximately^, and therefore the velocity = A/ ",
nearly. The result is the same if forces act on the fluid. Hence
the velocity must be approximately constant ; and therefore, the
fluid which came from the first vessel, instead of spreading out,
must keep to a canal of its own of uniform breadth. This is found
to agree with experiment. Hence we might expect that in the
case of the hyperbolas, if the end at which the fluid entered were
narrow, the entering fluid would have a tendency to keep to a
canal of its own, instead of spreading out.
In ordinary cases of steady motion, when the lines of motion
are open curves, the fluid is supplied from an expanse of fluid, and
consequently udx + vdy + wdz is an exact differential. Conse
quently, cases of open curves for which it is not an exact differen
tial do not ordinarily occur. We may, however, conceive such
cases to occur ; for we may suppose the velocity and direction of
motion, at each point of a section of the entering, and also of the
issuing stream, to be such as any case requires, by supposing the
fluid sent in and drawn out with the requisite velocity and in the
requisite direction through an infinite number of infinitely small
tubes.
In the case of closed curves however, in whatever manner the
fluid may have been put in motion, it seems probable that, if we
neglect the friction against the sides of the vessel, the fluid will
have a tendency to settle down into some steady mode of motion.
Consequently, taking account of the friction against the sides of
12 ON THE STEADY MOTION OF INCOMPRESSIBLE FLUIDS.
the vessel, it seems probable that the motion may in some cases
become approximately steady, before the friction has caused it to
cease altogether.
Motion symmetrical about an axis, the lines of motion being
in planes passing through the aods.
Before considering this case, it may be well to prove a prin
ciple which will a little simplify our equations.
The general equations of motion are,
I dp
yr dU
du
du
ni)
p dx
I dp _
dx
y dV
dy
dv
dz
dv
\ ij v>
(12)
pdy
Idp _
dx
~ dw
dy
dw
11 I/)
dz"
dw
n^\
And
p dz
the equation of
dx
continuity is
dy
dz
\ J
Putting tsTj, trr 2 , r 3 , for the last three terms in (11), (12), (13),
respectively, we have
^ = V  / (efjdx + *r t dy + OT 8 dz).
Hence the pressure consists of two parts, the firs,t, p V, the same
as if there were no motion, the second, the part due to the velocity.
Now the velocities are given by equation (14), and by the three
equations which result on eliminating p from (11), (12), and (13).
These latter equations, as well as (14), will be the same as if there
were no forces since
_^ = ^^.
dy ~ dx dz ~ dx dz dy
and therefore we shall not lose generality by omitting the forces
in (11), (12) and (13), since we shall only have to add pV to the
value of p so determined.
When the motion is symmetrical about an axis, and in planes
passing through that axis, let z be measured along the axis, and
ON THE STEADY MOTION OF INCOMPRESSIBLE FLUIDS. 13
r be the perpendicular distance from the axis, and s be the ve
locity perpendicular to the axis. Then, transforming the coordi
nates to z and r, and omitting the forces, it will be found that
equations (11), (12) and (13) are equivalent to only two separate
equations, which are
1 dp ds ds
~r = srwr
p dr dr dz
1 dp dw dw
T = SJ WT
p dz dr dz
and the equation of continuity becomes
J+ s  + ^ = 0.. ,..(17).
dr r dz
In the case where udx + vdy + wdz is an exact differential, it
will be found that the three equations
du _ dv du _dw dv _ dw
dy ~ dx dz dx dz dy
t/ i7
are equivalent to only one equation, which is
ds dw
In the general case we get, by eliminating p from (15)
and (16),
d f ds ds\ _ d f dw
dz \ dr dz) dr\ dr w dz ,
ds ds ds dw d?t
or ~~ ~~
jj ^^j
dr dz dz dz drdz
dw dw dw ds d*w d z io
The differential equation, between z and r, to a line of
motion is
dz _w
dr s
Let ju be a factor which renders sdz wdr an exact differential,
,, diis _
then ~ +  = 0,
dr dz
14 ON THE STEADY MOTION OF INCOMPRESSIBLE FLUIDS.
/n ^_v du, da S
or, using (17), s + w = f ,..
whence we easily see that //, = r is one such factor.
Let then dU= rsdz rwdr,
IdU IdU
so that s = 7 , w =  , .
r dz r a/
The equation which U is to satisfy will be got by expressing s
and w in terms of U, and substituting in (19) in the general case,
or by substituting in (18), in the case where udx + vdy + wdz is
an exact differential.
In the latter case the equation which Z7is to satisfy is
idu
7 Q T o 7
dz dr r dr
In the general case, the equation is what I shall write
d L _dUd\(l(d^7 d^U_ldJA\_
dz dr dr dz) \r z \dz z * dr* r dr)} ~ " ( >
The value of p is given by the equation
p ((( ds ds\ 7 f dw dw\ 7 )
*L\\{s^+wj}dr + \8 r + to T \dz\.
p J [\ dr dz) \ dr dz) }
Now
1 , , 2N ^5 7 dw j ds j dw 7
id (s 2 + w ) = s j dr + w j dz + s j dz + w j dr ;
dr dz dz < dr
and therefore
ds ds\ , / dw dw
= J d (s 2 + w 2 ) + j (wdr  sc?z) + T
/P 1/2 2\ [fds dw\ 1 7 77
whence = % (s + w } f I ( j = 7 1  au
p J V^ ar/ r
ON THE STEADY MOTION OF INCOMPRESSIBLE FLUIDS. 15
Hence the quantity under the integral sign must be a function
of U. And in fact, we can easily shew by trial that
d*U Id
is a first integral of (21). The last term of (22) is the value of
the constant in (1).
By expanding U in a series ascending according to integral
powers of z, which may be done as long as the origin is arbitrary,
it will be found that the integral of (20) may be written under the
form
U= cos ( V *) F (r) + sin ( v *) vYM,
where y a .F(r) denotes (35  *jF(r) t and y 2n F (r) denotes
that the operation ^ 2 ^ is repeated n times on F (r).
We may employ equations (21) or (20) just as before, to
determine whether the motion in a proposed system of lines is
possible. If F(r, z) = U t = C be the equation to the system, we
must have, as before, U (f> ( U^ ; whence we get, in the general
case,
^
dz y ^ 2 dr* ~r dr J] j
and in the more restricted case where udx f vdy + wdz is an exact
differential, we get
ff U ffU Id U\ .
As before, the ratio of the coefficients of 0" ( U^ and $ ( U^) must
be a function of U^ alone, when 3, r and C^ are connected by the
equation F (r, z) = Z7 r If the motion be possible, it will in general
be determinate, U being of the form Af (r, z] + B. If U^ = r how
ever, the form of remains arbitrary. In this case the fluid may
be conceived to move in cylindrical shells parallel to the axis, the
velocity being any function of the distance from the axis.
16 ON THE STEADY MOTION OF INCOMPRESSIBLE FLUIDS.
Particular cases are, where the lines of motion are right lines
directed to a point in the axis, and where they are equal parabolas
having the axis of z for a common axis. In these cases
udx + vdy + wdz
is an exact differential.
We may employ equations (20) and (21) to determine whether
the hypothesis of parallel sections can be strictly true in any case.
In this case, the sections being perpendicular to the axis of z, we
must have
IdU
w  T F(is) ;
r dr
dU , N
y rlW;
U1 + *(*)+/(*).
Substituting this value in (21), we find, by equating to zero
coefficients of different powers of r, that the most general case cor
responds to
If udoc \ vdy 4 wdz be an exact differential, the most general
case corresponds to
U= (a + bz)
[From the Transactions of the Cambridge Philosophical Society,
Vol. vin. p. 105.]
ON SOME CASES OF FLUID MOTION.
[Read May 29, 1843.]
THE equations of Hydrostatics are founded on the principles
that the mutual action of two adjacent elements of a fluid is normal
to the surface which separates them, and that the pressure is equal
in all directions. The latter of these is a necessary consequence
of the former, as has been shewn by Mr Airy*. An exactly simi
lar proof may be employed in Hydrodynamics, by which it may
be shewn that, if the mutual action of two adjacent elements of a
fluid in motion is normal to their common surface, the pressure
must be equal in all directions, in order that the accelerating force
which acts on the centre of gravity of an element may not become
infinite, when we suppose the dimensions of the element indefi
nitely diminished. In Hydrostatics, the accurate agreement of the
results of our calculations with experiments, (those phenomena
which depend on capillary attraction being excepted), fully justifies
our fundamental assumption. The same assumption is made in
Hydrodynamics, and from it are deduced the fundamental equa
tions of fluid motion. But the verification of our fundamental law
in the case of a fluid at rest, does not at all prove it to be true
in the case of a fluid in motion, except in the very limited case of
a fluid moving as if it were solid. Thus, oil is sufficiently fluid to
obey the laws of fluid equilibrium, (at least to a great extent),
yet no one would suppose that oil in motion ought to be considered
a perfect fluid. It would appear from the following consideration,
that the fluidity of water and other such fluids is not quite perfect.
* See also Professor Miller s Hydrostatics, page 2.
S. 2
18 ON SOME CASES OF FLUID MOTION.
When a mass of water contained in a vessel of the form of a solid
of revolution is stirred round, and then left to itself, it presently
comes to rest. This, no doubt, is owing to the friction against the
sides of the vessel. But if the fluidity of water were perfect, it
does not appear how the retardation due to this friction could be
transmitted through the mass. It would appear that in that case
a thin film of fluid close to the sides of the vessel would remain at
rest, the remaining part of the fluid being unaffected by it. And
in this respect, that part of Poisson s solution of the problem of an
oscillating sphere, which relates to friction, appears to me in some
degree unsatisfactory. A term enters into the equation of motion
of the sphere depending on the friction of the fluid on the sphere,
while no such term enters into the equations of motion of the
fluid, to express the equal and opposite friction of the sphere on
the fluid. In fact, as long as we regard the fluidity of the fluid as
perfect, no such term can enter. The only way by which to esti
mate the extent to which the imperfect fluidity of fluids may
modify the laws of their motion, without making any hypothesis
as to the molecular constitution of fluids, appears to be, to calculate
according to the hypothesis of perfect fluidity some cases of fluid
motion, which are of such a nature as to be capable of being accu
rately compared with experiment. The cases of that nature which
have hitherto been calculated, are by no means numerous. My
object in the present paper which I have the honour to lay before
the Society, has been partly to calculate some such cases which
may be useful in determining how far we are justified in regarding
fluids as perfectly fluid, and partly to give examples of the methods
by which the solution of problems depending on partial differential
equations may be effected.
In the first seven articles, I have mentioned and explained
some general principles, which are afterwards applied. Some of
these are not new, but it was convenient to state them for the
sake of reference. Others are I believe new, at least in their
development. In the remaining articles, I have given different
problems, of which I have succeeded in obtaining the solutions.
As the problem to be solved is usually stated at the head of each
article, I shall here only mention some of the results. As a parti
cular case of the problem given in Art. 8, I find that, when a
cylinder oscillates in an infinitely extended fluid, the effect of the
inertia of the fluid is to increase the mass of the cylinder by that of
ON SOME CASES OF FLUID MOTION. 10
the fluid displaced. In part of Art. 9, I find that when a ball pen
dulum oscillates in a concentric spherical envelope, the effect of the
b s 4 2a 3
inertia of the fluid is to increase the mass of the ball by ^jJ 3T
2i(Jb a )
times that of the fluid displaced, a being the radius of the ball, b
that of the envelope. Poisson, in his solution of the problem of the
sphere, arrives at the strange result that the envelope does not at
all retard the oscillating sphere. I have pointed out the errone
ous step by which he was led to this conclusion, which I am clearly
called upon to do, in venturing to differ from so high an authority.
Of the different cases of fluid motion which I have given, that
which appears to be capable of the most accurate and varied com
parison with experiment, is the motion of fluid in a rectangular
box which is closed on all sides, given in Art. 13. The experiment
consists in comparing the calculated and observed times of oscil
lation. I find that when the motion is small, the effect of the
fluid on the motion of the box is the same as that of a solid
having the same mass, centre of gravity, and principal axes, but
having different moments of inertia, these moments being given
by infinite series, which converge with great rapidity. I have also
in Art. 11, given some cases of progressive motion, deduced on the
supposition that the same particles of fluid remain in contact with
the solid, which do not at all agree with experiment.
In almost all the cases given in this paper, the problem of
finding the permanent state of temperature in the several solids
considered, supposing the surfaces of those solids kept up to con
stant temperatures varying from point to point, may be solved by
a similar analysis. I find that some of these cases have been
already solved by M. Duhamel in a paper inserted in the 22nd
Cahier of the Journal de lEcole Poly technique. The cases alluded
to are those of the temperature in a solid sphere, and in a rect
angular parallelepiped. Since, however, the application of the
formulae in the two cases of fluid motion and of the permanent
state of temperature is different, as well as the formulae themselves
to a certain extent, I thought it might be worth while to give
them.
1. The investigations in this paper apply directly to incom
pressible fluids, as the fluids spoken of will be supposed to be,
22
20 ON SOME CASES OF FLUID MOTION.
unless the contrary is stated. The motions of elastic fluids may
in most cases be divided into two classes, one consisting of those
condensations on which sound depends, the other, of those motions
which the fluid takes in consequence of the motion of solid bodies
in it. Those motions of the fluid, which take place in consequence of
very rapid motions of solids, (such as those of bullets), form a con
necting link between these two classes. The motions of the second
class are, it is true, accompanied by condensations, and propagated
with the velocity of sound, but if the motions of the solids are not
great we may, without sensible error, suppose the motions of the
fluid propagated instantaneously to distances where they cease to
be sensible, and may neglect the condensation. The investigations
in this paper will apply without sensible error to this kind of
motion of elastic fluids.
In all cases also the motion will be supposed to begin from
rest, which allows us to suppose that udx + vdy + wdz is an exact
differential d(f>, where u, v and w are the components, parallel to
the axes of x, y, and z, of the whole velocity of any particle. In
applying our investigations however to fluids such as they exist in
nature, this principle must not be strained too far. When a body
is made to revolve continually in a fluid, the parts of the fluid
near the body will soon acquire a rotatory motion, in consequence,
in all probability, of the mutual friction of the parts of the fluid ;
so that after a time udx + vdy + wdz could no longer be taken an
exact differential. It is true that in motion in two dimensions
there is one sort of rotatory motion for which that quantity is an
exact differential ; but if a close vessel, filled with fluid at first at
rest, be made to revolve uniformly round a fixed axis, the fluid
will soon do so too, and therefore that quantity will cease to be an
exact differential. For the same reason, in the progressive motion
of a solid in a fluid, the effect of friction continually accumulating,
the motion might at last be sensibly different from what it would
be if there were no friction, and that, even if the friction were
very small. In the case of small oscillatory motions however it
would appear that the effect of friction in the forward oscillation,
supposing that friction small, would be counteracted by its effect
in the backward oscillation, at least if the two were symmetrical.
In this case then we might expect our results to agree very nearly
with experiment, so far at least as the time of oscillation is con
cerned.
ON SOME CASES OF FLUID MOTION. 21
The forces which act on the fluid are supposed in the following
investigations to be such that Xdx + Ydy + Zdz is the exact dif
ferential of a function of x, y and z, where X, Y, Z, are the com
ponents, parallel to the axes, of the acccelerating force acting on
the particle whose coordinates are x, y, z. The only effect of such
forces, in the case of a homogeneous, incompressible fluid, being
to add the quantity pf(Xdx + Ydy +Zdz) to the pressure, the forces,
as well as the pressure due to them, will for the future be omitted
for the sake of simplicity.
2. It is a recognized principle, and one of great importance in
these investigations, that when a problem is determinate any solu
tion which satisfies all the requisite conditions, no matter how ob
tained, is the solution of the problem. In the case of fluid motion,
when the initial circumstances and the conditions with respect to
the boundaries of the fluid are given, the problem is determinate.
If it were required to find what sort of steady motion could take
place between given surfaces, the problem would not be determi
nate, since different kinds of steady motion might result from dif
ferent initial circumstances.
It may be well here to enumerate the conditions which must
be satisfied in the case of a homogeneous incompressible fluid
without a free surface, the case which is considered in this paper.
We have first the equations,
1 dp I dp 1 dp
p dx = w " pdy = ^ pd~ 2 = ^ ............ ^)J
... du du du du ,
putting ^fa + U +V + M an(i OT 2 > OT 3> for the cor
responding quantities for y and , and omitting the forces.
We have also the equation of continuity,
.. ,..
dx dy dz
(A) and (B) hold at all times for all points of the fluid mass.
If a be the velocity of the point (x, ?/, z) of the* surface of a
solid in contact with the fluid resolved along the normal, and v
the velocity, resolved along the same normal, of the fluid particle,
22 OX SOME CASES OF FLUID MOTION.
which at the time t is in contact with the above point of the solid,
we must have
v = * ........ ............................ (a)*,
at all times and for all points of the fluid which are in contact with
a solid.
If the fluid extend to infinity, and the motion at first be zero
at an infinite distance, we must have
u = 0, v = 0, w = 0, at an infinite distance ............. (b).
An analagous condition is, that the motion shall not become
infinitely great about a particular point, as the origin.
Lastly, if u 0) v ot w , be the initial velocities, subject of course
to satisfy equations (B) and (a), we must have
u u^ V = V Q , w = w , when = ..................... (c).
In the most general cases the equations which u, v and w are
to satisfy at every point of the mass and at every time are (B) and
the three equations
. ~,
~dy~~dx* dz ~~ dy dx ~ dz "
These equations being satisfied, the quantity
will be an exact differential, whence p may be determined by inte
grating the value of dp given by equations (A). Thus the condi
tion that these latter equations shall be satisfied is equivalent to
the condition that the equations ( C) shall be satisfied
In nearly all the cases considered in this paper, and in all those
of which the complete solution is given, the motion is such that
udx + vdy + ^udz is an exact differential dty. This being the case,
the equations (C) are, as it is well known, always satisfied, the
value of p being given by the equation
* For greater clearness, those equations which must hold for all values of the
variables within limits depending on the problem are denoted by capitals, while
those which hold only for certain values of the variables, or of some of them, are
denoted by small letters. The latter class serve to determine the forms of the
arbitrary functions contained in the integrals of the former.
ON SOME CASES OF FLUID MOTION. 23
being an arbitrary function of t, which may if we please be
included in (/>. In this case, therefore, the single condition which
has to be satisfied at all times, and at every point of the mass is
(j5), which becomes in this .case
(E).
In the case of impulsive motion, if U Q , v OJ w , be the velocities
just before impact, u, v, w, the velocities just after, and q the im
pulsive pressure, the equations (A) are replaced by the equations
1 da 1 da 1 da
 / = U + U Q , ~~ = V + V Q) j L = w + w 6 ....(F):
pdx p dy pdz
and in order that these equations may be satisfied it is necessary
and sufficient that (u u )dx + (v v ) dy f (w W Q ] dz be an exact
differential d<f), which gives
q = C pcf).
The only equation which must be satisfied at every point of the
mass is (B), which is equivalent to (E), since by hypothesis u , V Q ,
and w satisfy (B}. The conditions (a) and (b) remain the same
as before.
One observation however is necessary here. The values of u,
v and w are always supposed to alter continuously from one point
in the interior of a fluid mass to another. At the extreme boun
daries of the fluid they may however alter abruptly. Suppose now
values of u, v and w to have been assigned, which do not alter
abruptly, which satisfy equations (5) and ( C) as well as the con
ditions (a), (b) and (c), or, to take a particular case, values which
do not alter abruptly, which satisfy the equation (B) and the same
conditions, and which render udx + vdy + wdz an exact differential.
Then the values of dp/dx, dp/dy and dpjdz will alter continuously
from one point to another, but it does not follow that the value of
p itself cannot alter abruptly. Similarly in impulsive motion the
value of q may alter abruptly, although those of dq/dx, dq/dy and
dqjdz alter continuously. Such abrupt alterations are, however,
inadmissible; whence it follows as an additional condition to be
satisfied,
that the value of p or g, obtained by integrating j
equations (A) or (F), shall not alter abruptly > ........ (d).
from one point of the fluid to another. J
24 ON SOME CASES OF FLUID MOTION.
An example will make this clearer. Suppose a mass of fluid
to be at rest in a finite cylinder, whose axis coincides with that of
z t the cylinder being entirely filled, and closed at both ends. Sup
pose the cylinder to be moved by impact with an initial velocity C
in the direction of x ; then shall
u = C, v = 0, 10 = 0.
For these values render udx + vdy + wdz an exact differential d<f>,
where </> satisfies (E) ; they also satisfy (a) ; and, lastly, the value
of q obtained by integrating equations (F), namely, C 1 Cpx, does
not alter abruptly. But if we had supposed that <f> was equal
to Cx + C O, where 6 = tan" 1 yjx t the equation (E) and the con
dition (a) would still be satisfied, but the value of q would be
C"  p ( Cx + C ff), in which the term pG 6 alters abruptly from
%7rpC to 0, as 6 passes through the value 2?r. The condition (d)
then alone shews that the former and not the latter is the true
solution of the problem.
The fact that the analytical conditions of a problem in fluid
motion, as far as those conditions depend on the velocities, may be
satisfied by values of those velocities, which notwithstanding cor
respond to a pressure which alters abruptly, may be thus explained.
Conceive two masses of the same fluid contained in two similar
and equal close vessels A and B. For more simplicity, suppose
these vessels and the fluid in them to be at first at rest. Conceive
the fluid in B to be divided by an infinitely thin lamina which is
capable of assuming any form, and, at the same time, of sustaining
pressure. Suppose the vessels A and B to be moved in exactly
the same manner, the lamina in B being also moved in any arbi
trary manner. It is clear that, except for one particular motion
of the lamina, the motion of the fluid in B will be different from
that of the fluid in A. The velocities u, v, w, will in general be
different on opposite sides of the lamina in B. For particular
motions of the lamina however the velocities u, v, w, may be the
same on opposite sides of it, while the pressures are different.
The motion which takes place in B in this case might, only for
the condition (d) t be supposed to take place in A.
It is true that equations (A) or (F), could not strictly speaking
be said to hold good at those surfaces where such a discontinuity
should exist. Still, to avoid the liability to error, it is well to
state the condition (d) distinctly.
ON SOME CASES OF FLUID MOTION. 25
When the motion begins from rest, not only must udx+vdy+wdz
be an exact differential d(f>, and u, v, w, not alter abruptly, but
also (j> must not alter abruptly, provided the particles in contact
with the several surfaces remain in contact with those surfaces ;
for if this condition be not fulfilled, the surface for which it is not
fulfilled will as it were cut the fluid into two. For it follows from
the equation (D) that d<f>/dt must not alter abruptly, since other
wise p would alter abruptly from one point of the fluid to another;
and d<p/dt neither altering abruptly nor becoming infinite, it fol
lows that (/> will not alter abruptly. Should an impact occur at
any period of the motion, it follows from equations (F) that that
cannot cause the value of (f> to alter abruptly, since such an abrupt
alteration would give a corresponding abrupt alteration in the
value of qr.
3. A result which follows at once from the principle laid down
in the beginning of the last article is this, that when the motion
of a fluid in a close vessel which is at rest, and is completely filled,
is of such a kind that udx + vdy + wdz is an exact differential, it
will be steady. For let u, v, w, be the initial velocities, and let
us see if the velocities at the same point can remain u, v, w. First,
udx + vdy + wdz being an exact differential, equations (A) will be
satisfied by a suitable value of p, which value is given by equation
(D). Also equation (B) is satisfied since it is so at first. The con
dition (a) becomes v = 0, which is also satisfied since it is satis
fied at first. Also the value of p given by equation (U) will not
alter abruptly, for dfyjdt 0, or a function of t, and the velocities
d(j)/dx &c., are supposed not to alter abruptly. Hence, all the
requisite conditions are satisfied ; and hence, (Art. 2) the hypo
thesis of steady motion is correct*.
4. In the case of an incompressible fluid, either of infinite ex
tent, or confined, or interrupted in any manner by any solid bodies,
if the motion begin from rest, and if there be none of the cutting
motion mentioned in Art. 2, the motion at the time t will be the
* [N.B. It is only within a space which is at least doubly connected that such a
motion is possible. Thus in the example given in the preceding article, the axis of
the cylinder, where the velocity becomes infinite, may be regarded as an infinitely
slender core which we are forbidden to cross, and which renders the space within
the cylinder virtually ring shaped.]
26 ON SOME CASES OF FLUID MOTION.
same as if it were produced instantaneously by the impulsive
motion of the several surfaces which bound the fluid, including
among these surfaces those of any solids which may be immersed in
it. For let u t v, w, be the velocities at the time t. Then by a known
theorem udx + vdy + wdz will be an exact differential d<f>, and </>
will not alter abruptly (Art. 2). (f> must also satisfy the equation
(E}> and the conditions (a) and (b). Now if u, v , w , be the velo
cities on the supposition of an impact, these quantities must be
determined by precisely the same conditions as u, v and w. But
the problem of finding u , v and w , being evidently determinate, it
follows that the identical problem of finding u, v and w is also
determinate, and therefore the two problems have the same solu
tion ; so that
u = u , v v, w = w .
This principle has been mentioned by M. Cauchy, in a memoir
entitled Memoir e sur la Theorie des Ondes, in the first volume of
the Memoires des Savans Etrangers (1827), page 14. It will
be employed in this paper to simplify the requisite calculations by
enabling us to dispense with all consideration of the previous motion,
in finding the motion of the fluid at any time in terms of that of
the bounding surfaces. One simple deduction from it is that,
when all the bounding surfaces come to rest, each element of the
fluid will come to rest. Another is, that if the velocities of the
bounding surfaces are altered in any ratio the value of < will be
altered in the same ratio.
5. Superposition of different motions.
In calculating the initial motion of a fluid, corresponding to
given initial motions of the bounding surfaces, we may resolve the
latter into any number of systems of motions, which when com
pounded give to each point of each bounding surface a velocity,
which when resolved along the normal is equal to the given
velocity resolved along the same normal, provided that, if the
fluid be enclosed on all sides, each system be such as not to alter
its volume. For let u , v f , w, v , or , be the values of u, v, &c., corre
sponding to the first system of motions ; u", v", &c., the values of
those quantities corresponding to the second system, and so on ;
so that
v v
a cr + a" f ... .
ON SOME CASES OF FLUID MOTION. 27
Then since we have by hypothesis u dx + vdy + wdz an exact
differential d<j> , u"dx + v dy + w dz an exact differential d<f>" t and
so on, it follows that udx + vdy + wdz is an exact differential. Again
by hypothesis v = a, v" = a", &c., whence v = cr. Also, if the fluid
extend to an infinite distance, u, v, and w must there vanish, since
that is the case with each of the systems u, v, w\ &c. Lastly, the
quantities < , <", &c., not altering abruptly, it follows that <,
which is equal to < + </>"+ ... , will not alter abruptly. Hence the
compounded motion will satisfy all the requisite conditions, and
therefore (Art. 2) it is the actual motion.
It will be observed that the pressure p will not be obtained
by adding together the pressures due to each of the above systems
of velocities. To find p we must substitute the complete value of
(f) in equation (D). If, however, the motion be very small, so that
the square of the velocity is neglected, it will be sufficient to add
together the several pressures just mentioned.
In general the most convenient systems into which to decom
pose the motion of the bounding surfaces are those formed by
considering the motion of each surface, or of a certain portion of
each surface, separately". Such a portion may be either finite or
infinitesimal. In fact, in some of the cases of motion that will be
presently given, where (f> is expressed by a double integral with a
function under the integral sign expressing the motion of the
bounding surfaces, it will be found that each element of the inte
gral gives a value of (/> such that, except about the corresponding
element of the bounding surface, the motion of all particles in
contact with those surfaces is tangential.
A result which follows at once from this principle, and which
appears to admit of comparison with experiment, is the following.
Conceive an ellipsoid, or any body which is symmetrical with
respect to three planes at right angles to each other, to be made
to oscillate in a fluid in the direction of each of its three axes in
succession, the oscillations being very small. Then, in each case,
as may be shewn by the same sort of reasoning as that employed
in Art. 8, in the case of a cylinder, the effect of the inertia of the
fluid will be to increase the mass of the solid by a mass having a
certain unknown ratio to that of the fluid displaced. Let the axes
of coordinates be parallel to the axes of the solid; let x, y, z t be
28 ON SOME CASES OF FLUID MOTION.
the coordinates of the centre of the solid, and let M, M , M", be
the imaginary masses which we must suppose added to that of the
solid when it oscillates in the direction of the axes of x, y, z, respec
tively. Let it now be made to oscillate in the direction of a line
making angles a, /3, 7, with the axes, and let s be measured along
this line. Then the motions of the fluid due to the motions of
the solid in the direction of the three axes will be superimposed.
The motion being supposed to be small, the resultant of the pres
sures of the fluid on the solid will be three forces, equal to
2 <? ,7 2 <? /7 2 <?
a .ft f~.lv o AIT 1 1 lit a
M cos) M co S
respectively, in the directions of the three axes. The resultant of
these in the direction of the motion will be M t d 2 s/df where
M t = if cos 2 a + M f cos 2 /3 + M" cos 2 7 .
Each of the quantities M, M , M" and M ft may be determined
by observation, and we may find whether the above relation holds
between them. Other relations of the same nature may be de
duced from the principle explained in this article.
6. Reflection.
Conceive two solids, A and B, immersed in a fluid of infinite
extent, the whole being at rest. Suppose A to be moved in any
manner by impulsive forces, while B is held at rest. Suppose the
solids A and B of such forms that, if either were removed, and
the several points of the surface of the other moved instantaneously
in any given manner, the motion of the fluid could be determined :
then the actual motion can be approximated to in the following
manner. Conceive the place of B to be occupied by fluid, and A
to receive its given motion ; then by hypothesis the initial motion
of the fluid can be determined. Let the velocity with which the
fluid in contact with that which is supposed to occupy B s place
penetrates into the latter be found, and then suppose that the
several points of the surface of B are moved with normal velocities
equal and opposite to those just found, A a place being supposed
to be occupied by fluid. The motion of the fluid corresponding to
the velocities of the several points of the surface of B can then be
found, and A must now be treated as B has been, and so on. The
system of velocities of the particles of the fluid corresponding to
ON SOME CASES OF FLUID MOTION. 29
the first system of velocities of the particles of the surface of B,
form what may be called the motion of A reflected from B , the
motion of the fluid arising from the second system of velocities of
the particles of the surface of A may be called the motion of A
reflected from B and again from A, and so on. It must be re
membered that all these motions take place simultaneously. It
is evident that these reflected motions will rapidly decrease, at
least if the distance between A and B is considerable compared
with their diameters, or rather with the diameter of either. In
this case the calculation of one or two reflections will give the
motion of the fluid due to that of A with great accuracy. It is
evident that the principle of reflection will extend to any number
of solid bodies immersed in a fluid ; or again, the body B may be
supposed to be hollow, and to contain the fluid and A, or else A
to contain B. In some cases the series arising from the successive
reflections can be summed, in which case the motion will be deter
mined exactly. The principle explained in this article has been
employed in other subjects, and appears likely to be of great use
in this. It is the same for instance as that of successive influences
in Electricity.
7. If a mass of fluid be at rest or in motion in a close vessel
which it entirely fills, the vessel being either at rest or moving in
any manner, any additional motion of translation communicated
to the vessel will not affect the relative motion of the fluid. For
it is evident that on the supposition that the relative motion is
not affected the equation (B) and the condition (a) will still be
satisfied. Also, if Wj, tn 2 , v? 3 , be the components of the effective force
of any particle in the first case, and U, V, W, be the components
of the velocity of translation, then
dU dV dW
will be the components of the effective force of the same particle
in the second case. Now since by hypothesis vr^dx + vr z dy f vr s dz
is an exact differential, as follows from equations ((7), and U, V } W,
are functions of t only, it follows at once that
dU\, dV
30 ON SOME CASES OF FLUID MOTION.
is an exact differential, where x, y, z, are the coordinates of any
particle referred to the old axes, which are themselves moving in
space with velocities U, V, W. But if x lt y^z^ be the coordinates
of the same particle referred to parallel axes fixed in space, we
have
a^x + fUdt, y^y+fVdt, z^z+fWdt,
whence, supposing the time constant, dx=dx 1 , dy = dy lt dz = dz 1 ,
and therefore
dU\ 7 . f . d
) & >
is an exact differential. Hence, equations (A) can be satisfied by
a suitable value of p. Denoting by p the pressure about the par
ticle whose coordinates are x, y, z, in the first case, the pressure
about the same particle in the second case will be
{dU dV dW
It * dt
none of the terms of which will alter abruptly, since by hypothesis
p does not.
Since then the present hypothesis satisfies all the requisite
conditions, it follows from Art. 2 that that hypothesis is correct.
If F be the additional effective force of any particle of the vessel
in consequence of the motion of translation, and we take new axes
of x, y, z , of which the first is in the direction of F, the additional
term introduced into the value of the pressure will be pFx,
omitting the arbitrary function of the time. The resultant of the
additional pressures on the sides of the vessel will be equal to F
multiplied by the mass of the fluid, and will pass through the
centre of gravity of the fluid, and act in the directon of x.
8. Motion between two cylindrical surfaces having a common
axis.
Let us conceive a mass of fluid at rest, bounded by two cylin
drical surfaces having a common axis, these surfaces being either
infinite or bounded by two planes perpendicular to their axis. Let
us suppose the several generating lines of these cylindrical surfaces
to be moved parallel to themselves in any given manner consistent
with the condition that the volume of the fluid be not altered :
ON SOME CASES OF FLUID MOTION. 31
it is required to determine the initial motion at any point of the
mass.
Since the motion will take place in two dimensions, let the
fluid be referred to polar coordinates r, 0, in a plane perpendicular
to the axis, r being measured from the axis. Let a be the radius
of the inner surface, 6 that of the outer, f(6) the normal velocity
of any point of the inner surface, F(6) the corresponding quantity
for the outer.
Since for any particular radius vector between a and b the
value of (j> is a periodic function of 6 which does not become in
finite, (for the motion at each point of each bounding surface
is supposed to be finite), and which does not alter abruptly, it
may be expanded in a converging series of sines and cosines of
6 and its multiples. Let then
= P + 2r (P n cos nO + Q n sin nff) (1).
Substituting the above value in the equation
d
which <f) is to satisfy, and equating to zero the coefficients of
corresponding sines and cosines, which is allowable, since a given
function can be expanded in only one series of the form (1), we
find that P must satisfy the equation
A
of which the general integral is
the base being e, and P n and Q n must both satisfy the same
equation, viz.
dr dr
of which the general integral is
P n = Cr*+C r*.
We have then, omitting the arbitrary constant in </>, as will
32 ON SOME CASES OF FLUID MOTION.
be done for the future, since we have occasion to use only the
differential coefficients of (,
= A \og r + Sr {(A n r~ n + A n r n ] cos W
+ (J? n r" + J5>")sin^} ......... (3),
with the conditions
when r = a ..................... (4),
when r=6 ............ (5).
Let / (&) = + ( n cos n< + n sn w
j^(6>) = + SrC^cos n0 + & n sin TI
so that
with similar expressions for , &c. Then the condition (4)
gives
cos TZ^
whence,
Similarly, from the condition (5), we get
ON SOME CASES OF FLUID MOTION. 33
It will be observed that aC Q = bC Q , by the condition that the
volume of fluid remains unchanged, which gives
o Jo
From the above equations we easily get
and, changing the sign of n,
with similar expressions for B n and B M involving D in place of C.
We have then
</> = a(7 log r + Sr  (& 2W  a 2 ") 1 {[(b~ n+l C n  a~ n+1 CJ cos nO
fi
+ (b~ n+l D n  a n+l DJ sin nff] a?" b^r^
+ [(b n+1 C n a n "C n )cosn6
+ (b n+l D n a n+1 D n )siun0]r n } ..................... (6),
which completely determines the motion.
It will be necessary however, (Art. 2), to shew that this value
of (f> does not alter abruptly for points within the fluid, as may
be easily done. For the quantities C n> D n cannot be greater than
} where each element of the integral is taken posi
tively ; and since by hypothesis / (&) is finite for all values of 9
from to 2?r, it follows that neither G n nor D n can be numerically
greater than a constant quantity which is independent of n. The
same will be true of C n and D n . Remembering then that r>a
and < 6, it can be easily shewn that the series which occur in (6)
have their terms numerically less than those of eight geometric
series respectively whose ratios are less than unity; and since
moreover the terms of the former set of series do not alter abruptly,
it follows that < cannot alter abruptly. The same may be proved
in a similar manner of the differential coefficients of (/>. The other
infinite series expressing the value of < which occur in this paper
may be treated in the same way : and in Art. 10, where (/> is
expressed by a definite integral, the value of (/> and its differential
s. 3
84 ON SOME CASES OF FLUID MOTION.
coefficients will alter continuously, since that is the case with each
element of the integral. It will be unnecessary therefore to
refer again to the condition (d),
If the fluid be infinitely extended, we must suppose C n and
D n to vanish in (6), since the velocity vanishes at an infinite
distance ; we must then make b infinite, which reduces the above
equation to
..... (7).
This value of (/> may be put under the form of a definite
integral : for, replacing C Q , C n and D n by their values, it becomes
(0ff) dff,
which becomes on summing the series
log r*f(ff)W + **loe l  2 a  cos (6  ff] + / (P) dff;
whence
_^
~irr J
,
dr ~irr 2 r 2  2ar cos (0 ff)+
If we suppose r to become equal to a the quantity under the
integral sign vanishes, except for values of 6 , which are indefinitely
near to 6. The value of the integral itself becomes irf(0)*. Hence
it appears, that to the disturbance of each element of the surface,
there corresponds a normal velocity of the particles in contact
with the surface, which is zero, except just about the disturbed
element. The whole disturbance of the fluid will be the aggregate
of the disturbances due to those of the several elements of the
surface. The case of the initial motion of fluid within a cylinder,
and the analogous cases of motion within and without a sphere,
which will be given in the next article, may be treated in the
same manner.
The velocity in the direction of r given by the equation (7),
= > + 2 {G n cos ne + D n sin nO} t
* Poisson, TMorie de la Chalcur, Chap. vn.
ON SOME CASES OF FLUID MOTION. 35
and that perpendicular to r, and reckoned positive in the same
direction as 0, (= d(f)/rd0),
n+l
{C n smn0D n cosn6}.
Conceive a mass of fluid comprised between two infinite
parallel planes, and suppose that a certain portion of this fluid
contains solid bodies bounded by cylindrical surfaces perpendicular
to these planes. The whole being at first at rest, suppose that
the surfaces of these solids are moved in any manner, the motion
being in two dimensions. Conceive a circular cylindrical surface
described perpendicular to the parallel planes, and with a radius so
large that all the solids are comprised with it. Then, (Art. 4), we
may suppose the motion of the fluid at any time to have been
produced directly by impact. On this supposition the initial
motion of the part of the fluid without the above cylindrical
surface will be determined in terms of the normal motion of the
fluid forming that surface, as has just been done. If (7 be different
from zero, then, at a great distance in the fluid, the velocity will
be ultimately aCJr, and directed to or from the axis of the
cylinder, and alike in all directions. Since the rate of increase
of volume of a length I of the cylinder is equal to
it appears that the velocity at a great distance is proportional
to the expansion or contraction of a unit of length of the solids.
If however there should be no expansion or contraction, or if
the expansion of some of the solids should make up for the con
traction of the rest, then in general the most important part of
the motion at a great distance will consist of a velocity O cos l . /r 2
directed to or from the centre, and another C sin l . /r z perpen
dicular to the radius vector, the value of C and the direction from
which 0j is measured varying from one instant to another. The
resultant of these velocities will vary inversely as the square of
the distance.
Resuming the value of <f> given by equation (6), let us suppose
that the interior cylindrical surface is rigid, and moved with a
velocity C in the direction from which 6 is measured, the outer
32
36 ON SOME CASES OF FLUID MOTION.
surface being at rest: then / (6) = G cos 0, F (0) = Q ; whence
C l = C, and the other coefficients are each zero. We have then
Suppose now that the inner cylinder has a small oscillatory
motion about an axis parallel to the axis of the cylinders, the
cylinders having their axes coincident in the position of equi
librium. Let ty be the angle which a plane drawn through the
axis of rotation, and that of the solid cylinder at any time makes
with a vertical plane drawn through the former. The motion
of translation of the axis of the cylinder will differ from a recti
linear motion by quantities depending on iff: the motion of
rotation about its axis will be of the order ^, but will have no
effect on the fluid. Therefore in considering the motion of the
fluid we niay, if we neglect squares of ^, consider the motion
of the cylinder rectilinear. The expression given for </> by equa
tion (8) will be accurately true only for the instant when the
axes of the cylinders coincide ; but since the whole resultant
pressure on the solid cylinder in consequence of the motion is
of the order ty, we may, if we neglect higher powers of vjr than the
first, employ the approximate value of </> given by equation (8).
Neglecting the square of the velocity, we have
d$
P~ P dt
In finding the complete value of d(f>/dt it would be necessary to
express (f> by coordinates referred to axes fixed in space, which
after differentiation we might suppose to coincide with others
fixed in the body. But the additional terms so introduced de
pending on the square of the velocity, which by hypothesis is
neglected, we may differentiate the value of $ given by equation
(8) as if the axes were fixed in space. We have then, to the first
order of approximation,
ci
n =  ?~ 2 \ + r l cos &
dt b or (r J
If I be the length of the cylinder, the pressure on the element
ladO, resolved parallel to x and reckoned positive when it acts
in the direction of x,
ON SOME CASES OF FLUID MOTION. 87
P la *^
and integrating from = to = 2?r, we have the whole resultant
pressure parallel to x
tf + a? 2 dC
= 72 9 Vpla j .
If or dt
Since dC/dt is the effective force of the axis, parallel to x, and
that parallel to y is of the order ir 2 , we see that the effect of
the inertia of the fluid is to increase the mass of the cylinder
by TTT~ 2 A 6 * where p is the mass of the fluid displaced. This
imaginary additional mass must be supposed to be collected at the
axis of the cylinder.
If the cylinder oscillate in an infinitely extended fluid b = & ,
and the additional mass becomes equal to that of the fluid dis
placed. This appears to be a result capable of being compared
with experiment, though not with very great accuracy. Two
cylinders of the same material, and of the same radius, but whose
lengths differ by several radii, might be made to oscillate in
succession in a fluid, at a depth sufficiently great to allow us
to neglect the motion of the surface of the fluid. The time of
oscillation of each might then be calculated as if the cylinder
oscillated in vacuum, acted on by a moving force equal to its
weight minus that of the fluid displaced, acting downwards
through its centre of gravity, and having its mass increased by an
unknown mass collected in the axis. Equating the time of oscil
lation so calculated to that given by observation, we should
determine the unknown mass. The difference of these masses
would be very nearly equal to the mass which must be added
to that of a cylinder whose length is equal to the difference of
the lengths of the first two, when the motion is in two dimensions.
This evidently comes to supposing that, at a distance from the
middle of the longer cylinder not greater than half the difference
of the lengths of the two, the motion may be taken as in two
dimensions. The ends of the cylinders may be of any form,
provided that they are all of the same. They may be suspended
by fine equal wires, in which case we should have a compound
38 ON SOME CASES OF FLUID MOTION.
pendulum, or attached to a rigid body oscillating above the fluid
by means of thin flat bars of metal, whose plane is in the plane of
motion. Another way of getting rid of the motion in three
dimensions about the ends would be, to make those ends plane,
and to fix two rigid planes parallel to the plane of motion, which
should be almost in contact with the ends of the cylinder.
9. Motion between two concentric spherical surfaces. Motion
of a ball pendulum enclosed in a spherical case.
Let a mass of fluid be at rest, comprised between two con
centric spherical surfaces. Let the several points of these surfaces
be moved in any manner consistent with the condition that the
volume of the fluid be not changed : it is required to determine
the initial motion at any point of the mass.
Let a, b, be the radii of the inner and outer spherical surfaces
respectively ; then employing the coordinates r } 0, o>, where r
is the distance from the centre, 6 the angle which r makes with
a fixed line passing through the centre, G> the angle which a plane
passing through these two lines makes with a fixed plane through
the latter, the value of </> corresponding to any radius vector
comprised between a and b can be expanded in a converging
series of Laplace s coefficients. Let then
V n being a Laplace s coefficient of the n th order.
Substituting in the equation,
dV0 1 d ( . a d$\ , 1
r TT +  Q jh sm JQ + =T
dr sin 6 dv \ dvj sin
which </> is to satisfy, employing the equation
and then equating to zero the Laplace s coefficients of the several
orders, we find
The general integral of this equation is
ON SOME CASES OF FLUID MOTION. 39
where C and are functions of 6 and G>. Substituting in the
equation (9), and equating coefficients of the two powers of r
which enter into it separately to zero, we find that both C and G
satisfy it, and therefore are both Laplace s coefficients of the n ih
order. We have then
where Y n and Z n are each Laplace s coefficients of the n ih order,
and do not contain r. Let f(0, w) be the normal velocity of the
point of the inner surface corresponding to 6 and co, F(6, o>) the
corresponding quantity for the outer ; then the conditions which
<j) is to satisfy are that
~ = f(Q } G)) when r = a,
dr
^0 T^/a \ T_ 7
r = if (u, &)) when r b.
Let /(#, ft)), expanded in a series of Laplace s coefficients, be
P.+P....+P.+...
which expansion may be performed by the usual formula,, if not
by inspection: then the first condition gives
STw Y n n ~ l (n 1 1\ ^ /7( n + 2U ^P
t 7i \ * / n ** j ^o 71 >
and equating Laplace s coefficients of the same order, we get
V ^nl Inn i "T \ ^ ^(n+2) P f~\T\
71 2 n d \n\ i)Zj n d v = JL n \^^/
Let F(0, ft)), expanded in a series of Laplace s coefficients, be
F t +P t ...P,+ ...!
then from the second condition, we get
7 iF w ^l_( w+ l)^(n+2) == p n (12).
From (11) and (12) we easily get
P f lkn+2 P s*n+2
Tr JL u _/: (jj
provided n be greater than 0. If n 0, we have
40 ON SOME CASES OF FLUID MOTION.
But the condition that the volume of the fluid be not altered,
gives
tf I I *f(0, ) sin OdBdto = &*[* f F(0, to) sin 0d0da,
J OJQ JO ./O
or 47ra 2 P =47r& 2 P ,
which reduces the two equations just given to one.
We have then, omitting the constant Y QJ
i (P n 6J;2  P w a + 2 ) r n
which determines the motion.
When the fluid is infinitely extended, we have P n = since
the velocity vanishes at an infinite distance, and b = GO , whence
It may be proved, precisely as was done, (Art. 8), for motion
in two dimensions, that if any portion of an infinitely extended
fluid be disturbed by the motion of solid bodies, or otherwise,
if all the fluid beyond a certain distance from the part disturbed
were at first at rest, the velocity at a great distance will ultimately
be directed to or from the disturbed part, and will be the same
in all directions, and will vary as r~ 2 . The coefficient of r~ 2 will
be proportional to the rate of gain or loss of volume of the part
disturbed. If however this rate should be zero, then the most
important part of the velocity at a great distance will in general
be that depending on the term ^a s P l .r~ z in <, Since the
general form of P l is
J.cos 0+ J9sin0cosa> + sin0sin,
we easily find, by making use of rectangular coordinates, changing
the direction of the axes, and then again adopting polar co
ordinates, that the above term in <f> takes the form D cos 6^ . r~ 2 ,
0j being measured from some line passing through the origin.
The motion will therefore be the same as that round a ball
pendulum in an incompressible fluid, the centre of the ball being
in the origin; a case of motion which will be considered im
mediately. In order to represent the motion at different times,
ON SOME CASES OF FMJID MOTION. 41
we must suppose the velocity and direction of motion of the
ball to change with the time.
The value of </> given by equation (13) is applicable to the
determination of the motion of a ball pendulum enclosed in a
spherical case which is concentric with the ball in its position of
equilibrium. If G be the velocity of the centre of the ball at
the instant when the centres of the ball and case coincide, and
if 6 be measured from the direction in which it is moving, we
shall have
/. P = 0, P^C cos 0, P 2 = 0, &c., P = 0, &c.,
and the value of (/> for this instant is accurately
Co 3 b*
which, when b = oo , becomes
Co? cos
2r z
which is the known expression for the value of </> for a sphere
oscillating in an infinitely extended, incompressible fluid.
It may be shewn, by precisely the same reasoning as was
employed in the case of the cylinder, that in calculating the
small oscillations of the sphere the value of d(f>/dt to be employed is
and from the equation p = p d<f>/dt, we easily find that the whole
resultant pressure on the sphere in the direction of its centre, and
tending to retard it is
4 Trpa 8 t V_\dCL
and that perpendicular to this direction is zero. Since dC/dt is
the effective force of the centre in the direction of the motion, and
that perpendicular to this direction is of the second order, the
effect of the inertia of the fluid will be to increase the mass of the
sphere by a mass
42 ON SOME CASES OF FLUID MOTION.
IJL being the mass of the fluid displaced ; so that the effect of the
case is, to increase the mass which we must suppose added to
that of the ball in the ratio of b 3 + 2a 3 to V  a\
Poisson, in his solution of the problem of the oscillating sphere
given in the Memoir es de I Acade mie, Tome XL arrives at a different
conclusion, viz. that the case does not at all affect the motion of
the sphere. When the elimination which he proposes at p. 563
is made, the last term of equation (/), p. 550, becomes
where a is the velocity of propagation of sound, and 8 the ratio
of the density of air to that of the ball, f and " being functions
derived from others which enter into the value of <f> by putting
r = c, where c is the radius of the ball. He then argues that
this term may be neglected as insensible, since it involves 8 in
the numerator and a 2 in the denominator, tacitly assuming that
jy + is not large since is not large. Now for the disturb
ed dt
ances of the air which have the same period as those of the
pendulum d$/dt is not large compared with <, as it is for those on
which sound depends. Let then Poisson s solution of equation (a),
p. 547 of the volume already mentioned, be put under the form
/ and F denoting the derived functions, and all the Laplace s
coefficients except those of the first order being omitted, the value
of </> just given being supposed to be a Laplace s coefficient of that
order. Then if we expand the above functions in series ascending
according to powers of r/a, we find
and in order that when a = oo this equation may coincide with
(10), when all the Laplace s coefficients except those of the first
order are omitted in that equation, it will be seen that it is
ON SOME CASES OF FLUID MOTION. 43
necessary to suppose f"(t)F "(t) t and therefore f(t)F(),
to be of the order a/ 5 , while f(t) + F (t) is not large. Putting then
=x (0
l?(*)X()
we shall have
.
so that ^73^ will contain a term of the order a 2 , and the
Cut
term which Poisson proposes to leave out will be of the same
order of magnitude as those retained.
In making the experiment of determining the resistance of
the air to an oscillating sphere, it would appear to be desirable
to enclose the sphere in a concentric spherical case, which would
at the same time exclude currents of air, and facilitate in some
measure the experiment by increasing the small quantity which is
the subject of observation. The radius of the case however ought
not to be nearly as small as that of the ball, for if it were, in
the first place a small error in the position of the centre of the
ball when at rest might not be insensible, and in the second place
the oscillations would have to be inconveniently small, in order
that the value of <f> which has been given might be sufficiently
approximate. The effect of a small slit in the upper part of the
case, sufficient to allow the wire by which the ball is supported
to oscillate, would evidently be insensible, for the condensation
being insensible in a vertical plane passing through the axis of
rotation, since the alteration of pressure in that plane is insensible,
the air would not have a tendency alternately to rush in and out
at the slit.
10. Effect of a distant rigid plane on the motion of a ball
pendulum.
Although this problem may be more easily solved by an arti
fice, it may be well to give the direct solution of it by the method
mentioned in Article 6. In order to calculate the motion re
flected from the plane, it will be necessary to solve the following
problem :
44 ON SOME CASES OF FLUID MOTION.
To find the initial motion at any point of a mass of fluid in
finitely extended, except where it is bounded by an infinite solid but
not rigid plane, the initial motion of each point of the solid plane
being given.
It is evident that motion directed to or from a centre situated
in the plane, the velocity being the same in all directions, and
varying inversely as the square of the distance from that centre,
would satisfy the condition that udx + vdy + wdz is an exact
differential, and would give to the particles in contact with the
plane a velocity directed along the plane, except just about the
centre. Let us see if the required motion can be made up of an
infinite number of such motions directed to or from an infinite
number of such centres.
Let x, y, z, be the coordinates of any particle of fluid, the
plane xy coinciding with the solid plane, and the axis of z being
directed into the fluid. Let x, y y be the coordinates of any point
in the solid plane : then the part of < corresponding to the motion
of the element dxdy of the plane will be
ty(x, y )dx dy
and therefore the complete value of </> will be given by the equa
tion
4 = f f _ *fr.yWfr ( H).
* /(/ \a i /. ^. \* i .2) V X
The velocity parallel to z at any point = dcf>/dz
Now when z vanishes the quantity under the integral signs
vanishes, except for values of x and y indefinitely near to x arid y
respectively, the function ty(x t y ) being supposed to vanish when
x or y is infinite. Let then x = x + f , y = y + 77, then, and 77,
being as small as we please, the value of the above expression
when z = becomes
the limit off
/
Now if ^r(x, y } does not alter abruptly between the limits x
ON SOME CASES OF FLUID MOTION. 45
and x + % , of x, and y 77, and ;y + 77, of y , the above expression
may be replaced by
 x the limit
which is = 27Ti/r(a?, y}.
If now/(# , y") be the given normal velocity of any point (a? , y}
of the solid plane, the expression for < given by equation (14) may
be made to give the required normal velocity of the fluid particles
in contact with the solid plane by assuming
whence
A = IT f f(
2vr J _ J .. {(aj aj ) a + (y
This expression will be true for any point at a finite distance from
the plane xy even when / (x, y } does alter abruptly; for we may
first suppose it to alter continuously, but rapidly, and may then
suppose the rapidity of alteration indefinitely increased : this will
not cause the value of just given to become illusory for points
situated without the plane xy.
If it be convenient to use polar coordinates in the plane xy,
putting x = q cos co, y = q sin co, x = q cos co , y q sin &/, and re
placing/^ , y } by/(/, < ), the equation just given becomes
2?r o o {q 2 + <f Zqq cos (co  co ) +
To apply this to the case of a sphere oscillating in a fluid per
pendicularly to a fixed rigid plane, let a be the radius of the sphere,
and let its centre be moving towards the plane with a velocity C
at the time t. Then, (Art. 4), we may calculate the motion as if
it were produced directly by impact. Let h be the distance of the
centre of the sphere from the fixed plane at the time t, and let
the line h be taken for the axis of z, and let r, 0, be the polar co
ordinates of any point of the fluid, r being the distance from the
centre of the sphere, and 6 the angle between the lines r and h.
Then if the fluid were infinitely extended around the sphere we
should have
(7a s cos
40 ON SOME CASES OF FLUID MOTION.
The velocity of any particle, resolved in a direction towards the
plane, = d$/dr . cos d$/rd& . sin 6
LsCL r n *
For a particle in the plane xy we have
r cos 6 h, rsm6 = q>
and the above velocity becomes
We must now, according to the method explained in (Art. 6), sup
pose the several points of the plane xy moved with the a,bove
velocity parallel to z. We have then
whence, for the motion of the sphere reflected from the plane,
* a* a rf (wMw*
47r J o J tf+ < +  2 cos a,  + z 2 * "
 2 qq cos (a, 
We must next find the velocity, corresponding to this value of
(f>, with which the fluid penetrates the surface of the sphere. We
have in general
z h r cos 0, q r sin 0,
whence
[f + f  2qq cos (o>  < ) + z 2 }^
= {h 2 + r 2 + # 2  2hr cos  2q r sin cos (co  a> )}~*.
Now supposing the ratio of a to h to be very small, and retaining
the most important term, the value of d(f>/dr when r = a will be
equal to the coefficient of r when is expanded in a series ascend
ing according to powers of r,
_Ca?_ r ^ (2A 2  q z ) {h COS + tf sin 6 cos (a  a) )} q dq da)
47rJ J " (h* + q *)*
Ca 5 cos 9
In order now to determine the motion reflected from the
plane and again from the sphere, we must suppose the several
points of the sphere to be moved with a normal velocity
ON SOME CASES OF FLUID MOTION. 47
Ca 3 cos 6 . /8h 3 , or, which is the same, we must suppose the whole
sphere to be moved towards the plane with a velocity Ca s /8h 3 .
Hence the value of <f> corresponding to this motion will be given
by the equation
Ca 6 cos
For points at a great distance from the centre of the sphere,
the motion which is twice reflected will be very small compared
with that which is but once reflected. For points close to the
sphere however, with which alone we are concerned, those motions
will be of the same order of magnitude, and if we take account
of the one we must take account of the other.
Putting 2=rsin#, z = hrcos0 in (16), expanding, and
retaining the two most important terms, we have
K being a constant, the value of which is not required, and the
second term being evidently found by multiplying the quantity
at the second side of (17) by r. Adding together the parts of $
given by equations (15), (18) and (19), putting r = a t replacing
G by dC/dt, and taking for h the value which it has in equili
brium, just as in the case of the oscillating cylinder in Article 8,
we have for the small motion of the sphere
cty jrdC a/ Sa 3 \dC
The resultant of the part of the pressure due to the first term
is zero : that due to the second term is greater than if the plane
were removed in the ratio of l + Sa 3 /8h 3 to 1. Consequently, if
we neglect quantities of the order a 4 //*, 4 , the effect of the inertia
of the fluid is, to add a mass equal to (1 + 3a 3 /8h?) . \p to that of
the sphere, without increasing the moment of inertia of the latter
about its diameter. The effect therefore of a large spherical case
is eight times as great as that of a tangent plane to the case,
perpendicular to the direction of the motion of the ball.
The effect of a distant rigid plane parallel to the direction
of motion of an oscillating sphere might be calculated in the
same manner, but as the method is sufficiently explained by the
48 ON SOME CASES OF FLUID MOTION.
first case, it will be well to employ the artifice before alluded to,
an artifice which is frequently employed in this subject. It con
sists in supposing an exactly symmetrical motion to take place
on the opposite side of a rigid plane, by which means we may
evidently conceive the plane removed.
Let the sphere be oscillating in the direction of the axis of oc,
the oscillations in this case, as in the last, being so small that
they may be taken as rectilinear in calculating the motion of the
fluid ; and instead of a rigid plane conceive an equal sphere to exist
at an equal distance on the opposite side of the plane xy, moving
in the same direction and with the same velocity as the actual
sphere. Let r, 0, ew, be the polar coordinates of any particle
measured from the centre of the sphere, 6 being the angle between
r and a line drawn through the centre parallel to the axis of x,
and w the angle which the plane passing through these lines makes
with the plane ocz. Let r t , o> 3 be the corresponding quantities
symmetrically measured from the centre of the imaginary sphere.
If the fluid were infinite we should have for the motion cor
responding to that of the given sphere
The motion reflected from the plane is evidently the same as
that corresponding to the motion of the imaginary sphere in an
infinite mass of fluid, for which we have
Co? cos &
Now r cos = r cos 6, r sin & sin to = r sin 6 sin to,
/ sin & cos o> + r sin 6 cos o> = 2h ;
whence r 2 = r 2 + 4A 2 4<hr sin cos co,
and equation (21) is reduced to
<7aV cos 6
2 {r 2 + 4 2  4/tr sin cos o>} f
Retaining only the terms of the order a*r/h a or r 4 /^ 3 , so as to get
the value of d<j>/dr to the order a 3 /h 3 , the above equation is re
duced to
Ca*r cos ,
...................... (22)
ON SOME CASES OF FLUID MOTION. 49
and the value of dfy/dr when r = a is, to the required degree of
approximation,
Ca 3 cos 6
For the value of </> corresponding to the motion of the imaginary
sphere reflected from the real sphere, we shall therefore have
, Co? cos
Adding together the values of $ given by (20), (22) and (23),
putting r = a, and replacing C by dC/dt, we have, to the requisite
degree of approximation,
3 a s \ dO
Hence in this case the motion of the sphere will be the same as
if an additional mass equal to (l + 3a 8 /16& 8 ) . \p were collected
at its centre. The effect therefore of a distant rigid plane which
is parallel to the direction of the motion of a ball pendulum will
be half that of a plane at the same distance, and perpendicular
to that direction. It would seem from Poisson s words at page 562
of the eleventh volume of the Memoir es de VAcademie, that he
supposed the effect in the former case to depend on a higher
order of small quantities than that in the latter.
If the ball oscillate in a direction inclined to the plane, the
motion may be easily deduced from that in the two cases just
given, by means of the principle of superposition.
11. The values of </> which have been given for the motion
of translation of a sphere and cylinder do not require us to
suppose that either the velocity, or the distance to which the
centre of the sphere or axis of the cylinder has been moved, is
small, provided the same particles remain in contact with the
surface. The same indeed is true of the values corresponding to
a motion of translation combined with a motion of contraction
or expansion which is the same in all directions, but varies in any
manner with the time. The value of </> corresponding to a motion
of translation of the cylinder is  Ca 2 cos 9 . r~\ C being the velo
city of the axis, and 6 being measured from a line drawn in the
direction of its motion. The whole resultant of the part of the
pressure due to the square of the velocity is zero, since the velocity
at the point whose coordinates are r, 0, is the same as that at
S. 4
50 ON SOME CASES OF FLUID MOTION.
the point whose coordinates are r and rr6. To find the re
sultant of the part depending on d(j>/dt, it will be necessary to
express ^ by means of coordinates referred to axes fixed in space.
Let Ox, Oy, be rectangular axes passing through the centre of
any section of the cylinder, OT the angle which the direction of
motion of the axis makes with Ox, & the inclination of any radius
vector to Ox \ then
Co?
<f>=  jj (r cos 6 cos w + r sin 6 sin BT)
a*(C x+C"y)
x* + y*
putting Q and C" for the resolved parts of the velocity G along
the axes of x and y respectively. Taking now axes Ax, Ay,
parallel to the former and fixed in space, putting a and /3 for the
coordinates of 0, differentiating <j> with respect to t, and replacing
da/dt by C , and d/3/dt by C", and then supposing a and /3 to
vanish, we have
,/ dC?_ dC"\
d = a 2 2 _ 2o^( * + (Ty)! _ * V dt V dt
dt ~
The resultant of the part of the pressure due to the first two
terms is zero, since the pressure at the point (x, y) depending on
these terms is the same as that at the point ( x, y). It will
be easily found that the resultant of the whole pressure parallel
to x, and acting in the negative direction, on a length I of the
cylinder, is equal to irpla? . d C /dt, and that parallel to y equal to
Trpla 2 . dC"ldt. The resultant of these two will be TrplcfF, where
F is the effective force of a point in the axis of the cylinder, and
will act in a direction opposite to that of F. Hence the only
effect of the motion of the fluid will be, to increase the mass of
the cylinder by that of the fluid displaced. In a similar manner
it may be proved that, when a solid sphere moves in any manner
in an infinite fluid, the only effect of the motion of the fluid is to
increase the mass of the sphere by half that of the fluid displaced.
A similar result may be proved to be true for any solid sym
metrical with respect to two planes at right angles to each other,
and moving in the direction of the line of their intersection in
an infinitely extended fluid, the solid and fluid having been at
first at rest. Let the planes of symmetry be taken for the planes
of xy and xz, the origin being fixed in the body : then it is evident
ON SOME CASES OF FLUID MOTION. 51
that the resultant of the pressure on the solid due to the motion
will be in the direction of the axis of x, and that there will be
no resultant couple. Let C be the velocity of the solid at any
time ; then the value of < at that time will be of the form
Cifr (x, y, 2), where G alone contains t (Art. 4), and the velocity
of the particle whose coordinates are #, y, z, being proportional
to (7, the vis viva of the solid and the fluid together will be
proportional to C*. Now if no forces act on the fluid and solid,
except the pressure of the fluid, this vis viva must be constant * ;
therefore G must be constant ; therefore the resultant of the fluid
pressure on the solid must be zero. If now G be a function of t
we shall have
p = _p^ (a, #*) +/,
p being the pressure when G is constant. Since therefore the
resultant of the fluid pressure varies for the same solid and fluid
as dC/dt the effective force, and for different fluids varies as p,
the effect of the inertia of the fluid will be, to increase the mass
of the solid by n times that of the fluid displaced, n depending
only on the particular solid considered.
Let us consider two such solids, similar to each other, and
having the coordinate planes similarly situated, and moving with
the same velocities. Let the linear dimensions of the second
be greater than those of the first in the ratio of m to 1. Let
* If an incompressible fluid which is homogeneous or heterogeneous, and con
tains in it any number of rigid bodies, be in motion, the rigid bodies being also
in motion, if the rigid bodies are perfectly smooth, and no contacts are formed or
broken among them, and if no forces act except the pressure of the fluid, the
principle of vis viva gives
where v is the whole velocity of the mass m, and the sign 2 extends over the whole
fluid and the rigid bodies spoken of, and where dS is an element of the surface
which bounds the whole, p / the pressure about the element dS, and v the normal
velocity of the particles in that element, reckoned positive when tending into the
fluid, and where the sign ff extends to all points of the bounding surface. To apply
equation (a) to the case of motion at present considered, let us first confine our
selves to a spherical portion of the fluid, whose radius is r, and whose centre is near
the solid, so that dS refers to the surface of this portion. Let us now suppose r to
become infinite : then the second side of (a) will vanish, provided^ remain finite,
and v decrease in a higher ratio than r~ 2 . Both of these will be true, (Art. 9) ; for
v will vary ultimately as r~ 3 , since there is no alteration of volume. Hence if the
sign S extend to infinity, we shall have 2/mv 2 constant.
. 42
52 ON SOME CASES OF FLUID MOTION.
u, v, w, be the velocities, parallel to the axes, of the particle (x, y, z]
in the fluid about the first ; then shall the corresponding velocities
at the point (mx, my, mz) in the fluid about the second be also
u, v, to. For
udmx + vdmy + wdmz = m (udx + vdy f wdz) (24),
and is therefore an exact differential, since udx + vdy+wdz is
one : also the normal at the point (x, y, z) in the first surface will
be inclined to the axes at the same angles as the normal at the
point (mx, my, mz) of the second surface is inclined to its axes,
and therefore the normal velocities of the two surfaces at these
points are the same ; and the velocities of the fluid at these two
points parallel to the axes being also the same, it follows that the
normal velocity of each point of the second surface is equal to
that of the fluid in contact with it. Lastly, the motion about
the first solid being supposed to vanish at an infinite distance
from it, that about the second will vanish alsU Hence the sup
position made with respect to the motion of the fluid about the
second surface is correct. Now putting for $(udx + vdy + wda}}
for the fluid in the first case, the corresponding integral for the
fluid in the second case will be ???<, if the constant be properly
chosen, as follows from equation (24). Consequently the value of
that part of the expression for the pressure, on which the resist
ance depends, will be m times as great for any point in the second
case as it is for the corresponding point in the first. Also, each
element of the surface of the second solid will be m 2 times as
great as the corresponding element of the surface of the first.
Hence the whole resistance on the second solid will be m 3 times
as great as that on the first, and therefore the quantity n depends
only on the form, and not on the size of the solid.
When forces act on the fluid, it will only be necessary to add
the corresponding pressure. Hence when a sphere descends from
rest in a fluid by the action of gravity, the motion will be the same
as if a moving force equal to that of the sphere minus that of
the fluid displaced acted on a mass equal to that of the sphere
plus half that of the fluid displaced. For a cylinder which is
so long that we may suppose the length infinite, descending hori
zontally, every thing will be the same, except that the mass to be
moved will be equal to that of the cylinder plus the whole of the
fluid displaced. In these cases, as well as in that of any solid
ON SOME CASES OF FLUID MOTION. 53
which is symmetrical with respect to two vertical planes at right
angles to each other, the motion will be uniformly accelerated,
and similar solids of the same material will descend with equal
velocities. These results are utterly opposed even to the com
monest observation, which shews that large solids descend much
more rapidly than small ones of the same shape and material,
and that the velocity of a body falling in a fluid (such as water),
does not sensibly increase after a little time. It becomes then
of importance in the theory of resistances to enquire what may be
ths cause of this discrepancy between theory and observation.
The following are the only ways of accounting for it which suggest
themselves to me.
First. It has been supposed that the same particles remain in
contact with the solid throughout the motion. It must be re
membered that we suppose the ultimate molecules of fluids (if
such exist), to be so close that their distance is quite insensible, a
supposition of the truth of which there can be hardly any doubt.
Consequently we reason on a fluid as if it were infinitely divisible.
Now if the motion which takes place in the cases of the sphere
and cylinder be examined, supposing for simplicity their motions
to be rectilinear, it will be found that a particle in contact with
the surface of either moves along that surface with a velocity which
at last becomes infinitely small, and that it does not reach the
end of the sphere or cylinder from which the whole is moving
until after an infinite time, while any particle not in contact with
the surface is at last left behind. It seems difficult to conceive of
what other kind the motion can be, without supposing a line
(or rather surface) of particles to make an abrupt turn. If it
should be said that the particles may come off in tangents, it must
be remembered that this sort of motion is included in the con
dition which has been assumed with respect to the surface.
Secondly. The discrepancy alluded to might be supposed to
arise from the friction of the fluid against the surface of the solid.
But, for the reason mentioned in the beginning of this paper, this
explanation does not appear to me satisfactory.
Thirdly. It appears to me very probable that the spreading
out motion of the fluid, which is supposed to take place behind
the middle of the sphere or cylinder, though dynamically possible,
nay, the only motion dynamically possible when the conditions
54 ON SOME CASES OF FLUID MOTION.
which have been supposed are accurately satisfied, is unstable ;
so that the slightest cause produces a disturbance in the fluid,
which accumulates as the solid moves on, till the motion is quite
changed. Common observation seems to shew that, when a solid
moves rapidly through a fluid at some distance below the surface,
it leaves behind it a succession of eddies in the fluid. When the
solid has attained its terminal velocity, the product of the resist
ance, or rather the mean resistance, and any space through which the
solid moves, will be equal to half the via viva of the corresponding
portion of its tail of eddies, so that the resistance will be measured
by the vis viva in the length of two units of that tail. So far
therefore as the resistance which a ship experiences depends
on the disturbance of the water which is independent of its
elevation or depression, that ship which leaves the least wake
ought, according to this view, to be cceteris paribus the best sailer.
The resistance on a ship differs from that on a solid in motion
immersed in a fluid in the circumstance, that part of the resist
ance is employed in producing a wave.
Fourthly. The discrepancy alluded to may be due to the
mutual friction, or imperfect fluidity of the fluid.
12. Motion alout an elliptic cylinder of small eccentricity*.
The value of <, which has been deduced (Art. 8), for the
motion of the fluid about a circular cylinder, is found on the
supposition that for each value of r there exists, or may be
[* This particular problem, so far at least as concerns motion of translation,
is of little interest in itself, because Green (see Transactions of, the Eoijal Society
of Edinburgh, Vol. xm. p. 5.4, or p. 315 of his collected works) has determined the
motion of a fluid about an ellipsoid moving in any manner with a motion of trans
lation only; and the ellipsoid includes of course as a particular case an elliptic
cylinder of any eccentricity. The problem in the text will however serve as an
example of the mode of proceeding in the case of a cylinder of any kind differing
little from a circular cylinder.
In the case of such a cylinder, supposed to be free from abrupt changes of form,
it might safely be assumed that the expression for which applies to the fluid
beyond the greatest radius vector of any point of the surface might also be used
for some distance within, as explained in the text. By starting with this assumption,
which would be verified in the end, the process of solution would of course be
shortened. We should simply have to take the expression (31 ), form the expression
(26 ) for the velocity normal to the surface, putting r = c (1 + e cos 20), and expand
ing as far as the first power of e, and equate the result to the expression (26). We
should thus determine the arbitrary constants in (31 ), which would complete the
solution of the problem.]
ON SOME CASES OF FLUID MOTION. 55
supposed to exist, a real and finite value of <. This will be true,
in any case of motion in two dimensions where udx + vdy is an
exact differential, for those values of r for which the fluid is not
interrupted, but will be true for values of r for which it is in
terrupted by solids only when it is possible to replace those solids
at any instant by masses of fluid, without affecting the motion
of the fluid exterior to them, those masses moving in such a
manner that the motion of the whole fluid might have been
produced instantaneously by impact. In some cases such a
substitution could be made, while in others it probably could not.
In any case however we may try whether the expansion given
by equation (3) will enable us to get a result, and if it will, we
need be in no fear that it is wrong (Art. 2). The same remarks
will apply to the question of the possibility of the expansion of <
in the series of Laplace s coefficients given in equation (10), for
values of r for which the fluid is interrupted. They will also
apply to such a question as that of finding the permanent tempe
rature of the earth due to the solar heat, the earth being supposed
to be a homogeneous oblate spheroid, and the points of the
surface being supposed to be kept up to constant temperatures,
given by observation, depending on the latitude.
In cases of fluid motion such as those mentioned, the motion
may be determined by conceiving the whole mass of fluid divided
into two or more portions, taking the most general value of </> for
each portion, this value being in general expressed in a different
manner for the different portions, then limiting the general value
of (f> for each portion so as to satisfy the conditions with respect to
the surfaces of solids belonging to that portion, and lastly in
troducing the condition that the velocity arid direction of motion
of each pair of contiguous particles in any two of the portions are
the same. The question first proposed will afford an example
of this method of solution.
Let an elliptic cylinder be moving with a velocity (7, in the
direction of the major axis of a section of it made by a plane
perpendicular to its axis. The motion being supposed to be in
two dimensions, it will be sufficient to consider only this section.
Let
r = c (1 + e cos 20)
be the approximate equation to the ellipse so formed, the centre
56 ON SOME CASES OF FLUID MOTION.
being the pole, and powers of e above the first being neglected.
Let a circle be described about the same centre, and having a
radius 7 equal to (1 + k) c, k being ^ e, and being a small quantity
of the order e. Let the portions of fluid within and without the
radius 7 be considered separately, and putting
r = c + z t
let the value of (f> corresponding to the former portion be
P, Q and R being functions of 0, and the term in 2 being retained,
in order to get the value of dfyjdr true to the order e, while the
terms in z s , &c. are omitted. Substituting this value of <f> in
equation (2), and equating to zero coefficients of different powers
of z, we have
_
2c 2c 2 d6*
which is the only condition to be satisfied, since the other equations
would only determine the coefficients of z 3 , &c. in terms of the
preceding ones. We have then
Now if be the angle between the normal at any point of the
ellipse, and the major axis, we have
and the velocity of the ellipse resolved along the normal
= (7 cos f = G (I  e) cos + Ce cos 30 ......... (26).
The velocity of the fluid at the same point resolved along the
normal is
Let P and Q be expanded in series of cosines of 6 and its mul
tiples, so that
P = 2 " P H cos n9, Q = ^ Q n cos w0,
ON SOME CASES OF FLUID MOTION. 57
there being no sines in the expansions of P and Q, since the
motion is symmetrical with respect to the major axis ; then
j  ~ (Q, J P.) cos n6 . . ..(28) ;
(29);
For a point in the ellipse, z cecos 20, whence from (27), (29) and
(30), we find that the normal velocity of the fluid
= 2 " Q n cos 7i0 + I n (n  2) *  Q ;i cos (n  2)
which is the same thing as
}[< 2) *L
+ n (n + 2) Q,, +2 cos0....(31),
if we suppose P and Q to be zero when affected with a negative
suffix. This expression will have to be equated to the value of
C cos given by equation (26).
For the part of the fluid without the radius 7 we have
<=.^ logr + 2r cosn0* ............. (31 ),
since there will be no sines in the expression for <, because the
motion is symmetrical with respect to the major axis, and no
positive powers of r, because the velocity vanishes at an infinite
distance.
From the above value of </> we have, for the points at a distance
7 from the centre,
* The first term of this expression is accurately equal to zero, since there is
no expansion or contraction of the solid (Art. 8). I have however retained it, in
order to render the solution of the problem in the present article independent of
the proposition referred to.
58 ON SOME CASES OF FLUID MOTION.
d<f> A Q ^<*nA n
y = ^ Zj j&i cos rc#,
dr 7 1 7"
e_ S w4, 
~~* S
Equating the above expressions to the velocities along and per
pendicular to the radius vector given by equations (29) and (30),
when z is put = kc, and then equating coefficients of corresponding
sines and cosines, we have
(!&)&+* S = _^ .............. (32),
(33),
when n > 0, and equating constant terms we have
from which equation with (32) and (33) we have, putting
T) A M A A
^ = t?, <3 = ^? when>0, and , = =*.
p C v t/
Substituting these values in the expression (31), it becomes
2: (n + !)( 2)  + 1 ( + 1) ( + 2) g? cos
p c
In the case of a circular cylinder the quantities A , A, 2 , A a , &c. are
each zero. In the present case therefore they are small quantities
depending on e. Hence, neglecting quantities of the order e 2
in the above expression, it becomes
which must be equal to <7{(1 e) cos ^ + e cos 3^}. Equating
coefficients of corresponding cosines, we have
and the other quantities A , A a , &c. are of an order higher than e.
ON SOME CASES OF FLUID MOTION. 59
Hence, for the part of the fluid which lies without the radius 7,
we have
(34),
and for the part which lies between that radius and the ellipse we
have from (28)
< = _ Cc {(I  e) cos + cos 30} + C {(I  e) cos + 3e cos 30} z
 cos 6z z .. ..(35).
c v
The value of <f> given by equation (So) may be deduced from
that given by equation (3 4) by putting r c + z, and expanding as
far as to 2 2 . In the case of the elliptic cylinder then it appears
that the same value of <f> serves for the part of the fluid without,
and the part within the radius jy. If the cylinder i be moving with
a velocity C in the direction of the minor axis of a section, the
value of (f> will be found from that given by equation (34) by
changing the sign of e, putting C for C, and supposing 6 to be
measured from the minor axis.
If the cylinder revolve round its axis with an angular velocity
&), the normal velocity of the surface at any point will be 2coec sin 20.
Since e 2 is neglected, we may suppose this normal velocity to
take place on the surface of a circular cylinder whose radius is c ;
whence (Art. 8) the corresponding value of will be
 ~ sin 20.
If we suppose all these motions to take place together, we have
only (Art. 5) to add together the values of < corresponding to
each. If we suppose the motion very small, so as to neglect
the square of the velocity, we need only retain the terms depend
ing on dw/dt, dC/dt and dC /dt, in the value of d(f>/dt, and we
may calculate the pressure due to each separately. The resultant
of the pressure due to the term dco/dt will evidently be zero, on
account of the symmetry of the corresponding motion, while the
resultant couple will be of the order e 2 , since the pressure on
any point of the surface, and the perpendicular from the centre on
the normal at that point, are each of the order e. The pressure
due to the term dC/dt will evidently have a resultant in the
direction of the major axis of a section of the cylinder ; and it will
60 ON SOME CASES OF FLUID MOTION.
be easily proved that the resultant pressure on a length I of the
cylinder is TrpcH (1  2e) dC/dt. That due to the term dC /dt will
be 7rpc z l (1 f 2e) dC /dt, acting along the minor axis. If the
cylinder be constrained to oscillate so that its axis oscillates in a
direction making an angle a with the major axis, and if C" be
its velocity, which is supposed to be very small, the resultant
pressures along the major and minor axes will be
a<^ \ cL\j i ,., / \ a(j
2e) cos a , and yu, (1 + 2e) sin VTT
respectively, where ft is the mass of the fluid displaced. Resolving
these pressures in the direction of the motion, the resolved part
will be p(I2ecoa2y)dC"/dt 9 or p (1  Je* cos 2a) dC"/dt, e
being the eccentricity ; so that the effect of the inertia of the fluid
will be, to increase the mass of the solid by a mass equal to
//,(! Je 2 cos 2 a), which must be supposed to be collected at the
axis.
A similar method of calculation would apply to any given solid
differing little either from a circular cylinder or from a sphere.
In the latter case it would be necessary to use expansions in series
of Laplace s coefficients, instead of expansions in series of sines
and cosines.
13. Motion of fluid in a closed box whose interior is of the form
of a rectangular parallelepiped.
The motion being supposed to begin from rest, the motion
at any time may be supposed to have been produced by impact
(Art. 4). The motion of the box at any instant f may be resolved
into a motion of translation and three motions of rotation about
three axes parallel to the edges, and passing through the centre
of gravity of the fluid, and the part of </> due to each of these
motions may be calculated separately. Considering any one
of the motions of rotation, we shall see that the normal velocity
of each face in consequence of it will ultimately be the same
as if that face revolved round an axis passing through its centre,
and that the latter motion would not alter the volume of the
fluid. Consequently, in calculating the part of $ due to any one
of the angular velocities, we may calculate separately the part
due to the motion of each face.
Let the origin be in a corner of the box, the axes coinciding
ON SOME CASES OF FLUID MOTION. 61
with its edges. Let a, b, c, be these edges, U, V, W, the velocities,
parallel to the axes, of the centre of gravity of the interior of the
box, w , CD", &) ", the angular velocities of the box about axes
through this point parallel to those of #, y, z. Let us first con
sider the part of < due to the motion of the face xz in conse
quence of the angular velocity ft) ".
The value of $ corresponding to this motion must satisfy the
equation
with the conditions
y == 0> when x = Q or a ................. (.37),
Cv JC
g=0, when y = b ........................ (33),
^ = o> "(a;ia), wheny=0 ............ (39),
within limits corresponding to those of the box.
Now, for a given value of y t the value of (/> between x = and
x = a can be expanded in a convergent series of cosines of irx/a
and its multiples ; and, since (37) is satisfied, the series by which
d(j>/dx will be expressed will also hold good for the limiting values
of x, and will be convergent. The general value of </> then will be
of the form 2" Y n cos mrx/a. Substituting in (36), and equating
coefficients of corresponding cosines, which may be done, since any
function of x can be expanded in but one such series of cosines
between the limits and a, we find that the general value of
Y n is Ce n *ul a + C e n7r ^ a , or, changing the constants,
Y n =A n (e n * J>yV* + e mr(by)l
when n > 0, and for n = Q,
From the condition (38) we have
A + TroT 1 2"nB n (e 6 /*  e V a ) cos mrx/a = :
whence A Q = 0, B n 0, and, omitting B ,
62 ON SOME CASES OF FLUID MOTION.
From the condition (39), we have
TToT 1 ^nA n (e nirb/a e~ nirb/a ) cos nirxja = &/"(# 2 a )
Determining the coefficients in the usual manner, we have
whence
e nir(by)/a,
COS
putting 2 , for shortness, to denote the sum corresponding to odd
integral values of n from 1 to oo .
It is evident that the value of corresponding to the motion of
the opposite face in consequence of the angular velocity a/" will be
found from that just given by putting b y for y, and changing
the sign of a/"; whence the value corresponding to the motion
of these two faces in consequence of &&gt; " will be
4o/V ^   ,
. _ y i       =.    cos nirx a.
^ 72, 3 e mrb/a _ e #irb/a
Let this expression be denoted by &&gt;" ^(#, a, y, b). It is
evident that the part of $ due to the motion of the two faces
parallel to the plane yz will be got by interchanging x and y,
a and b, and changing the sign of " in the last expression, and
will therefore be  w"^r (y, b, x, a). The parts of < corresponding
to the angular velocities a/, a/ , will be got by interchanging the
requisite quantities. Also the part of </> due to the velocities
U, V, W, will be Ux + Vy + Wz (Art. 7), and therefore we have
for the complete value of </>
Ux + Vy + Wz + G) "{^(aj, a, y, 6) ^(y, b, x, a)} + a/ ty(y, b, z, c)
 ^ (si, c, y, 6)) + a>" {^ (z, c, x,a)ir (x, a, , c)).
According to Art. 7 we may consider separately the motion of
translation of the box and fluid, and the motion of rotation about
the centre of gravity of the latter ; and the whole pressure will be
compounded of the pressures due to each. The pressures at the
several points of the box due to the motion of translation will have
a single resultant, which will be the same as if the mass of the
fluid were collected at its centre of gravity. Those due to the
ON SOME CASES OF FLUID MOTION. 63
motion of rotation will have a single resultant couple, to calculate
which we have
= to " [^ (x, a, y, b)  ^ (y, b, x, a)) + &c.
Since for the motion of rotation there is no resultant force,
we may find the resultant couple of the pressures round any
origin, that for instance which has been chosen. If now we
suppose the motion very small, so as to neglect the square of
the velocity, we may find d(f)/dt as if the axes were fixed in space.
We have then for the motion of rotation
j rrt
a >y> &)^(y> > # a)}&c.
Hence we may calculate separately the couples due to each of
the quantities da> "/dt, dco /dt and dco"/dt. It is evident from the
symmetry of the motion that that due to dco "/dt will act round
the axis of z, and that the pressures on the two faces perpendicular
to that axis will have resultants which are equal and opposite.
Also, since ^ (a, a, y, 6) =  ty (0, a, y,. b) and ^ (x, a, 6, &,) =  ^
(x, a, 0, 5), it will be seen that the couples due to the pressures
on the faces perpendicular to the axes of x and y will be twice
as great respectively as those due to the pressures on the planes
yz and xz. The pressure on the element dydz of the plane yz will
be p x==Q dydz, and the moment of this pressure round the axis of z,
reckoned positive when it tends to turn the box from x to y,
will be
 P " y W" (0, a, V, 6)  f (y, &, O, a)} dydz.
Substituting the values of the functions, integrating from y to
y = b, and from z = to z c, replacing 2 l/n 5 by its value 7r 4 /96,
and reducing the other terms, it will be found that the couple
due to the pressure on the plane yz is
ptfbcda" _ S/oaWcTg 1 l
24 dt 7T 5 dt *l +
_
7T 5 dt t l +
We shall get the couple due to the pressure on the plane xz
by interchanging a and b, changing the sign of to ", and measuring
the couple in the opposite direction, or, which is the same, by
merely interchanging a and b. Adding together these two couples
64 ON SOME CASES OF FLUID MOTION.
and doubling their sum we shall find that the couple due to
do) "/dt is  Cda>"ldt, where
~ ~6~ ^? ) ^ 1 J. fnvb/a + 1 _i_ e mra/l>\
a +j) ............ (40).
Similarly, the couple due to dw /dt will be J. dco /dt, tending
to turn the box from y to z, and that due to dco"/dt will be
Bda)"/dt, tending to turn the box from z to x, where A and B
are derived from C by interchanging the requisite quantities.
Hence, considering the motions both of translation and rotation of
the box, we see that the small motions of the box will take place
as if the fluid were replaced by a solid having the same mass,
centre of gravity, and principal axes, and having A, B and G
for its principal moments. This will be true whether forces act
on the fluid or not, provided that if there are any they are of
the kind mentioned in Art. 1.
Patting A tt B f > C /t for the principal moments of inertia of the
solidified fluid, we have
Taking the ratio of C to (7,, replacing each term such as
2
] 
its approximate value 1 "260497, and for 384/Tr 5 its approximate
value 1254821, and employing subsidiary angles, we have
where tan e n =
so that
L tan O n = 10  k nl/a, L tan 6 . n = 10  k na/b,
where ^ = 0821882.
* [It will be shewn further on, in a supplement to this paper, that either of
these two infinite series may be expressed by means of the other, so that we shall
have only one of the infinite series to calculate in any case, for which we may
choose the more rapidly convergent.]
ON SOME CASES OF FLUID MOTION. G5
The numerical calculation of this ratio is very easy, on account
of the great rapidity with which the series contained in it con
verge, both on account of the coefficients, and on account of the
rapid diminution of the angles 6 n and n . The values of A/A,
and B/B t will be derived from that of C/C, by putting c for a in
the first case, and c for b in the second. The calculation of the
small motions of the box will thus be reduced to a question of
ordinary rigid dynamics*.
When one of the quantities a, 6, becomes infinitely great com
pared with the other, the ratio C/C t becomes 1, as will be seen
from equation (40). This result might have been expected. When
a = 6 the value of C/C, is 156537t
The experiment of the box appears capable of great variety
as well as accuracy. We may take boxes in which the edges have
* [Corresponding to the two simple cases of steady motion referred to in the
footnote to p. 7, are two in which the motion of the fluid within a box of simple
form can be expressed in finite terms, the box and the fluid being initially at rest,
and the box being then moved about its axis.
The first is that in which the box is of the form of a right prism, having for
its base an equilateral triangle. If as before a be the perpendicular from the
centre of the triangle on one side, and 6 be measured from this perpendicular,
we shall have
0=7^r 3 sin 30;
btt
and by performing the integrations we shall find that if fc be the radius of gyration
of what we may call the equivalent solid, that is, the solid, of the same mass as
the fluid, by which the fluid may be replaced without affecting the motion of the
box under given forces,
fc 2 = fa 2 ;
and as a is the radius of gyration for the fluid supposed solidified, the moment of
inertia of the equivalent solid is twofifths of that of the solidified fluid.
The other is that of a box of the form of a right elliptic prism. In this case <
is of the form cr 2 sin 20, 6 being measured from the major axis ; and determining c
so as to suit an ellipse of which a and 6 are the semiaxes, we find
k having the same meaning as before, it will be found that
so that the ratio of the moment of inertia of the equivalent solid to that of the
solidified fluid is that of (a 2  b 2 ) 2 to (a 2 + b 2 ) 2 .]
f [A passage containing a proposal to compare this result with experiment is
here omitted, as the experiment is described, in the form in which it was actually
carried out, in the supplement before referred to.]
s. 5
66 ON SOME CASES OF FLUID MOTION.
various ratios to each other, and may make the same box oscillate
in various positions.
14. Initial motion in a rectangular box, the several points of
the surface of which are moved with given velocities, consistent with
the condition that the volume of the fluid is not altered.
Employing the same notation as in the last case, let F (x, y}
be the given normal velocity at any point of the face in the plane
xy. Let I I F(x, y) dxdy = Wab, and let
Jo Jo
then, since the normal motion of the above face due to the function
f(x,y) does not alter the volume of the fluid, we may consider
separately the part of ^ due to this quantity. For this part we have
, _
z9 H r~2 n 7~a
dx* d d*r
with the conditions
= 0, when x = or a ............... (42),
dx
^ =0, when y = or I ............... (43),
dy
^=0, whence ...................... (44),
dz
^ =/K 2/)> when ^ = ., ............ (45),
within limits corresponding to those of the box.
For a given value of z the value of $ from x = to x = a and
from y = to y = b may be expanded in a series of the form
. cosmry/b,
the sign X referring to m, and S to n : and since the values of
<, d<j>/dx and d$/dy do not alter abruptly, and equations (42) and
(43) are satisfied, it follows that the series by which <, d(j>/d^ and
dfyjdy are expressed are convergent, and hold good for the limiting
values of x and y. Substituting the value of (/> just given in (41),
equating to zero coefficients of corresponding cosines, and intro
ON SOME CASES OF FLUID MOTION. 67
ducing the condition (44), we have, omitting the constant, or
supposing A 0,0=0,
(f) = 2 2 *4 m ^e pw(  c ~ z ^ c { e~ pir ^ c ~ z ^ c } cos mirxja . cos niry/b,
9 n Q
i p m n
where % = + y^ .
c a o
Determining the coefficients such as A m>n from the condition
(45) in the usual manner we have, m and n being > 0,
P _ Q PTT^  1 I I f^ y^ cos m7r jr/a . cos mry/b . c?^ (Zy ,
Jo Jo
A = 
Trpab
2
_. (atmro/o pmrc/o\  1 i
\ /
o Jo
nTc/6_ e mrc/6)i f" f y^ ^) cosniry/b . dxdy*,
Jo Jo
with a similar expression for J. m , whence the value of <f> corre
sponding to / (a?, y) is known. In a similar manner we may find
the values corresponding to the similar functions belonging to
each of the other faces. If W be the quantity corresponding to
W for the face opposite to the plane xy, and U, U , correspond to
W t W, for the faces perpendicular to the axis of x, and if V, V,
be the corresponding quantities for y, there remains only to be
found the part of </> due to these six quantities. Since U, U , are
the velocities parallel to the axis of x of the faces perpendicular
to that axis, and so for V, V, &c., the motion corresponding to
these six quantities may be resolved into three motions of trans
lation parallel to the three axes, the velocities being U, V and W,
and that motion which is due to the motions of the faces opposite
to the planes yz, zoo, ocy, moving with velocities U U, V V,
W W, parallel to the axes of x, y, z, respectively. The condition
that the volume of the fluid remains the same requires that
It will be found that the velocities
a^ b^ c v
satisfy all the requisite conditions. Hence the part of < due to
* The function f(x,y) in these integrals may be replaced by F(x,y), since
P cos mry/b . cosmrx/a . dxdy~Q, unless m = ?j = 0.
52
68 ON SOME CASES OF FLUID MOTION.
the six quantities U, U , V, V, W, W, is
 .
This quantity, added to the six others which have already been
given, gives the value of </> which contains the complete solution
of the problem.
The case of motion which has just been given seems at first
sight to be an imaginary one, capable of no practical application.
It may however be applied to the determination of the small
motion of a ball pendulum oscillating in a case in the form of
a rectangular parallelepiped, the dimensions of the case being
great compared with the radius of the ball. For this purpose it
will be necessary to calculate the motion of the ball reflected from
the case, by means of the formulae just given, and then the motion
again reflected from the sphere, exactly as has been done iu the
case of a rigid plane, Art. 10. In the present instance however
the result contains definite integrals, the numerical calculation of
which would be very troublesome.
[From the Cambridge Mathematical Journal, Vol. iv. p. 28. (Nov. 1843).]
ON THE MOTION OF A PlSTON AND OF THE AlR IN A CYLINDER.
WHEN a piston is in motion in a cylinder which also contains
air, if the motion of the piston be not very rapid, so that its
velocity is inconsiderable compared with the velocity of pro
pagation of sound, the motions of the air may be divided into
two classes, the one consisting of those which depend directly on
the motion of the piston, the other, of those which are propagated
with the velocity of sound, and depend on the initial state of the
air, or on a breach of continuity in the motion of the piston.
If we suppose the initial velocity and condensation of the air in
each section of the cylinder to be given, and also the initial
velocity of the piston, both kinds of motion will in general take
place, and the solution of the problem will be complicated. If,
however, we restrict ourselves to motions of the first class, the
approximate solution, though rather long, will be simple. In this
case we must suppose the inital velocity and condensation of the
air not to be given arbitrarily, but to be connected, according to
a certain law which is yet to be found, with the motion of the
piston. The problem as so simplified may perhaps be of some
interest, as affording an example of the application of the partial
differential equations of fluid motion, without requiring the em
ployment of that kind of analysis which is necessary in most
questions of that sort. It is, moreover, that motion of the air
which it is proposed to consider, which principally affects the
motion of the piston.
Conceive an airtight piston to move in a cylinder which is
closed at one end, and contains a mass of air between the closed
end and the piston. For more simplicity, suppose the rest of the
70 ON THE MOTION OF A PISTON AND OF
cylinder to contain no air. Let a point in the closed end be
taken for origin, and let x be measured along the cylinder. Let
% l be the abscissa of the piston ; a the initial value of x^ ; u the
velocity parallel to x of any particle of air whose abscissa is x ;
p the pressure, p the density about that particle; II the initial
mean pressure ; p l the value of p when x = x^\ X, a function of x,
the accelerating force acting on the air ; then for the motion of the
air we have
1 dp ^ r du du 1
  = Xj7 uj t
p dx dt dx
dp dpu _ I (1),
~T7 "I T ^j
dt dx
and p kp, j
neglecting the variation of temperature.
We have also the conditions
u = when x = Q (2);
dx
u= ~dl when x=x < (3)>
for positive values of t, and
ITa when ^ = (4).
ra
pdx =
Jo
o
Eliminating p from equations (1), we have
1 dp __! ( x _du du
p dx~k\* dt U dx
t + ^ = ........................... (6).
dt dx
Now, k being very large, for a first approximation let y be
neglected ; then, integrating (5),
j>0).
Substituting in (6), and integrating,
THE AIR IN A CYLINDER. 71
The conditions (2) and (3) give
C
whence <f> (t) =
x i
Substituting in (4) the value ofp when = 0, we have
dx = C Ha ;
o **
a
whence J
Let now, for a second approximation,
,. a % dec*
p = U+p U=  37
v x l ^ #! dt
so that y and u are small quantities of the order \jk ; then, sub
stituting these values in (5) and (6), remembering that the quan
tities which are not small must destroy each other, and retaining
only small quantities of the first order, we have
dp
* = 7 
dx kx
/ v x d*x\
I A  y, 2 I
l \ ^ dr J
dp 1 dx, dp^x a ^ =0 (8)
~dt + x t dt dx x, dx
and the conditions (2), (3) and (4) give
w = when aj = 0, or x = x^ t and t is positive ...(9);
( a pdx = when = ......... . ........ (10).
Jo
Integrating (7), we have
2 , . n , v
" .........
Substituting the values of p and of its differential coemcients
in (8), and integrating, we obtain
x 3 d d z x\ 1 dx. x , x d, \
(12).
72 ON THE MOTION OF A PISTON
The conditions (9) give f (t) ;
1 d ( d*x\ 1 dx^ (** , 1 d
7TT TL #1 ~TT 1  vs 7^ 2&W ==: jr
6& cfa V d^ / &i ft J o Ha dt
and integrating, we get
. tfx. Ila r*i f [ Xi
afTJ. (Jo
Putting/ for the initial value of tfxjdf we have, from (10) and
(11),
o o
and substituting the value of &&gt; (0) given by this equation in (13),
after having made t = 0, x l = a, dfxJd? =/in the latter, we have
TT C a C x
C=T dx Xdx.
K Jo Jo
Substituting this value of C in that of a (t), and substituting in
(11) and (12), and then substituting the values of p and u in
those of p and u t we have
= n^+S?(f~ w 
c&j tf^j VJo 2#, dt
d*x l
O/C ttt KX^Ja \J / ^ x K^JQ \J
(i*);
^t
5? dx v x_( __ x?\ d_ f d*x\
x,~drt~Qk\ xfjdt T 1 df)
Let A be the area of a section of the cylinder, and let TIAa/k=iJL,
so that fjL is the mass of the air ; then we have
[* It is best at once to get rid of the double integrals by integration by parts,
^Yhich simplifies the expression, converting the last two terms into
AND OF THE AIR IN A CYLINDER. 73
If there were no motion, the term J//, d*xjdt* would disappear.
But in that case the value of p^^ the pressure on the piston,
might be deduced immediately from the equation of equilibrium
of an elastic fluid
1 dp^X
p dx~~ k
Integrating this equation, determining the constant by the con
r*i
dition that I pdx=Tla, multiplying by 4, and putting x = x v
we have, neglecting 1/& 2 ,
A =
 4 r ( r
&i Jo VJo
Comparing this expression with the above, when the second term
of the latter is left out, we have
Xdx,
f^i / f#i \ J 1 ra rx l rx L rx
( J&teJ =? +=/ dx\ Xdx = \ dx\
J a \JO J i UJO JO ^i^O JO
a formula which may also be proved directly. We have then
. TT . a IJL d 2 .^ d ( 1 [* , [* v ,
p t A = UA  K srHMj I I dx.\ Xdx
x^ 3 dr dx t \xj^ I JQ
The first term would be the value of the pressure on the piston
if the air had no inertia and were acted on by no external forces ;
the second term is that due to the inertia of the air; the last
term is that due to the external forces, and in the case of gravity
expresses the effect of the weight of the air. If M be the mass
of the piston, P the accelerating force parallel to x acting on it,
not including the pressure of the air, its equation of motion is
d f 1 r* 1 7 [ Xl v 7 \ /i/>\
 dx. Xdx). ..(16)
x l \ajjJo Jo /
dx l
Hence the effect of the inertia of the air is to increase the mass
of the piston by one third of that of the air, without increasing
the moving force acting on it. If we could integrate equation (16)
twice, we should determine the arbitrary constants by means of
the initial values of ^ and dxjdt, and thus get ^ in terms of t :
then, substituting in (14) and (15), we should obtain p and u as
functions of x and t.
74 MOTION OF A PISTON AND OF THE AIR IN A CYLINDER.
If the cylinder be vertical and smooth and turned upwards,
we have P X = g ; and if, moreover, the motion be very small,
putting oc l a + y, and neglecting ?/ 2 , we have
The term at the second side of this equation is by hypothesis
small, and if we suppose the mean value of x to be taken for a,
it is zero. On this supposition II^l = \M+ ^J g, and the time
/
of a small oscillation will be 2?r y .  , which becomes,
If +2
since yu, 2 is neglected throughout, 2?r f 1 ^ . J \J  .
The reader who wishes to see the complete solution of the
problem, in the case where no forces act on the air, and the air
and piston are at first at rest, may consult a paper of Lagrange s
with additions made by Poisson in the Journal de VEcole Poly
technique. T. xin. (21 e Cah.) p. 187.
[From the Transactions of the Cambridge Philosophical Society,
Vol. VIIL p. 287.]
ON THE THEORIES OF THE INTERNAL FRICTION OF FLUIDS
IN MOTION, AND OF THE EQUILIBRIUM AND MOTION OF
ELASTIC SOLIDS.
[Eead April 14, 1845.]
THE equations of Fluid Motion commonly employed depend
upon the fundamental hypothesis that the mutual action of two
adjacent elements of the fluid is normal to the surface which
separates them. From this assumption the equality of pressure
in all directions is easily deduced, and then the equations of
motion are formed according to D Alembert s principle. This
appears to me the most natural light in which to view the sub
ject ; for the two principles of the absence of tangential action,
and of the equality of pressure in all directions ought not to be
assumed as independent hypotheses, as is sometimes done, inas
much as the latter is a necessary consequence of the former*.
The equations of motion so formed are very complicated, but yet
they admit of solution in some instances, especially in the case
of small oscillations. The results of the theory agree on the
whole with observation, so far as the time of oscillation is con
cerned. But there is a whole class of motions of which the
common theory takes no cognizance whatever, namely, those
which depend on the tangential action called into play by the
sliding of one portion of a fluid along another, or of a fluid along
the surface of a solid, or of a different fluid, that action in fact
which performs the same part with fluids that friction does with
solids.
* This may be easily shewn by the consideration of a tetrahedron of the fluid,
as in Art. 4.
76 ON THE FRICTION OF FLUIDS IN MOTION,
Thus, when a ball pendulum oscillates in an indefinitely ex
tended fluid, the common theory gives the arc of oscillation
constant. Observation however shews that it diminishes very
rapidly in the case of a liquid, and diminishes, but less rapidly,
in the case of an elastic fluid. It has indeed been attempted to
explain this diminution by supposing a friction to act on the ball,
and this hypothesis may be approximately true, but the imper
fection of the theory is shewn from the circumstance that no
account is taken of the equal and opposite friction of the ball on
the fluid.
Again, suppose that water is flowing down a straight aqueduct
of uniform slope, what will be the discharge corresponding to
a given slope, and a given form of the bed ? Of what magnitude
must an aqueduct be, in order to supply a given place with
a given quantity of water ? Of what form must it be, in order
to ensure a given supply of water with the least expense of
materials in the construction ? These, and similar questions are
wholly out of the reach of the common theory of Fluid Motion,
since they entirely depend on the laws of the transmission of that
tangential action which in it is wholly neglected. In fact, accord
ing to the common theory the water ought to flow on with
uniformly accelerated velocity ; for even the supposition of a
certain friction against the bed would be of no avail, for such
friction could not be transmitted through the mass. The practical
importance of such questions as those above mentioned has made
them the object of numerous experiments, from which empirical
formulae have been constructed. But such formulas, although
fulfilling well enough the purposes for which they were con
structed, can hardly be considered as affording us any material
insight into the laws of nature; nor will they enable us to pass
from the consideration of the phenomena from which they were
derived to that of others of a different class, although depending
on the same causes.
In reflecting on the principles according to which the motion
of a fluid ought to be calculated when account is taken of the
tangential force, and consequently the pressure not supposed the
same in all directions, I was led to construct the theory explained
in the first section of this paper, or at least the main part of it,
which consists of equations (13), and of the principles on which
AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 77
they are formed. I afterwards found that Poisson had written
a memoir on the same subject, and on referring to it I found that
he had arrived at the same equations. The method which he em
ployed was however so different from mine that I feel justified in
laying the latter before this Society*. The leading principles of my
theory will be found in the hypotheses of Art. 1, and in Art. 3.
The second section forms a digression from the main object of
this paper, and at first sight may appear to have little connexion
with it. In this section I have, I think, succeeded in shewing
that Lagrange s proof of an important theorem in the ordinary
theory of Hydrodynamics is untenable. The theorem to which I
refer is the one of which the object is to shew that udx+vdy+wdz,
(using the common notation,) is always an exact differential when
it is so at one instant. I have mentioned the principles of
M. Cauchy s proof, a proof, I think, liable to no sort of objection.
I have also given a new proof of the theorem, which would have
served to establish it had M. Cauchy not been so fortunate as to
obtain three first integrals of the general equations of motion.
As it is, this proof may possibly be not altogether useless.
Poisson, in the memoir to which I have referred, begins with
establishing, according to his theory, the equations of equilibrium
and motion of elastic solids, and makes the equations of motion
of fluids depend on this theory. On reading his memoir, I was
led to apply to the theory of elastic solids principles precisely
analogous to those which I have employed in the case of fluids.
The formation of the equations, according to these principles,
forms the subject of Sect. III.
The equations at which I have thus arrived contain two arbi
trary constants, whereas Poisson s equations contain but one. In
Sect. IV. I have explained the principles of Poisson s theories of
elastic solids, and of the motion of fluids, and pointed out what
appear to me serious objections against the truth of one of the
hypotheses which he employs in the former. This theory seems
to be very generally received, and in consequence it is usual to
deduce the measure of the cubical compressibility of elastic solids
from that of their extensibility, when formed into rods or wires,
* The same equations have also been obtained by Navier in the case of an in
compressible fluid (Mem. de V Academic, t. vi. p. 389), but his principles differ from
mine still more than do Poisson s.
78 ON THE FEICTION OF FLUIDS IN MOTION,
or from some quantity of the same nature. If the views which
I have explained in this section be correct, the cubical compres
sibility deduced in this manner is too great, much too great in
the case of the softer substances, and even the softer metals.
The equations of Sect. III. have, I find, been already obtained by
M. Cauchy in his Exercises Mathematiques, except that he has not
considered the effect of the heat developed by sudden compression.
The method which I have employed is different from his, although
in some respects it much resembles it.
The equations of motion of elastic solids given in Sect. in.
are the same as those to which different authors have been led,
as being the equations of motion of the luminiferous ether in
vacuum. It may seem strange that the same equations should
have been arrived at for cases so different ; and I believe this has
appeared to some a serious objection to the employment of those
equations in the case of light. I think the reflections which
I have made at the end of Sect. IV., where I have examined the
consequences of the law of continuity, a law which seems to per
vade nature, may tend to remove the difficulty.
SECTION I.
Explanation of the Theory of Fluid Motion proposed. Formation
of the Differential Equations, Application of these Equations
to a few simple cases.
1. Before entering on the explanation of this theory, it will
be necessary to define, or fix the precise meaning of a few terms
which I shall have occasion to employ.
In the first place, the expression " the velocity of a fluid at
any particular point" will require some notice. If we suppose
a fluid to be made up of ultimate molecules, it is easy to see that
these molecules must, in general, move among one another in an
irregular manner, through spaces comparable with the distances
between them, when the fluid is in motion. But since there is
no doubt that the distance between two adjacent molecules is
quite insensible, we may neglect the irregular part of the velocity,
compared with the common velocity with which all the molecules
in the neighbourhood of the one considered are moving. Or, we
may consider the mean velocity of the molecules in the neigh
bourhood of the one considered, apart from the velocity due to
AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 79
the irregular motion. It is this regular velocity which I shall
understand by the velocity of a fluid at any point, and I shall
accordingly regard it as varying continuously with the coordinates
of the point.
Let P be any material point in the fluid, and consider the
instantaneous motion of a very small element E of the fluid
about P. This motion is compounded of a motion of translation,
the same as that of P, and of the motion of the several points of
E relatively to P. If we conceive a velocity equal and opposite
to that of P impressed on the whole element, the remaining
velocities form what I shall call the relative velocities of the points
of the fluid about P ; and the motion expressed by these velocities
is what I shall call the relative motion in the neighbourhood of P.
It is an undoubted result of observation that the molecular
forces, whether in solids, liquids, or gases, are forces of enormous
intensity, but which are sensible at only insensible distances.
Let E be a very small element of the fluid circumscribing E, and
of a thickness greater than the distance to which the molecular
forces are sensible. The forces acting on the element E are the
external forces, and the pressures arising from the molecular
action of E 1 . If the molecules of E were in positions in which
they could remain at rest if E were acted on by no external force
and the molecules of E f were held in their actual positions, they
would be in what I shall call a state of relative equilibrium. Of
course they may be far from being in a state of actual equilibrium.
Thus, an element of fluid at the top of a wave may be sensibly
in a state of relative equilibrium, although far removed from its
position of equilibrium. Now, in consequence of the intensity of
the molecular forces, the pressures arising from the molecular
action on E will be very great compared with the external
moving forces acting on E. Consequently the state of relative
equilibrium, or of relative motion, of the molecules of E will not
be sensibly affected by the external forces acting on E. But the
pressures in different directions about the point P depend on that
state of relative equilibrium or motion, and consequently will not
be sensibly affected by the external moving forces acting on E.
For the same reason they will not be sensibly affected by any
motion of rotation common to all the points of E\ and it is
a direct consequence of the second law of motion, that they will
80 ON THE FRICTION OF FLUIDS IN MOTION,
not be affected by any motion of translation common to the whole
element. If the molecules of E were in a state of relative equi
librium, the pressure would be equal in all directions about P,
as in the case of fluids at rest. Hence I shall assume the follow
ing principle :
That the difference between the pressure on a plane in a given
direction passing through any point P of a fluid in motion and the
pressure which would exist in all directions about P if the fluid in
its neighbourhood were in a state of relative equilibrium depends
only on the relative motion of the fluid immediately about P ; and
that the relative motion due to any motion of rotation may be elimi
nated without affecting the differences of the pressures above men
tioned.
Let us see how far this principle will lead us when it is
carried out.
2. It will be necessary now to examine the nature of the
most general instantaneous motion of an element of a fluid.
The proposition in this article is however purely geometrical, and
may be thus enunciated : " Supposing space, or any portion of
space, to be filled with an infinite number of points which move
in any continuous manner, retaining their identity, to examine
the nature of the instantaneous motion of any elementary portion
of these points."
Let u, v, w be the resolved parts, parallel to the rectangular
axes, Ox, Oy, Oz, of the velocity of the point P, whose coordinates
at the instant considered are x, y, z. Then the relative velocities
at the point P , whose coordinates are x + x, y + y , z + z, will be
dw , dw , dw ^ ^
dx dy dz
neglecting squares and products of SB , y, z. Let these velocities
be compounded of those due to the angular velocities w , o>", to"
about the axes of x, y, z, and of the velocities U, F, W parallel
AND THE EQUILIBR\UM AND MOTION OF ELASTIC SOLIDS. 81
to x, y, z. The linear velocities due to the angular velocities
being w"z <*>" y , G> "X wz, wy f a>"x parallel to the axes of
r, ?/, z, we shall therefore have
du , fdu , f \ , fdu
dx \dy J \dz
dv ,,\ . dv . fdv .

dy ) * dz
Since &&gt; , <w", &&gt; " are arbitrary, let them be so assumed that
dU _dV dV _dW dW_dU
dy ~ dx ~dz ~ dy W ~ dz 7
which gives
dw dv\ ,,  du dw\ , fdv du
du ,
=&* +
= fdv u\ , + v / ...... (2).
* \dx dy) dy * * \dz dy )
TT7 . (dw du\ , . fdw dv\ , dw ,
W= ,  +  I a? +   +J) y + j * ,
The quantities &&gt; , &&gt;", w" are what I shall call the angular
velocities of the fluid at the point considered. This is evidently
an allowable definition, since, in the particular case in which the
element considered moves as a solid might do, these quantities
coincide with the angular velocities considered in rigid dynamics.
A further reason for this definition will appear in Sect. III.
Let us now investigate whether it is possible to determine x,
y , z so that, considering only the relative velocities U, V, W, the
line joining the points P, P shall have no angular motion. The
conditions to be satisfied, in order that this may be the case, are
evidently that the increments of the relative coordinates a? , y, z
of the second point shall be ultimately proportional to those co
ordinates. If e be the rate of extension of the line joining the two
points considered, we shall therefore have
Fx + liy 4 gz = ex , \
=ey A .............................. (3);
s.
82 ON THE FRICTION OF FLUIDS IN MOTION,
where
^ du n dv jj. dio  dv dw
F= dx G = dTy H = Tz V = dz + dj
2g^ + ^,2h^ + f.
dx dz dy ax
If we eliminate from equations (3) the two ratios which exist
between the three quantities x , y, z, we get the well known cubic
equation
which occurs in the investigation of the principal axes of a rigid
body, and in various others. As in these investigations, it may be
shewn that there are in general three directions, at right angles
to each other, in which the point P may be situated so as to
satisfy the required conditions. If two of the roots of (4) are
equal, there is one such direction corresponding to the third root,
and an infinite number of others situated in a plane perpendicular
to the former; and if the three roots of (4) are equal, a line
drawn in any direction will satisfy the required conditions.
The three directions which have just been determined I shall
call axes of extension. They will in general vary from one point
to another, and from one instant of time to another. If we denote
the three roots of (4) by e, e" } e ", and if we take new rectangular
axes Ox t) Oy t , Oz f) parallel to the axes of extension, and denote
by u tt U lt &c. the quantities referred to these axes corresponding
to u, U, &c., equations (3) must be satisfied by ?//= 0, z, = 0, e = e,
by <= 0, <= 0, e= e\ and by <= 0, y/= 0, e = c" ,, which requires
that/ 7 = 0, g t 0, 7^ / = 0, and we have
, & _du t _ _ dv, / _rr_^/
The values of U t , F, W /t which correspond to the residual
motion after the elimination of the motion of rotation correspond
ing to o> , &)" and &&gt; ", are
The angular velocity of which &&gt; , &&gt;", a/" are the components
is independent of the arbitrary directions of the coordinate axes :
the same is true of the directions of the axes of extension, and of
the values of the roots of equation (4). This might be proved in
AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 83
various ways ; perhaps the following is the simplest. The condi
tions by which co , ", ta>" are determined are those which express
that the relative velocities U, V, W, which remain after eliminating
a certain angular velocity, are such that Udx + Vdy + Wdz is
ultimately an exact differential, that is to say when squares and
products of x , ?/ and z are neglected. It appears moreover from
the solution that there is only one way in which these conditions
can be satisfied for a given system of coordinate axes. Let us
take new rectangular axes, OK, Oy, Oz, and let U, V, W be the
resolved parts along these axes of the velocities U, V, W, and
x , y , z , the relative coordinates of P ; then
U = Ucosicx + V cos xy \ Wcos xz,
dx = cosxxdx. + cosxydy + cosxzdz, &c. ;
whence, taking account of the well known relations between the
cosines involved in these equations, we easily find
Udx + Vdy + Wdz = Urfx + Vdy +Wdz.
It appears therefore that the relative velocities U, V, W, which
remain after eliminating a certain angular velocity, are such that
Ucx + Vdy + Wdz is ultimately an exact differential. Hence
the values of U, V, W are the same as would have been obtained
from equations (2) applied directly to the new axes, whence the
truth of the proposition enunciated at the head of this paragraph
is manifest.
The motion corresponding to the velocities U tt V t , W t may be
further decomposed into a motion of dilatation, positive or negative,
which is alike in all directions, and two motions which I shall call
motions of shifting, each of the latter being in two dimensions, and
not affecting the density. For let S be the rate of linear extension
corresponding to a uniform dilatation ; let <rx t cry/ be the velo
cities parallel to^, y y , corresponding to a motion of shifting parallel
to the plane x t y t , and let a f x lt a z t be the velocities parallel to
x iy z tt corresponding to a similar motion of shifting parallel to the
plane xz t . The velocities parallel to x lt y t , z t respectively corre
sponding to the quantities 8, a and <r will be (8 + <r + a ) xj, (8 <r)y ,
(8 v)z , and equating these to 7, V , W t we shall get
a = J(e + e" + e" / ) > ff = J(V + e "  2e"), <r = J ( + e"  2e").
Hence the most general instantaneous motion of an elementary
portion of a fluid is compounded of a motion of translation, a
02
84 ON THE FRICTION OF FLUIDS IN MOTION,
motion of rotation, a motion of uniform dilatation, and two motions
of shifting of the kind just mentioned.
3. Having determined the nature of the most general instan
taneous motion of an element of a fluid, we are now prepared to
consider the normal pressures and tangential forces called into
play by the relative displacements of the particles. Let p be the
pressure which would exist about the point P if the neighbouring
molecules were in a state of relative equilibrium: let p+p, be
the normal pressure, and t, the tangential action, both referred to
a unit of surface, on a plane passing through P and having a given
direction. By the hypotheses of Art. 1. the quantities p,, t t will
be independent of the angular velocities &&gt; , w" , " , depending
only on the residual relative velocities U, V, W, or, which comes
to the same, on e, e" and e", or on a, a and 8. Since this re
sidual motion is symmetrical with respect to the axes of extension,
it follows that if the plane considered is perpendicular to any one
of these axes the tangential action on it is zero, since there is no
reason why it should act in one direction rather than in the
opposite ; for by the hypotheses of Art. 1 the change of density
and temperature about the point P is to be neglected, the consti
tution of the fluid being ultimately uniform about that point.
Denoting then by p+p, p+p", p+p" the pressures on planes
perpendicular to the axes of a? y , ?/ /5 #,, we must have
p = <l>(e f ,e",e f "), p" = <j>(e", e", e \ p" = <b(e", e, e"),
</>(> , e" y e" } denoting a function of e , e and e" which is sym
metrical with respect to the two latter quantities. The question
is now to determine, on whatever may seem the most probable
hypothesis, the form of the function (p.
Let t;s first take the simpler case in which there is no dilata
tion, and only one motion of shifting, or in which e = e\ e" 0,
and let us consider what would take place if the fluid consisted of
smooth molecules acting on each other by actual contact. On
this supposition, it is clear, considering the magnitude of the pres
sures acting on the molecules compared with their masses, that
they would be sensibly in a position of relative equilibrium, except
when the equilibrium of any one of them became impossible from
the displacement of the adjoining ones, in which case the molecule
in question would start into a new position of equilibrium. This
start would cause a corresponding displacement in the molecules
AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 85
immediately about the one which started, and this disturbance
would be propagated immediately in all directions, the nature of
the displacement however being different in different directions,
and would soon become insensible. During the continuance of
this disturbance, the pressure on a small plane drawn through the
element considered would not be the same in all directions, nor
normal to the plane: or, which comes to the same, we may sup
pose a uniform normal pressure p to act, together with a normal
pressure p ti , and a tangential force t ljt p n and t /t being forces of
great intensity and short duration, that is being of the nature of
impulsive forces. As the number of molecules comprised in the
element considered has been supposed extremely great, we may
take a time r so short that all summations with respect to such
intervals of time may be replaced without sensible error by inte
grations, and yet so long that a very great number of starts shall
take place in it. Consequently we have only to consider the aver
age effect of such starts, and moreover we may without sensible
error replace the impulsive forces such as p n and t /f , which succeed
one another with great rapidity, by continuous forces. For planes
perpendicular to the axes of extension these continuous forces will
be the normal pressures p , p", p".
Let us now consider a motion of shifting differing from the
former only in having e increased in the ratio of m to 1. Then, if
we suppose each start completed before the starts which would be
sensibly affected by it are begun, it is clear that the same series of
starts will take place in the second case as in the first, but at
intervals of time which are less in the ratio of 1 to m. Conse
quently the continuous pressures by which the impulsive actions
due to these starts may be replaced must be increased in the ratio
of m to 1. Hence the pressures p t p" t p" must be proportional
to e, or we must have
p =Ce, p"=C e , p" =C"e .
It is natural to suppose that these formulae hold good for nega
tive as well as positive values of e. Assuming this to be true, let
the sign of e be changed. This comes to interchanging x and y,
and consequently p" must remain the same, and p and p" must
be interchanged. We must therefore have G" 0, G C. Put
ting then C= 2/j, we have
p =  SIM , p" = fye, p" = 0.
86 ON THE FRICTION OF FLUIDS IN MOTION,
It has hitherto been supposed that the molecules of a fluid are
in. actual contact. We have every reason to suppose that this is
not the case. But precisely the same reasoning will apply if they
are separated by intervals as great as we please compared with
their magnitudes, provided only we suppose the force of restitution
called into play by a small displacement of any one molecule to be
very great.
Let us now take the case of two motions of shifting which co
exist, and let us suppose e = <r + a , e" = <r, e" a . Let the
small time r be divided into 2n equal portions, and let us suppose
that in the first interval a shifting motion corresponding to e = 2cr,
e"= 2a takes place parallel to the plane x t y t , and that in the
second interval a shifting motion corresponding to e = 2cr , e "= 2o
takes place parallel to the plane xz^ and so on alternately. On
this supposition it is clear that if we suppose the time r/2?i to be
extremely small, the continuous forces by which the effect of the
starts may be replaced will be p = 2 //, (or + cr ), p"= 2/*o, p "= 2fia.
By supposing n indefinitely increased, we might make the motion
considered approach as near as we please to that in which the two
motions of shifting coexist ; but we are not at liberty to do so, for
in order to apply the above reasoning we must suppose the time
r/2n to be so large that the average effect of the starts which
occur in it may be taken. Consequently it must be taken as an
additional assumption, and not a matter of absolute demonstration,
that the effects of the two motions of shifting are superimposed.
Hence if 8 = 0, i.e. if e + e" + e" = 0, we shall have in general
/ = 2^ , p"=W, p" = 2f*e"! (5).
It was by this hypothesis of starts that I first arrived at these
equations, and the differential equations of motion which result
from them. On reading Poisson s memoir however, to which I
shall have occasion to refer in Section IV., I was led to reflect that
however intense we may suppose the molecular forces to be, and
however near we may suppose the molecules to be to their posi
tions of relative equilibrium, we are not therefore at liberty to
suppose them in those positions, and consequently not at liberty
to suppose the pressure equal in all directions in the intervals of
time between the starts. In fact, by supposing the molecular
forces indefinitely increased, retaining the same ratios to each
other, we may suppose the displacements of the molecules from
AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 87
their positions of relative equilibrium indefinitely diminished, but
on the other hand the force of restitution called into action by a
given displacement is indefinitely increased in the same proportion.
But be these displacements what they may, we know that the
forces of restitution make equilibrium with forces equal and oppo
site to the effective forces ; and in calculating the effective forces
we may neglect the above displacements, or suppose the molecules
to move in the paths in which they would move if the shifting
motion took place with indefinite slowness. Let us first consider
a single motion of shifting, or one for which e" = e, e" 0, and
let p t and t f denote the same quantities as before. If we now sup
pose e increased in the ratio of m to 1, all the effective forces will
be increased in that ratio, and consequently p t and t / will be in
creased in the same ratio. We may deduce the values of p p", and
p" just as before, and then pass by the same reasoning to the case
of two motions of shifting which coexist, only that in this case the
reasoning will be demonstrative, since we may suppose the time
r/2n indefinitely diminished. If we suppose the state of things
considered in this paragraph to exist along with the motions of
starting already considered, it is easy to see that the expressions
for p, p" and p " will still retain the same form.
There remains yet to be considered the effect of the dilatation.
Let us first suppose the dilatation to exist without any shifting :
then it is easily seen that the relative motion of the fluid at the
point considered is the same in all directions. Consequently the
only effect which such a dilatation could have would be to intro
duce a normal pressure p t , alike in all directions, in addition to
that due to the action of the molecules supposed to be in a state
of relative equilibrium. Now the pressure p t could only arise
from the aggregate of the molecular actions called into play by
the displacements of the molecules from their positions of relative
equilibrium ; but since these displacements take place, on an
average, indifferently in all directions, it follows that the actions
of which p t is composed neutralize each other, so that p t = 0. The
same conclusion might be drawn from the hypothesis of starts,
supposing, as it is natural to do, that each start calls into action
as much increase of pressure in some directions as diminution of
pressure in others.
If the motion of uniform dilatation coexists with two motions
88 ON THE FKICTION OF FLUIDS IN MOTION,
of shifting, I shall suppose, for the same reason as before, that the
effects of these different motions are superimposed. Hence sub
tracting S from each of the three quantities e, e" and e", and
putting the remainders in the place of e, e" and e" in equations
(5), we have
p = fj<e + e~e, p =
p" = $p(e +e"2e ") ............ (G).
If we had started with assuming </> (e, e" , e"} to be a linear func
tion of e , e" and e" , avoiding all speculation as to the molecular
constitution of a fluid, we should have had at onc&p =Ce+(J(e"+e"),
since p is symmetrical with respect to e" and e" \ or, changing the
constants, p f //, (e" + e" 2e) + K (e f e" + e "}. The expressions
for p" and p" would be obtained by interchanging the requisite
quantities. Of course we may at once put K = if we assume
that in the case of a uniform motion of dilatation the pressure at
any instant depends only on the actual density and temperature at
that instant, and not on the rate at which the former changes
with the time. In most cases to which it would be interesting to
apply the theory of the friction of fluids the density of the fluid is
either constant, or may without sensible error be regarded as con
stant, or else changes slowly with the time. In the first two cases
the results would be the same, and in the third case nearly the
same, whether K were equal to zero or not. Consequently, if
theory and experiment should in such cases agree, the experiments
must not be regarded as confirming that part of the theory which
relates to supposing K to be equal to zero.
4. It will be easy now to determine the oblique pressure, or
resultant of the normal pressure and tangential action, on any
plane. Let us first consider a plane drawn through the point P
parallel to the plane yz. Let Ox, make with the axes of #, y, z
angles whose cosines are I , m, n ; let I" , m", n" be the same for
Oy,, and l " t m" , n" the same for Oz r Let P 1 be the pressure,
and (xty), (xtz) the resolved parts, parallel to y, z respectively, of
the tangential force on the plane considered, all referred to a unit
of surface, (xty) being reckoned positive when the part of the
fluid towards  x urges that towards + x in the positive direction
of y, and similarly for (xtz). Consider the portion of the fluid
comprised within a tetrahedron having its vertex in the point P,
its base parallel to the plane yz, and its three sides parallel to the
AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 89
planes x t y lt yz ft zx t respectively. Let A be the area of the base,
and therefore I A, I" A, I" A the areas of the faces perpendicular
to the axes of x t , y t , z t . By D Alembert s principle, the pressures
and tangential actions on the faces of this tetrahedron, the moving
forces arising from the external attractions, not including the
molecular forces, and forces equal and opposite to the effective
moving forces will be in equilibrium, and therefore the sums of
the resolved parts of these forces in the directions of x, y and z
will each be zero. Suppose now the dimensions of the tetrahedron ,
indefinitely diminished, then the resolved parts of the external,
and of the effective moving forces will vary ultimately as the
cubes, and those of the pressures and tangential forces on the
sides as the squares of homologous lines. Dividing therefore the
three equations arising from equating to zero the resolved parts
of the above forces by A, and taking the limit, we have
the sign 2 denoting the sum obtained by taking the quantities
corresponding to the three axes of extension in succession. Putting
for p ,p" and p" their values given by (6), putting e +e"+e"=8S,
and observing that 2 2 = 1, 2W = 0, 2ZV = 0, the above equa
tions become
1\ =p  ZfjL^re + 2/xS, (xty) =  2^1 m e, (xtz) =  2/*2/W.
The method of determining the pressure on any plane from
the pressures on three planes at right angles to each other, which
has just been given, has already been employed by MM. Cauchy
and Poisson.
The most direct way of obtaining the values of 2?V &c. would
be to express I , m and n in terms of e by any two of equations
(3), in which x , y , z are proportional to I , m, n, together with
the equation I 2 + m 2 + n" 2 = 1, and then to express the resulting
symmetrical function of the roots of the cubic equation (4) in
terms of the coefficients. But this method would be excessively
laborious, and need not be resorted to. For after eliminating the
angular motion of the element of fluid considered the remaining
velocities are ex , e y, , e "z , parallel to the axes of x lt y t) z t .
The sum of the resolved parts of these parallel to the axis of
x is lex] + 1 e y + 1" e" z . Putting for x f , y , z t their values
I x + my + n z &c., the above sum becomes
x %1 V f y i in e + z ^l n e ;
90
OX THE FRICTION OF FLUIDS IN MOTION,
but this sum is the same thing as the velocity U in equation (2),
and therefore we have
du
du
du div\
dxj
It may also be very easily proved directly that the value of 38,
the rate of cubical dilatation, satisfies the equation
dz
(7).
dx dy
Let P 2 , (ytz), (ytr) be the quantities referring to the axis of y,
and P 3 , (ztx), (zty) those referring to the axis of z, which corre
spond to P l &c. referring to the axis of x. Then we see that
( y tz) = (zty), (ztx) = (xtz), (xty) = (ytx). Denoting these three
quantities by T 19 T 2 , T 3 , and making the requisite substitutions
and interchanges, we have
(8).
It may also be useful to know the components, parallel to
x, y, z t of the oblique pressure on a plane passing through the
point P, and having a given direction. Let /, m, n be the cosines
of the angles which a normal to the given plane makes with the
axes of x, y, z ; let P, Q, R be the components, referred to a unit
of surface, of the oblique pressure on this plane, P, Q, R being
reckoned positive when the part of the fluid in which is situated
the normal to which /, m and n refer is urged by the other part
in the positive directions of x t y, z, when I, m and n are positive.
Then considering as before a tetrahedron of which the base is
AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 91
parallel to the given plane, the vertex in the point P, and the
sides parallel to the coordinate planes, we shall have
(9).
In the simple case of a sliding motion for which u 0, v =f(x),
w = 0, the ojly forces, besides the pressure >, which act on planes
parallel to the coordinate planes are the two tangential forces T 9 ,
the value of which in this case is ^ dvjdx. In this case it is
easy to shew that the axes of extension are, one of them parallel
to Oz, and the two others in a plane parallel to xy, and inclined
at angles of 45 to Ox. We see also that it is necessary to suppose
JJL to be positive, since otherwise the tendency of the forces would
be to increase the relative motion of the parts of the fluid, and
the equilibrium of the fluid would be unstable.
5. Having found the pressures about the point F on planes
parallel to the coordinate planes, it will be easy to form the
equations of motion. Let X, Y, Z be the resolved parts, parallel
to the axes, of the external force, not including the molecular
force ; let p be the density, t the time. Consider an elementary
parallelepiped of the fluid, formed by planes parallel to the co
ordinate planes, and drawn through the point (x, y, z) and the
point (x + Aa 1 , y + Ay, z + A#). The mass of this element will be
ultimately pA^AyAz, and the moving force parallel to x arising
from the external forces will be ultimately pJTA^AyAz; the effec
tive moving force parallel too; will be ultimately p Du/Dt. A^AyAz,,
where D is used, as it will be in the rest of this paper, to denote
differentiation in which the independent variables are t and thre&
parameters of the particle considered, (such for instance as its
initial cordinates,) and not t, x, y, z. It is easy also to shew that
the moving force acting on the element considered arising from*
the oblique pressures on the faces is ultimately
dP dl\ dT n
c dy a
acting in the negative direction. Hence we have by D Alembert v s*
principle
ID X\ +* + * + %*=*, &c (10),.
92 ON THE FRICTION OF FLUIDS IN MOTION,
in which equations we must put for Du/Dt its value
du du du, du
j + U j + V j + W j ,
dt dx dy dz
and similarly for Dojdt and Dw/dt. In considering the general
equations of motion it will be needless to write down more than
one, since the other two may be at once derived from it by inter
changing the requisite quantities. The equations (10), the ordi
nary equation of continuity, as it is called,
i , dpu ^fjv ^^ " ,x /. . \
dt + ~da> + ~dy+~te = "I "
which expresses the condition that there is no generation or
destruction of mass in the interior of a fluid, the equation con
necting p and p, or in the case of an incompressible fluid the
equivalent equation Dp/Dt = 0, and the equation for the propa
gation of heat, if we choose to take account of that propagation,
are the only equations to be satisfied at every point of the interior
of the fluid mass.
As it is quite useless to consider cases of the utmost degree
of generality, I shall suppose the fluid to be homogeneous, and of
a uniform temperature throughout, except in so far as the
temperature may be raised by sudden compression in the case of
small vibrations. Hence in equations (10) //, may be supposed to
be constant as far as regards the temperature ; for, in the case
of small vibrations, the terms introduced by supposing it to vary
with the temperature would involve the square of the velocity,
which is supposed to be neglected. If we suppose /JL to be in
dependent of the pressure also, and substitute in (10) the values
of P l &c. given by (8), the former equations become
tDu ^\ dp _ fd 2 u d z u
P \I)t J dx \dx 2 dif
__ dfdu dv dw\
3 dx \dx dy dz)
Let us now consider in what cases it is allowable to suppose
p to be independent of the pressure. It has been concluded by
Dubuat, from his experiments on the motion of water in pipes
and canals, that the total retardation of the velocity due to
friction is not increased by increasing the pressure. The total
AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 93
retardation depends, partly on the friction of the water against
the sides of the pipe or canal, and partly on the mutual friction,
or tangential action, of the different portions of the water. Now
if these two parts of the whole retardation were separately variable
with p, it is very unlikely that they should when combined give
a result independent of p. The amount of the internal friction
of the water depends on the value of //.. I shall therefore suppose
that for water, and by analogy for other incompressible fluids,
fju is independent of the pressure. On this supposition, we have
from equations (11) and (12)
dp fd*u d 2 u d 2 u
=
du dv dw
~7 1 7~ + > = 0.
ax dy dz
These equations are applicable to the determination of the motion
of water in pipes and canals, to the calculation of the effect of
friction on the motions of tides and waves, and such questions.
If the motion is very small, so that we may neglect the square
of the velocity, we may put Du/Dt = du/dt, &c. in equations (13).
The equations thus simplified are applicable to the determination
of the motion of a pendulum oscillating in water, or of that of
a vessel filled with water and made to oscillate. They are also
applicable to the determination of the motion of a pendulum
oscillating in air, for in this case we may, with hardly any error,
neglect the compressibility of the air.
The case of the small vibrations by which sound is propagated
in a fluid, whether a liquid or a gas, is another in which dp/dp
may be neglected. For in the case of a liquid reasons have been
shewn for supposing //, to be independent of p, and in the case
of a gas we may neglect dp/dp, if we neglect the small change
in the value of /*, arising from the small variation of pressure due
to the forces X, Y, Z.
6. Besides the equations which must hold good at any point
in the interior of the mass, it will be necessary to form also the
equations which must be satisfied at its boundaries. Let M be
a point in the boundary of the fluid. Let a normal to the surface
at M, drawn on the outside of the fluid, make with the axes
angles whose cosines are I, m, n. Let P , Q , R be the components
94 ON THE FRICTION OF FLUIDS IN MOTION,
of the pressure of the fluid about M on the solid or fluid with
which it is in contact, these quantities being reckoned positive
when the fluid considered presses the solid or fluid outside it in
the positive directions of #, y, 2, supposing 7, m and n positive.
Let S be a very small element of the surface about M, which
will be ultimately plane, S a plane parallel and equal to S, and
directly opposite to it, taken within the fluid. Let the distance
between S and S be supposed to vanish in the limit compared
with the breadth of S, a supposition which may be made if we
neglect the effect of the curvature of the surface at M; and let
us consider the forces acting on the element of fluid comprised
between S and S , and the motion of this element. If we suppose
equations (8) to hold good to within an insensible distance from
the surface of the fluid, we shall evidently have forces ultimately
equal to PS, QS, US, (P } Q and It being given by equations (9),)
acting on the inner side of the element in the positive directions
of the axes, and forces ultimately equal to P S, Q S, H S acting
on the outer side in the negative directions. The moving forces
arising from the external forces acting on the element, and the
effective moving forces will vanish in the limit compared with the
forces PS, &c. ; the same will be true of the pressures acting
about the edge of the element, if we neglect capillary attraction,
and all forces of the same nature. Hence, taking the limit, we
shall have
The method of proceeding will be different according as the
bounding surface considered is a free surface, tb,e surface of a
solid, or the surface of separation of two fluids, and it will be
necessary to consider these cases separately. Of course the surface
of a liquid exposed to the air is really the surface of separation
of two fluids, but it may in many cases be regarded as a free
surface if we neglect the inertia of the air : it may always be
so regarded if we neglect the friction of the air as well as its
inertia,
Let us first take the case of a free surface exposed to a pres
siare II, which is supposed to be the same at all points, but may
vary with the time ; and let L = be the equation to the surface
In this case we shall have P = IH, Q = rail, jR = wII; and
putting for P, Q, R their values given by (9), and for P l &c. their
AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 95
values given by (8), and observing that in this case 8 = 0, we
shall have
dx \ay dx) \dz
in which equations /, m, n will have to be replaced by dL/dx,
dL/dy, dL/dz, to which they are proportional.
If we choose to take account of capillary attraction, we have
only to diminish the pressure n by the quantity Hi I ) , where
H is a positive constant depending on the nature of the fluid, and
r lt r 2 , are the principal radii of curvature at the point considered,
reckoned positive when the fluid is concave outwards. Equations
(14) with the ordinary equation
dl. dL dL^ dL
~ r + Ur+V,+Wj = ..................... (15),
dt dx dy dz
are the conditions to be satisfied for points at the free surface.
Equations (14) are for such points what the three equations of
motion are for internal points, and (15) is for the former what (11)
is for the latter, expressing in fact that there is no generation or
destruction of fluid at the free surface.
The equations (14) admit of being differently expressed, in a
way which may sometimes be useful. If we suppose the origin to
be in the point considered, and the axis of z to be the external
normal to the surface, we have I m = 0, n 1, and the equations
become
dw du ~ dw dv dw
T +y = 0, + = (), np+2/AT = ......... (16).
dx dz dy dz dz
The relative velocity parallel to z of a point (# , y , 0) in the
free surface, indefinitely near the origin, is dw/dx . x f dwjdy . y :
hence we see that dw/dx, dw/dy are the angular velocities, reckoned
from x to z and from y to z respectively, of an element of the free
surface. Subtracting the linear velocities due to these angular
velocities from the relative velocities of the point (x, ?/, z), and
calling the remaining relative velocities U, V, W, we shall have
96 ON THE FRICTION OF FLUIDS IN MOTION,
jr_du , du , /du div\ ,
dx dy \dz dx)
dv , dv , . ido . dw
T , v , v , o
V= jx + jy + ,
dx c?v \ dz
W = ~z.
dz
Hence we see that the first two of equations (16) express the con
ditions that dU/dz = and dV/dz = Q, which are evidently the
conditions to be satisfied in order that there may be no sliding
motion in a direction parallel to the free surface. It would be
easy to prove that these are the conditions to be satisfied in order
that the axis of z may be an axis of extension.
The next case to consider is that of a fluid in contact with a
solid. The condition which first occurred to me to assume for
this case was, that the film of fluid immediately in contact with
the solid did not move relatively to the surface of the solid. I
was led to try this condition from the following considerations.
According to the hypotheses adopted, if there was a very large
relative motion of the fluid particles immediately about any imagi
nary surface dividing the fluid, the tangential forces called into
action would be very large, so that the amount of relative motion
would be rapidly diminished. Passing to the limit, we might sup
pose that if at any instant the velocities altered discontinuously
in passing across any imaginary surface, the tangential force called
into action would immediately destroy the finite relative motion
of particles indefinitely close to each other, so as to render the
motion continuous; and from analogy the same might be supposed
to be true for the surface of junction of a fluid and solid. But
having calculated, according to the conditions which I have men
tioned, the discharge of long straight circular pipes and rectangular
canals, and compared the resulting formulae with some of the
experiments of Bossut and Dubuat, I found that the formulae did
not at all agree with experiment. I then tried Poisson s conditions
in the case of a circular pipe, but with no better success. In fact,
it appears from experiment that the tangential force varies nearly
as the square of the velocity with which the fluid flows past the
surface of a solid, at least when the velocity is not very small. It
appears however from experiments on pendulums that the total
AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 97
friction varies as the first power of the velocity, and consequently
we may suppose that Poisson s conditions, which include as a
particular case those which I first tried, hold good for very small
velocities. I proceed therefore to deduce these conditions in a
manner conformable with the views explained in this paper.
First, suppose the solid at rest, and let L = be the equation
to its surface. Let M f be a point within the fluid, at an insensible
distance h from M. Let to be the pressure which would exist
about M if there were no motion of the particles in its neighbour
hood, and let p f be the additional normal pressure, and t t the tan
gential force, due to the relative velocities of the particles, both
with respect to one another and with respect to the surface of the
solid. If the motion is so slow that the starts take place independ
ently of each other, on the hypothesis of starts, or that the mole
cules are very nearly in their positions of relative equilibrium,
and if we suppose as before that the effects of different relative
velocities are superimposed, it is easy to shew that p t and t t are
linear functions of u, v, w and their differential coefficients with
respect to as, y and z\ u, v, &c. denoting here the velocities of the
fluid about the point M , in the expressions for which however the
coordinates of M may be used for those of M , since h is neglected.
Now the relative velocities about the points M and M depending
on du/dx, &c. are comparable with du/dx . h, while those depending
on u, v and w are comparable with these quantities, and therefore
in considering the action of the fluid on the solid it is only neces
sary to consider the quantities u, v and w. Now since, neglecting
h, the velocity at M is tangential to the surface at M, u, v, and w
are the components of a certain velocity V tangential to the sur
face. The pressure p t must be zero ; for changing the signs u, v,
and w the circumstances concerned in its production remain the
same, whereas its analytical expression changes sign. The tangen
tial force at M will be in the direction of V, and proportional to it,
and consequently its components along the axes of x, y, z will be
proportional to u, v, w. Reckoning the tangential force positive
when, I, m, and n being positive, the solid is urged in the positive
directions of x, y y z, the resolved parts of the tangential force will
therefore be vu, w, vw, where v must evidently be positive, since
the effect of the forces must be to check the relative motion of the
fluid and solid. The normal pressure of the fluid on the solid
being equal to r, its components will be evidently far, me?, ntr.
s. 7
98 ON THE FRICTION OF FLUIDS IN MOTION,
Suppose now the solid to be in motion, and let u, v, w be the
resolved parts of the velocity of the point M of the solid, and w ,
to", w " the angular velocities of the solid. By hypothesis, the
forces by which the pressure at any point differs from the normal
pressure due to the action of the molecules supposed to be in a
state of relative equilibrium about that point are independent of
any velocity of translation or rotation. Supposing then linear and
angular velocities equal and opposite to those of the solid impressed
both on the solid and on the fluid, the former will be for an
instant at rest, and we have only to treat the resulting velocities
of the fluid as in the first case. Hence P =lG>+v(u u), &c.;
and in the equations (8) we may employ the actual velocities u,
v, w, since the pressures P, Q, R are independent of any motion
of translation and rotation common to the whole fluid. Hence
the equations F = P, &c. gives us
l(sr p) + v(u u)
du ^\ fdu dv\ fdu dw
8 ) + m (T+T } + n(J +
dx J \dy ctxj \dz
which three equations with (15) are those which must be satisfied
at the surface of a solid, together with the equation L = 0. It
will be observed that in the case of a free surface the pressures
P , Q , R are given, whereas in the case of the surface of a solid
they are known only by the solution of the problem. But on the
other hand the form of the surface of the solid is given, whereas
the form of the free surface is known only by the solution of the
problem.
Dubuat found by experiment that when the mean velocity of
water flowing through a pipe is less than about one inch in a
second, the water near the inner surface of the pipe is at rest.
If these experiments may be trusted, the conditions to be satisfied
in the case of small velocities are those which first occurred to me,
and which are included in those just given by supposing j/= oo .
I have said that when the velocity is not very small the tan
gential force called into action by the sliding of water over the
inner surface of a pipe varies nearly as the square of the velocity.
This fact appears to admit of a natural explanation. When a cur
rent of water flows past an obstacle, it produces a resistance varying
nearly as the square of the velocity. Now even if the inner surface
AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 99
of a pipe is polished we may suppose that little irregularities
exist, forming so many obstacles to the current. Each little pro
tuberance will experience a resistance varying nearly as the square
of the velocity, from whence there will result a tangential action
of the fluid on the surface of the pipe, which will vary nearly as
the square of the velocity ; and the same will be true of the equal
and opposite reaction of the pipe on the fluid. The tangential
force due to this cause will be combined with that by which the
fluid close to the pipe is kept at rest when the velocity is suf
ficiently small*.
[* Except in the case of capillary tubes, or, in case the tube be somewhat wider,
of excessively slow motions, the main part of the resistance depends upon the
formation of eddies. This much appears clear; but the precise way in which the
eddies act is less evident. The explanation in the text gives probably the correct
account of what takes place in the case of a river flowing over a rough stony bed;
but in the case of a pipe of fairly smooth interior surface the minute protuberances
would be too small to produce much resistance of the same kind as that con
templated in the paragraph beginning near the foot of p. 53.
What actually happens appears to be this. The rolling motion of the fluid
belonging to the eddies is continually bringing the more swiftly moving fluid which
is found nearer to the centre of the pipe close to the surface. And in consequence
the gliding or shifting motion of the fluid in the immediate neighbourhood of the
surface in such places is very greatly increased, and with it the tangential pressure.
Thus while in some respects these two classes of resistances are similar, in
others they are materially different. As typical examples of the two classes we
may take, for the first, that of a polished sphere of glass of some size descending
by its weight in deep water ; for the second, that of a very long circular glass pipe
down which water is flowing. In both cases alike eddies are produced, and the
eddies once produced ultimately die away. In both cases alike the internal friction
of the fluid, and the friction between the fluid and the solid, are intimately
connected with the formation of eddies, and it is by friction that the eddies die
away, and the kinetic energy of the mass is converted into molecular kinetic
energy, that is, heat. But in the first case the resistance depends mainly on the
clilerence of the pressure p in front and rear, the resultant of the other forces of which
the expressions are given in equations (8) being comparatively insignificant, while
in the second case it is these latter pressures that we are concerned with, the
resultant of the pressure p in the direction of the axis of the tube being practically
nil, even though the polish of the surface be not mathematically perfect.
Hence if, the motion being what it actually is, the fluidity of the fluid were
suddenly to become perfect, the immediate effect on the resistance in the first case
would be insignificant, while in the second case the resistance would practically
vanish. Of course if the fluidity were to remain perfect, the motion after some
time would be very different from what it had been before ; but that is not a point
under consideration.
Some questions connected with the effect of friction in altering the motion of
a nearly perfect fluid will be considered further on in discussing the case of motion
given in Art. 55 of a paper On the Critical Values of the Sums of Periodic Series. ]
72
100 ON THE FRICTION OF FLUIDS IN MOTION,
There remains to be considered the case of two fluids having a
common surface. Let u y v , w , /A , 8 denote the quantities belong
ing to the second fluid corresponding to u, &c. belonging to the
first. Together with the two equations .L = and (15) we shall
have in this case the equation derived from (15) by putting u, v t w
for u, v, w ; or, which comes to the same, we shall have the two
former equations with
l(uu )+m(vv ) + n(ww ) =0 (18).
If we consider the principles on which equations (17) were formed
to be applicable to the present case, we shall have six more equa
tions to be satisfied, namely (17), and the three equations derived
from (17) by interchanging the quantities referring to the two
fluids, and changing the signs I, m, n. These equations give the
value of CT, and leave five equations of condition. If we must
suppose v oo , as appears most probable, the six equations above
mentioned must be replaced by the six u u> v = v, w = w, and
lp pf(u, v, w} = lp f pf(u t v , w ), &c.,
f(u, v,w) denoting the coefficient of // in the first of equations (17).
We have here six equations of condition instead of five, but then
the equation (18) becomes identical.
7. The most interesting questions connected with this subject
require for their solution a knowledge of the conditions which
must be satisfied at the surface of a solid in contact with the fluid,
which, except perhaps in case of very small motions, are unknown.
It may be well however to give some applications of the preceding
equations which are independent of these conditions. Let us then
in the first place consider in what manner the transmission of sound
in a fluid is affected by the tangential action. To take the simplest
case, suppose that no forces act on the fluid, so that the pressure
and density are constant in the state of equilibrium, and conceive
a series of plane waves to be propagated in the direction of the
axis of x, so that u =/(a?, <), v = 0, w = 0. Let p / be the pressure,
and p, the density of the fluid when it is in equilibrium, and put
p=p / +p. Then we have from equations (11) and (12), omitting
the square of the disturbance,
I dp du du dp 4 d*u_
+ =0 +
AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 101
Let A&p be the increment of pressure due to a very small incre
ment Ap of density, the temperature being unaltered, and let m
be the ratio of the specific heat of the fluid when the pressure is
constant to its specific heat when the volume is constant; then
the relation between p and p will be
p = mA(p P/ ) .............................. (20).
Eliminating > and p from (19) and (20) we get
~ 0
df da* S P/ dtdx z "
To obtain a particular solution of this equation, let
. x ,,,..
u = <j) (t) cos   1 ^r (f) sin
.
A A.
Substituting in the above equation, we see that $ (t) and i/r (t)
must satisfy the same equation, namely,
<t>" ^ + y
the integral of which is
* fn ^fa nt 27T&A
= e ~ ct (C cos   + C sm  
\ A A /
A y
where
9XV/
(7 and being arbitrary constants. Taking the same expression
with different arbitrary constants for ty(t), replacing products of
sines and cosines by sums and differences, and combining the
resulting sines and cosines two and two, we see that the resulting
value of u represents two series of waves propagated in opposite
directions. Considering only those waves which are propagated
in the positive direction of x, we have
V" *~J i ^ 2 I \^*)
We see then that the effect of the tangential force is to make
the intensity of the sound diminish as the time increases, and to
render the velocity of propagation less than what it would other
wise be. Both effects are greater for high, than for low notes;
but the former depends on the first power of p, while the latter
depends only on //, 2 . It appears from the experiments of M. Biot,
made on empty water pipes in Paris, that the velocity of propaga
102 ON THE FRICTION OF FLUIDS IN MOTION,
tion of sound is sensibly the same whatever be its pitch. Hence
it is necessary to suppose that for air fjf/\*p* is insensible com
pared with A or pjp r I am not aware of any similar experiments
made on water, but the ratio of (/a/Xp,) 2 to A would probably be
insensible for water also. The diminution of intensity as the time
increases is, in the case of plane waves, due entirely to friction ;
but as we do not possess any means of measuring the intensity of
sound the theory cannot be tested, nor the numerical value of fju
determined, in this way.
The velocity of sound in air, deduced from the note given by
a known tube, is sensibly less than that determined by direct
observation. Poisson thought that this might be due to the
retardation of the air by friction against the sides of the tube.
But from the above investigation it seems unlikely that the effect
produced by that cause would be sensible.
The equation (21) may be considered as expressing in all
cases the effect of friction; for we may represent an arbitrary
disturbance of the medium as the aggregate of series of plane
waves propagated in all directions.
8. Let us now consider the motion of a mass of uniform
inelastic fluid comprised between two cylinders having a common
axis, the cylinders revolving uniformly about their axis, and the
fluid being supposed to have attained its permanent state of
motion. Let the axis of the cylinders be taken for that of z, and
let q be the actual velocity of any particle, so that u q sin 0,
v = q cos 0, w = 0, r and 6 being polar coordinates in a plane
parallel to xy.
Observing that
^" J __ J I _ %/ I
daf^dtf dr 2 * r dr r* d&
where/ is any function of x and y, and that dp/d& = 0, we have
from equations (13), supposing after differentiation that the axis
of x coincides with the radius vector of the point considered, and
omitting the forces, and the part of the pressure due to them,
dr ^ r
d*q 1 dq q _ / 22 >,
SH ~7 2 "* \"*if
dr r dr r
and the equation of continuity is satisfied identically.
AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 103
The integral of (22) is
q= + C r.
r
If a is the radius of the inner, and b that of the outer cylinder,
and if q l , q 2 are the velocities of points close to these cylinders
respectively, we must have q = q l when r = a, and q = q 2 when
r = b } whence
a ,  a?,) r} (23).
J. A *.&t M
If the fluid is infinitely extended, b oo , and
These cases of motion were considered by Newton (Principia,
Lib. II. Prop. 51). The hypothesis which I have made agrees in
this case with his, but he arrives at the result that the velocity
is constant, not, that it varies inversely as the distance. This
arises from his having taken, as the condition of their being no
acceleration or retardation of the motion of an annulus, that the
force tending to turn it in one direction must be equal to that
tending to turn it in the opposite, whereas the true condition is
that the moment of the force tending to turn it one way must
be equal to the moment of the force tending to turn it the other.
Of course, making this alteration, it is easy to arrive at the above
result by Newton s reasoning. The error just mentioned vitiates
the result of Prop. 52. It may be shewn from the general equa
tions that in this case a permanent motion in annuli is impossible,
and that, whatever may be the law of friction between the solid
sphere and the fluid. Hence it appears that it is necessary to
suppose that the particles move in planes passing through the
axis of rotation, while they at the same time move round it. In
fact, it is easy to see that from the excess of centrifugal force in
the neighbourhood of the equator of the revolving sphere the
particles in that part will recede from the sphere, and approach
it again in the neighbourhood of the poles, and this circulating
motion will be combined with a motion about the axis. If how
ever we leave the centrifugal force out of consideration, as Newton
has done, the motion in annuli becomes possible, but the solution
is different from Newton s, as might have been expected.
104 ON THE FRICTION OF FLUIDS IN MOTION,
The case of motion considered in this article may perhaps
admit of being compared with experiment, without knowing the
conditions which must be satisfied at the surface of a solid. A
hollow, and a solid cylinder might be so mounted as to admit of
being turned with different uniform angular velocities round their
common axis, which is supposed to be vertical. If both cylinders
are turned, they ought to be turned in opposite directions, if only
one, it ought to be the outer one ; for if the inner were made to
revolve too fast, the fluid near it would have a tendency to fly
outwards in consequence of the centrifugal force, and eddies would
be produced. As long as the angular velocities are not great, so
that the surface of the liquid is very nearly plane, it is not of much
importance that the fluid is there terminated ; for the conditions
which must be satisfied at a free surface are satisfied for any sec
tion of the fluid made by a horizontal plane, so long as the motion
about that section is supposed to be the same as it would be if the
cylinders were infinite. The principal difficulty would probably be
to measure accurately the time of revolution, and distance from the
axis, of the different annuli. This would probably be best done by
observing motes in the fluid. It might be possible also to discover
in this way the conditions to be satisfied at the surface of the
cylinders ; or at least a law might be suggested, which could be
afterwards compared more accurately with experiment by means
of the discharge of pipes and canals.
If the rotations of the cylinders are in opposite directions,
there will be a certain distance from the axis at which the fluid
will not revolve at all. Writing  q l for q t in equation (23), we
have for this distance
9. Although the discharge of a liquid through a long straight
pipe or canal, under given circumstances, cannot be calculated
without knowing the conditions to be satisfied at the surface of
contact of the fluid and solid, it may be well to go a certain way
towards the solution.
Let the axis of z be parallel to the generating lines of the
pipe or canal, and inclined at an angle a to the horizon ; let the
plane yz be vertical, and let y and z be measured downwards.
AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 105
The motion being uniform, we shall have u = 0, v = 0, w =f(, y),
and we have from equations (13)
dp _ dp dp fd^w d*w\
7 = 0, f  gp cos a, f = #0 sin a + p,( j^ + .
d dz \dx 2 d* J
7 ,  , ,
dx dy dz \dx 2 dy* J
In the case of a canal dp/dz = Q; and the calculation of the
motion in a pipe may always be reduced to that of the motion
in the same pipe when dpjdz is supposed to be zero, as may be
shewn by reasoning similar to Dubuat s. Moreover the motion
in a canal is a particular case of the motion in a pipe. For
consider a pipe for which dp/dz = Q, and which is divided sym
metrically by the plane xz. From the symmetry of the motion,
it is clear that we must have dw/dy = when z = ; but this is
precisely the condition which would have to be satisfied if the
fluid had a free surface coinciding with the plane xz ; hence we
may suppose the upper half of the fluid removed, without affect
ing the motion of the rest, and thus we pass to the case of a canal.
Hence it is the same thing to determine the motion in a canal,
as to determine that in the pipe formed by completing the canal
symmetrically with respect to the surface of the fluid.
We have then, to determine the motion, the equation
d*w d z w gp sin a _
w 1 "^ 4 ~T~
In the case of a rectangular pipe, it would not be difficult to
express the value of w at any point in terms of its values at the
several points of the perimeter of a section of the pipe. In the
case of a cylindrical pipe the solution is extremely easy : for if
we take the axis of the pipe for that of z, and take polar co
ordinates r, 6 in a plane parallel to xy, and observe that dw/d6 = 0,
since the motion is supposed to be symmetrical with respect to
the axis, the above equation becomes
d z w 1 dw gp sin a
7 u H ~~7   = 0.
ar r ar /JL
Let a be the radius of the pipe, and U the velocity of the fluid
close to the surface ; then, integrating the above equation, and
determining the arbitrary constants by the conditions that w shall
be finite when r = 0, and w = U when r = a, we have
_
v
106 ON THE FRICTION OF FLUIDS IN MOTION,
SECTION II.
Objections to Lagrange s proof of the theorem that if udx+vdy+wdz
is an exact differential at any one instant it is always so, the
pressure being supposed equal in all directions. Principles of
M. Cauchys proof. A new proof of the theorem. A physical
interpretation of the circumstance of the above expression
being an exact differential.
10. The proof of this theorem given by Lagrange depends
on the legitimacy of supposing u, v and w capable of expansion
according to positive integral powers of t, for a sufficiently small
finite value of t. It is clear that the expansion cannot contain
negative powers of t, since u, v and w are supposed to be finite
when t = ; but it may be objected to Lagrange s proof that there
are functions of t of which the expansion contains fractional
powers of t } and that we do not know but that u, v and w may
be such functions. This objection has been considered by Mr
Power*, who has shewn that the theorem is true if we suppose
u, v and w capable of expansion according to any powers of t.
Still the proof remains unsatisfactory, in fact inconclusive, for
these are functions of t, (for instance, e~ l / t2 , t log,) which do not
admit of expansion according to powers of t, integral or fractional,
and we do not know but that u, v and w may be functions of this
nature. I do not here mention the proof which Poisson has
given of the theorem in his Traite de Mecanique, because it
appears to me liable to an objection to which I shall presently
have occasion to refer : in fact, Poisson himself did not think the
theorem generally true.
It is remarkable that Mr Power s proof, if it were legitimate,
would establish the theorem even when account is taken of the
variation of pressure in different directions, according to the
theory explained in Section I., if we suppose that d^/dp = 0. To
shew this we have only got to treat equations (12) as Mr Power
has treated the three equations of fluid motion formed on the
ordinary hypothesis. Yet in this case the theorem is evidently
untrue. Thus, conceive a mass of fluid which is bounded by
a solid plane coinciding with the plane yz, and which extends
* Cambridge Philosophical Transactions, Vol. vn. (Part 3) p. 455.
AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 107
infinitely in every direction on the positive side of the axis of x,
and suppose the fluid at first to be at rest. Suppose now the
solid plane to be moved in any manner parallel to the axis of y\
then, unless the solid plane exerts no tangential force on the fluid,
(and we may suppose that it does exert some,) it is clear that at
a given time we shall have u = Q, v=f(x) t t0 = 0, and therefore
udx + vdy + wdz will not be an exact differential. It will be
interesting then to examine in this case the nature of the function
of t which expresses the value of v.
Supposing X, Y, Z to be zero in equations (12), and observing
that in the case considered we have dp/dy 0, we get
dv _n d^v . ,.
dt~p dx*" ( * ]
Differentiating this equation n 1 times with respect to t, we
easily get
d*v
dt n \pj dx in
but when t 0, v = when x > 0, and therefore for a given value
of x all the differential coefficients of v with respect to t are zero.
Hence for indefinitely small values of t the value of v at a given
point increases more slowly than if it varied ultimately as any
power of t, however great ; hence v cannot be expanded in a series
according to powers of t. This result is independent of the con
dition to be satisfied at the surface of the solid plane.
I think what has just been proved shews clearly that La
grange s proof of the theorem considered, even with Mr Power s
improvement of it, is inadmissible.
11. The theorem is however true, and a proof of it has been
given by M. Cauchy*, which appears to me perfectly free from
objection, and which is very simple in principle, although it
depends on rather long equations. M. Cauchy first eliminates p
from the three equations of motion by means of the conditions
that d*p/dxdy = d*p/dydx, &c., he then changes the independent
variables from x, y, z, t to a, b, c, t, where a, b, c are the initial
* Memoire sur la Theorie des Ondcs, in the first volume of the Memoires des
savans Etrangers. M. Cauchy has not had occasion to enunciate the theorem, but
it is contained in his equations (16). This equation may be obtained in the same
manner in the more general case in which p is supposed to be a function of p.
108 ON THE FRICTION OF FLUIDS IN MOTION,
coordinates of the particles. The three transformed equations
admit each of being once integrated with respect to t ; and deter
mining the arbitrary functions of a, 6, c by the initial values of
u, v and w, the three integrals have the form
G>; = Fto + Ga>" + Ha> ", &c.,
a) , w" and co" f denoting here the same as in Art. 2, and &&gt; , &c.
denoting the initial values of to , &c. for the same particle. Solving
the above equations with respect to &&gt; , w" and o> ", the resulting
equations are
, 1 dx , dx ,, dx
CO = TV
where S is a function of the differential coefficients of x, y and z
with respect to a, b and c, which by the condition of continuity is
shewn to be equal to pjp, p being the initial density about the
particle whose density at the time considered is p. Since dx/da, &c.
are finite, (for to suppose them infinite would be equivalent to
supposing a discontinuity to exist in the fluid,) it follows at once
from the preceding equations that if &&gt; = 0, &&gt; " = 0, &&gt; " = 0, that
is if u da + V db + w Q dc be an exact differential, either for the whole
fluid or for any portion of it, then shall &&gt; = 0, &&gt;" = 0, &&gt; " = 0, i.e.
udx + vdy + wdz will be an exact differential, at any subsequent
time, either for the whole mass or for the above portion of it.
12. It is not from seeing the smallest flaw in M. Cauchy s
proof that I propose a new one, but because it is well to view the
subject in different lights, and because the proof which I am about
to give does not require such long equations. It will be necessary
in the first place to prove the following lemma.
LEMMA. If o^, ft> 2 ,...a> n are n functions of t, which satisfy the
n differential equations
[ (25),
^=P. t +/,... + F..,J
where P,, Q l . . . V n may he functions of <,<,.. .co,, and if when <,= 0,
o) 2 = 0...w H = 0, none of the quantities P I; ... F. is infinite for any
AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 109
value of t from to T, and if o^...^ are each zero when = 0,
then shall each of these quantities remain zero for all values of t
from to T.
DEMONSTRATION. Let T be a finite value of t, then by hypo
thesis T may be taken so small that the values of co l ...a) n are suf
ficiently small to exclude all values which might render any one
of the quantities ^...F^ infinite. Let L be a superior limit to
the numerical values of the several quantities P l ...V n for all
values of t from to T; then it is evident that a) l ...a) n cannot
increase faster than if they satisfied the equations
dw, T , . i
(26),
at J
vanishing in this case also when t = 0. But if co l f o> 2 . . . + &&gt; ?
we have by adding together the above equations
czn
if now fl be not equal to zero, dividing this equation by H and
integrating, we have
but no value of C different from zero will allow 1 to vanish
when t = 0, whereas by hypothesis it does vanish ; hence H = ;
but fl is the sum of n quantities which evidently cannot be nega
tive, and therefore each of these must be zero. Since then co l ...(o n
would have to be equal to zero for all values of t from to T even
if they satisfied equations (26), they must d fortiori be equal to
zero in the actual case, since they satisfy equations (25). Hence
there is no value of t from to T at which any one of the
quantities co l ...co n can begin to differ from zero, and therefore
these quantities must remain equal to zero for all values of t
from to T.
This lemma might be extended to the case in which n = oo ,
with certain restrictions as to the convergency of the series. We
may also, instead of the integers 1, 2...W, have a continuous
variable a which varies from to a, so that o> is a function of
110 ON THE FRICTION OF FLUIDS IN MOTION,
the independent variables a and t, satisfying the differential
equation
day
where ^r(a, 0, t) does not become infinite for any value of a from
to a combined with any value of t from to T. It may be shewn,
just as before, that if <w = when = for all values of a from
to a, then must a> = for all values of t from to T. The proposi
tion might be further extended to the case in which a = oo , with
a certain restriction as to the convergency of the integral, but
equations (25) are already more general than I shall have occa
sion to employ.
It appears to me to be sometimes assumed as a principle that
two variables, functions of another, t, are proved to be equal for
all values of t when it is shewn that they are equal for a certain
value of t, and that whenever they are equal for the same value
of t their increments for the same increment of t are ultimately
equal. But according to this principle, if two curves could be
shewn always to touch when they meet they must always coincide,
a conclusion manifestly false. I confess I cannot see that Newton
in his Principia, Lib. I., Prop. 40, has proved more than that if
the velocities of the two bodies are equal at equal distances, the in
crements of those velocities for equal increments of the distances
are ultimately equal : at least something additional seems re
quired to put the pjoof quite out of the reach of objection.
Again it is usual to speak of the condition, that the motion of
a particle of fluid in contact with the surface of a solid at rest
is tangential to the surface, as the same thing as the condition
that the particle shall always remain in contact with the surface.
That it is the same thing might be shewn by means of the lemma
in this article, supposing the motion continuous; but independ
ently of proof I do not see why a particle should not move in
a curve not coinciding with the surface, but touching it where
it meets it. The same remark will apply to the condition that
a particle which at one instant lies in a free surface, or is in
contact with a solid, shall ultimately lie in the free surface, or be
in contact with the solid, at the consecutive instant. I refer here
to the more general case in which the solid is at rest or in motion.
For similar reasons Poisson s proof of the Hydrodynamical theorem
AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. Ill
which forms the principal subject of this section has always ap
peared to me unsatisfactory, in fact far less satisfactory than
Lagrange s. I may add that Poisson s proof, as well as Lagrange s,
would apply to the case in which friction is taken into account,
in which case the theorem is not true.
13. Supposing p to be a function of p, I// (p), the ordinary
equations of Hydrodynamics are
df( p) _ Y tftPl V ~ *tf_(P) 7 _ P w /.o7\
"~dx~ 1H dy Dt dz ~Dt
The forces X, Y, Z will here be supposed to be such that
Xdx + Ydy + Zdz is an exact differential, this being the case
for any forces emanating from centres, and varying as any func
tions of the distances. Differentiating the first equations (27)
with respect to y, and the second with respect to x, subtracting,
putting for Du/Dt and Dv/Dt their values, adding and subtracting
du/dz . dv/dz, and employing the notation of Art. 2, we obtain
Dw" _du , dv (du dv\ ,,, , ,
~DT~~dz K dz ~(dx + dy) a: ^ "
By treating the first and third, and then the second and third of
equations (27) in the same manner, we should obtain two more
equations, which may be got at once from that which has just
been found by interchanging the requisite quantities. Now for
points in the interior of the mass the differential coefficients
du/dz, &c. will not be infinite, on account of the continuity of the
motion, and therefore the three equations just obtained are a
particular case of equations (25). If then udx + vdy + wdz is an
exact differential for any portion of the fluid when = 0, that is,
if w , a)" and CD " are each zero when t = 0, it follows from the
lemma of the last article that ft/, co" and co " will be zero for any
value of t, and therefore udx + vdy + wdz will always remain an
exact differential. It will be observed that it is for the same
portion of fluid, not for the fluid occupying the same portion of
space, that this is true, since equations (28), &c. contain the
differential coefficients Dco /Dt, &c., and not dw /dt, &c.
14. The circumstance of udx + vdy + wdz being an exact
differential admits of a physical interpretation which may be
112 ON THE FRICTION OF FLUIDS IN MOTION,
noticed, as it is well to view a subject of this nature in different
lights.
Conceive an indefinitely small element of a fluid in motion
to become suddenly solidified, and the fluid about it to be suddenly
destroyed ; let the form of the element be so taken that the re
sulting solid shall be that which is the simplest with respect to
rotatory motion, namely, that which has its three principal
moments about axes passing through the centre of gravity equal
to each other, and therefore every axis passing through that point
a principal axis, and let us enquire what will be the linear and
angular motion of this element just after solidification.
By the instantaneous solidification, velocities will be suddenly
generated or destroyed in the different portions of the element,
and a set of mutual impulsive forces will be called into action.
Let x, y, z be the coordinates of the centre of gravity G of the
element at the instant of solidification, x + x, y + y , z + z those
of any other point in it. Let u t v, w be the velocities of G along
the three axes just before solidification, u, v , w the relative velo
cities of the point whose relative coordinates are x, y, z . Let
u, v, w be the velocities of G, u t , v,, w, the relative velocities of the
point above mentioned, and o> , o>", w " the angular velocities just
after solidification. Since all the impulsive forces are internal, we
have
u = u, v v, w = w.
We have also, by the principle of conservation of areas,
%m {y (w / w ) z (v t v)} = 0, &c.,
m denoting an element of the mass of the element considered.
But u t = o/Y (o "y , u is ultimately equal to
du , du , du ,
r~ x + 7 v f = z ,
dx dy y dz
and similar expressions hold good for the other quantities. Sub
stituting in the above equations, and observing that
= 0, and ^mx* = %m 2 = 2m/ 2 ,
dw dv
i
dy
We see then that an indefinitely small element of the fluid,
of which the three principal moments about the centre of gravity
. ,  fdw dv\
we nave o> = i  r >
2 \d dzj
AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 113
are equal, if suddenly solidified and detached from the rest of the
fluid will begin to move with a motion simply of translation,
which may however vanish, or a motion of translation combined
with one of rotation, according as udx + vdy 4 wdz is, or is not an
exact differential, and in the latter case the angular velocities
will be the same as in Art. 2.
The principle which forms the subject of this section might
be proved, at least in the case of a homogeneous incompressible
fluid, by considering the change in the motion of a spherical
element of the fluid in the indefinitely small time dt. This
method of proving the principle would shew distinctly its inti
mate connexion with the hypothesis of normal pressure, or the
equivalent hypothesis of the equality of pressure in all directions,
since the proof depends on the impossibility of an angular velo
city being generated in the element in the indefinitely small
time dt by the pressure of the surrounding fluid, inasmuch as the
direction of the pressure at any point of the surface ultimately
passes through the centre of the sphere. The proof I speak of
is however less simple than the one already given, and would
lead me too far from my subject.
SECTION III.
Application of a method analogous to that of Sect. I. to the
determination of the equations of equilibrium and motion of
elastic solids.
15. All solid bodies are more or less elastic, as is shewn by
the capability they possess of transmitting sound, and vibratory
motions in general. The solids considered in this section are
supposed to be homogeneous and uncrystallized, so that when in
their natural state the average arrangement of their particles is
the same at one point as at another, and the same in one direction
as in another. The natural state will be taken to be that in which
no forces act on them, from which it may be shewn that the pres
sure in the interior is zero at all points and in all directions,
neglecting the small pressure depending on attractions of the
nature of capillary attraction.
Let x, y, 2 be the coordinates of any point P in the solid con
sidered when in its natural state, a, /3, 7 the increments of those
S. 8
114 ON THE FRICTION OF FLUIDS IN MOTION,
coordinates at the time considered, whether the body be in a state
of constrained equilibrium or of motion. It will be supposed that
a, @ and 7 are so small that their squares and products may be
neglected. All the theorems proved in Art. 2 with reference to
linear and angular velocities will be true here with reference to
linear and angular displacements, since these two sets of quantities
are resolved according to the same laws, as long as the angular
displacements are supposed to be very small. Thus, the most
general displacement of a very small element of the solid consists
of a displacement of translation, an angular displacement, and three
displacements of extension in the direction of three rectangular
axes, which may be called in this case, with more propriety than in
the former, axes of extension. The three displacements of extension
may be resolved into two displacements of shifting, each in two
dimensions, and a displacement of uniform dilatation, positive or
negative. The pressures about the element considered will depend
on the displacements of extension only; there may also, in the
case of motion, be a small part depending on the relative velocities,
but this part may be neglected, unless we have occasion to consider
the effect of the internal friction in causing the vibrations of solid
bodies to subside. It has been shewn (Art. 7) that the effect of
this cause is insensible in the case of sound propagated through
air; and there is no reason to suppose it greater in the case of
solids than in the case of fluids, but rather the contrary. The
capability which solids possess of being put into a state of isochro
nous vibration shews that the pressures called into action by small
displacements depend on homogeneous functions of those displace
ments of one dimension. I shall suppose moreover, according to
the general principle of the superposition of small quantities, that
the pressures due to different displacements are superimposed, and
consequently that the pressures are linear functions of the dis
placements. Since squares of or, (3 and 7 are neglected, these
pressures may be referred to a Unit of surface in the natural state
or after displacement indifferently, and a pressure which is normal
to any surface after displacement may be regarded as normal to
the original position of that surface. Let AS be the pressure
corresponding to a uniform linear dilatation & when the solid is in
equilibrium, and suppose that it becomes  mAS, in consequence
of the heat developed, when the solid is in a state of rapid vibra
tion. Suppose also that a displacement of shifting parallel to
AND THE EQUILIBKIUM AND MOTION OF ELASTIC SOLIDS. 115
the plane xy, for which a. = kx, /3 = ky, 7 = 0, calls into action a
pressure Bk on a plane perpendicular to the axis of x, and a
pressure Bk on a plane perpendicular to that of y\ the pressures
on these planes being equal and of opposite signs, that on a plane
perpendicular to the axis of z being zero, and the tangential forces
on those planes being zero, for the same reasons as in Sect. I. It
may also be shewn as before that it is necessary to suppose B
positive, in order that the equilibrium of the solid medium may
be stable, and it is easy to see that the same must be the case
with A for the same reason.
It is clear that we shall obtain the expressions for the pressures
from those already found for the case of a fluid by merely putting
a, /3, 7, B for u, v, w, //, and AS or mAB for p, according as we
are considering the case of equilibrium or of vibratory motion, the
body being in the latter case supposed to be constrained only in
so far as depends on the motion.
For the case of equilibrium then we have from equations (8)
* U i i /^ a djB dy\ , , ,. .... .
o being here = k ( = + , \r\l and the equations ot equilibrium
3 \dx dy dzj
will be obtained from (12) by putting Du/Dt = 0, p = AS, making
the same substitution as before for u, v, w and p. We have there
fore, for the equations of equilibrium,
Y if A m <L l drj  ^ d 1\
P iJ \ * ) ~l~ I 7 ~7 i T~ I
ax \ax dy dzj
In the case of a vibratory motion, when the body is in its
natural state except so far as depends on the motion, we have
from equations (8)
and the equations of motion will be derived from (12) as before,
only Du/Dt &c. must be replaced by d*x/d(? &c., and X, Y, Z put
equal to zero. The equations of motion, then, are
82
116 ON THE FRICTION OF FLUIDS IN MOTION,
d f d * , d & i <
  + +
16. The conditions to be satisfied at the surface of the solid
may be easily deduced from the analogous conditions in the case
of a fluid with a free surface, only it will be necessary to replace
the normal pressure II by an oblique pressure, of which the com
ponents will be denoted by X lt Y I} Z^ We have then, making
the necessary changes in the quantities involved in (14),
X, + IAS + B + + + n + = 0, &c.,
{ dx \dy dx) \dz dxj)
for the case of equilibrium, and for the case of motion such as that
just considered it will only be necessary to replace A by mA in
these equations. If we measure the angles of which I, m, n are
the cosines from the external normal, the forces Jf 1? Y lt Z 1 must be
reckoned positive when, Z, m and n being positive, the surface of
the solid is urged in the negative directions of x, y, z, and in other
cases the signs must be taken conformably.
If the solid considered is in a state of constraint when at rest,
and is moreover put into a state of vibration, the pressures and
displacements due to these two causes must be calculated separately
and added together. If m were equal to 1, they could be calcu
lated together from the same equations.
r f
SECTION IV.
Principles of Poissons theory of elastic solids, and of the oblique
pressures existing in fluids in motion. Objections to one of his
hypotheses. Reflections on the constitution, and equations of
motion of the luminiferous ether in vacuum.
17. In the twentieth Colder of the Journal de I Ecole Polytech
nique may be found a memoir by Poisson, entitled Memoire sur les
Equations generates de VEquilibre et du Mouvement des Corps
solides tlastiques et des Fluides, which contains the substance of
two memoirs presented by him to the Academy, brought together
with some additions. In this memoir the author treats principally
AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 117
of the equations of equilibrium and motion of elastic solids, of the
equations of equilibrium of fluids, with reference especially to
capillary attraction, and of the equations of motion of fluids, sup
posing the pressure not to be equal in all directions,
It is supposed by Poisson that all bodies, whether solid or
fluid, are composed of ultimate molecules, separated from each
other by vacant spaces. In the cases of an uncrystallized solid
in its natural state, and of a fluid in equilibrium, he supposes
that the molecules are arranged irregularly, and that the average
arrangement is the same in all directions. These molecules he
supposes to act on each other with forces, of which the main
part is a force in the direction of the line joining the centres of
gravity, and varying as some function of the distance of these
points, and the remainder a secondary force, or it may be two
secondary forces, depending on the molecules not being mathe
matical points. He supposes that it is on these secondary forces
that the solidity of solid bodies depends. He supposes however
that in calculating the pressures these secondary forces may be
neglected, partly because they become insensible at much smaller
distances than the main part of the forces, and partly because they
act, on the average, alike in all directions. He supposes that the
molecular force decreases very rapidly as the distance increases,
yet not so rapidly but that the sphere in which the molecular
action is sensible contains an immense number of molecules. He
supposes consequently that in estimating the resultant force of a
hemisphere of the medium on a molecule in the centre of its base
the action of the neighbouring molecules, which are situated
irregularly, may be neglected compared with the action of those
more remote, of which the average may be taken. The consequence
of this supposition of course is that the total action is normal to
the base of the hemisphere, and sensibly the same for one molecule
as for an adjacent one.
The rest of the reasoning by which Poisson establishes the
equations of motion and equilibrium of elastic solids is purely
mathematical, sufficient data having been already assumed. It
might appear that the reasoning in Art. 16 of his memoir, by
which the expression for N is simplified, required the fresh hypo
thesis of a symmetrical arrangement of the molecules ; but it really
does not, being admissible according to the principle of averages.
118 ON THE FRICTION OF FLUIDS IN MOTION,
Taking for the natural state of the body that in which the pressure
is zero, the equations at which Poisson arrives contain only one
unknown constant k, whereas the equations of Sect. in. of this
paper contain two, A or mA and B. This difference depends on
the assumption made by Poisson that the irregular part of the
force exerted by a hemisphere of the medium on a molecule in the
centre of its base may be neglected in comparison with the whole
force. As a result of this hypothesis, Poisson finds that the change
in direction, and the proportionate change in length, of a line
joining two molecules are continuous functions of the coordinates
of one of the molecules and the angles which determine the direc
tion of the line ; whereas in Sect, ill., if we adopt the hypothesis
of ultimate molecules at all, it is allowable to suppose that these
quantities vary irregularly in passing from one pair of molecules
to an adjacent pair. Of course the equations of Sect. ill. ought to
reduce themselves to Poisson s equations for a particular relation
between A and B. Neglecting the heat developed by compression,
as Poisson has done, and therefore putting m = 1, this relation is
4 = 55.
18. Poisson s theory of fluid motion is as follows. The time
t is supposed to be divided into a number n of equal parts, each
equal to r. In the first of these the fluid is supposed to be dis
placed as an elastic solid would be, according to Poisson s previous
theory, and therefore the pressures are given by the same equa
tions. If the causes producing the displacement were now to
cease, the fluid would rearrange itself, so that the average arrange
ment about each point should be the same in all directions after
a very short time. During this time, the pressures would have
altered, in an unknown manner, from those corresponding to a
displaced solid to a normal pressure equal to p + Dp/Dt . r, the
pressures during the alteration involving an unknown function of
the time elapsed since the end of the interval r. Another dis
placement and another rearrangement may now be supposed to
to take place, and so on. But since these very small relative mo
tions will take place independently of each other, we may suppose
each displacement to begin at the expiration of the time during
which the preceding one is supposed to remain, and we may sup
pose each rearrangement to be going on during the succeeding
displacements. Supposing now n to become infinite, we pass to
AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 119
the case in which the fluid is supposed to be continually beginning
to be displaced as a solid would, and continually rearranging itself
so as to make the average arrangement about each point the same
in all directions.
Poisson s equations (9), page 152, which are applicable to the
motion of a liquid, or of an elastic fluid in which the change of
density is small, agree with equations (12) of this paper. For the
quantity fyt is the pressure p which would exist at any instant if
the motion were then to cease, and the increment, j r or ~ r,
Cut JJt
of this quantity in the very small time r will depend only on
the increment, ?? T or  T, of the density yt or p. Consequently
the value of J r will be the same as if the density of the par
ctt
tide considered passed from %t to %t +  T in the time T by a
uniform motion of dilatation. I suppose that according to Pois
son s views such a motion would not require a re arrangement of
the molecules, since the pressure remains equal in all directions.
On this supposition we shall get the value of J from that of
ut
R t Kin the equations of page 140 by putting
du dv dw 1 dyt
. . A/
doc dy dz 3%t dt
We have therefore
_ _
a ~dt ~3 (K blG 
Putting now for ft + ft its value 2a&, and for J* its value given
%
by equation (2), the expression for r, page 152, becomes
j. a /JFJ. f^ J.2E j. dw \
= ^ + o(^ + ^)r~ + ^ r T~ /
" 3 v \dx dy dz)
Observing that a (K + Jc) = /9, this value of CT reduces Poisson s
equations (9) to the equations (12) of this paper.
Poisson himself has not made this reduction of his equations,
nor any equivalent one, so that his equations, as he has left them,
120 ON THE FRICTION OF FLUIDS IN MOTION,
involve two arbitrary constants. The reduction of these two to
one depends on the assumption that a uniform expansion of any
particle does not require a rearrangement of the molecules, as it
leaves the pressure still equal in all directions. If we do not
make this assumption, but retain the two arbitrary constants, the
equations will be the same as those which would be obtained by
the method of this paper, supposing the quantity K of Art. 3 not
to be zero.
19. There is one hypothesis made in the common theory of
elastic solids, the truth of which appears to me very questionable.
That hypothesis is the one to which I have already alluded in
Art. 17, respecting the legitimacy of neglecting the irregular part
of the action of the molecules in the immediate neighbourhood of
the one considered, in comparison with the total action of those
more remote, which is regular. It is from this hypothesis that it
follows as a result that the molecules are not displaced among one
another in an irregular manner, in consequence of the directive
action of neighbouring molecules. Now it is obvious that the
molecules of a fluid admit of being displaced among one another
with great readiness. The molecules of solids, or of most solids
at any rate, must admit of new arrangements, for most solids
admit of being bent, permanently, without being broken. Are we
then to suppose that when a solid is constrained it has no tendency
to relieve itself from the state of constraint, in consequence of its
molecules tending towards new relative positions, provided the
amount of constraint be very small ? It appears to me to be much
more natural to suppose a priori that there should be some such
tendency.
In the case of a uniform dilatation or contraction of a particle,
a rearrangement of its molecules would be of little or no avail
towards relieving it from constraint, and therefore it is natural to
suppose that in this case there is little or no tendency towards such
a rearrangement. It is quite otherwise, however, in the case of
what I have called a displacement of shifting. Consequently B
will be less than if there were no tendency to a rearrangement.
On the hypothesis mentioned in this article, of which the absence
of such tendency is a consequence, I have, said that a relation has
been found between A and B, namely A = oB. It is natural
then to expect to find the ratio of A to B greater than 5, ap
AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 121
preaching more nearly to 5 as the solid considered is more hard
and brittle, but differing materially from 5 for the softer solids,
especially such as India rubber, or, to take an extreme case, jelly.
According to this view the relation A 5B belongs only to an
ideal elastic solid, of which the solidity, or whatever we please to
call the property considered, is absolutely perfect.
To shew how implicitly the common theory of elasticity seems
to be received by some, I may mention that MM. Lame* and
Clapeyron mention Indian rubber among the substances to which
it would seem they consider their theory applicable*. I do not
know whether the coefficient of elasticity, according to that theory,
has been determined experimentally for India rubber, but one
would fancy that the cubical compressibility thence deduced, by a
method which will be seen in the next article, would turn out com
parable with that of a gas.
20. I am not going to enter into the solution of equations (30),
but I wish to make a few remarks on the results in some simple
cases.
If k be the cubical contraction due to a uniform pressure P,
then will
If a wire or rod, of which the boundary is any cylindrical sur
face, be pulled in the direction of its length by a force of which
the value, referred to a unit of surface of a section of the rod, in P,
the rod will extend itself uniformly in the direction of its length,
and contract uniformly in the perpendicular direction ; and if e
be the extension in the direction of the length, and c the contraction
in any perpendicular direction, both referred to a unit of length,
we shall have
~ SAB GAB
p
also, the cubical dilatation = e 2c = r .
If a cylindrical wire of radius r be twisted by a couple of which
* Memoires dcs savam Etranyers, Tom. iv. p. 4G9.
122 ON THE FRICTION OF FLUIDS IN MOTION,
the moment is M y and if be the angle of torsion for a length z of
the wire, we shall have
ZM*

The expressions for k, c, e and 0, and of course all expressions
of the same nature, depend on the reciprocals of A and B. Sup
pose now the value of e, or 0, or any similar quantity not depending
on A alone, be given as the result of observation. It will easily
be conceived that we might find very nearly the same value for B
whether we supposed A = 5B or A = nB, where n may be consider
ably greater than 5, or even infinite. Consequently the observation
of two such quantities, giving very nearly the same value of B,
might be regarded as confirming the common equations.
If we denote by E the coefficient of elasticity when A is
supposed to be equal to 5B we have, neglecting the atmospheric
pressure*,
_2P fl _2M*
~ ~
If now we denote by E l the value of E deduced from observation
of the value of e, and by E 9 the value of E obtained by observing
the value of 6, or else, which comes to the same, by observing the
time of oscillation of a known body oscillating by torsion, we shall
have
If A be greater than oB, E l ought to be a little greater than E z .
This appears to agree with observation. Thus the following num
bers are given by M. Lame f E^ = 8000, E t = 7500 for iron; E^= 2510,
E z 2250 for brass J. The difference between the values of E t and
E 2 is attributed by M. Lame to the errors to which the obser
vation of the small quantity e is liable. If the above numbers
may be trusted, we shall have
A
A = 60000, B = 7500, = = 8 for iron;
A = 29724, B = 2250, ^ = 13 21 for brass.
* Lam6, Cours de Physique, Tom. i.
t Lamd, Cotirs de Physique, Tom. i.
$ These numbers refer to the French units of length and weight.
AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 123
The cubical contraction k is almost too small to be made the
subject of direct observation*, it is therefore usually deduced from
the value of e y or from the coefficient of elasticity E found in some
other way. On the supposition of a single coefficient E, we have
Jc/e = f, but retaining the two, A and B, we have
* U* *(*
e A + B \ A
which will differ greatly from f if A/B be much greater than 5.
The whole subject therefore requires, I think, a careful examina
tion, before we can set down the values of the coefficients of cubical
contraction of different substances in the list of well ascertained
physical data. The result, which is generally admitted, that the
ratio of the velocity of propagation of normal, to that of tangential
vibrations in a solid is equal to \/3, is another which depends en
tirely on the supposition that A = oB. The value of m, again, as
deduced from observation, will depend upon the ratio of A to B ;
and it would be highly desirable to have an accurate list of the
values of m for different substances, in hopes of thereby discover
ing in what manner the action of heat on those substances is
related to the physical constants belonging to them, such as their
densities, atomic weights, &c.
The observations usually made on elastic solids are made on
slender pieces, such as wires, rods, and thin plates. In such pieces,
all the particles being at no great distance from the surface, it is easy
to see that when any small portion is squeezed in one direction it
has considerable liberty of expanding itself in a direction perpen
dicular to this, and consequently the results must depend mainly
on the value of B, being not very different from what they would
be if A were infinite. This is not so much the case with thick,
stout pieces. If therefore such pieces could be put into a state of
isochronous vibration, so that the musical notes and nodal lines
could be observed, they would probably be better adapted than
slender pieces for determining the value of mA. The value of
* I find however that direct experiments have been made by Prof. Oersted.
According to these experiments the cubical compressibility of solids which would
be obtained from Poisson s theory is in some cases as much as 20 or 30 times too
great. See the Report of the British Association for 1833, p. 353, or Archives des
decouvertes, &c. for 1834, p. 94. [It is to be noted that Oersted s method gives only
differences of compressibility.]
124 ON THE FRICTION OF FLUIDS IN MOTION,
m might be determined by comparing the value of mA, deduced
from the observation of vibrations, with the value of A, deduced
from observations made in cases of equilibrium, or, perhaps, of very
slow motion.
21. The equations (32) are the same as those which have
been obtained by different authors as the equations of motion of
the luminiferous ether in vacuum. Assuming for the present
that the equations of motion of this medium ought to be deter
mined on the same principles as the equations of motion of an
elastic solid, it will be necessary to consider whether the equations
(32) are altered by introducing the consideration of a uniform
pressure II existing in the medium when in equilibrium; for we
have evidently no right to assume, either that no such pressure
exists, or, supposing it to exist, that the medium would expand
itself but very slightly if it were removed. It will now no longer
be allowable to confound the pressure referred to a unit of surface
as it was, in the position of equilibrium of the medium, with
the pressure referred to a unit of surface as it actually is. The
latter mode of referring the pressure is more natural, and will
be more convenient. Let the pressure, referred to a unit of
surface at it is, be resolved into a normal pressure H+p l and a
tangential pressure ^. All the reasoning of Sect. in. will apply
to the small forces p^ and t l ; only it must be remembered that
in estimating the whole oblique pressure a normal pressure II
must be compounded with the pressures given by equations (31).
In forming the equations of motion, the pressure II will not
appear, because the resultant force due to it acting on the element
of the medium which is considered is zero. The equations (32)
will therefore be the equations of motion required.
If we had chosen to refer the pressure to a unit of surface in
the original state of the surface, and had resolved the whole
pressure into a pressure II + p l normal to the original position
of the surface, and a pressure ^ tangential to that position, the
reasoning of Sec. III. would still have applied, and we should
have obtained the same expressions as in (31) for the pressures
P lt T l) &c., but the numerical value of A would have been
different. According to this method, the pressure II would have
appeared in the equations of motion. It is when the pressures
are measured according to the method which I have adopted that
AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 125
it is true that the equilibrium of the medium would be unstable
if either A or B were negative. I must here mention that from
some oversight the righthand sides of Poisson s equations, at
page 68 of the memoir to which I have referred, are wrong. The
*. A .. . . Tl(d*u d*u
first ot these equations ought to contain I = 2 f j ^
p \dx z dy* dz*.
instead of =5 , and similar changes must be made in the other
p dx
two equations.
It is sometimes brought as an objection to the equations of
motion of the luminiferous ether, that they are the same as those
employed for the motion of solid bodies, and that it seems un
natural to employ the same equations for substances which must
be so differently constituted. It was, perhaps, in consequence
of this objection that Poisson proposes, at page 147 of the memoir
which I have cited, to apply to the calculation of the motion of
the lurniferous ether the same principles, with a certain modifica
tion, as those which he employed in arriving at his equations (9)
page 152, i.e. the equations (12) of this paper. That modification
consists in supposing that a certain function of the time < (t) does
not vary very rapidly compared with the variation of the pressure.
Now the law of the transmission of a motion transversal to the
direction of propagation depending on equations (12) of this paper
is expressed, in the simplest case, by the equation (24) ; and we
see that this law is the same as that of the transmission of heat,
a law extremely different from that of the transmission of vi
bratory motions. It seems therefore unlikely that these principles
are applicable to the calculation of the motion of light, unless
the modification which I have mentioned be so great as wholly
to alter the character of the motion, that is, unless we suppose the
pressure to vary extremely fast compared with the function cj> (t),
whereas in ordinary cases of the motion of fluids the function <f> (t)
is supposed to vary extremely fast compared with the pressure.
Another view of the subject may be taken which I think
deserves notice. Before explaining this view however it will be
necessary to define what I mean in this paragraph by the word
elasticity. There are two distinct kinds of elasticity ; one, that
by which a body which is uniformly compressed tends to
regain its original volume, the other, that by which a body
which is constrained in a manner independent of compres
126 ON THE FRICTION OF FLUIDS IN MOTION,
sion tends to assume its original form. The constants A and
B of Sec. Hi. may be taken as measures of these two kinds
of elasticity. In the present paragraph, the word will be used
to denote the second kind. Now many highly elastic substances,
as iron, copper, &c., are yet to a very sensible degree plastic. The
plasticity of lead is greater than that of iron or copper, and, as
appears from experiment, its elasticity less. On the whole it
is probable that the greater the plasticity of a substance the less
its elasticity, and vice versa, although this rule is probably far
from being without exception. When the plasticity of the sub
stance is still further increased, and its elasticity diminished,
it passes into a viscous fluid. There seems no line of demarcation
between a solid and a viscous fluid. In fact, the practical dis
tinction between these two classes of bodies seems to depend on
the intensity of the extraneous force of gravity, compared with
the intensity of the forces by which the parts of the substance
are held together. Thus, what on the Earth is a soft solid might,
if carried to the Sun, and retained at the same temperature, be
a viscous fluid, the force of gravity at the surface of the Sun
being sufficient to make the substance spread out and become
level at the top : while what on the Earth is a viscous fluid might
on the surface of Pallas be a soft solid. The gradation of viscous,
into what are called perfect fluids seems to present as little ab
ruptness as that of solids into viscous fluids; and some experiments
which have been made on the sudden conversion of water and
ether into vapour, when enclosed in strong vessels and exposed
to high temperatures, go towards breaking down the distinction
between liquids and gases.
According to the law of continuity, then, we should expect
the property of elasticity to run through the whole series, only,
it may become insensible, or else may be masked by some other
more conspicuous property. It must be remembered that the
elasticity here spoken of is that which consists in the tangential
force called into action by a displacement of continuous sliding :
the displacements also which will be spoken of in this paragraph
must be understood of such displacements as are independent
of condensation or rarefaction. Now the distinguishing property
of fluids is the extreme mobility of their parts. According to
the views explained in this article, this mobility is merely an
extremely great plasticity, so that a fluid admits of a finite, but
AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 127
exceedingly small amount of constraint before it will be relieved
from its state of tension by its molecules assuming new positions
of equilibrium. Consequently the same oblique pressures can
be called into action in a fluid as in a solid, provided the amount
of relative displacement of the parts be exceedingly small. All
we know for certain is that the effect of elasticity in fluids,
(elasticity of the second kind be it remembered,) is quite insensible
in cases of equilibrium, and it is probably insensible in all ordinary
cases of fluid motion. Should it be otherwise, equations (8) and
(12) will not be true, or only approximately true. But a little
consideration will shew that the property of elasticity may be
quite insensible in ordinary cases of fluid motion, and may yet
be that on which the phenomena of light entirely depend. When
we find a vibrating string, the small extent of vibration of which
can be actually seen, filling a whole room with sound, and re
member how rapidly the intensity of the vibrations of the air
must diminish as the distance from the string increases*, we may
easily conceive how small in general must be the amount of the
relative motion of adjacent particles of air in the case of sound.
Now the extent of the vibration of the ether, in the case of light,
may be as small compared with the length of a wave of light
as that of the air is compared with the length of a wave of sound :
we have no reason to suppose it otherwise. When we remember
then that the length of a wave of sound in air varies from some
inches to several feet, while the greatest length of a wave of
light is about 00003 of an inch, it is easy to imagine that the
relative displacement of the particles of ether may be so small
as not to reach, nor even come near to the greatest relative dis
placement which could exist without the molecules of the medium
assuming new positions of equilibrium, or, to keep clear of the
idea of molecules, without the medium assuming a new arrange
ment which might be permanent.
It has been supposed by some that air, like the luminiferous
ether, ought to admit of transversal vibrations. According to
the views of this article such would, mathematically speaking,
be the case ; but the extent of such vibrations would be necessarily
so very small as to render them utterly insensible, unless we had
* [In all ordinary cases it is to the vibrations of the soundingboard, or of
the supporting body acting as a soundingboard, and not to those of the string
directly, that the sound is almost wholly due.]
128 ON THE FRICTION OF FLUIDS IN MOTION,
organs with a delicacy equal to that of the retina adapted to
receive them.
It has been shewn to be highly probable that the ratio of A
to B increases rapidly according as the medium considered is
softer and more plastic. For fluids therefore, and among them
for the luminiferous ether, we should expect the ratio of A to B
to be extremely great. Now if N be the velocity of propagation
of normal vibrations in the medium considered in Sect, in., and
T that of transversal vibrations, it may be shewn from equations
(32) that
AT2 _ m A + 4>B B
\ ,
3p p
This is very easily shewn in the simplest case of plane waves : for
if /3 = 7 = 0, a=f(x), the equations (32) give
whence a = <j> (Nt  a) + ty (Nt + a) ; and if a = 7 = 0, /3 =/(#),
A , .. d*/3 .,<?* ,
the same equations give p  = o  , whence
Consequently we should expect to find the ratio of N to T ex
tremely great. This agrees with a conclusion of the late Mr
Green s*. Since the equilibrium of any medium would be
unstable if either A or B were negative, the leas,t possible value
of the ratio of N 2 to T 2 is f, a result at which Mr Green also
arrived. As however it has been shewn to be highly probable
that A>5B even for the hardest solids, while for the softer ones A/B
is much greater than 5, it is probable that N/T is greater than ^3
for the hardest solids, and much greater for the softer ones.
If we suppose that in the luminiferous ether A/B may be con
sidered infinite, the equations of motion admit of a simplification.
For if we put mA ( d ,~ + ^ + ^] =p in equations (32), and
\dx dy dzJ
suppose m^4 to become infinite while p remains finite, the equa
tions become
* Cambridge Philosophical Transactions, Vol. vii. Part I. p. 2.
AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 129
d a dp , fi (d*a <Za d a
=  B
, .
* d0 dy
and j + j + j = 0.
a,c ay as;
When a vibratory motion is propagated in a medium of which
(33) are the equations of motion, it may be shewn that p ^(t)
if the medium be indefinitely extended, or else if there be no
motion at its boundaries. In considering therefore the trans
mission of light in an uninterrupted vacuum the terms involving
p will disappear from equations (33) ; but these terms are, I
believe, important in explaining Diffraction, which is the principal
phenomenon the laws of which depend only on the equations of
motion of the luminiferous ether in vacuum. It will be observed
that putting A = GO comes to the same thing as regarding the
ether as incompressible with respect to those motions which
constitute Light.
ON THE PROOF OF THE PROPOSITION THAT (Mx + Ny)~ l is AN
INTEGRATING FACTOR OF THE HOMOGENEOUS DIFFERENTIAL
EQUATION M + N dyjdx = 0.
[From the Cambridge Mathematical Journal, Vol. iv. p. 241. (May, 1845.)]
A FALLACIOUS proof is sometimes given of this proposition,
which ought to be examined. The substance of the proof is as
follows.
Let us see whether it is possible to find a multiplier V, a
homogeneous function of x and y, which shall render Mdx + Ndy
an exact differential. Let M and N be of n, and V of p dimen
sions; let
&U=V(Mdx + Ndy) ..................... (1);
then, on properly choosing the arbitrary constant in U,} ...
7 will be a homogeneous function of n + p + 1 dimensions,] ^
whence, by a known theorem,
....... (2);
therefore, dividing (1) by (2),
dU = Mdxj^Ndy .
(n +p + 1) U " MX + Ny~
and the first side of this equation being an exact differential, it
follows that the second side is so also, and consequently that
(Mas + Ny}~ 1 is an integrating factor.
Now the factor so found is of n l dimensions ; so that
the first side of (2) is zero. In fact, we shall see that the state
ment (A) is not true as applied to the case in question, unless
MX + Ny = 0.
FACTOR OF HOMOGENEOUS DIFFERENTIAL EQUATION. 131
The general form of a function of x of n dimensions is Ax n .
The general form of a homogeneous function of x and y of n di

. The integral of the first is in general
Ax n ^j(n + 1),. omitting the arbitrary constant; and consequently
the dimensions of the function are increased by unity by inte
gration. But in the particular case in which n = 1, the integral
is A logx, which is not a quantity of dimensions, at least accord
ing to the definition just given, according to which definition only
is the proposition with reference to homogeneous functions as
sumed in (2) true. Let us now examine in what cases U will be
of n \p +1 dimensions.
Putting M=M Q x n , N=N^x n , y = xz, M Q and N will be func
tions of z alone, and we shall have
Mdx + Ndy = x n {(M Q + JV dx + N x dz}.
If M Q + N Q z = 0, i.e. if MX + Ny = 0, we see that af*" 1 will be
an integrating factor. The integral, being a function of z, will
be of dimensions, and both sides of (2) will be zero.
If MX + Ny is not equal to 0, we may multiply and divide by
(M f N Q z) x, and we have
Hence we see that {x n+1 (M + N Q z)}~* or (Mx + Ny}
integrating factor. For this factor we have
(f> denoting the function arising from the integration with respect
to*.
dU dU
In this case we have x j \ y = = 1, not = 0.
dx y dy
It may be of some interest to enquire in what cases an exact
differential of any number of independent variables, in which
the differential coefficients are homogeneous functions of n dimen
sions, has an integral which is a homogeneous function of n + 1
dimensions.
92
132 INTEGRATING FACTOR OF
Let d U=Mdx \Ndy + Pdz f ... be the exact differential. Let
y = yx, z = z x . . . , M=M x*, N =N x n . . . , so that M 0) N Q . . . are
functions of y , z . . . only ; then
= x n {(M Q + N,y + P/. . .) dx + (N Q dy + Pdz . . .) x}.
First, suppose the coefficient of dx in this equation to be zero,
or Mx + Ny + Pz ... =0; then the expression for dU cannot be
an exact differential unless n 1. In this case U will be a
function of y, z ..., and will therefore be a homogeneous function
of n + 1 or dimensions.
Secondly, suppose the coefficient of dx not to be zero ; then
Now I say that jf^ > p / is the exact differential of
a function of the independent variables y , z ..., or, taking y, z ...
for the independent variables instead of y, z ..., x being supposed
constant, and putting for , N ,... their values, that
Mx + Ny + Pz ...
is an exact differential.
For, putting MX + Ny + Pz ... = D, in order that the quantity
considered should be an exact differential, it is necessary and
sufficient that the system of equations of which the type is
j J P
d D d D
 =  should be satisfied. This equation gives
dz dy
D (^p\ +P f N f^.
\dz dy) dy dz
Now, since dN/dz = dP/dy, by the conditions of M dx 4 Ndy
\Pdz .... being an exact differential, the above equation becomes
HOMOGENEOUS DIFFERENTIAL EQUATION. 133
Keplacing dM/dy, dP/dy ... by dN/dx, dN/dz... and dM/dz,
dN/dz ... by dP/dx, dP/dy . . ., this equation becomes
dN dN dN \ ^fdP dP dP
l x+^r y + Tz...}N(j~x
dx dy 9 d* J \dx
dN dN
Now up* +&* +
dP dP
and therefore the above equation is satisfied. Hence
z... orits
..>
is an exact differential dty(y,z ...}. Consequently equation (3)
becomes
which equation being by hypothesis integrable, it follows that
and Mx+Ny... being moreover a homogeneous function of
dimensions, it is clear that we must have </> (a) = Ae^ +l)a . Hence
we have
dU = Ax n+l e<+* d (log x + ^r).
If now 7i + 1 is not equal to 0, we have
omitting the constant ; but if n = 1, we have
We see then that if Mx + Ny + Pz... =0, which can only
happen when w = 1, U will be a homogeneous function of n + 1
or dimensions. If Mx + Ny + Pz ...... is not equal to 0, then,
if n + 1 is not equal to 0, and the constant in V is properly chosen,
U will be a homogeneous function of n + 1 dimensions, but if
n + l =0, 7 will not be a homogeneous function of dimensions,
but will contain log x. Of course it might equally have contained
the logarithm of y or z t &c.; in fact,
z ...
[From the Philosophical Magazine, Vol. xxvii. p. 9. (July, 1845.)]
ON THE ABEKRATION OF LIGHT.
THE general explanation of the phenomenon of aberration is
so simple, and the coincidence of the value of the velocity of
light thence deduced with that derived from the observations of
the eclipses of Jupiter s satellites so remarkable, as to leave no
doubt on the mind as to the truth of the explanation. But when
we examine the cause of the phenomenon more closely, it is far
from being so simple as it appears at first sight. On the theory
of emissions, indeed, there is little difficulty ; and it would seem
that the more particular explanation of the cause of aberration
usually given, which depends on the consideration of the motion
of a telescope as light passes from its objectglass to its cross
wires, has reference especially to this theory ; for it does not apply
to the theory of undulations, unless we make the rather startling
hypothesis that the luminiferous ether passes freely through the
sides of the telescope and through the earth itsetf. The undu
latory theory of light, however, explains so simply and so beauti
fully the most complicated phenomena, that we are naturally led
to regard aberration as a phenomenon unexplained by it, but not
incompatible with it.
The object of the present communication is to attempt an
explanation of the cause of aberration which shall be in accordance
with the theory of undulations. I shall suppose that the earth
and the planets carry a portion of the ether along with them so
that the ether close to their surfaces is at rest relatively to those
surfaces, while, its velocity alters as we recede from the surface,
till, at no great distance, it is at rest in space. According to the
undulatory theory, the direction in which a heavenly body is seen
ON THE ABERRATION OF LIGHT. 135
is normal to the fronts of the waves which have emanated from
it, and have reached the neighbourhood of the observer, the ether
near him being supposed to be at rest relatively to him. If
the ether in space were at rest, the front of a wave of light at any
instant being given, its front at any future time could be found
by the method explained in Airy s tracts. If the ether were in
motion, and the velocity of propagation of light were infinitely
small, the wave s front would be displaced as a surface of parti
cles of the ether. Neither of these suppositions is however true,
for the ether moves while light is propagated through it. In the
following investigation I suppose that the displacements of a
wave s front in an elementary portion of time due to the two
causes just considered take place independently.
Let u, v, w be the resolved parts along the rectangular axes of
x, y, z, of the velocity of the particle of ether whose coordinates
are x, y, z, and let V be the velocity of light supposing the ether
at rest. In consequence of the distance of the heavenly bodies, it
will be quite unnecessary to consider any waves except those which
are plane, except in so far as they are distorted by the motion of
the ether. Let the axis of z be taken in, or nearly in the direction
of propagation of the wave considered, so that the equation of
a wave s front at any time will be
z = c+Vt + t; ........................... (i),
C being a constant, t the time, and f . a small quantity, a function
of x, y and t. Since u, v, w and f are of the order of the aberra
tion, their squares and products may be neglected.
Denoting by a, 0, 7 the angles which the normal to the wave s
front at the point (x, y, z) makes with the axes, we have, to the
first order of approximation,
~
cosa = ~ , co$p = ~, 0087 = 1 ............... (2);
and if we take a length Vdt along this normal, the coordinates
of its extremity will be
i,_KJt yvdt, z+Vdt.
dx dy
If the ether were at rest, the locus of these extremities would be
the wave s front at the time t + dt, but since it is in motion, the
136 ON THE ABERRATION OF LIGHT.
coordinates of those extremities must be further increased by udt,
vdt, wdt. Denoting then by x, y, z the coordinates of the point
of the wave s front at the time t + dt which corresponds to the
point (x, y, z] at the time t, we have
z = z f (w 4 V) dt ;
and eliminating x t y and z from these equations and (1), and de
noting % by f(x, y, t), we have for the equation to the wave s front
at the time t + dt,
C + Vt
or, expanding, neglecting dt 2 and the square of the aberration, and
suppressing the accents of x, y and z >
z=C+Vt + Z+(w+V)dt .................. (3).
But from the definition of it follows that the equation to the
wave s front at the time t + dt will be got from (1) by putting
t + dt for t, and we have therefore for this equation
Comparing the identical equations (3) and (4), we have
d
_ ? /i/i
dt"
This equation gives f = [?$; but in the small term f we may
replace I wdt by \wdz+ V\ this comes to taking the approximate
value of z given by the equation z = 0+ Ftf instead of * for the
parameter of the system of surfaces formed by the wave s front in
its successive positions. Hence equation (1) becomes
Combining the value of f just found with equations (2), we
get, to a first approximation,
ON THE ABERRATION OF LIGHT. 137
equations which might very easily be proved directly in a more
geometrical manner.
If random values are assigned to u, v and w, the law of aber
ration resulting from these equations will be a complicated one;
but if u, v and w are such that udx + vdy + wdz is an exact dif
ferential, we have,
dw _ du dw _dv t
dx dz dy dz
whence, denoting by the suffixes 1, 2 the values of the variables
belonging to the first and second limits respectively, we obtain
If the motion of the ether be such that udx + vdy + wdz is an
exact differential for one system of rectangular axes, it is easy to
prove, by the transformation of coordinates, that it is an exact
differential for any other system. Hence the formulae (6) will
hold good, not merely for light propagated in the direction first
considered, but for light propagated in any direction, the direc
tion of propagation being taken in each case for the axis of jg. If
we assume that udx + vdy + wdz is an exact differential for that
part of the motion of the ether which is due to the motion of
translation of the earth and planets, it does not therefore follow
that the same is true for that part which depends on their motions
of rotation. Moreover, the diurnal aberration is too small to be
detected by observation, or at least to be measured with any ac
curacy, and I shall therefore neglect it.
It is not difficult to shew that the formulas (6) lead to the
known law of aberration. In applying them to the case of a star,
if we begin the integrations in equations (5) at a point situated
at such a distance from the earth that the motion of the ether,
and consequently the resulting change in the direction of the
light, is insensible, we shall have u t = 0, v t = ; and if, moreover,
we take the plane xz to pass through the direction of the earth s
motion, we shall have
, u
and ~ a 
138 ON THE ABERRATION OF LIGHT.
that is, the star will appear displaced towards the direction in
which the earth is moving, through an angle equal to the ratio of
the velocity of the earth to that of light, multiplied by the sine of
the angle between the direction of the earth s motion and the line
joining the earth and the star.
ADDITIONAL NOTE.
[In what precedes waves of light are alone considered, and the
course of a ray is not investigated, the investigation not being
required. There follows in the original paper an investigation
having for object to shew that in the case of a body like the
moon or a planet which is itself in motion, the effect of the dis
tortion of the waves in the neighbourhood of the body in altering
the apparent place of the body as determined by observation is
insensible. For this, the orthogonal trajectory of the wave in its
successive positions from the body to the observer is considered,
a trajectory which in its main part will be a straight line, from
which it will not differ except in the immediate neighbourhood of
the body and of the earth, where the ether is distorted by their
respective motions. The perpendicular distance of the further
extremity of the trajectory from the prolongation of the straight
line which it forms in the intervening quiescent ether is shewn to
subtend at the earth an angle which, though not actually 0, is so
small that it may be disregarded.
The orthogonal trajectory of a wave in its successive positions
does not however represent the course of a ray, as it would do if
the ether were at rest. Some remarks made by Professor Challis
in the course of discussion suggested to me the examination of
the path of a ray, which in the case in which udx + vdy + wdz
is an exact differential proved to be a straight line, a result which
I had not foreseen when I wrote the above paper, which I may
mention was read before the Cambridge Philosophical Society on
the 18th of May, 1845 (see Philosophical Magazine, vol. XXIX.,
p. 62). The rectilinearity of the path of a ray in this case, though
not expressly mentioned by Professor Challis, is virtually con
tained in what he wrote. The problem is rather simplified by
introducing the consideration of rays, and may be treated from
the beginning in the following manner.
ON THE ABERRATION OF LIGHT. 139
The notation in other respects being as before, let a , /3 be the
small angles by which the direction of the wavenormal at the
point (xy y, z) deviates from that of Oz towards Ox, Oy, respec
tively, so that a , ft are the complements of a, /3, and let a,, ft / be
the inclinations to Oz of the course of a ray at the same point.
By compounding the velocity of propagation through the ether
with the velocity of the ether we easily see that
Let us now trace the changes of a /t @, during the time dt.
These depend first on the changes of a , /3 , and secondly on those
of u, v.
As regards the change in the direction of the wavenormal, we
notice that the seat of a small element of the wave in its suc
cessive positions is in a succession of planes of particles nearly
parallel to the plane of x, y. Consequently the direction of the
element of the wave will be altered during the time dt by the
motion of the ether as much as a plane of particles of the ether
parallel to the plane of the wave, or, which is the same to the
order of small quantities retained, parallel to the plane xy. Now
if we consider a particle of ether at the time t having for co
ordinates x, y, z, another at a distance dx parallel to the axis
of x y and a third at a distance dy parallel to the axis of y, we see
that the displacements of these three particles parallel to the axis
of z during the time dt will be
7 / dw .. / dw
wdt,
and dividing the relative displacements by the relation distances,
we have dw/dx. dt, dw/dy . dt for the small angles by which the
normal is displaced, in the planes of xz t yz t from the axes x t y, so
that
dw ., 7/v dw 7 ,
dy. = dt, dp=jdt.
dx dy
We have seen already that the changes of u, v are dujdz . Vdt,
dv/dz . Vdt, so that
dv
140 ON THE ABERRATION OF LIGHT.
Hence, provided the motion of the ether be such that
udx + vdy + wdz
is an exact differential, the change of direction of a ray as it
travels along is nil, and therefore the course of a ray is a straight
line notwithstanding the motion of the ether. The rectilinearity
of propagation of a ray of light, which a priori would seem very
likely to be interfered with by the motion of the ether produced by
the earth or heavenly body moving through it, is the tacit as
sumption made in the explanation of aberration given in treatises
of Astronomy, and provided that be accounted for the rest follows
as usual*. It follows further that the angle subtended at the
earth by the perpendicular distance of the point where a ray leaves
a heavenly body from the straight line prolonged which represents
its course through the intervening quiescent ether, is not merely
too small to be observed, but actually nil.]
* To make this explanation quite complete, we should properly, as Professor
Challis remarks, consider the light coming from the wires of the observing telescope,
in company with the light from the heavenly body.
[From the Philosophical Magazine, Vol. xxvm. p. 76. (Feb. 1846.)]
ON FRESNEL S THEORY OF THE ABERRATION OF LIGHT.
THE theory of the aberration of light, and of the absence of
any influence of the motion of the earth on the laws of refraction,
&c., given by Fresnel in the ninth volume of the Annales de
Chimie, p. 57, is really very remarkable. If we suppose the
diminished velocity of propagation of light within refracting media
to arise solely from the greater density of the ether within them,
the elastic force being the same as without, the density which it
is necessary to suppose the ether within a medium of refractive
index //, to have is yu, 2 , the density in vacuum being taken for unity.
Fresnel supposes that the earth passes through the ether without
disturbing it, the ether penetrating the earth quite freely. He
supposes that a refracting medium moving with the earth carries
with it a quantity of ether, of density yu, 2 1, which constitutes the
excess of density of the ether within it over the density of the
ether in vacuum. He supposes that light is propagated through
this ether, of which part is moving with the earth, and part is
at rest in space, as it would be if the whole were moving with the
velocity of the centre of gravity of any portion of it, that is, with
a velocity (1 /T 2 ) v, v being the velocity of the earth. It may
be observed however that the result would be the same if we
supposed the whole of the ether within the earth to move to
gether, the ether entering the earth in front, and being im
mediately condensed, and issuing from it behind, where it is
immediately rarefied, undergoing likewise sudden condensation or
rarefaction in passing from one refracting medium to another.
On this supposition, the evident condition that a mass v of the
ether must pass in a unit of time across a plane of area unity,
142 ON FRESNEL S THEORY OF THE ABERRATION OF LIGHT.
drawn anywhere within the earth in a direction perpendicular
to that of the earth s motion, gives (1 /^~ 2 ) v for the velocity
of the ether within a refracting medium. As this idea is rather
simpler than Fresnel s, I shall adopt it in considering his theory.
Also, instead of considering the earth as in motion and the ether
outside it as at rest, it will be simpler to conceive a velocity equal
and opposite to that of the earth impressed both on the earth and
on the ether. On this supposition the earth will be at rest ; the
ether outside it will be moving with a velocity v, and the ether
in a refracting medium with a velocity v//j?, in a direction contrary
to that of the earth s real motion. On account of the smallness of
the coefficient of aberration, we may also neglect the square of
the ratio of the earth s velocity to that of light ; and if we resolve
the earth s velocity in different directions, we may consider the
effect of each resolved part separately.
In the ninth volume of the Comptes Rendus of the Academy
of Sciences, p. 774, there is a short notice of a memoir by M.
Babinet, giving an account of an experiment which seemed to
present a difficulty in its explanation. M. Babinet found that
when two pieces of glass of equal thickness were placed across
two streams of light which interfered and exhibited fringes, in
such a manner that one piece was traversed by the light in the
direction of the earth s motion, and the other in the contrary
direction, the fringes were not in the least displaced. This result,
as M.. Babinet asserts, is contrary to the theory of aberration
contained in a memoir read by him before the Academy in 1829,
as well as to the other received theories on the subject. I have
not been able to meet with this memoir, but it is easy to shew
that the result of M. Babinet s experiment is in perfect accordance
with Fresnel s theory.
Let T be the thickness of one of the glass plates, V the ve
locity of propagation of light in vacuum, supposing the ether
at rest. Then V/p would be the velocity with which light would
traverse the glass if the ether were at rest; but the ether
moving with a velocity v/fjf, the light traverses the glass with a
velocity  + o , and therefore in a time
J ~
ON FRESNEL S THEORY OF THE ABERRATION OF LIGHT. 143
But if the glass were away, the light, travelling with a velocity
V v, would pass over the space T in the time
T
Hence the retardation, expressed in time, =(/u, 1) ^, the same
as if the earth were at rest. But in this case no effect would be
produced on the fringes, and therefore none will he produced in
the actual case.
I shall now shew that, according to Fresnel s theory, the laws
of reflexion and refraction in singly refracting media are un
influenced by the motion of the earth. The method which I
employ will, I hope, be found simpler than Fresnel s ; besides
it applies easily to the most general case. Fresnel has not given
the calculation for reflexion, but has merely stated the result;
and with respect to refraction, he has only considered the case
in which the course of the light within the refracting medium
is in the direction of the earth s motion. This might still leave
some doubt on the mind, as to whether the result would be the
same in the most general case.
If the ether were at rest, the direction of light would be that
of a normal to the surfaces of the waves. When the motion
of the ether is considered, it is most convenient to define the
direction of light to be that of the line along which the same
portion of a wave moves relatively to the earth. For this is in
all cases the direction which is ultimately observed with a tele
scope furnished with cross wires. Hence, if A is any point in
a wave of light, and if we draw AB normal to the wave, and
proportional to V or V/JJL, according as the light is passing through
vacuum or through a refracting medium, and if we draw EG in
the direction of the motion of the ether, and proportional to
v or v/fjf, and join AC, this line will give the direction of the ray.
Of course, we might equally have drawn AD equal and parallel to
BC and in the opposite direction, when DB would have given the
direction of the ray.
Let a plane P be drawn perpendicular to the reflecting or
refracting surface and to the waves of incident light, which in this
investigation may be supposed plane. Let the velocity v of the
ether in vacuum be resolved into p perpendicular to the plane P,
14t ON FRESNEL S THEORY OF THE ABERRATION OF LIGHT.
and q in that plane ; then the resolved parts of the velocity v/fj?
of the ether within a refracting medium will be pip?, y/f^ Let
us first consider the effect of the velocity p.
It is easy to see that, as far as regards this resolved part of
the velocity of the ether, the directions of the refracted and
reflected waves will be the same as if the ether were at rest.
Let BAG (fig. 1) be the intersection of the refracting surface
and the plane P\ DAE a normal to the refracting surface; AF,
A Gr, AH normals to the incident, reflected and refracted waves.
Hence AF, AG, AH will be in the plane P, and
^ GAD = FAD, p sin HAE = sin FAD.
Take
AH=AF.
Draw Gg> Hh perpendicular to the plane P, and iri the direction
of the resolved part p of the velocity of the ether, and Ff in the
opposite direction ; and take
Ff : Hh : FA :: p : 4 : V, and Gg = Ff,
and join A with / g and h. Then fA t Ag, Ah will be the di
rections of the incident, reflected and refracted rays. Draw FD,
HE perpendicular to DE, and join/D, hE. ThenfDF, hEH will
be the inclinations of the planes fAD t hAE to the plane P.
Now
tan
and wa.FADii.wa.HAE; therefore tavFDf=ianHEh, and
ON FRESNEL S THEORY OF THE ABERRATION OF LIGHT. 145
therefore the refracted ray A h lies in the plane of incidence
fAD. It is easy to see that the same is true of the reflected ray
Ag. Also t gAD =/AD; and the angles fAD, hAE are sensibly
equal to FAD, HAE respectively, and we therefore have without
sensible error, sin fAD = /j,smhAE. Hence the laws of reflexion
and refraction are not sensibly affected by the velocity p.
Let us now consider the effect of the velocity q. As far as
depends on this velocity, the incident, reflected and refracted
rays will all be in the plane P. Let AH, AK, AL be the in
tersections of the plane P with the incident, reflected and refracted
waves. Let ty, ^, >r be the inclinations of these waves to the
refracting surface ; let NA. be the direction of the resolved part
q of the velocity of the ether, and let the angle NA C = a.
The resolved part of q in a direction perpendicular to AH
is (7 sin (\r } a). Hence the wave AH travels with the velocity
Ft q sin (^r + a) ; and consequently the line of its intersection
with the refracting surface travels along AB with the velocity
coseCA/r [V+ q sin (^ + a)}. Observing that q/fj? is the velocitv
of the ether within the refracting medium, and V/fju the velocity
of propagation of light, we shall find in a similar manner that
the lines of intersection of the refracting surface with the reflected
and refracted waves travel along AB with velocities
coseci/rj V+ q sin (^ a.)}, cosec
sn
But since the incident, reflected and refracted waves intersect
the refracting surface in the same line, we must have
sin^ { V+q sin (^ + a)} = sin i/r {F + q sin (^ a)}
sin ^ { F+ q sin (\fr + a) }
sn
F+  sin
a)
...(A)
10
146 ON FRESNEL S THEORY OF THE ABERRATION OF LIGHT.
Draw HS perpendicular to AH, ST parallel to NA, take
ST : HS : : q : V, and join HT. Then HT is the direction of
the incident ray; and denoting the angles of incidence, reflexion
and refraction by <, <,, </> , we have
(f)ty = SIIT = J*? = y x resolved part of q along AH
Similarly,
whence
sin ^ = sin ^> ^cos <^> cos (</> + a),
sin 1^= sin (j> cos ^ cos (<, a),
sin ^r = sin < cos $ cos (</> + a).
On substituting these values in equations (A), and observing
that in the terms multiplied by q we may put </=<> p sin?<jb = sin0,
the small terms destroy each other, and we have sin $, = sin <,
yu,sin (j) =siia.(f). Hence the laws of reflexion and refraction at
the surface of a refracting medium will not be affected by the
motion of the ether.
In the preceding investigation it has been supposed that the
refraction is out of vacuum into a refracting medium. But the
result is the same in the general case of refraction out of one
medium into another, and reflexion at the common surface. For
all the preceding reasoning applies to this case if we merely
substitute p/p z , q/p* for p t q, V/p for V, and p/p for M, fi being
the refractive index of the first medium. Of course refraction
out of a medium into vacuum is included as a particular case.
It follows from the theory just explained, that the light coming
from any star will behave in all cases of reflexion and ordinary
refraction precisely as it would if the star were situated in the
place which it appears to occupy in consequence of aberration,
and the earth were at rest. It is, of course, immaterial whether
the star is observed with an ordinary telescope, or with a telescope
having its tube filled with fluid. It follows also that terrestrial
ON FRESNEL S THEORY OF THE ABERRATION OF LIGHT. 147
objects are referred to their true places. All these results would
follow immediately from the theory of aberration which I pro
posed in the July number of this Magazine ; nor have I been able
to obtain any result, admitting of being compared with experi
ment, which would be different according to which theory we
adopted. This affords a curious instance of two totally different
theories running parallel to each other in the explanation of phe
nomena. I do not suppose that many would be disposed to main
tain Fresnel s theory, when it is shewn that it may be dispensed
with, inasmuch as we would not be disposed to believe, without
good evidence, that the ether moved quite freely through the solid
mass of the earth. Still it would have been satisfactory, if it had
been possible, to have put the two theories to the test of some
decisive experiment.
102
[From the Cambridge and Dublin Mathematical Journal,
Vol. I. p. 183 (May, 1846).]
ON A FOKMULA FOR DETERMINING THE OPTICAL CONSTANTS
OF DOUBLY REFRACTING CRYSTALS.
IN order to explain the object of this formula, it will be neces
sary to allude to the common method of determining the optical
constants. Two plane faces of the crystal are selected, which
are parallel to one of the axes of elasticity; or if such do not
present themselves, they are obtained artificially by grinding.
A pencil of light is transmitted across these faces in a plane per
pendicular to them both, as in the case of an ordinary prism.
This pencil is by refraction separated into two, of which one is
polarized in the plane of incidence, and follows the ordinary law
of refraction, while the other is polarized in a plane perpendicular
to the plane of incidence, and follows a different law. It will
be convenient to call these pencils respectively the ordinary and
the extraordinary, in the case of biaxal, as well as uniaxal crystals.
The minimum deviation of the ordinary pencil is tnen observed,
and one of the optical constants, namely that which relates to
the axis of elasticity parallel to the refracting edge, is thus de
termined by the same formula which applies to ordinary media.
This formula will also give one of the other constants, by means
of the observation of the minimum deviation of the extraordinary
pencil, in the particular case in which one of the principal planes
of the crystal bisects the angle between the refracting planes :
but if this condition be not fulfilled it will be necessary to employ
either two or three prisms, according as the crystal is uniaxal
or biaxal, to determine all the constants. The extraordinary
pencil, however, need not in any case be rejected, provided only a
formula be obtained connecting the minimum deviation observed
ON A FORMULA FOR DETERMINING THE OPTICAL, ETC. 149
with the optical constants. It will thus be possible to determine
all the constants with a smaller number of prisms ; the necessity
of using artificial faces may often be obviated ; or if two faces
are cut as nearly as may be equally inclined to one of the axes of
elasticity lying in the plane of incidence, or one cut face is used
with a natural face, the errors of cutting may be allowed for.
Let AEB be a section of the prism by the plane of refraction,
(the reader will have no difficulty in drawing a figure,) E being
the refracting edge; let i be the refracting angle; OA, OB, OG
the directions of the axes of elasticity, being any point within
the prism, the two former of these lines being in, and the latter
perpendicular to, the plane of refraction ; a, b, c the optical con
stants referring to them, that is, according to Fresnel s theory,
the velocities of propagation of waves in which the vibrations
are parallel to the three axes respectively. Everything being
symmetrical with respect to the plane of incidence, we need only
consider what takes place in that plane. This plane will cut
the wave surface in a circle of radius c, and an ellipse whose
semiaxes are a along OB and b along OA. We have only got to
consider the ellipse, since it is it that determines the direction
of the extraordinary ray. The form of the crystal will very often
make known the directions of the axes of elasticity. Supposing
these directions known, let a, ft denote the inclinations of OA, OB
to the produced parts of EA, EB respectively ; a, /? and i being
of course connected by the equation a. + /3 = JTT + i.
Let 0, TT be the angles of incidence and emergence, the light
being supposed incident on the face EA ; $ the inclination of the
refracted wave to EA, ty its inclination to EB, D the deviation,
v the velocity of the wave within the crystal, u its velocity in
the outer medium, which may be supposed to be either air, or a
liquid of known refractive power. Then we have
D = <j> + ^ri* (1),
* + * = (2),
v sin $ = wsin </> (3),
vsin ty u sin^r (4),
v* = a? cos 2 (a </> )+ 6 2 sin 2 (a  < ) (5).
* I am indebted to the Rev. P. Frost for the suggestion of employing equations
(1)...(4), rather than making use of the ellipse in which the wave surface is cut by
the plane of incidence.
150 ON A FORMULA FOR DETERMINING THE
From (2), (3), (4),
u sin ijr = v sin \Jr = u sin (i ( ) = u sin i cos $ v cos i* sin <> ;
. . cos d> f = = . (sin ilr 4 cos i sin d>) :
u sin ^ x
, . ,, v . . . ,
and sin cf> = : . sin i sin 6 :
w sin i
substituting in (5),
w 2 sin 2 i = a 2 {cos a (sin ty 4 cos t sin </>) 4 sin a sin i sin </>) 2
.4 & 2 (sin a (sin ir 4 cos i sin 0) cos a sin 4 sin </>) 2 ,
or w* sin 2 * = a 2 (cos a sin ir 4 sin /? sin (/>) 2
4 & 2 (sin a sin ty 4 cos ^ sin (/)) 2 ...................... (6),
the relation between < and ^. Putting ty $ 0, and taking
account of (1), (6) becomes
2w* sin 8 i = [a 2 cos 2 a 4 6 2 sin 2 a} (1  cos (D + {+0)}
4 {a 2 sin 2 /3 4 6 2 cos 2 ] {1  cos (D 4 *  6>)}
4 2 (a 2 cos a sin y8 4 6 2 sin a cos /3) {cos ^ cos (D + i)},
or .Fcos04sin04# = .................. (7),
Avhere
F= a? {(cos* a 4 sin 2 /3) cos (D 4 i)  2 cos a sin /5}
4 6 2 {(sin 2 a 4 cos 2 j3) cos (D 4 )  2 sin a cos /5},
(7 = ( a _ &) ( S i n 2 ^ _ C os 2 a ) sin (D 4 i),
JI= 2w* sin 2 i a 2 {cos 2 a 4 sin 2 ft 2 cos a sin $ cos (Z> 4 i)}
b 2 {sin 2 a 4 cos 2 /3 2 sin a cos /3 cos (D 4 )}.
Now when D, regarded as a function of 0, is a maximum or mini
mum ^ = 0, whence from (7)
do
and eliminating 6 from this equation and (7), we have
Putting for F t G and H their values, and reducing, this equation
becomes
bin 2 (D 4 a*V  {cos 2 a 4 sin 2 /3  2 cos (D 4 cos a sin /3} wV
 {sin 2 a 4 cos 2  2 cos (D 4 1) sin a cos /3J iftf 4 sin 2 . %* = . . . (8).
This equation will be rendered more convenient for numerical
calculation by replacing products and powers of sines and cosines
OPTICAL CONSTANTS OF DOUBLY REFRACTING CRYSTALS. 151
by sums and differences. Treated in this manner, the equation
becomes
versin 2 (D + i) cfb 2 (A+B) u z c?  (A  B] iftf
+ versin 2i.w 4 = 0... (9),
where A = versin D + versin (D + 2i),
B = cos 2a  cos 2/3  cos (D + 2a) + cos (D + 2/3).
If the principal plane A OC of the crystal bisects the angle
between the refracting faces, we have
i Q 7T i
a = 2 P^Z+Z*
whence from (8), putting D + i = A,
(9 . 9 A 9 oA/79 9 A o 9 \
a 2 sm 2 g tr BUT ~ J f D cos 2 ^  u 2 cos 2 ^ ) = 0.
The former of these factors is evidently that which corresponds to
the problem ; the latter corresponds to refraction through a prism
having its faces parallel to those of the actual prism, and having
its refracting angle supplemental to /. We have therefore
. i
Sm 2
so that the constant a is given by the same formula that applies to
ordinary media, as it should.
If the refracting faces are perpendicular to the axes of elas
ticity which lie in the plane of incidence, the formula (8) or (9)
takes a very simple form. In this case we have a=/3 = i=%7r,
and therefore
cos 2 D . ct 2 6 2  wV tftf + w 4 = 0.
Mathematically speaking, one prism would be sufficient for
determining the three constants a, 6, c. For c would be deter
mined by means of the ordinary pencil; and by observing the
extraordinary pencil with the crystal in air, and again with the
crystal in some liquid, we should have two equations of the form
(8), by combining which we should obtain a 2 and 6 2 by the
solution of a quadratic equation. But since a is usually nearly
equal to 6, it is evident that the course of the extraordinary ray
within the crystal would be nearly the same in the two observa
152 ON A FORMULA FOR DETERMINING THE OPTICAL, ETC.
tions, being in each case inclined at nearly equal angles to the
refracting faces, and consequently the errors of observation would
be greatly multiplied in the result. Even if a differed greatly
from b, only one of these constants could be accurately determined
in this manner if the refracting angle were nearly bisected by
a principal plane. But two prisms properly chosen appear amply
sufficient for determining accurately the three constants by the
method of minimum deviations, even should neither prism have
its angle exactly bisected by a principal plane of the crystal.
It is not necessary to observe the deviation when it is a
minimum, as Professor Miller has remarked to me, since the angle
of incidence may be measured very accurately by moving the
telescope employed till the luminous slit, seen directly, appears
on the cross wires, and then turning it till the slit, seen by re
flection at the first face of the prism, again appears on the cross
wires, the prism meanwhile remaining fixed*. The angle through
which the telescope has been turned is evidently the supplement
of twice the angle of incidence. If this method of observation be
adopted, <, D, and i will be known by observation, whence ^
will be got immediately from (1). Thus all the coefficients in
(6) will be known quantities, and this equation furnishes a very
simple relation between a and b. The coefficients may easily be
calculated numerically by treating them like those in equation
(8), or else by employing subsidiary angles.
[* A method of measuring the refractive indices of isotropic media depending on
the measurement of the deviation and angle of incidence is described by Professor
Swan in the Edinburgh New Philos( phical Journal , Vol. xxxvi. (1844) p. 102.]
[From the Philosophical Magazine^ Vol. xxix. p. 6 (July, 1846)].
ON THE CONSTITUTION OF LUMINIFEROUS ETHER, VIEWED WITH
REFERENCE TO THE PHENOMENON OF THE ABERRATION OF
LIGHT.
IN a former communication to this Magazine (July, 1845),*
I shewed that the phenomenon of aberration might be explained
on the undulatory theory of light, without making the startling
supposition that the earth in its motion round the sun offers
no resistance to the ether. It appeared that the phenomenon
was fully accounted for, provided we supposed the motion of the
ether such as to make
udx + vdy + wdz (a)
an exact differential, Avhere u, v, w are the resolved parts, along
three rectangular axes, of the velocity of the particle of ether
whose coordinates are x, y, z. It appeared moreover that it
was necessary to make this supposition in order to account in
this way for the phenomenon of aberration. I did not in that
paper enter into any speculations as to the physical causes in
consequence of which (a) might be an exact differential. The
object of the present communication is to consider this question.
The enquiry naturally divides itself into two parts : First,
In what manner does one portion of ether act on another be
yond the limits of the earth s atmosphere ? Secondly, What
takes place in consequence of the mutual action of the air and
the ether ?
In order to separate these two questions, let us first conceive
the earth to be destitute of an atmosphere. Before considering
the motion of the earth and the ether, let us take the case of
* Ante, p. 134.
154 ON THE CONSTITUTION OF LUMINIFEROUS ETHER.
a solid moving in an ordinary incompressible fluid, which may
be supposed to be infinitely extended in all directions about the
solid. If we suppose the solid and fluid to be at first at rest,
and the solid to be then moved in any manner, it follows from
the three first integrals of the ordinary equations of fluid motion,
obtained by M. Cauchy, that the motion of the fluid at any
time will be such that (a) is an exact differential. From this
it may be easily proved, that if at any instant the solid be re
duced to rest, the whole of the fluid will be reduced to rest
likewise ; and that the motion of the fluid is the same as it would
have been if the solid had received by direct impact the motion
which it has at that instant. Practically however the motion
of the fluid after some time would differ widely from what would
be thus obtained, at least if the motion of the solid be progressive
and not oscillatory. This appears to be due to two causes : first,
the motion considered would probably be unstable in the part
of the fluid behind the solid; and secondly, a tangential force
is called into play by the sliding of one portion of fluid along
another ; and this force is altogether neglected in the common
equations of hydrodynamics, from which equations the motion
considered is deduced. If, instead of supposing the solid to
move continuously, we supposed it first to be in motion for a
very small interval of time, then to be at rest for another equal
interval, then to be in motion for a third interval equal to the
former, and so on alternately, theoretically the fluid ought to
be at rest at the expiration of the first, third, &c. intervals, but
practically a very slight motion would remain at the end of the
first interval, would last through the second and third, and would
be combined with a slight motion of the same kind, which would
have been left at the end of the third interval, even if the fluid
immediately before the commencement of it had been at rest ;
and the accumulation of these small motions would soon become
sensible.
Let us now return to the ether. We know that the trans
versal vibrations constituting light are propagated with a velocity
about 10,000 times as great as the velocity of the earth; and
Mr Green has shewn that the velocity of propagation of normal
vibrations is in all probability incomparably greater than that
of transversal vibrations (Cambridge Philosophical Transactions,
vol. VII. p. 2). Consequently, in considering the motion of the
ON THE CONSTITUTION OF LUMINIFEROUS ETHER. 155
ether due to the motion of the earth, we may regard the ether
as perfectly incompressible. To explain dynamically the pheno
mena of light, it seems necessary to suppose the motion of the
ether subject to the same laws as the motion of an elastic solid.
If the views which I have explained at the end of a paper On
the Friction of Fluids, &c. (Cambridge Philosophical Transactions,
vol. viii. part 8)* be correct, it is only for extremely small vi
bratory motions that this is the case, while if the motion be
progressive, or not very small, the ether will behave like an
ordinary fluid. According to these views, therefore, the earth
will set the ether in motion in the same way as a solid would
set an ordinary incompressible fluid in motion.
Instead of supposing the earth to move continuously, let us
first suppose it to move discontinuously, in the same manner
as the solid considered above, the ether being at rest just before
the commencement of the first small interval of time. By what
precedes, the ether will move during the first interval in the
same, or nearly the same, manner as an incompressible fluid
would ; and when, at the end of this interval, the earth is reduced
to rest, the whole of the ether will be reduced to rest, except
as regards an extremely small motion, of the same nature as
that already considered in the case of an ordinary fluid. But
in the present case this small motion will be propagated into
space with the velocity of light; so that just before the com
mencement of the third interval the ether may be considered
as at rest, and everything will be the same as before. Supposing
now the number of intervals of time to be indefinitely increased,
and their magnitude indefinitely diminished, we pass to the case
in which the earth is supposed to move continuously.
It appears then, from these views of the constitution of the
ether, that (a) must be an exact differential, if it be not pre
vented from being so by the action of the air on the ether. We
know too little about the mutual action of the ether and material
particles to enable us to draw any very probable conclusion
respecting this matter; I would merely hazard the following
conjecture. Conceive a portion of the ether to be filled with a
great number of solid bodies, placed at intervals, and suppose
these bodies to move with a velocity which is very small compared
* Ante, p. 125.
156 ON THE CONSTITUTION OF LUMINIFEROUS ETHER.
with the velocity of light, then the motion of the ether between
the bodies will still be such that (a) is an exact differential. But
if these bodies are sufficiently close and numerous, they must
impress either the whole, or a considerable portion of their own
velocity on the ether between them. Now the molecules of air
may act the part of these solid bodies. It may thus come to pass
that (a) is an exact differential, and yet the ether close to the
surface of the earth is at rest relatively to the earth. The latter
of these conditions is however not necessary for the explanation of
aberration*.
[* A short demonstration that the path of a ray in the moving ether is a
straight line, which here followed, is omitted, as the proposition has already been
proved in the additional note printed at p. 138.]
[From the Report of the British Association for 1846, Part I. p. 1.]
REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS.
AT the meeting of the British Association held at Cambridge last
year, the Committee of the Mathematical Section expressed a wish
that a Report on Hydrodynamics should be prepared, in continua
tion of the reports which Prof. Challis had already presented to
the Association on that subject. Prof. Challis having declined the
task of preparing this report, in consequence of the pressure of
other engagements, the Committee of the Association did me the
honour to entrust it to me. In accordance with the wishes of the
Committee, the object of the present report will be to notice re
searches in this subject subsequent to the date of the reports of
Prof. Challis. It will sometimes however be convenient, for the
sake of giving a connected view of certain branches of the subject,
to refer briefly to earlier investigations.
The fundamental hypothesis on which the science of hydro
statics is based may be considered to be, that the mutual action
of two adjacent portions of a fluid at rest is normal to the surface
which separates them. The equality of pressure in all directions
is not an independent hypothesis, but a necessary consequence of
the former. This may be easily proved by the method given in
the Exercises of M. Cauchy*, a method which depends on the con
sideration of the forces acting on a tetrahedron of the fluid, the
dimensions of which are in the end supposed to vanish. This
proof applies equally to fluids at rest and fluids in motion ; and
thus the hypothesis abovementioned may be considered as the
fundamental hypothesis of the ordinary theory of hydrodynamics,
as well as hydrostatics. This hypothesis is fully confirmed by
* Tom. ii. p. 42.
158 REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS.
experiment in the case of the equilibrium of fluids ; but the com
parison of theory and experiment is by no means so easy in the
case of their motion, on account of the mathematical difficulty of
treating the equations of motion. Still enough has been done to
shew that the ordinary equations will suffice for the explanation
of a great variety of phenomena; while there are others the
laws of which depend on a tangential force, which is neglected in
the common theory, and in consequence of which the pressure is
different in different directions about the same point. The linear
motion of fluids in uniform pipes and canals is a simple instance*.
In the following report I shall first consider the common theory
of hydrodynamics, and then notice some theories which take ac
count of the inequality of pressure in different directions. It
will be convenient to consider the subject under the following
heads :
I. General theorems connected with the ordinary equations of
fluid motion.
II. Theory of waves, including tides.
III. The discharge of gases through small orifices.
IV. Theory of sound.
V. Simultaneous oscillations of fluids and solids.
VI. Formation of the equations of motion when the pressure
is not supposed equal in all directions.
I. Although the common equations of hydrodynamics have
been so long known, their complexity is so great f that little has
been done with them except in the case in which the expression
usually denoted by
udx + vdy + wdz (A)
is the exact differential of a function of the independent variables
x, y, ( I* becomes then of the utmost importance to inquire in
what cases this supposition may be made. Now Lagrange enun
ciated two theorems, by virtue of which, supposing them true, the
supposition may be made in a great number of important cases,
in fact, in nearly all those cases which it is most interesting to
[* See the footnote at p. 99.]
t In nearly all the investigations of Mr Airy it will be found that (A) is an
exact differential, although he does not start with assuming it to be so.
REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 159
investigate. It must be premised that in these theorems the
accelerating forces X, Y, Z are supposed to be such that
Xdx + Ydy + Zd*
is an exact differential, supposing the time constant, and the
density of the fluid is supposed to be either constant, or a function
of the pressure. The theorems are
First, that (A) is approximately an exact differential when the
motion is so small that squares and products of u, v, w and their
differential coefficients may be neglected. By calling (A) approxi
mately an exact differential, it is meant that there exists an ex
pression u t dx + v t dy + w t dz, which is accurately an exact differential,
and which is such that u^ v^ w t differ from u, v, w respectively by
quantities of the second order only.
Secondly, that (A) is accurately an exact differential at all
times when it is so at one instant, and in particular when the
motion begins from rest.
It has been pointed out by Poisson that the first of these
theorems is not true*. In fact, the initial motion, being arbitrary,
need not be such as to render (A) an exact differential. Thus
those cases coming under the first theorem in which the assertion
is true are merged in those which come under the second, at least
if we except the case of small motions kept up by disturbing
causes, a case in which we have no occasion to consider initial
motion at all. This case it is true is very important.
The validity of Lagrange s proof of the second theorem depends
on the legitimacy of supposing u, v and w capable of expansion
according to positive, integral powers of the time t, for a sufficiently
small value of that variable. This proof lies open to objection ;
for there are functions of t the expansions of which contain frac
tional powers, and there are others which cannot be expanded
according to ascending powers of t, integral or fractional, even
though they may vanish when t = 0. It has been shewn by Mr
Power that Lagrange s proof is still applicable if u, v and w admit
of expansion according to ascending powers of t of any kindf. The
second objection however still remains : nor does the proof which
Poisson has substituted for Lagrange s in his { Traite de Me cani
que appear at all more satisfactory. Besides, it does not appear
* Memoires de VAcadCmie des Sciences, torn. x. p. 554.
t Transactions of the Cambridge Philosophical Society, vol. vii. p. 455.
160 REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS.
from these proofs what becomes of the theorem if it is only for a
certain portion of the fluid that (A) is at one instant an exact
differential.
M. Cauchy has however given a proof of the theorem *, which
is totally different from either of the former, and perfectly satis
factory. M. Cauchy first eliminates the pressure by differentiation
from the three partial differential equations of motion. He then
changes the independent variables in the three resulting equations
from x, y, z, t to a, b, c, t, where a, b, c are the initial coordinates
of the particle whose coordinates at the time t are x, y, z. The
three transformed equations admit each of being once integrated
with respect to t, and the arbitrary functions of a, b, c introduced
by integration are determined by the initial motion, which is sup
posed to be given. The theorem in question is deduced without
difficulty from the integrals thus obtained. It is easily proved
that if the velocity is suddenly altered by means of impulsive
forces applied at the surface of the fluid, the alteration is such as
to leave (A) an exact differential if it were such before impact.
M. Cauchy s proof shews moreover that if (A) be an exact diffe
rential for one portion of the fluid, although riot for the whole, it
will always remain so for that portion. It should be observed,
that although M. Cauchy has proved the theorem for an incom
pressible fluid only, the same method of proof applies to the more
general case in which the density is a function of the pressure.
In a paper read last year before the Cambridge Philosophical
Society, I have given a new proof of the same theorem f. This
proof is rather simpler than M. Cauchy s, inasmuch as it does not
require any integration.
In a paper published in the Philosophical Magazine J, Prof.
Challis has raised an objection to the application of the theorem
to the case in which the motion of the fluid begins from rest.
According to the views contained in this paper, we are not in
general at liberty to suppose (A) to be an exact differential when
u, v and w vanish : this supposition can only be made when the
limiting value of t~ a (udx + vdy + wdz) is an exact differential, where
a is so taken as that one at least of the terms in this expression
does not vanish when t vanishes.
* M6moires des Savans Etrangers, torn. i. p. 40.
t Transactions of the Cambridge Philosophical Society, vol. viii. p. 307.
J Vol. xxiv. New Series, p. 94.
REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 161
It is maintained by Prof. Challis that the received equations
of hydrodynamics are not complete, as regards the analytical prin
ciples of the science, and he has given a new fundamental equation,
in addition to those received, which he calls the equation of con
tinuity of the motion*. On this equation Prof. Challis rests a result
at which he has arrived, and which all must allow to be most
important, supposing it correct, namely that whenever (A) is
an exact differential the motion of the fluid is necessarily recti
linear, one peculiar case of circular motion being excepted. As I
have the misfortune to differ from Professor Challis on the points
mentioned in this and the preceding paragraph, for reasons which
cannot be stated here, it may be well to apprise the reader that
many of the results which will be mentioned further on as satis
factory lie open to Professor Challis s objections.
By virtue of the equation of continuity of a homogeneous
incompressible fluid, the expression udy vdx will always be the
exact differential of a function of x and y. In the Cambridge
Philosophical Transactions^ there will be found some applications
of this function, and of an analogous function for the case of
motion which is symmetrical about an axis, and takes place in
planes passing through the axis. The former of these functions
had been previously employed by Mr Earnshaw.
II. In the investigations which come under this head, it is to
be understood that the motion is supposed to be very small, so
that first powers only of small quantities are retained, unless the
contrary is stated.
The researches of MM. Poisson and Cauchy were directed to
the investigation of the waves produced by disturbing causes
acting arbitrarily on a small portion of the fluid, which is then left
to itself. The mathematical treatment of such cases is extremely
difficult ; and after all, motions of this kind are not those which
it is most interesting to investigate. Consequently it is the
simpler cases of wave motion, and those which are more nearly con
nected with the phenomena which it is most desirable to explain,
that have formed the principal subject of more recent investiga
tions. It is true that there is one memoir by M. Ostrogradsky,
* Transactions of the Cambridge Philosophical Society, vol. viii. p. 31; and
Philosophical Magazine,, vol. xxvi. New Series, p. 425.
t Vol. vii. p. 439. (Ante, p. 1.)
S. 11
162 REPORT OX RECENT RESEARCHES IN HYDRODYNAMICS.
read before the French Academy in 1826*, to which this character
does not apply. In this memoir the author has determined the
motion of the fluid contained in a cylindrical basin, supposing the
fluid at first at rest, but its surface not horizontal. The interest
of the memoir however depends almost exclusively on the mathe
matical processes employed ; for the result is very complicated,
and has not been discussed by the author. There is one circum
stance mentioned by M. Plana*) which increases the importance of
the memoir in a mathematical point of view, which is that Poisson
met with an apparent impossibility in endeavouring to solve the
same problem. I do not know whether Poisson s attempt w is
ever published.
Theory of Long Waves. When the length of the waves whose
motion is considered is very great compared with the depth of the
fluid, we may without sensible error neglect the difference between
the horizontal motions of different particles in the same vertical
line, or in other words suppose the particles once in a vertical
line to remain in a vertical line : we may also neglect the vertical,
compared with the horizontal effective force. These considerations
extremely simplify the problem ; and the theory of long waves is
very important from its bearing on the theory of the tides. La
grange s solution of the problem in the case of a fluid of uniform
depth is well known. It is true that Lagrange fell into error in
extending his solution to cases to which it does not apply ; but
there is no question as to the correctness of his result when
properly restricted, that is when applied to the case of long waves
only. There are however many questions of interest connected
with this theory which have not been considered by Lagrange.
For instance, what will be the velocity of propagation in a uniform
canal whose section is not rectangular ? How will the form of the
wave be altered if the depth of the fluid, or the dimensions of the
canal, gradually alter ?
In a paper read before the Cambridge Philosophical Society in
May 1837 + , the late Mr Green has considered the motion of long
waves in a rectangular canal whose depth and breadth alter very
slowly, but in other respects quite arbitrarily. Mr Green arrived
at the following results : If & be the breadth, and 7 the depth of
* Mtmoires des Savans Etrangers, torn. iii. p. 23.
t Turin Memoirs for 1835, p. 253.
J Transactions of the Cambridge Philosophical Society, vol. vi. p. 457.
REPORT OX RECEXT RESEARCHES IX HYDRODYXAMICS. 163
the canal, then the height of the wave ccyS ^ i, the horizontal
velocity of the particles in a given phase of their motion oc 4 7"^,
the length of the wave oc 7?, and the velocity of propagation = Jg^>
With respect to the height of the wave, Mr Russell was led by his
experiments to the same law of its variation as regards the breadth
of the canal, and with respect to the effect of the depth he observes
that the height of the wave increases as the depth of the fluid
decreases, but that the variation of the height of the wave is very
slow compared with the variation of the depth of the canal.
In another paper read before the Cambridge Philosophical
Society in February 1839*, Mr Green has given the theory of the
motion of long waves in a triangular canal with one side vertical.
Mr Green found the velocity of propagation to be the same as that
in a rectangular canal of half the depth.
In a memoir read before the Royal Society of Edinburgh in
April 1839 f, Prof. Kelland has considered the case of a uni
form canal whose section is of any form. He finds that the velo
city of propagation is given by the very simple formula AjT >
where A is the area of a section of the canal, and I the breadth
of the fluid at the surface. This formula agrees with the experi
ments of Mr Russell, and includes as a particular case the formula
of Mr Green for a triangular canal.
Mr Airy, the Astronomer Royal, in his excellent treatise on
Tides and Waves, has considered the case of a variable canal with
more generality than Mr Green, inasmuch as he has supposed the
section to be of any formj. If A, b denote the same things as in
the last paragraph, only that now they are supposed to vary slowly
in passing along the canal, the coefficient of horizontal displace
ment oc A~%$, and that of the vertical displacement oc A~^b~^ 9
while the velocity of propagation at any point of the canal is that
given by the formula of the preceding paragraph. Mr Airy has
proved the latter formula in a more simple manner than Prof.
Kelland, and has pointed out the restrictions under which it is
* Transactions of the Cambridge Philosophical Society, vol. vii. p. 87.
t Transactions of the Royal Society of Edinburgh, vol. xiv. pp. 524, 530.
J Encyclopedia Metropolitan^ article Tides and Waves. Art. 260 of the
treatise.
Art. 218, &c.
112
164 REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS.
true. Other results of Mr Airy s will be more conveniently con
sidered in connection with the tides.
Theory of Oscillatory Waves. When the surface of water is
covered with an irregular series of waves of different sizes, the
longer waves will be continually overtaking the shorter, and the
motion will be very complicated, and will offer no regular laws.
In order to obtain such laws we must take a simpler case: we
may for instance propose to ourselves to investigate the motion of
a series of waves which are propagated with a constant velocity,
and without change of form, in a fluid of uniform depth, the
motion being in two dimensions and periodical. A series of waves
of this sort may be taken as the type of oscillatory waves in
general, or at least of those for which the motion is in two dimen
sions: to whatever extent a series of waves propagated in fluid
of a uniform depth deviates from this standard form, to the same
extent they fail in the characters of uniform propagation and in
variable form.
The theory of these waves has long been known. In fact each
element of the integrals by which MM. Poisson and Cauchy ex
pressed the disturbance of the fluid denotes what is called by Mr
Airy a standing oscillation, and a progressive oscillation of the
kind under consideration will result from the superposition of two
of these standing oscillations properly combined. Or, if we merely
replace products of sines and cosines under the integral signs by
sums and differences, each element of the new integrals will denote
a progressive oscillation of the standard kind. The theory of these
waves however well deserves a more detailed investigation. The
most important formula connected with them is that which gives
the relation between the velocity of propagation, the length of the
waves, and the depth of the fluid. If c be the velocity of propa
gation, X the length of the waves, measured from crest to crest, h
9
the depth of the fluid, and ra = , then
m
If the surface of the fluid be cut by a vertical plane perpen
dicular to the ridges of the waves, the section of the surface will
be the curve of sines. Each particle of the fluid moves round and
round in an ellipse, whose major axis is horizontal. The particle
REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 165
is in its highest position when the crest of the wave is passing
over it, and is then moving in the direction of propagation of the
wave ; it is in its lowest position when the hollow of the wave is
passing over it, and is then moving in a direction contrary to the
direction of propagation. At the bottom of the fluid the ellipse is
reduced to a right line, along which the particle oscillates. When
the length of waves is very small compared with the depth of the
fluid, the motion at the bottom is insensible, and all the expres
sions will be sensibly the same as if the depth were infinite. On
this supposition the expression for c reduces itself to A / >~~ The
ellipses in which the particles move are replaced by circles, and
the motion in each circle is uniform. The motion decreases with
extreme rapidity as the point considered is further removed from
the surface ; in fact, the coefficients of the horizontal and vertical
velocity contain as a factor the exponential e~ y , where y is the
depth of the particle considered below the surface. When the depth
of the fluid is finite, the law (as to time) of the horizontal and
vertical displacements of the particles is the same as when the depth
is infinite. When the length of the waves is very great compared
with the depth of the fluid, the horizontal motion of different
particles in the same vertical line is sensibly the same. The ex
pression for c reduces itself to Jgh, the same as would have been
obtained directly from the theory of long waves. The whole
theory is given very fully in the treatise of Mr Airy*. The nature
of the motion of the individual particles, as deduced from a rigor
ous theory, was taken notice of, I believe for the first time, by
Mr Green f, who has considered the case in which the depth is in
finite.
The oscillatory waves just considered are those which are pro
pagated uniformly in fluid of which the depth is everywhere the
same. When this condition is not satisfied, as for instance when
the waves are propagated in a canal whose section is not rectangu
lar, it is desirable to know how the velocity of propagation and
the form of the waves are modified by this circumstance. There
is one such case in which a solution has been obtained. In a
paper read before the Eoyal Society of Edinburgh in January 1841,
* Tides and Waves, art. 160, &c.
t Transactions of the Cambridge Philosophical Society, vol. vii. p. 95.
ICG REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS.
Prof. Kelland has arrived at a solution of the problem in the case
of a triangular canal whose sides are inclined at an angle of 45
to the vertical, or of a canal with one side vertical and one side
inclined at an angle of 45, in which the motion will of course be the
same as in one half of the complete canal*. The velocity of propa
gation is given by the formula (B), which applies to a rectangular
canal, or to waves propagated without lateral limitation, provided
we take for h half the greatest depth in the triangular canal, and
for X, or 27T/m, a quantity less than the length of the waves in the
triangular canal in the ratio of 1 to *J2. As to the form of the
waves, a section of the surface made by a vertical plane parallel
to the edges of the canal is the curve of sines ; a section made by
a vertical plane perpendicular to the former is the common cate
nary, with its vertex in the plane of the middle of the canal
(supposed complete), and its concavity turned upwards or down
wards according as the section is taken where the fluid is elevated
or where it is depressed. Thus the ridges of the waves do not
bend forwards, but are situated in a vertical plane, and they rise
higher towards the slanting sides of the canal than in the middle.
I shall write down the value of <, the integral of (A), and then any
one who is familiar with the subject can easily verify the preceding
results. In the following expression x is measured along the
bottom line of the canal, y is measured horizontally, and z verti
cally upwards :
(f> = A(ey}e a y)(e aZ +  aZ *)smj2z(xct) (C).
I have mentioned these results under the head, of oscillatory
waves, because it is to that class only that the investigation strictly
applies. The length of the waves is however perfectly arbitrary,
and when it bears a large ratio to the depth of the fluid, it seems
evident that the circumstances of the motion of any one wave can
not be materially affected by the waves which precede and follow
it, especially as regards the form of the middle portion, or ridge,
of the wave. Now the solitary waves of Mr Russell are long com
pared with the depth of the fluid ; thus in the case of a rect
angular canal he states that the length of the wave is about six
times the depth. Accordingly Mr Russell finds that the form of
the ridge agrees well with the results of Prof. Kelland.
* Transactions of the Royal Society of Edinburgh, vol. xv. p. 121.
REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 167
It appears from Mr Russell s experiments that there is a certain
limit to the slope of the sides of a triangular canal, beyond which
it is impossible to propagate a wave in the manner just considered.
Prof. Kelland has arrived at the same result from theory, but his
mathematical calculation does not appear to be quite satisfactory.
Nevertheless there can be little doubt that such a limit does
exist, and that if it be passed, the wave will be either continually
breaking at the sides of the canal, or its ridge will become bow
shaped, in consequence of the portion of the wave in the middle
of the canal being propagated more rapidly than the portions
which lie towards the sides. When once a wave has become suf
ficiently curved it may be propagated without further change, as
Mr Airy has shewn*. Thus the case of motion above considered
is in nowise opposed to the circumstance that the tide wave as
sumes a curved form when it is propagated in a broad channel in
which the water is deepest towards the centre.
It is worthy of remark, that if in equation (C), we transfer the
origin to either of the upper edges of the canal (supposed com
plete), and then suppose h to become infinite, having previously
written Ae~ iah for A, the result will express a series of oscillatory
waves propagated in deep water along the edge of a bank having
a slope of 45, the ridges of the waves being perpendicular to the
edge of the fluid. It is remarkable that the disturbance of the
fluid decreases with extreme rapidity as the perpendicular distance
from the edge increases, and not merely as the distance from the
surface increases. Thus the disturbance is sensible only in the
immediate neighbourhood of the edge, that is at a distance from
it which is a small multiple of X. The formula may be accommo
dated to the case of a bank having any inclination by merely
altering the coefficients of y and z, without altering the sum of the
squares of the coefficients. If i be the inclination of the bank to
the vertical, it will be easily found that the velocity of propagation
is equal to (  cos i ) . When i vanishes these waves pass into those
already mentioned as the standard case of oscillatory waves ; and
when i becomes negative, or the bank overhangs the fluid, a motion
of this sort becomes impossible.
I have had occasion to refer to what Mr Airy calls a standing
* Tides and Waves, art. 359.
168 REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS.
oscillation or standing wave. To prevent the possibility of con
fusion, it may be well to observe that Mr Airy uses the term in
a totally different sense from Mr Russell. The standing wave of
Mr Airy is the oscillation which would result from the coexistence
of two series of progressive waves, which are equal in every respect,
but are propagated in opposite directions. With respect to the
standing wave of Mr Russell, it cannot be supposed that the ele
vations observed in mountain streams can well be made the sub
ject of mathematical calculation. Nevertheless in so far as the
motion can be calculated, by taking a simple case, the theory does
not differ from that of waves of other classes. For if we only sup
pose a velocity equal and opposite to that of the stream impressed
both on the fluid and on the stone at the bottom which produces
the disturbance, we pass to the case of a forced wave produced in
still water by a solid dragged through it. There is indeed one
respect in which the theory of these standing waves offers a pecu
liarity, which is, that the velocity of a current is different at
different depths. But the theory of such motions is one of great
complexity and very little interest.
Theory of Solitary Waves. It has been already remarked that
the length of the solitary wave of Mr Russell is considerable com
pared with the depth of the fluid. Consequently we might expect
that the theory of long waves would explain the main phenomena
of solitary waves. Accordingly it is found by experiment that the
velocity of propagation of a solitary wave in a rectangular canal
is that given by the formula of Lagrange, the height of the wave
being very small, or that given by Prof. Kelland s, formula when
the canal is not rectangular. Moreover, the laws of the motion of
a solitary wave, deduced by Mr Green from the theory of long
waves, agree with the observations of Mr Russell. Thus Mr Green
found, supposing the canal rectangular, that the particles in a
vertical plane perpendicular to the length of the canal remain in
a vertical plane ; that the particles begin to move when the wave
reaches them, remain in motion while the wave is passing over
them, and are finally deposited in new positions ; that they move
in the direction of propagation of the wave, or in the contrary
direction, according as the wave consists of an elevation or a de
pression*. But when we attempt to introduce into our calculations
* Transactions of the Cambridge Philosophical Society, vol. vii. p. 87.
REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 169
the finite length of the wave, the problem becomes of great
difficulty. Attempts have indeed been made to solve it by the
introduction of discontinuous functions. But whenever such func
tions are introduced, there are certain conditions of continuity
to be satisfied at the common surface of two portions of fluid to
which different analytical expressions apply; and should these
conditions be violated, the solution will be as much in fault as it
would be if the fluid were made to penetrate the bottom of the
canal. No doubt, the theory is contained, to a first approximation,
in the formulas of MM. Poisson and Cauchy ; but as it happens
the obtaining of these formula? is comparatively easy, their discus
sion forms the principal difficulty. When the height of the wave
is not very small, so that it is necessary to proceed to a second
approximation, the theory of long waves no longer gives a velocity
of propagation agreeing with experiment. It follows, in fact, from
the investigations of Mr Airy, that the velocity of propagation of a
long wave is, to a second approximation, *g(h + 3k), where h is
the depth of the fluid when it is in equilibrium, and h + k the
height of the crest of the wave above the bottom of the canal *.
The theory of the two great solitary waves of Mr Russell forms
the subject of a paper read by Mr Earnshaw before the Cambridge
Philosophical Society in December lastf. Mr Russell found by
experiment that the horizontal motion of the fluid particles was
sensibly the same throughout the whole of a vertical plane per
pendicular to the length of the canal. He attributed the observed
degradation of the wave, and consequent diminution of the velocity
of propagation, entirely to the imperfect fluidity of the fluid, and
its adhesion to the sides and bottom of the canal. Mr Earnshaw
accordingly investigates the motion of the fluid on the hypotheses,
first, that the particles once in a vertical plane, perpendicular to
the length of the canal, remain in a vertical plane ; secondly, that
the wave is propagated with a constant velocity and without
* Tides and Waves, art. 208. In applying this formula to a solitary wave, it is
necessary to take for h the depth of the undisturbed portion of the fluid. In the
treatise of Mr Airy the formula is obtained for a particular law of disturbance, but
the same formula would have been arrived at, by the same reasoning, had the law
not been restricted. This formula is given as expressing the velocity of propagation
of the phase of high water, which it is true is not quite the same as the velocity of
propagation of the crest of the wave ; but the two velocities are the same to the
second order of approximation.
t Transactions of the Cambridge Philosophical Society, vol. viii. p. 326.
170 REPORT OX RECENT RESEARCHES IN HYDRODYNAMICS.
change of form. It is important to observe that these hypotheses
are used not as a foundation for calculation, but as a means of
selecting a particular kind of motion for consideration. The equa
tions of fluid motion admit of integration in this case in finite
terms, without any approximation, and it turns out that the motion
is possible, so far as the wave itself is concerned, and everything is
determined in the result except two constants, which remain arbi
trary. However, in order that the motion in question should
actually take place, it is necessary that there should be an instan
taneous generation or destruction of a finite velocity, and likewise
an abrupt change of pressure, at the junction of the portion of
fluid which constitutes the wave with the portions before and
behind which are at rest, both which are evidently impossible. It
follows of course that one at least of the two hypotheses must be
in fault. Experiment shewing that the first hypothesis is very
nearly true, while the second (from whatever cause) is sensibly
erroneous, the conclusion is that in all probability the degradation
of the wave is not to be attributed wholly to friction, but that it
is an essential characteristic of the motion. Nevertheless the
formula for the velocity of propagation of the positive wave, at
which Mr Earnshaw has arrived, agrees very well with the experi
ments of Mr Russell; the formula for the negative wave also agrees,
but not closely. These two formula can be derived from each
other only by introducing imaginary quantities.
It is the opinion of Mr Russell that the solitary wave is a
phenomenon swi generis, in nowise deriving its character from the
circumstances of the generation of the wave. His experiments
seem to render this conclusion probable. Should it be correct,
the analytical character of the solitary wave remains to be dis
covered. A complete theory of this wave should give, not only
its velocity of propagation, but also the law of its degradation,
at least of that part of the degradation which is independent of
friction, which is probably by far the greater part. With respect
to the importance of this peculiar wave however, it must be re
marked that the term solitary wave, as so defined, must not be
extended to the tide wave, which is nothing more (as far as
regards the laws of its propagation) than a very long wave, of
which the form may be arbitrary. It is hardly necessary to re
mark that the mechanical theories of the solitary wave and of the
aerial sound wave are altogether different.
REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 171
Theory of River and Ocean Tides. The treatise of Mr Airy
already referred to is so extensive, and so full of original matter,
that it will be impossible within, the limits of a report like the
present to do more than endeavour to give an idea of the nature
of the calculations and methods of explanation employed, and to
mention some of the principal results.
On account of the great length of the tide wave, the horizontal
motion of the water will be sensibly the same from top to bottom.
This circumstance most materially simplifies the calculation. The
partial differential equation for the motion of long waves, when
the motion is very small, is in the simplest case the same as that
which occurs in the theory of the rectilinear propagation of sound ;
and in Mr Airy s investigations the arbitrary functions which occur
in its integral are determined by the conditions to be satisfied at
the ends of the canal in which the waves are propagated, in a
manner similar to that in which the arbitrary functions are deter
mined in the case of a tube in which sound is propagated. When
the motion is not very small, the partial differential equation of
wave motion may be integrated by successive approximations, the
arbitrary functions being determined at each order of approxima
tion as before.
To proceed to some of the results. The simplest conceivable
case of a tidal river is that in which the river is regarded as a
uniform, indefinite canal, without any current. The height of the
water at the mouth of the canal will be expressed, as in the open
sea, by a periodic function of the time, of the form a sin (nt + a).
The result of a first approximation of course is that the disturb
ance at the mouth of the canal will be propagated uniformly up
it, with the velocity due to half the depth of the water. But on
proceeding to a second approximation*, Mr Airy finds that the
form of the wave will alter as it proceeds up the river. Its front
will become shorter and steeper, and its rear longer and more
gently sloping. When the wave has advanced sufficiently far up
the river, its surface will become horizontal at one point in the
rear, and further on the wave will divide into two. At the mouth
of the river the greatest velocities of the ebb and flow of the tide
are equal, and occur at low and high water respectively; the time
during which the water is rising is also equal to the time during
* Art. 198, &c.
172 REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS.
which it is falling. But at a station up the river the velocity of
the ebbstream is greater than that of the flowstream, and the
rise of the water occupies less time than its fall. If the station
considered is sufficiently distant from the mouth of the river, and
the tide sufficiently large, the water after it has fallen some way
will begin to rise again : there will in fact be a double rise and
fall of the water at each tide. This explains the double tides
observed in some tidal rivers. The velocity with which the phase of
high water travels up the river is found to be Jgk(I + ob), k being
the depth of the water when in equilibrium, and bk the greatest
elevation of the water at the mouth of the river above its mean
level. The same formula will apply to the case of low water if we
change the sign of b. This result is very important, since it shews
that the interval between the time of the moon s passage over the
meridian of the river station and the time of high water will be
affected by the height of the tide. Mr Airy also investigates the
effect of the current in a tidal river. He finds that the difference
between the times of the water s rising and falling is increased by
the current.
When the canal is stopped by a barrier the circumstances are
altered. When the motion is supposed small, and the disturbing
force of the sun and moon is neglected, it is found in this case
that the tidewave is a stationary wave*, so that there is high or
low water at the same instant at every point of the canal; but
if the length of the canal exceeds a certain quantity, it is high
water in certain parts of the canal at the instant when it is low
water in the remainder, and vice versa. The height of high water
is different in different parts of the canal : it increases from the
mouth of the canal to its extremity, provided the canal s length
does not exceed a certain quantity. If four times the length of
the canal be any odd multiple of the length of a free wave whose
period is equal to that of the tide, the denominator of the expres
sion for the tidal elevation vanishes. Of course friction would
prevent the elevation from increasing beyond a certain amount,
but still the tidal oscillation would in such cases be very large.
When the channel up which the tide is propagated decreases
in breadth or depth, or in both, the height of the tide increases in
ascending the channel. This accounts for the great height of the
* Art. 307.
REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 173
tides observed at the head of the Bristol Channel, and in such
places. In some of these cases however the great height may
be partly due to the cause mentioned at the end of the last
paragraph.
When the tidewave is propagated up a broad channel, which
becomes shallow towards the sides, the motion of the water in the
centre will be of the same nature as the motion in a free canal, so
that the water will be flowing up the channel with its greatest
velocity at the time of high water. Towards the coasts however
there will be a considerable flow of water to and from the shore ;
and as far as regards this motion, the shore will have nearly the
same effect as a barrier in a canal, and the oscillation will be of
the nature of a stationary wave, so that the water will be at rest
when it is at its greatest height. If, now, we consider a point at
some distance from the shore, but still not near the middle of the
channel, the velocity of the water up and down the channel will
be connected with its height in the same way as in the case of a
progressive wave, while the velocity to and from the shore will be
connected with the height of the water in the same way as in
a stationary wave. Combining these considerations, Mr Airy is
enabled to explain the apparent rotation of the water in such
localities, which arises from an actual rotation in the direction of
its motion*.
When the motion of the water is in two dimensions the mathe
matical calculation of the tidal oscillations is tolerably simple, at
least when the depth of the water is uniform. But in the case of
nature the motion is in .three dimensions, for the water is distri
buted over the surface of the earth in broad sheets, the boundaries
of which are altogether irregular. On this account a complete
theory of the tides appears hopeless, even in the case in which the
depth is supposed uniform. Laplace s theory, in which the whole
earth is supposed to be covered with water, the depth of which
follows a very peculiar law, gives us no idea of the effect of the
limitation of the ocean by continents. Mr Airy consequently in
vestigates the motion of the water on the supposition of its being
confined to narrow canals of uniform depth, which in the calcula
tion are supposed circular. The case in which the canal forms a
great circle is especially considered. This method enables us in
* Art. 360, &c.
174 REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS.
some degree to estimate the effect of the boundaries of the sea ;
aad it has the great advantage of leading to calculations which
can be worked out. There can be 110 doubt, too, that the con
clusions arrived at will apply, as to their general nature, to the
actual case of the earth.
With a view to this application of the theory, Mr Airy calcu
lates the motion of the water in a canal when it is under the
action of a disturbing force, which is a periodic function of the
time. The disturbing force at a point whose abscissa, measured
along the canal from a fixed point, is x, is supposed to be expressed
by a function of the form A sin (nt mx f a). This supposition is
sufficiently general for the case of the tides, provided the canal on
the earth be supposed circular. In all cases the disturbing force
will give rise to an oscillation in the water having the same period
as the force itself. This oscillation is called by Mr Airy a forced
wave. It will be sufficient here to mention some of the results of
this theory as applied to the case of the earth.
In all cases the expression for the tidal elevation contains as a
denominator the difference of the squares of two velocities, one
the velocity of propagation of a free wave along the canal, the
other the velocity with which a particular phase of the disturbing
force travels along the canal, or, which is the same, the velocity of
propagation of the forced wave. Hence the height of the tides
will not depend simply on the magnitude of the disturbing force,
but also on its period. Thus the mass of the moon cannot be in
ferred directly from the comparison of spring and neap tides, since
the heights of the solar and lunar tides are affected by the different
motions of the sun arid moon in right ascension, and consequently
in hourangle. When the canal under consideration is equatorial
the diurnal tide vanishes. The height of high water is the same
at all points of the canal, and there is either high or low water at
the point of the canal nearest to the attracting body, according as
the depth of the water is greater or less than that for which a
free wave would be propagated with the same velocity as the
forced wave. In the general case there is both a diurnal and a
semidiurnal tide, and the height of high water, as well as the
interval between the transit of the attracting body over the meri
dian of the place considered and the time of high water, is different
at different points of the canal. When the canal is a great circle
passing through the poles, the tidewave is a stationary wave.
REPORT ON RECENT RESEARCHES IX HYDRODYNAMICS. 175
When the coefficient of the disturbing force is supposed to vary
slowly, in consequence of the change in declination, &c. of the
disturbing body, it is found that the greatest tide occurs on the
day on which the disturbing force is the greatest.
The preceding results have been obtained on the supposition
of the absence of all friction ; but Mr Airy also takes friction into
consideration. He supposes it to be represented by a horizontal
force, acting uniformly from top to bottom of the water, and vary
ing as the first power of the horizontal velocity. Of course this
supposition is not exact : still there can be no doubt that it
represents generally the effect of friction. When friction is taken
into account, the denominator of the expressions for the tidal
elevation is essentially positive, so that the motion can never
become infinite. In the case of a uniform tidal river stopped by
a barrier, the high wa,ter is no longer simultaneous at all points,
but the phase of high water always travels up the river. But of
all the results obtained by considering friction, the most important
appears to be, that when the slow variation of the disturbing
force is taken into account, the greatest tide, instead of happening
on the day when the disturbing force is greatest, will happen later
by a certain time p v Moreover, in calculating the tides, we must
use, not the relative positions of the sun and moon for the instant
for which the tide is calculated, but their relative positions for a
time earlier by the same interval p l as in the preceding case. The
expression for p l depends both on the depth of the canal and on
the period of the tide, and therefore its value for the diurnal tide
cannot be inferred from its value for the semidiurnal. It appears
also that the phase of the tide is accelerated by friction.
The mechanical theory of the tides of course belongs to hydro
dynamics; but I do not conceive that the consideration of the
reduction and discussion of tidal observations falls within the
province of this report.
Before leaving the investigations of Mr Airy, I would call at
tention to a method which he sometimes employs very happily in
giving a general explanation of phenomena depending on motions
which are too complicated to admit of accurate calculation. It is
evident that any arbitrary motion may be assigned to a fluid,
(with certain restrictions as to the absence of abruptness,) provided
we suppose certain forces to act so as to produce them. The
values of these forces are given by the equations of motion. In
176 REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS.
some cases the forces thus obtained will closely resemble some
known forces ; while in others it will be possible to form a clear
conception of the kind of motion which must take place in the
absence of such forces. For example, supposing that there is pro
pagated a series of oscillatory waves of the standard kind, except
that the height of the waves increases proportionably to their
distance from a fixed line, remaining constant at the same point
as the time varies, Mr Airy finds for the force requisite to maintain
such a motion an expression which may be assimilated to the force
which wind exerts on water. This affords a general explanation
of the increase in the height of the waves in passing from a wind
ward to a lee shore*. Again, by supposing a series of waves, as
near the standard kind as circumstances will admit, to be pro
pagated along a canal whose depth decreases slowly, and examin
ing the force requisite to maintain this motion, he finds that a
force must be applied to hold back the heads of the waves. In
the absence, then, of such a force the heads of the waves will have
a tendency to shoot forwards. This explains the tendency of waves
to break over a sunken shoal or along a sloping beach~f*. The
word tendency is here used, because when a wave comes at all
near breaking, but little reliance can be placed in any investigation
which depends upon the supposition of the motion being small.
To take one more example of the application of this method, by
supposing a wave to travel, unchanged in form, along a canal, with
a velocity different from that of a free wave, and examining the
force requisite to maintain such a motion, Mr Airy is enabled to
give a general explanation of some very curious circumstances
connected with the motion of canal boats J, which have been ob
served by Mr Russell.
III. In the 16th volume of the Journal de 1 Ecole Polytech
nique , will be found a memoir by MM. Barre* de Saint Venant
and Wantzel, containing the results of some experiments on the
discharge of air through small orifices, produced by considerable
differences of pressure. The formula for the velocity of efflux
derived from the theory of steady motion, and the supposition
that the mean pressure at the orifice is equal to the pressure at a
distance from the orifice in the space into which the discharge
* Art. 265, &e. + Art. 238, &c.
+ Art. 405, &c. Cahier xxvii. p. 85.
REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 177
takes place, leads to some strange results of such a nature as to
make us doubt its correctness. If we call the space from which
the discharge takes place i\iQ first space, and that into which it
takes place the second space, and understand by the term reduced
velocity the velocity of efflux diminished in the ratio of the density
in the second space to the density in the first, so that the reduced
velocity measures the rate of discharge, provided the density in
the first space remain constant, it follows from the common for
mula that the reduced velocity vanishes when the density in the
second space vanishes, so that a gas cannot be discharged into a
vacuum. Moreover, if the density of the first space is given, the
reduced velocity is a maximum when the density in the second
space is rather more than half that in the first. The results
remain the same if we take account of the contraction of the
vein, and they are not materially altered if we take into account
the cooling of the air by its rapid dilatation. The experiments
above alluded to were made by allowing the air to enter an ex
hausted receiver through a small orifice, and observing simul
taneously the pressure and temperature of the air in the receiver,
and the time elapsed since the opening of the orifice. It was
found that when the exhaustion was complete the reduced velocity
had a certain value, depending on the orifice employed, and that
the velocity did not sensibly change till the pressure of the air in
the receiver became equal to about Jths of the atmospheric pres
sure. The reduced velocity then began to decrease, and finally
vanished when the pressure of the air in the receiver became
equal to the atmospheric pressure.
These experiments shew that when the difference of pressure
in the first and second spaces is considerable, we can by no means
suppose that the mean pressure at the orifice is equal to the
pressure at a distance in the second space, nor even that there
exists a contracted vein, at which we may suppose the pressure to
be the same as at a distance. The authors have given an empiri
cal formula, which represents very nearly the reduced velocity,
whatever be the pressure of the air in the space into which the
discharge takes place.
The orifices used in these experiments were generally about
one millimetre in diameter. It was found that widening the
mouth of the orifice, so as to make it funnelshaped, produced a
much greater proportionate increase of velocity when the velocity
S. 12
178 REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS.
of efflux was small than when it was large. The authors have
since repeated their experiments with air coming from a vessel in
which the pressure was four atmospheres: they have also tried
the effect of using larger orifices of four or five millimetres
diameter. The general results were found to be the same as
before*.
IV. In the 6th volume of the Transactions of the Cambridge
Philosophical Society, p. 403, will be found a memoir by Mr Green
on the reflection and refraction of sound, which is well worthy of
attention. This problem had been previously considered by Pois
son in an elaborate memoir f\ Poisson treats the subject with
extreme generality, and his analysis is consequently very compli
cated. Mr Green, on the contrary, restricts himself to the case of
plane waves, a case evidently comprising nearly all the phenomena
connected with this subject which are of interest in a physical
point of view, and thus is enabled to obtain his results by a very
simple analysis. Indeed Mr Green s memoirs are very remarkable,
both for the elegance and rigour of the analysis, and for the ease
with which he arrives at most important results. This arises in a
great measure from his divesting the problems he considers of all
unnecessary generality: where generality is really of importance
he does not shrink from it. In the present instance there is one
important respect in which Mr Green s investigation is more general
than Poisson s, which is, that Mr Green has taken the case of any
two fluids, whereas Poisson considered the case of two elastic fluids,
in which equal condensations produce equal increments of pressure.
It is curious, that Poisson, forgetting this restriction, applied his
formulae to the case of air and water. Of course his numerical
result is altogether erroneous. Mr Green easily arrives at the
ordinary laws of reflection and refraction. He obtains also a very
simple expression for the intensity of the reflected sound. If A is
the ratio of the density of the second medium to that of the first,
and B the ratio of the cotangent of the angle of refraction to the
cotangent of the angle of incidence, then the intensity of the
reflected sound is to the intensity of the incident as A B to
A + B. In this statement the intensity is supposed to be mea
sured by the first power of the maximum displacement. When
* Comptes Rendus, torn. xvii. p. 1140.
t Memoires de VAcadtmie des Sciences, torn. x. p. 317.
REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 179
the velocity of propagation in the first medium is less than in the
second, and the angle of incidence exceeds what may be called the
critical angle, Mr Green restricts himself to the case of vibrations
following the cycloidal law. He finds that the sound suffers total
internal reflection. The expression for the disturbance in the
second medium involves an exponential with a negative index,
and consequently the disturbance becomes quite insensible at a
distance from the surface equal to a small multiple of the length
of a wave. The phase of vibration of the reflected sound is also
accelerated by a quantity depending on the angle of incidence.
It is remarkable, that when the fluids considered are ordinary
elastic fluids, or rather when they are such that equal condensa
tions produce equal increments of pressure, the expressions for
the intensity of the reflected sound, and for the acceleration of
phase when the angle of incidence exceeds the critical angle, are
the same as those given by Fresnel for light polarized in a plane
perpendicular to the plane of incidence.
V. Not long after the publication of Poisson s memoir on the
simultaneous motions of a pendulum and of the surrounding air*,
a paper by Mr Green was read before the Royal Society of Edin
burgh, which is entitled Researches on the Vibration of Pendulums
in Fluid Media [. Mr Green does not appear to have been at that
time acquainted with Poisson s memoir. The problem which he
has considered is one of the same class as that treated by Poisson.
Mr Green has supposed the fluid to be incompressible, a suppo
sition, however, which will apply without sensible error to air, in
considering motions of this sort. Poisson regarded the fluid as
elastic, but in the end, in adapting his formula to use, he has
neglected as insensible the terms by which the effect of an elastic
differs from that of an inelastic fluid. The problem considered by
Mr Green is, however, in one respect much more general than
that solved by Poisson, since Mr Green has supposed the oscil
lating body to be an ellipsoid, whereas Poisson considered only a
sphere. Mr Green has obtained a complete solution of the pro
blem in the case in which the ellipsoid has a motion of translation
only, or in which the small motion of the fluid due to its motion
* M&moires de V Academic des Sciences, torn. xi. p. 521.
t This paper was read in December, 1833, and is printed in the 13th volume of
the Society s Transactions, p. 54, &c.
122
180 REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS.
of rotation is neglected. The result is that the resistance of the
fluid will be allowed for if we suppose the mass of the ellipsoid
increased by a mass bearing a certain ratio to that of the fluid
displaced. In the general case this ratio depends on three trans
cendental quantities, given by definite integrals. If, however,
the ellipsoid oscillates in the direction of one of its principal axes,
the ratio depends on one only of these transcendents. When the
ellipsoid passes into a spheroid, the transcendents above mentioned
can be expressed by means of circular or logarithmic functions.
When the spheroid becomes a sphere, Mr Green s result agrees
with Poisson s. It is worthy of remark, that Mr Green s formula
will enable us to calculate the motion of an ellipse or circle oscil
lating in a fluid, in a direction perpendicular to its plane, since a
material ellipse or circle may be considered as a limiting form of
an ellipsoid. In this case, however, the motion would probably
have to be extremely small, in order that the formula should apply
with accuracy.
In a paper On the Motion of a small Sphere acted on by the
Vibrations of an Elastic Medium, read before the Cambridge
Philosophical Society in April 1841*, Prof. Challis has considered
the motion of a ball pendulum, retaining in his solution small
quantities to the second order. The principles adopted by Prof.
Challis in the solution of this problem are at variance with those
of Poisson, and have given rise to a controversy between him and
Mr Airy, which will be found in the 17th, 18th, and 19 volumes
of the Philosophical Magazine (New Series). In the paper just
referred to, Prof. Challis finds that when the fluid is incompressible
there is no decrement in the arc of oscillation, except what arises
from friction and capillary attraction. In the case of air there is
a slight theoretical decrement ; but it is so small that Prof. Challis
considers the observed decrement to be mainly owing to friction.
This result follows also from Poisson s solution. Prof. Challis also
finds that a small sphere moving with a uniform velocity experi
ences no resistance, and that when the velocity is partly uniform
and partly variable, the resistance depends on the variable part
only. The problem, however, referred to in the title of this paper,
is that of calculating the motion of a small sphere situated in an
elastic fluid, and acted on by no forces except the pressure of the
* Transactions of the Cambridge Philosophical Society, vol. vii. p. 333.
REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 181
fluid, in which an indefinite series of plane condensing and rarefy
ing waves is supposed to be propagated. This problem is solved
by the author on principles similar to those which he has adopted
in the problem of an oscillating sphere. The views of Prof. Challis
with respect to this problem, which he considers a very important
one, are briefly stated at the end of a paper published in the
Philosophical Magazine*.
In a paper On some Cases of Fluid Motion, published in the
Transactions of the Cambridge Philosophical Society^, I have
considered some modifications of the problem of the ball pendu
lum, adopting in the main the principles of Poisson, of the
correctness of which I feel fully satisfied, but supposing the fluid
incompressible from the first. In this paper the effect of a distant
rigid plane interrupting the fluid in which the sphere is oscillating
is given to the lowest order of approximation with which the
effect is sensible. It is shewn also that when the ball oscillates
in a concentric spherical envelope, the effect of the resistance of
the fluid is to add to the mass of the sphere a mass equal to
b* + 2a* m
~F^ 2
where a is the radius of the ball, b that of the envelope, and m
the mass of the fluid displaced. Poisson, having reasoned on the
very complicated case of an elastic fluid, had come to the con
clusion that the envelope would have no effect.
One other instance of fluid motion contained in this paper will
here be mentioned, because it seems to afford an accurate means
of comparing theory and experiment in a class of motions in
which they have not hitherto been compared, so far as I am
aware. When a box of the form of a rectangular parallelepiped,
filled with fluid and closed on all sides, is made to perform small
oscillations, it appears that the motion of the box will be the.
same as if the fluid were replaced by a solid having the same
mass, centre of gravity, and principal axes as the solidified fluid
but different principal moments of inertia. These moments are
given by infinite series, which converge with extreme rapidity, so
that the numerical calculation is very easy. The oscillations most
convenient to employ would probably be either oscillations by
torsion, or bifilar oscillations.
* Vol. xviii., New Series, p. 481. t Vol. viii. p. 105.
182 REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS.
VI. M. Navier was, I believe, the first to give equations for
the motion of fluids without supposing the pressure equal in all
directions. His theory is contained in a memoir read before
the French Academy in 1822*. He considers the case of a
homogeneous incompressible fluid. He supposes such a fluid
to be made up of ultimate molecules, acting on each other by
forces which, when the molecules are at rest, are functions simply
of the distance, but which, when the molecules recede from, or
approach to each other, are modified by this circumstance, so
that two molecules repel each other less strongly when they are
receding, and more strongly when they are approaching, than
they do when they are at rest")*. The alteration of attraction or
repulsion is supposed to be, for a given distance, proportional to
the velocity with which the molecules recede from, or approach
to each other; so that the mutual repulsion of two molecules
will be represented by f(r) VF(r), where r is the distance of
the molecules, V the velocity with which they recede from each
other, and f(r), F(r) two unknown functions of r depending on
the molecular force, and as such becoming insensible when r
has become sensible. This expression does not suppose the
molecules to be necessarily receding from each other, nor their
mutual action to be necessarily repulsive, since V and F (r) may
be positive or negative. It is not absolutely necessary that f(r)
and F (r) should always have the same sign. In forming the
equations of motion M. Navier adopts the hypothesis of a sym
metrical arrangement of the particles, or at least, which leads
to the same result, neglects the irregular part <of the mutual
action of neighbouring molecules. The equations at which he
arrives are those which would be obtained from the common
dp 4 (d"u d*u d*u\ . , f dp .
equations by wntmg ~A ^ + ^ + ^ m place of ^ m
the first, and making similar changes in the second and third.
A is here an unknown constant depending on the nature of the
fluid.
The same subject has been treated on by PoissonJ, who has
adopted hypotheses which are very different from those of M.
* Memoires de V Academic des Sciences, torn. vi. p. 389.
t This idea appears to have been borrowed from Dubuat. See his Principes
d Hydraulique, torn. ii. p. 60.
J Journal de VEcole Poly technique, torn. xiii. cah. 20, p. 139.
REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 183
Navier. Poisson s theory is of this nature. He supposes the
time t to be divided into n equal parts, each equal to r. In
the first of these he supposes the fluid to be displaced in the same
manner as an elastic solid, so that the pressures in different
directions are given by the equations which he had previously
obtained for elastic solids. If the causes producing the dis
placement were now to cease to act, the molecules would very
rapidly assume a new arrangement, which would render the
pressure equal in all directions, and while this rearrangement
was going on, the pressure would alter in an unknown manner
from that belonging to a displaced elastic solid to the pressure
belonging to the fluid in its new state. The causes of dis
placement are however going on during the second interval r;
but since these different small motions will take place inde
pendently, the new displacement which will take place in the
second interval r will be the same as if the molecules were not
undergoing a rearrangement. Supposing now n to become in
finite, we pass to the case in which the fluid is continually be
ginning to be displaced like an elastic solid, and continually
rearranging itself so as to make the pressure equal in all direc
tions. The equations at which Poisson arrived are, in the cases
of a homogeneous incompressible fluid, and of an elastic fluid
in which the change of density is small, those which would be
derived from the common equations by replacing dp/dx in the
first by
dp , fd\i d*u d*u\ p d fdu dv dw\
dx \dx* dy* dz 2 J dx \dx dy dzj
and making similar changes in the second and third. In these
equations A and B are two unknown constants. It will be
observed that Poisson s equations reduce themselves to Navier s
in the case of an incompressible fluid.
The same subject has been considered in a quite different
point of view by M. Barre de Saint Venant, in a communication
to the French Academy in 1843, an abstract of which is contained
in the Comptes Rendus*. The principal difficulty is to connect
the oblique pressures in different directions about the same point
with the differential coefficients dujdx, du/dy, &c., which express
the relative motion of the fluid particles in the immediate neigh
* Tom. xvii. p. 1240.
184 REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS.
bourhood of that point. This the author accomplishes by as
suming that the tangential force on any plane passing through
the point in question is in the direction of the principal sliding
(glissement] along that plane. The sliding along the plane xy
, , dw du . , ,. , dw dv .
is measured by j h j in the direction of x, and j + y in the
7 ax dz dy dz
direction of y. These two slidings may be compounded into one,
which will form the principal sliding along the plane xy. It
is then shewn, by means of M. Cauchy s theorems connecting
the pressures in different directions in any medium, that the
tangential force on any plane passing through the point considered,
resolved in any direction in that plane, is proportional to the
sliding along that plane resolved in the same direction, so that
if T represents the tangential force, referred to a unit of surface,
and 8 the sliding, T=eS. The pressure on a plane in any direc
tion is then found. This pressure is compounded of a normal
pressure, alike in all directions, and a variable oblique pressure,
the expression for which contains the one unknown quantity e.
If the fluid be supposed incompressible, and e constant, the
equations which would be obtained by the method of M. Barre
de Saint Venant agree with those of M. Navier. It will be
observed that this method does not require the consideration of
ultimate molecules at all.
When the motion of the fluid is very small, Poisson s equations
agree with those given by M. Cauchy for the motion of a solid
entirely destitute of elasticity*, except that the latter do not
contain the pressure p. These equations have been obtained
by M. Cauchy without the consideration of molecules. His
method would apply, with very little change, to the case of
fluids.
In a paper read last year before the Cambridge Philosophical
Society "f 1 , I have arrived at the equations of motion in a different
manner. The method employed in this paper does not neces
sarily require the consideration of ultimate molecules. Its prin
cipal feature consists in eliminating from the relative motion
of the fluid about any particular point the relative motion which
corresponds to a certain motion of rotation, and examining the
nature of the relative motion which remains. The equations
* Exercices de Mathematiques, torn. iii. p. 187.
f Transactions of the Cambridge Philosophical Society, vol. viii. p. 287.
REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 185
finally adopted in the cases of a homogeneous incompressible
fluid, and of an elastic fluid in which the change of density is
small, agree with those of Poisson, provided we suppose in the
latter A ^B. It is shewn that this relation between A and B
may be obtained on Poisson s own principles.
The equations hitherto considered are those which must be
satisfied at any point in the interior of the fluid mass ; but there
is hardly any instance of the practical application of the equations,
in which we do not want to know also the particular conditions
which must be satisfied at the surface of the fluid. With respect
to a free surface there can be little doubt : the condition is simply
that there shall be no tangential force on a plane parallel to the
surface, taken immediately within the fluid. As to the case
of a fluid in contact with a solid, the condition at which Navier
arrived comes to this : that if we conceive a small plane drawn
within the fluid parallel to the surface of the solid, the tangential
force on this plane, referred to a unit of surface, shall be in the
same direction with, and proportional to the velocity with which
the fluid flows past the surface of the solid. The condition ob
tained by Poisson is essentially the same.
Dubuat stated, as a result of his experiments, that when the
velocity of water flowing through a pipe is less than a certain
quantity, the water adjacent to the surface of the pipe is at rest*.
This result agrees very well with an experiment of Coulomb s.
Coulomb found that when a metallic disc was made to oscillate
very slowly in water about an axis passing through its centre
and perpendicular to its plane, the resistance was not altered
when the disc was smeared with grease; and even when the
grease was covered with powdered sandstone the resistance was
hardly increased f. This is just what one would expect on the
supposition that the water close to the disc is carried along with
it, since in that case the resistance must depend on the internal
friction of the fluid ; but the result appears very extraordinary on
the supposition that the fluid in contact with the disc flows
past it with a finite velocity. It should be observed, however,
that this result is compatible with the supposition that a thin
film of fluid remains adhering to the disc, in consequence of
capillary attraction, and becomes as it were solid, and that the
* See the Table given in torn. i. of his Principes d Hydr antique, p. 93.
t Memoires de VInstitut, 1801, torn. iii. p. 286.
186 REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS.
fluid in contact with this film flows past it with a finite velocity.
If we consider Dubuat s supposition to be correct, the condition
to be assumed in the case of a fluid in contact with a solid is
that the fluid does not move relatively to the solid. This con
dition will be included in M. Navier s, if we suppose the coefficient
of the velocity when M. Navier s condition is expressed analy
tically, which he denotes by E, to become infinite. It seems
probable from the experiments of M. Girard, that the condition to
be satisfied at the surface of fluid in contact with a solid is
different according as the fluid does or does not moisten the
surface of the solid.
M. Navier has applied his theory to the results of some ex
periments of M. Girard s on the discharge of fluids through
capillary tubes. His theory shews that if we suppose E to be
finite, the discharge through extremely small tubes will depend
only on E, and not on A. The law of discharge at which he
arrives agrees with the experiments of M. Girard, at least when
the tubes are extremely small. M. Navier explained the differ
ence observed by M. Girard in the discharge of water through
tubes of glass and tubes of copper of the same size by supposing
the value of E different in the two cases. This difference was
explained by M. Girard himself by supposing that a thin film
of fluid remains adherent to the pipe, in consequence of molecular
action, and that the thickness of this film differs with the sub
stance of which the tube is composed, as well as with the liquid
employed*. If we adopt Navier s explanation, we may reconcile
it with the experiments of Coulomb by supposing that E is very
large, so that unless the fluid is confined in a very narrow pipe,
the results will depend mainly on A, being sensibly the same as
they would be if E were infinite.
There is one circumstance connected with the motion of a
ballpendulum oscillating in air, which has not yet been ac
counted for, the explanation of which seems to depend on this
theory. It is found by experiment that the correction for the
inertia of the air is greater for small than for large spheres,
that is to say, the mass which we must suppose added to that
of the sphere bears a greater ratio to the mass of the fluid dis
placed in the former, than in the latter case. According to the
common theory of fluid motion, in which everything is supposed
* M6moires de VAcademie des Sciences, torn i. pp. 203 and 234.
REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 187
to be perfectly smooth, the ratio ought to be independent of the
magnitude of the sphere. In the imperfect theory of friction in
which the friction of the fluid on the sphere is taken into account,
while the equal and opposite friction of the sphere on the fluid is
neglected, it is shewn that the arc of oscillation is diminished,
while the time of oscillation is sensibly the same as before. But
when the tangential action of the sphere on the fluid, and the
internal friction of the fluid itself are considered, it is clear that
one consequence will be, to speak in a general way, that a portion
of the fluid will be dragged along with the sphere. Thus the
correction for the inertia of the fluid will be increased, since the
same moving force has now to overcome the inertia of the fluid
dragged along with the sphere, and not only, as in the former
case, the inertia of the sphere itself, and of the fluid pushed away
from before it, and drawn in behind it. Moreover the additional
correction for inertia must depend, speaking approximately, on
the surface of the sphere, whereas the first correction depended on
its volume, and thus the effect of friction in altering the time of
oscillation will be more conspicuous in the case of small, than in
the case of large spheres, other circumstances being the same.
The correction for inertia, when friction is taken into account, will
not, however, depend solely on the magnitude of the sphere, but
also on the time of oscillation. With a given sphere it will be
greater for long, than for short oscillations.
[From the Transactions of the Cambridge Philosophical Society, Vol. VIII.
p. 409.]
SUPPLEMENT TO A MEMOIR ON SOME CASES OF FLUID
MOTION.
Eead Nov. 3, 1846.
IN a memoir which the Society did me the honour to publish
in their Transactions*, I shewed that when a box whose interior
is of the form of a rectangular parallelepiped is filled with fluid
and made to perform small oscillations the motion of the box
will be the same as if the fluid were replaced by a solid having
the same mass, centre of gravity, and principal axes as the
solidified fluid, but different moments of inertia about those axes.
The box is supposed to be closed on all sides, and it is also
supposed that the box itself and the fluid within it were both
at rest at the beginning of the motion. The investigation was
founded upon the ordinary equations of Hydrodynamics, which
depend upon the hypothesis of the absence of any tangential
force exerted between two adjacent portions of a fluid in motion,
an hypothesis which entails as a necessary consequence the
equality of pressure in all directions. The particular case of
motion under consideration appears to be of some importance,
because it affords an accurate means of comparing with experiment
the common theory of fluid motion, which depends upon the
hypothesis just mentioned. In my former paper, I gave a series
by means of which the numerical values of the principal moments
of the solid which may be substituted for the fluid might be
calculated with facility. The present supplement contains a
different series for the same purpose, which is more easy of
numerical calculation than the former. The comparison of the
* Vol. YIII. Part i. p. 105. (Ante, p. 17.)
ON SOME CASES OF FLUID MOTION. 189
two series may also be of some interest in an analytical point
of view, since they appear under very different forms. I have
taken the present opportunity of mentioning the results of some
experiments which I have performed on the oscillations of a box,
such as that under consideration. The experiments were not
performed with sufficient accuracy to entitle them to be described
in detail.
The calculation of the motion of fluid in a rectangular box
is given in the 13th article of my former paper. I shall not
however in the first instance restrict myself to a rectangular
parallelepiped, since the simplification which I am about to give
applies more generally. Suppose then the problem to be solved
to be the following. A vessel whose interior surface is composed
of any cylindrical surface and of two planes perpendicular to the
generating lines of the cylinder is filled with a homogeneous,
incompressible fluid ; the vessel and the fluid within it having
been at first at rest, the former is then moved in any manner ;
required to determine the motion of the fluid at any instant,
supposing that at that instant the vessel has no motion of rotation
about an axis parallel to the generating lines of the cylinder.
I shall adopt the notation of my former paper, u, v, w are
the resolved parts of the velocity at any point along the rect
angular axes of x, y, z. Since the motion begins from rest we
shall have udx + vdy f wdz an exact differential d$. Let the
rectangular axes to which the fluid is referred be fixed relatively
to the vessel, and let the axis of x be parallel to the generating
lines of the cylindrical surface. The instantaneous motion of
the vessel may be decomposed into a motion of translation, and
two motions of rotation about the axes of y and z respectively ;
for by hypothesis there is no motion of rotation about the axis
of x. According to the principles of my former paper, the in
stantaneous motion of the fluid will be the same as if it had
been produced directly by impact, the impact being such as
to give the vessel the velocity which it has at the instant con
sidered. We may also consider separately the motion of trans
lation of the vessel, and each of the motions of rotation ; the
actual motion of the fluid will be compounded of those which
correspond to each of the separate motions of the vessel. For
my present purpose it will be sufficient to consider one of the
190 . SUPPLEMENT TO A MEMOIR
motions of rotation, that which takes place round the axis of
z for instance. Let co be the angular velocity about the axis
of z, co being considered positive when the vessel turns from
the axis of x to that of y. It is easy to see that the instantaneous
motion of the cylindrical surface is such as not to alter the volume
of the interior of the vessel, supposing the plane ends fixed,
and that the same is true of the instantaneous motion of the
ends. Consequently we may consider separately the motion of
the fluid due to the motion of the cylindrical surface, and to that
of the ends. Let cf) c be the part of < due to the motion of the
cylindrical surface, <j) e the part due to the motion of the ends.
Then we shall have
*=*.+*. .......................... a)
Consider now the motion corresponding to a value of <, wxy.
It will be observed that wxy satisfies the equation, {(36) of my
former paper,} which (f> is to satisfy. Corresponding to this value
of </> we have
u wy, v = cox, w = 0.
Hence the velocity, corresponding to this motion, of a particle
of fluid in contact with the cylindrical surface of the vessel,
resolved in a direction perpendicular to the surface, is the same
as the velocity of the surface itself resolved in the same direction,
and therefore the fluid does not penetrate into, nor separate
from the cylindrical surface. The velocity of a particle in contact
with either of the plane ends, resolved in a direction perpendicular
to the surface, is equal and opposite to the velocity of the surface
itself resolved in the same direction. Hence we shall get the
complete value of </> by adding the part already found, namely
a>xy, to twice the part due to the motion of the plane ends. We
have therefore,
</> = nay + 2<^ = 2( c  vxy, by (1) ........... (2),
and $ c $ e = axy ............................. (3).
Hence whenever either <j> c or <f) e can be found, the complete
solution of the problem will be given by (2). And even when
both these functions can be obtained independently, (2) will
enable us to dispense with the use of one of them, and (3) will
give a relation between them. In this case (3) will express a
theorem in pure analysis, a theorem which will sometimes be
ON SOME CASES OF FLUID MOTION. 191
very curious, since the analytical expressions for <f> c and </><, will
generally be totally different in form. The problem admits of
solution in the case of a circular cylinder terminated by planes
perpendicular to its axis, and in the case of a rectangular paral
lelepiped. In the former case, the numerical calculation of the
moments of inertia of the solid by which the fluid may be re
placed would probably be troublesome, in the latter it is extremely
easy. I proceed to consider this case in particular.
Let the rectangular axes to which the fluid is referred coincide
with three adjacent edges of the parallelepiped, and let a, 6, c
be the lengths of the edges. The motion which it is proposed
to calculate is that which arises from a motion of rotation of the
box about an axis parallel to that of z and passing through the
centre of the parallelepiped. Consequently in applying (2) we
must for a moment conceive the axis of z to pass through the
centre of the parallelepiped, and then transfer the origin to the
corner, and we must therefore write co (x Ja) (y J b) for wxy.
In the present case the cylindrical surface consists of the four
faces which are parallel to the axis of x, and the remaining faces
form the plane ends. The motion of the face xy and the opposite
face has evidently no effect on the fluid, so that <j) G will be the
part of cf> due to the motion of the face xz and the opposite face.
The value of this quantity is given near the middle of page 62 in my
former paper. We have then by the second of the formulae (2)
(e ~ n7rb/a 
~ COS
the sign S denoting the sum corresponding to all odd integral
values of n from 1 to oo . This value of expresses completely
the motion of the fluid due to a motion of rotation of the box
about an axis parallel to that of z, and passing through the centre
of its interior.
Suppose now the motion to be very small, so that the square
of the velocity may be neglected. Then, p denoting the part of
the pressure due to the motion, we shall have p = p d<f>/dt.
Also in finding d$/dt we may suppose the axes to be fixed in
192 SUPPLEMENT TO A MEMOIR
space, since by taking account of their motion we should only
introduce terms depending on the square of the velocity. In fact,
if for the sake of distinction we denote the coordinates of a
fluid particle referred to the moveable axes by x, y, while a?, y
denote its coordinates referred to axes fixed in space, which
after differentiation with respect to t we may suppose to coincide
with the moveable axes at the instant considered, and if we
denote the differential coefficient of </> with respect to t by (d(f>/dt)
when x, y, t are the independent variables, and by d(f)/dt when
x , y, t are the independent variables, we shall have
(d(f)\ dcf> d(f> dx d(f> dy d(f> dx rfy *
\dtj dt dx dt dy dt ~ dt dt dt
for d<l>/dx, dcf)/dy mean absolutely the same as d<f>/dx, dcfr/dy, and
are therefore equal to u, v respectively. Now dx/dt, dy /dt, de
pending on the motion of the axes, are small quantities of the
order co ; their values are in fact coy, cox ; so that, omitting
small quantities of the order &&gt; 2 , we have
"dt
We shall therefore find the value of p from that of </> by merely
writing pdco/dt for o>. In order to determine the motion of
the box it will be necessary to find the resultant of the fluid
pressures on its several faces. As shewn in my former paper,
these pressures will have no resultant force, but only a resultant
couple, of which the axis will evidently be parallel to that of z.
In calculating this couple, it is immaterial whether we take the
moments about the axis of z, or about a line parallel to it passing
through the centre of the parallelepiped : suppose that we adopt
the latter plan. If we reckon the couple positive when it tends
to turn the box from the axis of x to that of y we shall evidently
have I I p y =Q ( x  ) dxdz for the part arising from the
J o Jo \ 2t)
* It may be very easily proved by means of this equation, combined with the
general equation which determines^, that whether the velocity be great or small
the fluid will have the same effect on the motion of the box as the solid of which the
moment of inertia is determined in this paper on the supposition that the motion
is small.
ON SOME CASES OF FLUID MOTION. 193
rb re / i\
pressure on the face xz, and p 9 *[y  5 J dydz for the part
Jo J o \ &/
arising from the pressure on the face ya. It is easily seen from
(4) that^ =a = ^ a . =0 , and py= b = p y = , so that the couples due
to the pressures on the faces xz t yz are equal to the couples due
to the pressures on the opposite faces respectively. In order,
therefore, to find the whole couple we have only got to double
the part already found. As the integrations do not present the
slightest difficulty, it will be sufficient to write down the result.
It will be found that the whole couple is equal to Cdw/dt,
where
This expression has been simplified after integration by putting
for S 1/n* its value 7r 4 /96.
It appears then that the effect of the inertia of the fluid is
to increase the moment of inertia of the box about an axis passing
through its centre and parallel to the edge c by the quantity C.
In equation (40) of my former paper, there is given an expression
for C which is apparently very different from that given by (5),
but the numerical values of the two expressions are necessarily
the same. If we denote the moment of inertia of the fluid sup
posed to be solidified by C,, we shall have C t = pabc (a 2 + 6 2 )/12 ;
and if we put
and treat (5) as equation (40) of my former paper was treated we
shall find
f(r) = (1 + r 2 ) 1 {1  3r 2 + 2r 3 (1.260497  1.254821 2 1 versin 26> n )}
fi
.................. (6),
where tab. log tan 6 n = 10  .6821882  .
The equation (6) is true, (except as regards the decimals
omitted,) whatever be the value of r; but for convenience of
calculation it will be proper to take r less than 1, that is, to
choose for a the smaller of the two a, b. The value of/(r) given
by (6) is apparently very different from that given at the bottom
s  13
194
SUPPLEMENT TO A MEMOIR
of page 64 of my former paper, but any one may easily satisfy
himself as to equivalence of the two expressions by assigning
to r a value at random, and calculating the value of f(r) from
the two expressions separately. The expression (6) is however
preferable to the other, especially when we have to calculate the
value of f (r) for small values of r. The infinite series contained
in (6) converges with such rapidity that in the most unfavourable
case, that is, when r = 1 nearly, the omission of all terms after the
first would only introduce an error of about .000003 in the value
of/(r).
For the sake of shewing the manner in which f (r) alters
with r, I have calculated the following values of the function.
The expression (6) shews that / (r) = 0, when r = 0; and f (r)
is also =0 when r l, since /(] =f(r).
r
f(r)
?
f(r]
0.0
1
0.6
0.3374
0.1
0.9629
0.7
0.2521
0.2
0.8655
0.8
0.1958
0.3
0.7322
0.9
0.1655
0.4
0.5873
1
0.1565
0.5
0.4512
The experiments to which I have alluded were made with a
wooden box measuring inside 8 inches by 4 square. The box
weighed not quite 1 lb., and contained about 4J Ibs. of water,
so that the inertia of the water which had to be overcome was
by no means small compared with that of the box. The box
was suspended by two parallel threads 3 inches apart and between
4 and 5 feet long : it was twisted a little, and then left to itself,
so that it oscillated about a vertical axis midway between the
threads. The points of attachment of the threads were in a line
drawn through the centre of the upper face parallel to one of its
sides, and were equidistant from the centre. The weight of the
box when empty, the length and distance of the threads, the time
ON SOME CASES OF FLUID MOTION. 195
of oscillation, and the known length of the seconds pendulum
are data sufficient for determining the moment of inertia of the
box about a vertical axis passing through its centre. When the
box is filled with water the same quantities determine the mo
ment of inertia of the box and the water it contains, whence the
moment of inertia of the water alone is obtained by subtraction.
It is supposed here that the centre of gravity of the box coincides
with the centre of gravity of its interior volume. In the following
experiments a different face of the box was uppermost each time.
In Nos. 1 and 2 the long edges of the box were vertical, in Nos. 3
and 4 they were horizontal. In all cases the inertia determined
by experiment was a little greater than that resulting from
theory : the difference will be given in fractional parts of the
latter. The difference was 1/21 in No. 1, 1/13 in No. 2, 1/17 in
No. 3, and 1/21 in No. 4. On referring to the table at the end
of the last paragraph, it will be seen that the ratio of the moment
of inertia of the fluid to what it would be if the fluid were solid
is about three times as great in the last two experiments as in
the first two.
I had expected beforehand to find the inertia determined by
experiment a little greater than that given by theory, for this
reason. In the theory, it is supposed that both the fluid itself
and the surface of the box are perfectly smooth. This however
is not strictly true. The box by its roughness exerts a tangential
force on the fluid immediately in contact with it, and this force
produces an effect on the fluid at a small distance from the surface
of the box, in consequence of the internal friction of the fluid
itself. We may conceive the effect of this force on the time of
oscillation in a general way by supposing a thin film of fluid
close to the surface of the box to be dragged along with it. Con
sequently, the moment of inertia determined by experiment will
be a little greater than it would have been had the fluid and
the surface of the box been perfectly smooth.
These experiments are sufficient to shew that in the case of
a vessel of about the size and shape of the one I used, filled
with water, and performing small oscillations of the duration of
about one second (as was the case in my experiments), the time
of oscillation is not much increased by friction; at least, if we
suppose, as there is reason for supposing, that the effect of friction
132
196 SUPPLEMENT TO A MEMOIR ON FLUID MOTION.
does not depend on the nature of the surface of the box. They
are not however sufficiently exact to allow us to place any reliance
on the accuracy of the small differences between the results of
experiment, and of the common theory of fluid motion, and con
sequently they are useless as tests of any theory of friction.
[From the Transactions of the Cambridge Philosophical Society,
Vol. vni. p. 441.]
ON THE THEORY OF OSCILLATORY WAVES.
[Read March 1, 1847.]
IN the Report of the Fourteenth Meeting of the British
Association for the Advancement of Science it is stated by Mr
Russell, as a result of his experiments, that the velocity of pro
pagation of a series of oscillatory waves does not depend on the
height of the waves*. A series of oscillatory waves, such as that
observed by Mr Russell, does not exactly agree with what it is
most convenient, as regards theory, to take as the type of oscil
latory waves. The extreme waves of such a series partake in
some measure of the character of solitary waves, and their height
decreases as they proceed. In fact it will presently appear that
it is only an indefinite series of waves which possesses the pro
perty of being propagated with a uniform velocity, and without
change of form : at least this is the case when the waves are
such as can be propagated along the surface of a fluid which was
previously at rest. The middle waves, however, of a series such
as that observed by Mr Russell agree very nearly with oscillatory
waves of the standard form. Consequently, the velocity of pro
pagation determined by the observation of a number of waves,
according to Mr Russell s method, must be very nearly the same
as the velocity of propagation of a series of oscillatory waves of
the standard form, and whose length is equal to the mean length
of the waves observed, which are supposed to differ from each
other but slightly in length.
* Page 369 (note), and page 370.
198 ON THE THEORY OF OSCILLATORY WAVES.
On this account I was induced to investigate the motion of
oscillatory waves of the above form to a second approximation,
that is, supposing the height of the waves finite, though small.
I find that the expression for the velocity of propagation is in
dependent of the height of the waves to a second approximation.
With respect to the form of the waves, the elevations are no
longer similar to the depressions, as is the case to a first ap
proximation, but the elevations are narrower than the hollows,
and the height of the former exceeds the depth of the latter.
This is in accordance with Mr Russell s remarks at page 448 of
his first Report*. I have proceeded to a third approximation
in the particular case in which the depth of the fluid is very
great, so as to find in this case the most important term, de
pending on the height of the waves, in the expression for the
velocity of propagation. This term gives an increase in the
velocity of propagation depending on the square of the ratio of
the height of the waves to their length.
There is one result of a second approximation which may
possibly be of practical importance. It appears that the forward
motion of the particles is not altogether compensated by their
backward motion ; so that, in addition to their motion of oscil
lation, the particles have a progressive motion in the direction
of propagation of the waves. In the case in which the depth of
the fluid is very great, this progressive motion decreases rapidly
as the depth of the particle considered increases. Now when a
ship at sea is overtaken by a storm, and the sky remains overcast,
so as to prevent astronomical observations, there, is nothing to
trust to for finding the ship s place but the dead reckoning. But
the estimated velocity and direction of motion of the ship are
her velocity and direction of motion relatively to the water. If
then the whole of the water near the surface be moving in the
direction of the waves, it is evident that the ship s estimated
place will be erroneous. If, however, the velocity of the water
can be expressed in terms of the length and height of the waves,
both which can be observed approximately from the ship, the
motion of the water can be allowed for in the dead reckoning.
As connected with this subject, I have also considered the
motion of oscillatory waves propagated along the common surface
of two liquids, of which one rests on the other, or along the upper
* Reports of the British Association, Vol. vi.
ON THE THEOKY OF OSCILLATORY WAVES. 199
surface of the upper liquid. In this investigation there is no
object in going beyond a first approximation. When the specific
gravities of the two fluids are nearly equal, the waves at their
common surface are propagated so slowly that there is time to
observe the motions of the individual particles. The second case
affords a means of comparing with theory the velocity of pro
pagation of oscillatory waves in extremely shallow water. For by
pouring a little water on the top of the mercury in a trough we
can easily procure a sheet of water of a small, and strictly uniform
depth, a depth, too, which can be measured with great accuracy
by means of the area of the surface and the quantity of water
poured in. Of course, the common formula for the velocity of
propagation will not apply to this case, since the motion of the
mercury must be taken into account.
1. In the investigations which immediately follow, the fluid
is supposed to be homogeneous and incompressible, and its depth
uniform. The inertia of the air, and the pressure due to a column
of air whose height is comparable with that of the waves are also
neglected, so that the pressure at the upper surface of the fluid
may be supposed to be zero, provided we afterwards" add the at
mospheric pressure to the pressure so determined. The waves
which it is proposed to investigate are those for which the motion
is in two dimensions, and which are propagated with a constant
velocity, and without change of form. It will also be supposed
that the waves are such as admit of being excited, independently of
friction, in a fluid which was previously at rest. It is by these
characters of the waves that the problem will be rendered de
terminate, and not by the initial disturbance of the fluid, supposed
to be given. The common theory of fluid motion, in which the
pressure is supposed equal in all directions, will also be em
ployed.
Let the fluid be referred to the rectangular axes of x, y, z,
the plane xz being horizontal, and coinciding with the surface
of the fluid when in equilibrium, the axis of y being directed
downwards, and that of x taken in the direction of propagation
of the waves, so that the expressions for the pressure, &c. do not
contain z. Let p be the pressure, p the density, t the time, u, v
the resolved parts of the velocity in the directions of the axes
200 ON THE THEORY OF OSCILLATORY WAVES.
of x, y ; g the force of gravity, h the depth of the fluid when in
equilibrium. From the character of the waves which was men
tioned last, it follows by a known theorem that udx + vdy is an
exact differential d(p. The equations by which the motion is to
be determined are well known. They are
= 0, wheny = A .................... (3);
_ 0)Wl
where (3) expresses the condition that the particles in contact with
the rigid plane on which the fluid rests remain in contact with
it, and (4) expresses the condition that the same surface of par
ticles continues to be the free surface throughout the motion,
or, in other words, that there is no generation or destruction of
fluid at the free surface.
If c be the velocity of propagation, u, v and p will be by
hypothesis functions of x ct and y. It follows then from the
equations u dfy/dx, v = dfyjdy and (1), that the differential
coefficients of (f> with respect to x, y and t will be functions of
x ct and y ; and therefore <f> itself must be of the form
f(xct, y)+Ct.
The last term will introduce a constant into (1) ; and if this
constant be expressed, we may suppose </> to be a function of
x ct and y. Denoting x ct by x , we have
dp _ dp dp _ dp
dx~d^ ft ~dt~ da/
and similar equations hold good for ^>. On making these sub
stitutions in (1) and (4), omitting the accent of x, and writing
gk for (7, we have
. c + = 0, wbenpO (G).
dx dy dy
ON THE THEORY OF OSCILLATORY WAVES. 201
Substituting in (6) the value of p given by (5), we have
d$_ ff (dff$ ,d d^\
J dy da? ~* {dot do? * dy dxdyl
_
dxdx* dxdydxdy \dydf~
......... (8).
The equations (7) and (8) are exact; but if we suppose the
motion small, and proceed to the second order only of approxima
tion, we may neglect the last three terms in (7), and we may
easily eliminate y between (7) and (8). For putting < , <,, &c.
for the values of d(f)/dx, dfy/dy, &c. when y = 0, the number of
accents above marking the order of the differential coefficient
with respect to x, and the number below its order with respect
to y, and observing that & is a small quantity of the first order
at least, we have from (8)
g (y + fc) + c (f + </>  i[ (f + <#>/) = 0,
whence y = *jU +%/(&+ f\ + _L
y y \ y * y
Substituting the first approximate value of y in the first two
terms of (7), putting # = in the next two, and reducing, we
have
+2c
<f> will now have to be determined from the general equation (2)
with the particular conditions (3) and (10). When $ is known,
?/, the ordinate of the surface, will be got from (9), and k will
then be determined by the condition that the mean value of y
shall be zero. The value of p, if required, may then be obtained
from (5).
2. In proceeding to a first approximation we have the equa
tions (2), (3) and the equation obtained by omitting the small
terms in (10), namely,
* The reader will observe that the y in this equation is the ordinate of the
surface, whereas the y in (1) and (2) is the ordinate of any point in the fluid. The
context will always shew in which sense y is employed.
202 ON THE THEORY OF OSCILLATORY WAVES.
The general integral of (2) is
the sign S extending to all values of A, m and n, real or imagi
nary, for which m 2 + n 2 = : the particular values of </>, Cx + G ,
Dy + D , corresponding respectively to n = Q, ra = 0, must also be
included, but the constants C , D may be omitted. In the
present case, the expression for < must not contain real ex
ponentials in x, since a term containing such an exponential
would become infinite either for x = GO , or for x = + oo , as well
as its differential coefficients which would appear in the ex
pressions for u and v; so that m must be wholly imaginary.
Replacing then the exponentials in x by circular functions, we
shall have for the part of < corresponding to any one value
of m,
(Ae mv +^ e~ wy ) sin mx + (Bt mv + B e~ mv ) cos mx,
and the complete value of < will be found by taking the sum of
all possible particular values of the above form and of the par
ticular value Cx + Dy. When the value so formed is substituted
in (3), which has to hold good for all values of x, the coefficients
of the several sines and cosines, and the constant term must be
separately equated to zero. We have therefore
D = 0, A = e* mh A, B = &*B ;
so that if we change the constants we shall have
<p = Cue + S (e m ( h ~rt + e~ m ( h rt] (A sin mx + B cos mx)...(12),
the sign S extending to all real values of m, A and B, of which
in may be supposed positive.
3. To the term Cx in (12) corresponds a uniform velocity
parallel to x, which may be supposed to be impressed on the
fluid in addition to its other motions. If the velocity of pro
pagation be defined merely as the velocity with which the wave
form is propagated, it is evident that the velocity of propagation
is perfectly arbitrary. For, for a given state of relative motion
of the parts of the fluid, the velocity of propagation, as so defined,
can be altered by altering the value of C. And in proceeding to
the higher orders of approximation it becomes a question what
we shall define the velocity of propagation to be. Thus, we might
define it to be the velocity with which the wave form is propa
ON THE THEORY OF OSCILLATORY WAVES. 203
gated when the mean horizontal velocity of a particle in the
upper surface is zero, or the velocity of propagation of the wave
form when the mean horizontal velocity of a particle at the
bottom is zero, or in various other ways. The following two
definitions appear chiefly to deserve attention.
First, we may define the velocity of propagation to be the
velocity with which the wave form is propagated in space, when
the mean horizontal velocity at each point of space occupied by the
fluid is zero. The term "mean" here refers to the variation of
the time. This is the definition which it will be most convenient
to employ in the investigation. I shall accordingly suppose (7=0
in (12), and c will represent the velocity of propagation according
to the above definition.
Secondly, we may define the velocity of propagation to be the
velocity of propagation of the wave form in space, when the mean
horizontal velocity of the mass of fluid comprised between two
very distant planes perpendicular to the axis of x is zero. The
mean horizontal velocity of the mass means here the same thing
as the horizontal velocity of its centre of gravity. This appears
to be the most natural definition of the velocity of propagation,
since in the case considered there is 110 current in the mass of
fluid, taken as a whole. I shall denote the velocity of propagation
according to this definition by c. In the most important case
to consider, name]y, that in which the depth is infinite, it is
easy to see that c = c, whatever be the order of approximation.
For when the depth becomes infinite, the velocity of the centre
of gravity of the mass comprised between any two planes parallel
to the plane yz vanishes, provided the expression for u contain
no constant term.
4. We must now substitute in (11) the value of <.
</> =2 ( m tt0> + etf) (A siumx + I? cos m#)... (13);
but since (11) has to hold good for all values of #, the coefficients
of the several sines and cosines must be separately equal to zero :
at least this must be true, provided the series contained in (11)
are convergent. The coefficients will vanish for any one value
of m, provided
fj gm _ ~
t *!
204 ON THE THEORY OF OSCILLATORY WAVES.
Putting for shortness 2mh = p, we have
^logc a = _l 2
dp fl e*  *
which is positive or negative, p being supposed positive, according
as
and is therefore necessarily negative. Hence the value of c given
by (14) decreases as jj, or m increases, and therefore (11) cannot
be satisfied, for a given value of c, by more than one positive
value of m. Hence the expression for <f> must contain only one
value of m. Either of the terms A cos mcc, B sin mx may be
got rid of by altering the origin of sc. We may therefore take,
for the most general value of <,
< = ^(e w ^^+e^2/>)sinra^ (15).
Substituting in (8), we have for the ordinate of the surface
D? A (*
y = ( h + e h )cosmx (16),
u
k being = 0, since the mean value of y must be zero. Thus
everything is known in the result except A and m, which are
arbitrary.
5. It appears from the above, that of all waves for which
the motion is in two dimensions, which are propagated in a fluid
of uniform depth, and which are such as could be propagated
into fluid previously at rest, so that udx + vdy is an exact differ
ential, there is only one particular kind, namely, that just con
sidered, which possesses the property of being propagated with
a constant velocity, and without change of form ; so that a
solitary wave cannot be propagated in this manner. Thus the
degradation in the height of such waves, which Mr Russell ob
served, is not to be attributed wholly, (nor I believe chiefly,) to
the imperfect fluidity of the fluid, and its adhesion to the sides
and bottom of the canal, but is an essential characteristic of a
solitary wave. It is true that this conclusion depends on an
investigation which applies strictly to indefinitely small motions
only : but if it were true in general that a solitary wave could be
propagated uniformly, without degradation, it would be true in
ON THE THEORY OF OSCILLATORY WAVES. 205
the limiting case of indefinitely small motions ; and to disprove a
general proposition it is sufficient to disprove a particular case.
6. In proceeding to a second approximation we must sub
stitute the first approximate value of <j>, given by (15), in the
small terms of (10). Observing that k = to a first approximation,
and eliminating g from the small terms by means of (14), we
find
9<t>,c 2 <j>"  6A*m*c sin 2mx= ............ (17).
The general value of < given by (13), which is derived from (2)
and (3), must now be restricted to satisfy (17). It is evident that
no new terms in < involving sin mx or cos mx need be introduced,
since such terms may be included in the first approximate value,
and the only other term which can enter is one of the form
Substituting this term in (17), and simplifying by means of (14),
we find
~ C (
Moreover since the term in <p containing sin. ma must disappear
from (17), the equation (14) will give c to a second approxi
mation.
If we denote the coefficient of cosmic in the first approximate
value of y, the ordinate of the surface, by a, we shall have
A _ go* ca
me (e mh + e ~ mh ) ( 6 w&_ e mh) >
and substituting this value of A in that of <j>, we have
e m(h  y) __ 6  m(h  y) 2m(h  y) __ e  2m(h  y)
=   sin
...... (18).
The ordinate of the surface is given to a second approximation
by (9). It will be found that
I e mh\ ( f 2mh i f 2mh _i_ /f\
ma
7. The equation to the surface is of the form
mx Ka? cos 2w# (20),
206 ON THE THEORY OF OSCILLATORY WAVES.
where K is necessarily positive, and a may be supposed to be
positive, since the case in which it is negative may be reduced to
that in which it is positive by altering the origin of x by the
quantity TT/W or X/2, X being the length of the waves. On re
ferring to (20) we see that the waves are symmetrical with respect
to vertical planes drawn through their ridges, and also with
respect to vertical planes drawn through their lowest lines. The
greatest depression of the fluid occurs when % = or = + X, &c.,
and is equal to aa?K: the greatest elevation occurs when
# = X/2 or = + 3X/2, &c., and is equal to a + a*K. Thus the
greatest elevation exceeds the greatest depression by Za*K. When
the surface cuts the plane of mean level, cos mx aKcos 2mx = 0.
Putting in the small term in this equation the approximate value
mx = 7T/2, we have cos mx  aK= cos (?r/2 + aK], whence
x = + (x/4 + a/a/2<7r), = (5X/4 + aKX/Zir), &c.
We see then that the breadth of each hollow, measured at the
height of the plane of mean level, is X/2 + aK\/7r, while the
breadth of each elevated portion of the fluid is X/2  aK\/7r.
It is easy to prove from the expression for K, which is given
in (19), that for a given value of X or of m, K increases as h
decreases. Hence the difference in form of the elevated and
depressed portions of the fluid is more conspicuous in the case
in which the fluid is moderately shallow than in the case in
which its depth is very great compared with the length of the
waves.
8. When the depth of the fluid is very great Compared with
the length of a wave, we may without sensible error suppose h to
be infinite. This supposition greatly simplifies the expressions
already obtained. W T e have in this case
sin mx .................................... (21),
y =a cos mx
m TT
(22),
the y in (22) being the ordinate of the surface.
It is hardly necessary to remark that the state of the fluid at
any time will be expressed by merely writing xct in place of x
in all the preceding expressions.
ON THE THEORY OF OSCILLATORY WAVES. 207
9. To find the nature of the motion of the individual par
ticles, let x + f be written for x, y + 77 for y, and suppose x and y
to be independent of t, so that they alter only in passing from one
particle to another, while f and 77 are small quantities depending
on the motion. Then taking the case in which the depth is in
finite, we have
~sau 7wace~ m ^ +1? )cosm(# + f ct) mace~ my cosm(# ct)
dt
+ m?ac~ my sin m(x ct).% + m*ace~ my cos m (x ct) . 77, nearly,
^ = v = mace~ m(  y+ ^ sin m (x + f ctf) = mace~ my sin 7?2(# c)
+ m*ace~ my cos m(x ct).% m 2 ace~ my sin m(# c) . 77, nearly.
To a first approximation
% = ae~ my sin w (a? ct), rj = ae~ my cos m (x ct),
the arbitrary constants being omitted. Substituting these values
in the small terms of the preceding equations, and integrating
again, we have
= ae~ my sin m(x ct) + m*a?cte~ 2my ,
77 = ae~ my cos m(x ct).
Hence the motion of the particles is the same as to a first
approximation, with one important difference, which is that in
addition to the motion of oscillation the particles are transferred
forwards, that is, in the direction of propagation, with a constant
velocity depending on the depth, and decreasing rapidly as the
depth increases. If U be this velocity for a particle whose depth
below the surface in equilibrium is y, we have
re?. (23).
The motion of the individual particles may be determined in
a similar manner when the depth is finite from (18). In this case
the values of f and r t contain terms of the second order, involving
respectively sin 2m (x ct) and cos %m(x ct), besides the term in
f which is multiplied by t. The most important thing to consider
is the value of U, which is
h) \ e 1m(yli)
_ mh (24).
208 ON THE THEORY OF OSCILLATORY WAVES.
Since U is a small quantity of the order a 2 , and in proceeding
to a second approximation the velocity of propagation is given to
the order a only, it is immaterial which of the definitions of velo
city of propagation mentioned in Art. 3 we please to adopt.
10. The waves produced by the action of the wind on the
surface of the sea do not probably differ very widely from those
which have just been considered, and which may be regarded as
the typical form of oscillatory waves. On this supposition the
particles, in addition to their motion of oscillation, will have a
progressive motion in the direction of propagation of the waves,
and consequently in the direction of the wind, supposing it not to
have recently shifted, and this progressive motion will decrease
rapidly as the depth of the particle considered increases. If the
pressure of the air on the posterior parts of the waves is greater
than on the anterior parts, in consequence of the wind, as un
questionably it must be, it is easy to see that some such progres
sive motion must be produced. If then the waves are not break
ing, it is probable that equation (23), which is applicable to deep
water, may give approximately the mean horizontal velocity of
the particles ; but it is difficult to say how far the result may be
modified by friction. If then we regard the ship as a mere parti
cle, in the first instance, for the sake of simplicity, and put f/ for
the value of U when y = 0, it is easy to see that after sailing for
a time t, the ship must be a distance U Q t to the lee of her estimated
place. It will not however be sufficient to regard the ship as a
mere particle, on account of the variation of the factor e~ 2w % as y
varies from to the greatest depth of the ship below the surface
of the water. Let 8 be this depth, or rather a depth something
less, in order to allow for the narrowing of the ship towards the
keel, and suppose the effect of the progressive motion of the water
on the motion of the ship to be the same as if the water were
moving with a velocity the same as all depths, and equal to the
mean value of the velocity U from y = to y = 8. If U l be this
mean velocity,
ma?c
On this supposition, if a ship be steered so as to sail in a direc
tion making an angle 6 with the direction of the wind, supposing
the water to have no current, and if F be the velocity with which
ON THE THEORY OF OSCILLATORY WAVES. 209
the ship moves through the water, her actual velocity will be the
resultant of a velocity V in the direction just mentioned, which,
for shortness, I shall call the direction of steering, and of a velocity
Z7 X in the direction of the wind. But the ship s velocity as esti
mated by the logline is her velocity relatively to the water at the
surface, and is therefore the resultant of a velocity V in the direc
tion of steering, and a velocity U U t in a direction opposite to
that in which the wind is blowing. If then E be the estimated
velocity, and if we neglect U 2 ,
But the ship s velocity is really the resultant of a velocity V+
in the direction of steering, and a velocity U l sin 6 in the perpen
dicular direction, while her estimated velocity is E in the direction
of steering. Hence, after a time t, the ship will be a distance
U t cos 6 ahead of her estimated place, and a distance Uj sin 6
aside of it, the latter distance being measured in a direction per
pendicular to the direction of steering, and on the side towards
which the wind is blowing.
I do not suppose that the preceding formula can be employed
in practice ; but I think it may not be altogether useless to call
attention to the importance of having regard to the magnitude
and direction of propagation of the waves, as well as to the wind,
in making the allowance for leeway.
11. The formula of Art. 6 are perfectly general as regards the
ratio of the length of the waves to the depth of the fluid, the only
restriction being that the height of the waves must be sufficiently
small to allow the series to be rapidly convergent. Consequently,
they must apply to the limiting case, in which the waves are sup
posed to be extremely long. Hence long waves, of the kind con
sidered, are propagated without change of form, and the velocity
of propagation is independent of the height of the waves to a
second approximation. These conclusions might seem, at first
sight, at variance with the results obtained by Mr Airy for the
case of long waves *. On proceeding to a second approximation,
Mr Airy finds that the form of long waves alters as they proceed,
and that the expression for the velocity of propagation contains a
* Encyclopedia Metropolitana, Tides and Waves, Articles 198, &c.
S. 14
210 ON THE THEORY OF OSCILLATORY WAVES.
term depending on the height of the waves. But a little attention
will remove this apparent discrepancy. If we suppose mh very
small in (19), and expand, retaining only the most important
terms, we shall find for the equation to the surface
3a 2
a cos mx . r cos zmx.
Now, in order that the method of approximation adopted may be
legitimate, it is necessary that the coefficient of cos Zmx in this
equation be small compared with a. Hence a/m z h 3 , and therefore
X 2 a/A 3 , must be small, and therefore a/h must be small compared
with (h/\y. But the investigation of Mr Airy is applicable to the
case in which \/h is very large ; so that in that investigation a/h
is large compared with (/t/\) 2 . Thus the difference in the results
obtained corresponds to a difference in the physical circumstances
of the motion.
12. There is no difficulty in proceeding to the higher orders
of approximation, except what arises from the length of the for
mulas. In the particular case in which the depth is considered
infinite, the formulae are very much simpler than in the general
case. I shall proceed to the third order in the case of an infinite
depth, so as to find in that case the most important term, depend
ing on the height of the waves, in the expression for the velocity
of propagation.
For this purpose it will be necessary to retain the terms of
the third order in the expansion of (7). Expanding this equation
according to powers of y, and neglecting terms of the fourth, &c.
orders, we have
t  c 2 </>,") y + (g^r <ty,,") + 2c (f f
(25).
In the small terms of this equation we must put for <f> and y
their values given by (21) and (22) respectively. Now since the
value of <f> to a second approximation is the same as its value to a
first approximation, the equation g$ C 2 <"=0 is satisfied to terms
of the second order. But the coefficients of y and y 2 / 2 , ^ n tne
first line of (25), are derived from the lefthand member of the
ON THE THEORY OF OSCILLATORY WAVES. 211
preceding equation by inserting the factor G~ m v, differentiating
either once or twice with respect to y, and then putting y = 0.
Consequently these coefficients contain no terms of the second
order, and therefore the terms involving y in the first line of (25)
are to be neglected. The next two terms are together equal to
But
which does not contain oc, so that these two terms disappear. The
coefficient of y in the second line of (25) may be derived from the
two terms last considered in the manner already indicated, and
therefore the terms containing y will disappear from (25). The
only small terms remaining are the last three, and it will easily
be found that their sum is equal to raVc 3 sin mx, so that (25) be
comes
#(/>, c 2 </> + ra Vc 3 sin m# = ..................... (26).
The value of < will evidently be of the form Ae~ my sin mx. Sub
stituting this value in (26), we have
(mV  mg} A + mVc 3 = 0.
Dividing by mA, and putting for A and c 2 their approximate values
ac, g/m respectively in the small term, we have
g + mV<7,
The equation to the surface may be found without difficulty. It
is
y = a cos mx J ma 2 cos 2mx + f mV cos Smx* ......... (27) :
we have also
k = 0, < = ac (1 fmV)
* It is remarkable that this equation coincides with that of the prolate cycloid,
if the latter equation be expanded according to ascending powers of the distance of
the tracing point from the centre of the rolling circle, and the terms of the fourth
order be omitted. The prolate cycloid is the form assigned by Mr Russell to waves
of the kind here considered. Reports of the British Association, Vol. vi. p. 448.
When the depth of the fluid is not great compared with the length of a wave, the
form of the surface does not agree with the prolate cycloid even to a second
approximation.
142
212 ON THE THEORY OF OSCILLATORY WAVES.
The following figure represents a vertical section of the waves
propagated along the surface of deep water. The figure is drawn
for the case in which a = . The term of the third order in (27)
is retained, but it is almost insensible. The straight line represents
a section of the plane of mean level.
13. If we consider the manner in which the terms introduced
by each successive approximation enter into equations (7) and (8),
we shall see that, whatever be the order of approximation, the
series expressing the ordinate of the surface will contain only
cosines of mx and its multiples, while the expression for <f> will
contain only sines. The manner in which y enters into the
coefficient of cos rmx in the expression for <f> is determined in the
case of a finite depth by equations (2) and (3). Moreover, the
principal part of the coefficient of cos rmx or sin rmx will be of
the order a r at least. We may therefore assume
< = T>,"a r A r ("<*> + ermUivY) sm rmXt
y = a cos mx + 2 2 a r .Z? r cos rmx,
and determine the arbitrary coefficients by means of equations
(7) and (8), having previously expanded these equations according
to ascending powers of y. The value of c 2 will be determined by
equating to zero the coefficient of sin mx in (7).
Since changing the sign of a comes to the same thing as
altering the origin of x by \ X, it is plain that the expressions
for A r , B r and c 2 will contain only even powers of a. Thus
the values of each of these quantities will be of the form
o o+ cx + cx + 
It appears also that, whatever be the order of approximation,
the waves will be symmetrical with respect to vertical planes
passing through their ridges, as also with respect to vertical planes
passing through their lowest lines.
14 Let us consider now the case of waves propagated at
the common surface of two liquids, of which one rests on the
ON THE THEORY OF OSCILLATORY WAVES. 213
other. Suppose as before that the motion is in two dimensions,
that the fluids extend indefinitely in all horizontal directions,
or else that they are bounded by two vertical planes parallel to
the direction of propagation of the waves, that the waves are
propagated with a constant velocity, and without change of form,
and that they are such as can be propagated into, or excited
in, the fluids supposed to have been previously at rest. Suppose
first that the fluids are bounded by two horizontal rigid planes.
Then taking the common surface of the fluids when at rest for
the plane xz, and employing the same notation as before, we
have for the under fluid
= wheny=a ................ (29),
neglecting the squares of small quantities. Let h / be the depth
of the upper fluid when in equilibrium, and let p t , p,, <f>,, C, be
the quantities referring to the upper fluid which correspond to
P, p> & referring to the under : then we have for the upper
fluid
d* df  ( 3 )>
^P = when y = h, (31),
We have also, for the condition that the two fluids shall not
penetrate into, nor separate from each other,
Lastly, the condition answering to (11) is
he
214 ON THE THEORY OF OSCILLATORY WAVES.
Since C C is evidently a small quantity of the first order at
least, the condition is that (33) shall be satisfied when # = 0.
Equation (34) will then give the ordinate of the common surface
of the two liquids when y is put = in the last two terms.
The general value of <j> suitable to the present case, which
is derived from (28) subject to the condition (29), is given by (13)
if we suppose that the fluid is free from a uniform horizontal
motion compounded with the oscillatory motion expressed by (18).
Since the equations of the present investigation are linear, in
consequence of the omission of the squares of small quantities,
it will be sufficient to consider one of the terms in (13). Let
then
m(h ^smmx ......... (35).
The general value of <f> t will be derived from (13) by merely
writing h l for h. But in order that (32) may be satisfied, the
value of </> y must reduce itself to a single term of the same form
as the second side of (35). We may take then for the value
offc
y = A,(e t(h + rt + e m V t + v } )&in.mx ............ (36).
Putting for shortness
and taking $,, D, to denote the quantities derived from 8, D by
writing A, for h, we have from (32)
DA + D 4 A t = ...................... (37),
and from (33)
P (gDmc*S)Aip l (gD, + mc*S)A, = ........ (38).
Eliminating A and A t from (37) and (38), we have
The equation to the common surface of the liquids will be
obtained from (34). Since the mean value of y is zero, we have
in the first place
C = C .................................. (40).
We have then, for the value of y,
mx ............................. (41),
ON THE THEORY OF OSCILLATORY WAVES. 215
where
_

g pp,
Substituting in (35) and (36) the values of A and A f derived from
(37) and (42), we have
= ^ (* > +*>) sin wa? ....... .,...(43),
rt *
(44).
Equations (39), (40), (41), (43) and (44) contain the solution
of the problem. It is evident that C remains arbitrary. The
values of p and p t may be easily found if required.
If we differentiate the logarithm of c 2 with respect to m, and
multiply the result by the product of the denominators, which
are necessarily positive, we shall find a quantity of the form
Pp+P t p,, where P and P t do not contain p or p r It may be
proved in nearly the same manner as in Art. 4, that each of the
quantities P, P t is necessarily negative. Consequently c will
decrease as m increases, or will increase with X. It follows from
this that the value of </> cannot contain more than two terms,
one of the form (35), and the other derived from (35) by replacing
sin mx by cos mx, and changing the constant A : but the latter
term may be got rid of by altering the origin of x.
The simplest case to consider is that in which both h and ti
are regarded as infinite compared with X. In this case we have
<j> =  ace ~ y sin mx, <, = ace my sin mx,
P P Q
c 2 = r \LL <L. y a cos mx,
p + p,m
the latter being the equation to the surface.
15. The preceding investigation applies to two incompressible
fluids, but the results are applicable to the case of the waves
propagated along the surface of a liquid exposed to the air, pro
vided that in considering the effect of the air we neglect terms
which, in comparison with those retained, are of the order of
the ratio of the length of the waves considered to the length of
216 ON THE THEORY OF OSCILLATORY WAVES.
a wave of sound of the same period in air. Taking then p for
the density of the liquid, p t for that of the air at the time, and
supposing h t = oo , we have
If we had considered the buoyancy only of the air, we should
have had to replace g in the formula (14) by  g* We should
have obtained in this manner
t_s_(?PJV
m pS mS\
/A
p)
Hence, in order to allow for the inertia of the air, the correction
for buoyancy must be increased in the ratio of 1 to 1 + D/S.
The whole correction therefore increases as the ratio of the length
of a wave to the depth of the fluid decreases. For very long
waves the correction is that due to buoyancy alone, while in
the case of very short waves the correction for buoyancy is
doubled. Even in this case the velocity of propagation is altered
by only the fractional part pjp of the whole ; and as this quantity
is much less than the unavoidable errors of observation, the effect
of the air in altering the velocity of propagation may be neglected.
16. There is a discontinuity in the density of the fluid mass
considered in Art. 14, in passing from one fluid into the other;
and it is easy to shew that there is a corresponding discontinuity
in the velocity. If we consider two fluid particles in contact
with each other, and situated on opposite sides of the surface
of junction of the two fluids, we see that the velocities of these
particles resolved in a direction normal to that surface are the
same ; but their velocities resolved in a direction tangential to
the surface are different. These velocities are, to the order of
approximation employed in the investigation, the values of d<j>/dx
and dfyjdcc when y = 0. We have then from (43) and (44),
for the velocity with which the upper fluid slides along the
under,
8 S\
mac I W f =: cos moc.
ON THE THEORY OF OSCILLATORY WAVES. 21?
17. When the upper surface of the upper fluid is free, the
equations by which the problem is to be solved are the same
as those of Art. 14, except that the condition (31) is replaced by
=  h > .......... (45);
and to determine the ordinate of the upper surface, we have
where y is to be replaced by h t in the last term. Let us con
sider the motion corresponding to the value of $ given by (35).
We must evidently have
<, = (A f e m v f B t e m y) sin ma?,
where A t and B t have to be determined. The conditions (32),
(33) and (45) give
p (gD  mc*S) A+p,(g + mc z ) A, p,(g me 2 ) B t = 0,
(g + me 2 ) e~ mh A t (g me 2 ) e"*>B t = 0.
Eliminating A, A f and B t from these equations, and putting
m
we find
The equilibrium of the fluid being supposed to be stable, we
must have p, < p. This being the case, it is easy to prove that
the two roots of (46) are real and positive. These two roots
correspond to two systems of waves of the same length, which
are propagated with the same velocity.
In the limiting case in which p/p t = oo , (46) becomes
SSf  (8D t + SD) +DD t = 0,
the roots of which are D/8 and D]S t , as they evidently ought
to be, since in this case the motion of the under fluid will not
be affected by that of the upper, and the upper fluid can be in
motion by itself.
When p, = p one root of (46) vanishes, and the other becomes
_ f m(h+h t )
_ ! . The former of these roots cor
88
218 ON THE THEORY OF OSCILLATORY WAVES.
responds to the waves propagated at the common surface of the
fluids, while the latter gives the velocity of propagation belonging
to a single fluid having a depth equal to the sum of the depths
of the two considered.
When the depth of the upper fluid is considered infinite,
we must put DJS, = \ in (46). The two roots of the equation
f f \ 7~)
so transformed are 1 and ^ITTJJ) > the former corresponding to
waves propagated at the upper surface of the upper fluid, and the
latter agreeing with Art. 15.
When the depth of the under fluid is considered infinite, and
that of the upper finite, we must put D/S=l in (46). The two
roots will then become 1 and ^~^ . The value of the
former root shews that whatever be the depth of the upper fluid,
one of the two systems of waves will always be propagated with
the same velocity as waves of the same length at the surface of a
single fluid of infinite depth. This result is true even when the
motion is in three dimensions, and the form of the waves changes
with the time, the waves being still supposed to be such as could
be excited in the fluids, supposed to have been previously at rest,
by means of forces applied at the upper surface. For the most
general small motion of the fluids in this case may be regarded
as the resultant of an infinite number of systems of waves of the
kind considered in this paper. It is remarkable that when the
depth of the upper fluid is very great, the root f = 1 is that which
corresponds to the waves for which the upper fluid is disturbed,
while the under is sensibly at rest; whereas, when the depth of
the upper fluid is very small, it is the other root which corresponds
to those waves which are analogous to the waves which would
be propagated in the upper fluid if it rested on a rigid plane.
When the depth of the upper fluid is very small compared
with the length of a wave, one of the roots of (46) will be very
small ; and if we neglect squares and products of mh i and f , the
equation becomes %pD 2 (p />,) mhfl = 0, whence
(47).
These formulae will not hold good if mh be very small as well as
inh /t and comparable with it, since in that case all the terms of
ON THE THEORY OF OSCILLATORY WAVES. 219
(46) will be small quantities of the second order, mh, being re
garded as a small quantity of the first order. In this case, if we
neglect small quantities of the third order in (46), it becomes
4pf 2  4mp (h + \ ) f + 4 (p  Pt ) ra 2 M, = 0,
whence
(48).
Of these values of c 2 , that in which the radical has the negative
sign belongs to that system of waves to which the formula (47)
apply when Ji t is very small compared with h.
If the two fluids are water and mercury, p/p, is equal to about
13*57. If the depth of the water be very small compared both
with the length of the waves and with the depth of the mercury,
it appears from (47) that the velocity of propagation will be less
than it would have been, if the water had rested on a rigid plane,
in the ratio of 9624 to 1, or 26 to 27 nearly.
APPENDIX.
[A. On the relation of the preceding investigation to a case of wave
motion of the oscillatory kind in which the disturbance can be
expressed in finite terms.
In the Philosophical Transactions for 1863, p. 127, is a paper
by the late Professor Rankine in which he has shewn that it is
possible to express in finite terms, without any approximation,
the motion of a particular class of waves of the oscillatory kind.
It is remarkable that the results for waves of this kind were
given as long ago as in 1802, by Gerstner*, whose investigation
however seems to have been but little noticed for a long time.
This case of motion has latterly attracted a good deal of atten*
tion, partly no doubt from the facility of dealing with it, but
partly, it would seem, from misconceptions as to its intrinsic
importance.
* See Weber s Wcllenlehrc auf Experimente gcgriindet, p. 338,
220 ON THE THEORY OF OSCILLATORY WAVES.
The investigation may be presented in very short compass in
the following manner.
Let us confine our attention to the case of a mass of liquid, re
garded as a perfect fluid of a depth practically infinite, in which
an indefinite series of regular periodic waves is propagated along
the surface, the motion being in two dimensions, and vanishing at
an infinite depth. Taking the plane of motion for the plane of xy,
y being measured vertically downwards, let us seek to express the
actual coordinates x, y of any particle in terms of two parameters
h, k particularising that particle, and of the time t. Let us assume
for trial
x = h + Ksm m(h ct), y = k + Kcos m(h ct) (49),
where m, c are two constants, and K a function of k only. It
will be easily seen that these equations, regarded merely as
expressing the geometrical motion of points, and apart from the
physical possibility of the motion, represent a wave disturbance
of periodic character travelling in the direction of OX with a
velocity of propagation c.
As the disturbance is in two dimensions, we may speak of areas
as representing volumes. Let us consider first the condition of
constancy of the mass. The four loci corresponding to constant
values h, h + dh, k, k + dk, of the two parameters respectively en
close a quadrangular figure which is ultimately a parallelogram,
the area of which must be independent of the time. Now the
area is Sdhdk where
*
~ _ dx dy dx dy
dh dk dk dh
On performing the differentiations we find
S=l + (mk + K )cQ$m(hct) + mKK (51),
where K stands for dKfdk. In order that this may be indepen
dent of the time it is necessary and sufficient that
mK + K = (52),
whence
K= ae mk (52 ^
and
S= 1 m 2 # 2 = 1 mVe 2 ^ (53).
ON THE THEORY OF OSCILLATORY WAVES. 221
The dynamical equations give
dp , (d*x d
gdy + m*c*K (sin in (h ct) dx f cos m (h ct) dy}
= gdy + wV {(xh)d(a)h) + (yk)d(y k)}
+ m*c 2 {(x h)dh+(y k) dk}.
The last line becomes by (49) and (52),
mc 2 {mKsm m(h ct) dh K cos m(h ct) dk},
or mc*d . K cos m (h ct).
The dynamical equations are therefore satisfied, the expression for
dp being a perfect differential, and we have
V {(as  h)* + (y  k) 2 }  mc*K cos m(hct) + C
 mc z ) K cos m(h ct) + G.
It remains to consider the equations of condition at the boun
daries of the fluid. The expression for K satisfies the condition of
giving a disturbance which decreases indefinitely as the depth in
creases, and we have only to see if it be possible to satisfy the
condition at the free surface. Now the particles at the free sur
face differ only by the value of the parameter h, as follows from
the fundamental conception of wave motion, and therefore for some
one value of k we must have p = independently of the time.
This requires that
*=_=&.
m 27T
and if we please to take k = at the surface, and determine C
accordingly, we have
(I **) ......... (54).
Since p is independent of the time, not merely for k = 0, but
for any constant value of k, it follows that when the wave motion
is converted into steady motion by superposing a velocity equal
and opposite to that of propagation, it is not merely the line of
motion or streamline which forms the surface but all the stream
lines that are lines of constant pressure. This is undoubtedly no
necessary property of wavemotion converted into steady motion,
which only requires that the particular streamline at the surface
222 ON THE THEORY OF OSCILLATORY WAVES.
shall be one for which the pressure is constant, though Gerstner
has expressed himself as if he supposed it necessarily true ; it is
merely a character of the special case investigated by Gerstner
and Kankine. Nevertheless in the case of deep water it must be
very approximately true. For in the first place it is strictly true
at the surface, and in the second place, it must be sensibly true
at a very moderate depth and for all greater depths, since the
disturbance very rapidly diminishes on passing from the surface
downwards; so that unless the amount of disturbance be excessive
the supposition that all the streamlines are lines of constant
pressure will not be much in error.
In the case investigated by the mathematicians just mentioned,
each particle returns periodically to the position it had at a given
instant ; there is no progressive motion combined with a periodic
disturbance, such as was found in the case investigated in the pre
sent paper : and for deep water the absence of progressive motion
is doubtless peculiar to the former case, as will presently more
clearly appear.
If we suppose a regular periodic wave motion to be going on,
and then suppose small suitable pressures applied to the surface in
such a manner as to check the motion, we may evidently produce
a secular subsidence of the wave disturbance while still leaving it
at any moment regular and periodic, save as to secular change,
provided the opposing pressures are suitably chosen. The wave
length will be left unchanged, but not so, in general, the periodic
time. If the amount of disturbance in one wave period be insen
sible, the particles which at one time have a common mean depth
must at any future time have a common mean depth, and must
ultimately lie in a horizontal plane when the wave motion has
wholly subsided. In this condition therefore there can be no
mption except a horizontal flow with a velocity which is some
function of the depth. By a converse process we may imagine a
regular periodic wave motion of given wavelength excited in a
fluid in which there previously was none; and according to the
nature of the arbitrary flow with which we start, we shall obtain
as the result a wave motion of such or such a kind*.
In any given case of wave motion, the flow which remains
* To prevent possible misconception I may observe that I am not here con
templating the actual mode of excitement of waves by wind, which in some respects
is essentially different.
ON THE THEORY OF OSCILLATORY WAVES. 223
when the waves have been caused to subside in the manner above
explained is easily determined, since we know that in the motion
of a liquid in two dimensions the angular velocity is not affected
by forces applied to thd surface. If a) be the angular velocity
dv du _ 1 (dy dv dy dv dx du dx du}
~ dx dy~ S (dk dh dh dk dk dh dh dk)
S being denned by (50). In Gerstner and Rankine s solution
u = mace~ mk cos m(h ct), v = mace~ mk sin m(h ct),
and on effecting the differentiations and substituting for S from
(53) we find
Let y be the depth and u the horizontal velocity, after the
wavemotion has been destroyed as above explained, of the line of
particles which had k for a parameter ; then we must have
,.
(oC) 
Since in a horizontal length which may be deemed infinite com
pared with X the area between the ordinates y , y + dy must
be the same as between the lines of particles which have k } k + dk
for their ^parameter
dy = Sdk,
S being defined by (50). Putting for S its value given by (53)
we have
dy =(lm*a?* mk )dk ..................... (57),
y f = klma?(le* mk } .................. (58).
We have then from (56) by (55) and (57),
u = 2wVc Je 2m *cta =  wiVce" 21 "* ............ (59),
since u vanishes when k oo .
It appears then that in order that it should be possible to
excite these waves in deep water previously free from wave dis
turbance, by means of pressures applied to the surface, a prepara
tion must be laid in the shape of a horizontal velocity decreasing
from the surface downwards according to the value of e~ 2mk , where
k is a function of the depth y determined by the transcendental
equation in k (58), and moreover a velocity decreasing downwards
according to this law will serve for waves of the present kind of
224 ON THE THEORY OF OSCILLATORY WAVES.
only one particular height depending on the coefficient of the ex
ponential in the expression for the flow. Under these conditions
the horizontal velocity depending (when we adopt approximations)
on the square and higher powers of the elevation, which belongs to
the wavemotion, is exactly neutralized by the preexisting hori
zontal velocity in a contrary direction, preexisting, that is, when
we think of the waves as having been excited in a fluid previously
destitute of wavemotion, not as having gone on as they are from
a time indefinitely remote. The absence of any forward horizontal
motion of the individual particles in waves of this kind, though
attractive at first sight, is not of any real physical import,
because we are not concerned with the biographies so to speak of
the individual particles.
The oscillatory waves which most naturally present themselves
to our attention are those which are excited in the ocean or on
a lake by the action of the wind, or those which having been so
excited are propagated into (practically, though not in a rigorous
mathematical sense) still water. Of the latter kind are the surf
which breaks upon our western coasts as a result of storms out in
the Atlantic, or the grand rollers which are occasionally observed
at St Helena and Ascension Island. The motion in these cases
having been produced from rest, by forces applied to the surface,
there is no molecular rotation, and therefore the investigation of
the present paper strictly applies. Moreover, if we conceive the
waves gradually produced by suitable forces applied to the surface,
in the manner explained at p. 222, the investigation applies to the
waves (secular change apart) at any period of their growth, and
not merely when they have attained one particular height.
There can be no question, it seems to me, that this is the class
of oscillatory waves which on merely physical grounds we should
naturally select for investigation. The interest of the solution first
given by Gerstner, and it is of great interest, arises not from any
physical preeminence of the class of waves to which it relates, but
from the imperfection of our analysis, which renders it important
to discuss a case in which all the circumstances of the motion can
be simply expressed in mathematical terms without any approxima
tion. And though this motion is not exactly that which on purely
physical grounds we should prefer to investigate, namely, that in
which the molecular rotation is nil, yet unless the height of the
ON THE THEORY OF OSCILLATORY WAVES. 225
waves be extravagant, it agrees so nearly with it that for many
purposes the simpler expressions of Rankine may be used without
material error, even when we are investigating wave motion of
the irrotational kind.
B. Considerations relative to the greatest height of oscillatory
irrotational waves which can be propagated without change
of form.
In a paper published in the Philosophical Magazine, Vol. xxix.
(1865), p. 25, Rankine gave an investigation which led him to
the conclusion that in the steepest possible oscillatory waves of
the irrotational kind, the crests become at the vertex infinitely
curved in such a manner that a section of the crest by the plane
of motion presents two branches of a curve which meet at a right
angle*.
In this investigati6n it is assumed in the first place that the
steepness may be pushed to the limit of an infinite curvature
at a particular point, and in the second place that the variations
* It is not quite clear whether Rankine supposed his proposition, that "all
waves in which molecular rotation is null, begin to break when the two slopes of
the crest meet at right angles," to apply only to free waves, or to forced waves as
well. One would have supposed the former, were it not that a figure is referred to
representing forced waves of one particular kind. It is readily shewn that the
contour of a forced wave is arbitrary, even though the motion be restricted to be
irrotational. Let U=C (p. 4) be the general equation of the stream lines when the
wave motion is converted into steady motion. Then in the general case of a finite
depth, which includes as a limiting and therefore particular case that of an infinite
depth, the parameter C has one constant value at the upper surface, and another at
the bottom, and it satisfies the partial differential equation (5) of p. 4. Hence the
problem of finding U is the same as that of determining the permanent tem
perature, varying in two dimensions only, of a homogeneous isotropic solid the
section of which is bounded below by a horizontal line at a finite or infinite depth,
and above by a given arbitrary contour, the bounding surfaces being at two given
constant temperatures. The latter problem is evidently determinate, and therefore
also the former, so that forced waves may present in their contour sharp angles,
not merely of 90, but of any value we please to take.
s. 15
226 ON THE THEORY OF OSCILLATOKY WAVES.
of the components of the velocity, in passing from the crest to
a point infinitely close to it, may be obtained by differentiation,
or in other words from the second terms of the expansion by
Taylor s Theorem applied to infinitely small increments of the
variables.
The first assumption might perhaps be called in question,
but it would appear likely to give at any rate a superior limit
to the steepest form possible, if not the steepest form itself.
But as regards the second it would seem a priori very likely
that the crest might just be one of those singular points where
Taylor s Theorem fails; and that such must actually be the case
may be shewn by simple considerations.
Let us suppose that a fluid of either finite or infinite depth
is disturbed by a wave motion which is propagated uniformly
without change, the motion of the fluid being either rotational
or not, and let us suppose further that the crests are perfectly
sharp, so that a crest is formed by two branches of a curve which
either meet at a finite angle (their prolongations belonging to the
region of space where the fluid is not), or else touch, forming
a cusp.
Reduce the wave motion to steady motion by superposing
a velocity equal and opposite to that of propagation. Then
a particle at the surface may be thought of as gliding along a
fixed smooth curve: this follows directly from physical considera
tions, or from the ordinary equation of steady motion. On
arriving at a crest the particle must be momentarily at rest, and
on passing it must be ultimately in the condition of a particle
starting from rest down an inclined or vertical plane. Hence the
velocity must vary ultimately as the square root of the distance
from the crest.
Hitherto the motion has been rotational or not, Jet us now
confine ourselves to the case of irrotational motion. Place the
origin at the crest, refer the function $ to polar coordinates r, 6 ;
6 being measured from the vertical, and consider the value of </>
very near the origin, where </> may be supposed to vanish, as the
arbitrary constant may be omitted. In general <j> will be of the
form ^A n r n s\unO + ^B n cosn0. In the present case < must con
tain sines only on account of the symmetry of the motion, as
ON THE THEORY OF OSCILLATORY WAVES. 227
already shewn (p. 212), so that retaining only the most important
term we may take (j> = Ar n sin n6. Now for a point in the section
of the profile we must have dfy/dO = 0, and dfyjdr varying
ultimately as ?A This requires that n = %, and for the profile
that \Q ^TT, so that the two branches are inclined at angles of
60 to the vertical, and at an angle of 120 to each other, not
of 90 as supposed by Rankine.
This however leaves untouched the question whether the
disturbance can actually be pushed to the extent of yielding crests
with sharp edges, or whether on the other hand there exists
a limit, for which the outline is still a smooth curve, beyond which
no waves of the oscillatory irrotational kind can be propagated
without change of form.
After careful consideration I feel satisfied that there is no
such earlier limit, but that we may actually approach as near
as we please to the form in which the curvature at the vertex
becomes infinite, and the vertex becomes a multiple point where
the two branches with which alone we are concerned enclose an
angle of 120. But whether in the limiting form the inclination
of the wave to the horizon continually increases from the trough
to the summit, and is consequently limited to 30, or whether on
the other hand the points of inflexion which the profile presents
in the general case remain at a finite distance from the summit
when the limiting form is reached, so that 011 passing from the
trough to the summit the inclination attains a maximum from
which it begins to decrease before the summit is reached, is a
question which I cannot certainly decide, though I feel little doubt
that the former alternative represents the truth.
In Rankine s case of wave motion the limiting form presents
crests which are cusped. For the maximum wave ma = 1 or
a = A/27T. We see from (55) that in this case the angular velo
city becomes infinite at the surface, where 7c vanishes; and if
we suppose such waves excited in the manner already explained
in a fluid initially destitute of wave motion, the horizontal velocity
u r which must exist in preparation for the waves must be such that
dujdy becomes infinite at the surface. It appears to be this cir
cumstance which renders it possible for even rotational waves to
attain in the limit to an infinite thinness of crest without losing
the property of uniform propagation.
152
228 ON THE THEORY OF OSCILLATORY WAVES.
When swells are propagated towards a smooth, very gently
shelving shore, the height increases when the finiteness of depth
begins to take effect. Presently the limiting height for uniformly
propagated irrotational waves is passed, and then the form of the
wave changes independently of the mere secular change due to
diminishing depth. The tendency is now for the high parts to
overtake the less high in front of them, and thereby to become
higher still, until at last the crest topples over and the wave
finally breaks. The breaking is no doubt influenced by friction
against the bottom (denoting by " friction" the effect of the eddies
produced), but I do not believe that it is wholly or even mainly
due to this cause. Before the wave breaks altogether the top
gets very thin, but the maximum height for uniform propaga
tion is probably already passed by a good deal, so that we must
guard against being misled by this observation as to the character
of the limiting form.
In watching many years ago a grand surf which came rolling
in on a sandy beach near the Giant s Causeway, without any storm
at the place itself, I recollect being struck with the blunt wedge
like form of the waves where they first lost their flowing outline,
and began to show a little broken water at the very summit. It is
only I imagine on an oceanic coast, and even there on somewhat
rare occasions, that the form of waves of this kind, of nearly the
maximum height, can be studied to full advantage. The observer
must be stationed nearly in a line with the ridges of the waves
where they begin to break.
C. Remark on the method of Art. I.
There appears to be a slight advantage in employing the
function U or ^ (= j(udy  vdx) ) instead of <, the wave motion
having been reduced to steady motion as is virtually done in
Art. 1. The general equation for i/r is the same as for </>, (2), and
the general expression for ty answering to that given for < on
p. 212 is
C e rm h ri  (kti cos rmx.
The expression for p in terms of ty is almost identical with that in
terms of c/>. So far there is nothing to choose between the two. But
ON THE THEORY OF OSCILLATORY WAVES. 229
for the two equations which have to be satisfied simultaneously at
the surface, instead of p = and the somewhat complicated equa
tion (7), we have ^ = and ^r = const., which constant we may
take =0 if we leave open the origin of y. The substitution of
this equation of simpler form for (7) is a gain in proceeding to
higher orders of approximation. I remember however thinking
as I was working at the paper that as far as the approximation
there went the gain was not such as to render it worth while to
make the change.
But while these sheets were going through the press I devised
a totally different method of conducting the approximation, which
I find possesses very substantial advantages in proceeding to
higher orders of approximation. The reader will find this new
method after the paper "on the critical values of the sums of
periodic series."]
[From the Report of the British Association for 1847, Part n. p. 6.]
ON THE RESISTANCE OF A FLUID TO Two OSCILLATING
SPHERES.
THE object of this communication was to shew the application
of Professor Thomson s method of images to the solution of certain
problems in hydrodynamics. Suppose that there exists in an in
finite mass of incompressible fluid a point from which, or to which
the fluid is flowing with a velocity alike in all directions. Con
ceive now two such points, of intensities equal in magnitude and
opposite in sign, to coexist in the fluid ; and then suppose these
points to approach, and ultimately coalesce, their intensities varying
inversely as the distance between them. Let the resulting point be
called a singular point of the second order. The motion of a fluid
about a solid, oscillating sphere is the same as if the solid sphere
were replaced by fluid, in the centre of which existed such a point.
It is easy to shew that the motion of the fluid due to a point of
this kind, when the fluid is interrupted by a sphere having its
centre in the axis of the singular point, is the same as if the
sphere s place were occupied by fluid containing one singular point
of the second order. By the application of this principle may be
found the resistance experienced by a sphere oscillating in presence
of a fixed sphere or plane, or within a spherical envelope, the
oscillation taking place in the line joining the centres, or perpen
dicular to the plane. In a similar manner may be found the resist
ance to two spheres which touch, or are connected by a rod, or to
the solid made up of two spheres which cut, provided the exterior
angle of the surfaces be a submultiple of two right angles, the
oscillation in these cases also taking place in the line joining the
centres. The numerical calculation is very simple, and may be
carried to any degree of accuracy.
RESISTANCE OF A FLUID TO TWO OSCILLATING SPHERES. 231
The investigation mentioned in the preceding paper arose out
of the communication to me by Sir William Thomson of his
beautiful method of electrical images before he had published it.
Having myself paid more attention to the motion of fluids than
to electricity, I endeavoured to find if it would in any manner
apply to the solution of problems in the motion of fluids. I found
that what is called above a singular point of the second order had
a perfect image in a sphere when its axis was in the direction of
a radius, which led to a complete solution of the problem men
tioned in the paper when one sphere lay wholly outside or inside
the other. I shewed this to Professor Thomson, who pointed out
to me that a solution was also attainable, and that in finite
terms, when the spheres intersected, provided the angle of inter
section was a submultiple of two right angles. He saw that the
property of a singular point of the second order of giving a perfect
image in the case mentioned, admitted of an application to the
theory of magnetism, which he has published in a short paper in
the second volume of the Cambridge and Dublin Mathematical
Journal, (1847) p. 240.
Although the mathematical result is contained in the paper
just mentioned, I subjoin the process by which I found it out.
The expression (see p. 41) for the function < around a sphere
which moves in a perfect fluid previously at rest may be thought
of as applying to the whole of an infinite mass of fluid, provided
we conceive what has here been called a singular point of the
second order to exist at the origin. Let us conceive a spherical
surface S with its centre at and having a radius a to exist in
the fluid ; let P be the singular point, lying either within or with
out the sphere S, and having its axis in the line OP. Let /, & be
polar coordinates originating at P, & being measured from OP
produced, and let r, 6 be polar coordinates originating at ; let
m be a constant, and OP c, then </> being the function due to the
singular point we have
mcos# m.r cos0 rcos# c
Now if e be less than 1,
mj (r 2 2cr cos 6 + c 2 /
etc
232 ON THE RESISTANCE OF A FLUID
where P , P x , P 2 ... are Laplace s, or in this case more properly
Legendre s, coefficients*. Hence by expanding and differentiating
with respect to c, we have
(1),
We are not of course concerned with the constant term in the
latter of these two expressions. For the normal velocity (v) at
the surface of the sphere we get by differentiating with respect
to r, and then putting r a
/1 . 2P t 2 . 3cP 2 3 . 4c 2 P 8 , \ . , , Q ,
v = m( ^ L ] 1 H 5 + ... ), if a> c (3),
\ a a CL /
First suppose the point P outside the sphere, let the sphere be
thought of as a solid sphere, and consider the motion "reflected "
(p. 28) from it. The reflected motion being symmetrical about
the axis, we must have for it
where Q , Q lf Q t ... are Laplace s functions involving 6 only. This
gives for the normal velocity (v) in the reflected motion at the
surface of the sphere
and since we must have v = v we get from (4) and (6)
la P. 2a 5 P 2 3a 7 P 3
Q = 0, Q^wj H, Q 2 = m^ 2 , Q 9 = n*gp
which reduces (5) to
a 3 /!P 2a 2 P 3a 4 P
* The functions which in Art. 9 of the paper " On some Cases of Fluid Motion"
(p. 38) I called "Laplace s coefficients," following the nomenclature of Pratt s
Mechanical Philosophy, are more properly called " Laplace s functions ;" the term
"Laplace s coefficients" being used to mean the coefficients in the expansion of
[1  2e { cos 6 cos & + sin sin ff cos (w  w ) } + c 2 ]"^,
to be understood according to the usual notation and not as in the text.
TO TWO OSCILLATING SPHERES. 233
This is identical with what (1) becomes on writing w , c for m, C
provided that
a 3 , a 2
m = m  3) c=.
Hence the reflected motion is perfectly represented by sup
posing the sphere s place occupied by fluid within which, at the
point P in the line OP determined by OP = c, there exists a
singular point of the same character as P, but of opposite sign,
and of intensity less in the ratio of a 3 to c 3 .
The case of a spherical mass of fluid within a rigid enclosure
and containing a singular point of the second order with its axis
in a radial direction might be treated in a manner precisely similar,
by supposing the space exterior to the sphere filled with fluid,
taking to represent the reflected motion in this case, instead of (5),
the corresponding expression according to ascending powers of r,
and comparing the resulting normal velocity at the surface of the
sphere with (3) instead of (4). This is however unnecessary, since
we see that the relation between the two singular points P, P is
reciprocal, so that either may be regarded as the image of the
other.
Suppose now that we have two solid spheres, S, S , exterior to
each other, immersed in a fluid. Suppose that S is at rest, and
that S moves in the direction of the line joining the centres, the
fluid being at rest except as depends on the motion of S. The
motion of the fluid may be determined by the method of successive
reflections (p. 28), which in this case becomes greatly simplified
in consequence of the existence of a perfect image representing
each reflected motion, so that the process is identical with that of
Thomson s method of images, except that the decrease of intensity
of the successive images takes place according to the cubes of the
ratios of the successive quantities such as a, c, instead of the first
powers.
If a sphere move inside a spherical envelope, in the direction
of the line joining the centres, the space between being filled with
fluid which is otherwise at rest, the motion may be determined in
a precisely similar manner.
If two spheres outside each other, or just touching, be con
nected by an infinitely thin rod, and move in a fluid in the direction
of the line joining their centres, we have only to find the motion
234 ON THE RESISTANCE OF A FLUID
due to the motion of each sphere supposing the other at rest, and
to superpose the results.
I should probably not have thought of applying the method
to the solid bounded by the outer portions of two intersect
ing spheres, had not Professor Thomson shewn me that it was
not limited to the cases in which each sphere is complete ; and
that although it fails from nonconvergence when the spheres
intersect, yet when the exterior angle of intersection is a sub
multiple of two right angles the places of the successive images
recur in a cycle, and a solution of the problem may be obtained
in finite terms by placing singular points of the second order at
the places of the images in a complete cycle.
The simplest case is that in which the spheres are generated
by the revolution round their common axis of two circles which
intersect at right angles. In this case if $, S f are the spheres,
0, their centres, O l the middle point of the common chord of
the circles, the image of in $ will be at O lt and the image
of O in S will be at .
Let a, b be the radii of the spheres ; c the distance V( 2
of their centres ; e, f the distances a?/c, tf/c of O l from 0, ;
C the velocity of the spheres ; r, 6 the polar coordinates of any
point measured from 0; r\, t the coordinates measured from 4 ;
r, & the coordinates measured from ; 0, 6 lt & being all
measured from the line 00 . If S were away, we should have
for the fluid exterior to 8
n 3 cos 6
+ <&&
For the image of this in S we have a singular point at 6) x for
which
, Ca*b* cosfl,
* = ~<T ^>
and for the image of this again in 8 we have a singular point at
for which
3COS0 7
9 =  oo gpr >
which is precisely what is required to give the right normal
velocity at the surface of S . Moreover all the singular points
lie inside the space bounded by the exterior portions of the inter
TO TWO OSCILLATING SPHERES. 235
secting spheres. Hence the three motions together satisfy all the
conditions of the problem, so that for the complete solution we
have
. , (a* cos d a s b 3 cos O l cos ff\
0= *V\?r ff* ?r}
Just as in the case of a sphere, if a force act on the solid in
the direction of its axis, causing a change in the velocity C, the
only part of the expression for the resistance of the fluid which
will have a resultant will be that depending upon dC/dt. This
follows at once, as at pp. 50, 51, from the consideration that when
there is no change of C the vis viva is constant, and therefore the
resultant pressure is nil. If we denote by M dC/dt the resultant
pressure acting backwards, we get for the part of M due to the
pressure of the fluid on the exposed portion of the surface of S t
72 ffa 3 cos0 cWcosfl, , ,,/! .
vrpb I j  2  372 + # cos f cos # sin
taken between proper limits. Putting b cos & x t we have
r cos 6 = c + x, ?\ cos A =/ + x,
Expressing cos 0, cos0 t , cos & in terms of x and r, x and r lt x,
and changing the independent variable, first to x y and then in the
first term to r and in the second to r lt we have for the indefinite
integral with sign changed
which is to be taken between the limits r = a to r = c + b, r^ ab/c
to f+b, x f to 6. The part of M due to the integral over
the exposed part of the surface of 8 will be got from the above
by interchanging; and on adding the two expressions together,
and putting /= b 2 /c, c = V( 2 + & 2 )> we get for the final result
 26 6 }.
When one of the radii, as b, vanishes, we get M = f Trpa 3 as
it ought to be.
[From the Transaction* of the Cambridge Philosophical Society,
Vol. vin. p. 533.]
ON THE CRITICAL VALUES OF THE SUMS OF PERIODIC SERIES.
[Read December 6, 1847.]
THERE are a great many problems in Heat, Electricity, Fluid
Motion, &c., the solution of which is effected by developing an
arbitrary function, either in a series or in an integral, by means of
functions of known form. The first example of the systematic
employment of this method is to be found in Fourier s Theory
of Heat. The theory of such developements has since become an
important branch of pure mathematics.
Among the various series by which an arbitrary function f(x)
can be expressed within certain limits, as and a, of the variable
#, may particularly be mentioned the series which proceeds accord
ing to sines of TTX/O, and its multiples, and that which proceeds
according to cosines of the same angles. It has been rigorously
demonstrated that an arbitrary, but finite function of #, /(#), may
be expanded in either of these series. The function is not
restricted to be continuous in the interval, that is to say, it may
pass abruptly from one finite value to another ; nor is either the
function or its derivative restricted to vanish at the limits and a.
Although however the possibility of the expansion of an arbitrary
function in a series of sines, for instance, when the function does
not vanish at the limits and a, cannot but have been contem
plated, the utility of this form of expansion has hitherto, so far as
I am aware, been considered to depend on the actual evanescence
of the function at those limits. In fact, if the conditions of the
problem require that /(O) and f(a) be equal to zero, it has been
CRITICAL VALUES OF THE SUMS OF PERIODIC SERIES. 237
considered that these conditions were satisfied by selecting the
form of expansion referred to. The chief object of the following
paper is to develope the principles according to which the expan
sion of an arbitrary function is to be treated when the conditions
at the limits which determine the particular form of the expansion
are apparently violated ; and to shew, by examples, the advantage
that frequently results from the employment of the series in such
cases.
In Section I. I have begun by proving the possibility of the
expansion of an arbitrary function in a series of sines. Two
methods have been principally employed, at least in the simpler
cases, in demonstrating the possibility of such expansions. One,
which is that employed by Poisson, consists in considering the
series as the limit of another formed from it by multiplying its
terms by the ascending powers of a quantity infinitely little less
than 1 ; the other consists in summing the series to n terms, that
is, expressing the sum by a definite integral, and then considering
the limit to which the sum tends when n becomes infinite. The
latter method certainly appears the more direct, whenever the
summation to n terms can be effected, which however is not always
the case; but the former has this in its favour, that it is thus
that the series present themselves in physical problems. The
former is the method which I have followed, as being that which
I employed when I first began the following investigations, and
accordingly that which best harmonizes with the rest of the paper.
I should hardly have ventured to bring a somewhat modified
proof of a wellknown theorem before the notice of this Society,
were it not for the doubts which some mathematicians seem to
feel on this subject, and because there are some points which
Poisson does not seem to have treated with sufficient detail.
I have next shewn how the existence and nature of the dis
continuity of /(& ) and its derivatives may be ascertained merely
from the series, whether of sines or cosines, in which f(x) is
developed, even though the summation of the series cannot be
effected. I have also given formulae for obtaining the develope 
ments of the derivatives of f(x) from that of f(x) itself. These
developements cannot in general be obtained by the immediate
differentiation of the several terms of the developement of f(x),
or in other words by differentiating under the sign of summa
tion.
238 ON THE CRITICAL VALUES OF
It is usual to restrict the expanded function to be finite. This
restriction however is not necessary, as is shewn towards the end
of the section. It is sufficient that the integral of the function be
finite.
Section II. contains formulae applicable to the integrals which
replace the series considered in Section I. when the extent a of
the variable throughout which the function is considered is sup
posed to become infinite.
Section III. contains some general considerations respecting
series and integrals, with reference especially to the discontinuity
of the functions which they express. Some of the results obtained
in this section are referred to by anticipation in Sections I. and II.
They could not well be introduced in their place without too much
interrupting the continuity of the subject.
Section IV. consists of examples of the application of the pre
ceding results. These examples are all taken from physical
problems, which in fact afford the best illustrations of the applica
tion of periodic series and integrals. Some of the problems
considered are interesting on their own account, others, only as
applications of mathematical processes. It would be unnecessary
here to enumerate these problems, which will be found in their
proper place. It will be sufficient to make one or two remarks.
The problem considered in Art. 52, which is that of determin
ing the potential due to an electrical point in the interior of a
hollow conducting rectangular parallelepiped, and to the electricity
induced on the surface, is given more for the sake of the artifice
by which it is solved than as illustrating the methods of this paper.
The more obvious mode of solving this problem would lead to a
very complicated result.
The problem solved in Art. 54 affords perhaps the best example
of the utility of the methods given in this paper. The problem
consists in determining the motion of a fluid within the sector of a
cylinder, which is made to oscillate about its axis, or a line parallel
to its axis. The expression for the moment of inertia of the
fluid which would be obtained by the methods generally employed
in the solution of such problems is a definite integral, the numeri
cal calculation of which would be very laborious; whereas the
expression obtained by the method of this paper is an infinite series
which may be summed, to a sufficient degree of approximation,
without much trouble.
THE SUMS OF PERIODIC SERIES. (SECT. I. SERIES.) 239
The series for the developement of an arbitrary function con
sidered in this paper are two, a series of sines and a series of
cosines, together with the corresponding integrals ; but similar
methods may be applied in other cases. I believe that the follow
ing statement will be found to embrace the cases to which the
method will apply.
Let u be a continuous function of any number of independent
variables, which is considered for values of the variables lying
within certain limits. For facility of explanation, suppose u a
function of the rectangular coordinates x, y, z, or of x, y, z and t,
where t is the time, and suppose that u is considered for values of
x, y, z, t lying between and a, and b, and c, and T, respec
tively. For such values suppose that u satisfies a linear partial
differential equation, and suppose it to satisfy certain linear equa
tions of condition for the limiting values of the variables. Let
Z7=0, U = Q be two of the equations of condition, corresponding
to the two limiting values of one of the variables, as x. Then
the expansion of u to which these equations lead may be applied
to the more general problem which leads to the corresponding
equations of condition U F, U = F t where F and F are any
functions of all the variables except x, or of any number of
them.
SECTION I.
Mode of ascertaining the nature of the discontinuity of a function
which is expanded in a series of sines or cosines, and of obtain
ing the developements of the derived functions.
1. By the term, function I understand in this paper a quantity
whose value depends in any manner on the value of the variable,
or on the values of the several variables of which it is composed.
Thus the functions considered need not be such as admit of being
expressed by any combination of algebraical symbols, even between
limits of the variables ever so close. I shall assume the ordinary
rules of the differential and integral calculus as applicable to such
functions, supposing those rules to have been established by the
method of limits, which does not in the least require the
possibility of the algebraical expression of the functions con
sidered.
240 ON THE CRITICAL VALUES OF
The term discontinuous, as applied to a function of a single
variable, has been used in two totally different senses. Sometimes
a function is called discontinuous when its algebraical expression
for values of the variable lying between certain limits is different
from its algebraical expression for values of the variable lying
between other limits. Sometimes a function of x, f(x), is called
continuous when, for all values of #, the difference between f (x)
and f(xh) can be made smaller than any assignable quantity by
sufficiently diminishing h, and in the contrary case discontinuous.
If / (x) can become infinite for a finite value of x, it will be con
venient to consider it as discontinuous according to the second
definition. It is easy to see that a function may be discontinuous
in the first sense and continuous in the second, and vice versa.
The second is the sense in which the term discontinuous is I
believe generally employed in treatises on the differential calculus
which proceed according to the method of limits, and is the sense
in which I shall use the term in this paper. The terms continuous
and discontinuous might be applied in either of the above senses
to functions of two or more independent variables. If I have
occasion to employ them as applied to such a function, I shall
employ them in the second sense; but for the present I shall
consider only functions of one independent variable.
In the case of the functions considered in this paper, the value
of the variable is usually supposed to be restricted to lie within
certain limits, as will presently appear. I exclude from considera
tion all functions which either become infinite themselves, or have
any of their differential coefficients of the orders considered
becoming infinite, within the limits of the variable within which
the function is considered, or at the limits themselves, except
when the contrary is expressly stated. Thus in an investigation
into which / (x) and its first n differential coefficients enter, and
in which f(x) is considered between the limits x=0 and a? = a,
those functions are excluded, at least at first, which are such that
any one of the quantities /(#), / (x) ...f n (x) is infinite for a
value of x lying between and a, or for x = or x = a ; but the
differential coefficients of the higher orders may become infinite.
The quantities /(#), / (#) ../ n (x) may however alter discon
tinuously between the limits x = and x = a, but I exclude
from consideration all functions which are such that any one of
the above quantities alters discontinuously an infinite number
THE SUMS OF PERIODIC SERIES. (SECT. I. SERIES.) 241
of times between the limits within which x is supposed to
lie.
The terms convergent and divergent, as applied to infinite series,
will be used in this paper in their usual sense ; that is to say, a
series will be called convergent when the sum to n terms
approaches a finite and unique limit as n increases beyond all
limit, and divergent in the contrary case. Series such as
1 1 + 1 . . . , sin x + sin Zx + sin 3 ic + . . . ,
(where x is supposed not to be or a multiple of TT,) will come
under the class divergent ; for, although the sum to n terms does
not increase beyond all limit, it does not approach a unique limit
as n increases beyond all limit. Of course the first n terms of a
divergent series may be the limits of those of a convergent series :
nor does it appear possible to invent a series so rapidly divergent
that it shall not be possible to find a convergent series which shall
have for the limits of its first n terms the first n terms respectively
of the divergent series. Of course we may employ a divergent
series merely as an abbreviated mode of expressing the limit of
the sum of a convergent series. Whenever a divergent series is
employed in this way in the present paper, it will be expressly
stated that the series is so regarded.
Convergent series may be divided into two classes, according
as the series resulting from taking all the terms of the given
series positively is convergent or divergent. It will be convenient
for the purposes of the present paper to have names for these two
classes. I shall accordingly call series belonging to the first class
essentially convergent, and series belonging to the second acci
dentally convergent, while the term convergent, simply, will be used
to include both classes. Thus, according to the definitions which
will be employed in this paper, the series
x + J z* 4 J x 9 + ...
is essentially convergent so long as x 2 < 1 ; it is divergent when
as* > 1, and when x = 1 ; and it is accidentally convergent when
x = l.
The same definitions may be applied to integrals, when one at
/oo
least of the limits of integration is oo . Thus, if a > 0, / of 2 djc
J a
s. 16
242 ON THE CRITICAL VALUES OF
t* 00
is essentially convergent at the limit oo , while I ^^ dx is only
J a &
accidentally convergent, andjsina;^, not being convergent,
J a
comes under the class of divergent integrals. These definitions
may be applied also to integrals taken between finite limits, when the
quantity under the integral sign becomes infinite within the limits
f a
of integration, or at one of the limits. Thus I log # dx is conver
J o
[ a dx
gent, but I  divergent, at the limit 0.
J o ^
2. Let f (x) be a function of x which is only considered
between the limits x = and x = a, and which can be expanded
between those limits in a convergent series of sines of irx/a and
its multiples, so that
To determine A n , multiply both sides of (1) by smmrx/a.dx and
integrate from x = to x a. Since the series in (1) is conver
gent, and sin mrx/a does not become infinite for any real value
of x, we may first multiply each term by sin mrx/a . dx and
integrate, and then sum, instead of first summing and then inte
grating*. But each term of the series in (1) except the n th will
produce in the new series a term equal to zero, and the n th will
produce J aA n . Hence
A 2 [ a , . . nirx 7
A  fix) sin  dx,
n aJ
and therefore
*f \ % \? f a ^/ \ n7rx 7 7?7r< ^ /o\
fW = % /Wsin doe. am ............ (2).
Uj J o tl U
3. Hence, whenever f(x) can be expanded in the convergent
series which forms the righthand side of (1), the value of A n can
be very readily found, and the expansion performed. But this
leaves us quite in the dark as to the degree of generality that a
function which can be so expanded admits of. In considering this
* Moigno, Lecons de Cdlcul DifferenUel, &c. Tom. n. p. 70.
THE SUMS OF PERIODIC SERIES. (SECT. I. SERIES.) 243
question it will be convenient, instead of endeavouring to develope
f(x), to seek the value of the infinite series
2<C f // \ n7rx j i n7r % /o\
 2, /() sm ax . sm  (3),
a J a a
provided the series be convergent ; for it is only in that case that
we can, without further definition, speak of the sum of the series
at all. Now if we had only a finite number n of terms in the
series (3) we might of course replace the series by
 I f(x ) sin sin + sin  sin . . ,
aJo [a a a a
. njrx . nirx] , , , . N
4 sin sin \ dx (4).
a a j
As it is however this transformation cannot be made, because, the
series within brackets in the expression which would replace (4)
not being convergent, the expression would be a mere symbol
without any meaning. If however the series (3) is essentially con
vergent, its sum is equal to the limit of the sum of the following
essentially convergent series
 2<7 n / f (V) sin dx .sin (5),
a y Jo a a
when g from having been less than 1 becomes in the limit 1. It
will be observed that if (3) were only accidentally convergent, we
could not with certainty affirm the sum of (3) to be the limit of
the sum of (5). For it is conceivable, or at least not at present
proved to be impossible, that the mode of the mutual destruction
of the terms of (3) in the infinitely remote part of the series
should be altered by the introduction of the factor g n , however
little^ might differ from 1. Let us now, instead of seeking the
sum of (3) in those cases in which the series is convergent, seek
the limit to which the sum of (5) approaches as g approaches to 1
as its limit.
4. The transformation already referred to, which could not be
effected on the series (3), may be effected on (5), that is to say,
instead of first integrating the several terms and then summing,
we may first sum and then integrate. We have thus, for the value
of the series,
9 Ca ( v/ ^\
(6).
162
244 ON THE CRITICAL VALUES OF
The convergent series within brackets can easily be summed. The
expression (6) thus becomes
(7).
1 2# cos TT (se f a?) /a + #
Now since the quantity under the integral sign vanishes when
g = 1, provided cos TT (a? + a?) /a be not = 1, the limit of (7) when
g 1 will not be altered if we replace the limits and a of x by
any other limits or groups of limits as close as we please, provided
they contain the values of x which render x x equal to zero or
any multiple of 2a. Let us first suppose that we are considering
a value of x lying between and a, and in the neighbourhood of
which /(#) alters continuously. Then, since x + x never becomes
equal to zero or any multiple of 2a within the limits of integra
tion, we may omit the second term within brackets in (7) ; and
since x x never becomes equal to any multiple of 2a, and
vanishes only when x = x } we may take for the limits of x two
quantities lying as close as we please to x, and therefore so close
as to exclude all values of x for which f(x) alters discontinuously.
Let g=~Lh, x = x + f , expand cos irf/a by the ordinary formula,
and put f(x) =/ (x) + R. Then the limit of (7) will be the same
as that of
the limits of f being as small as we please, the first negative and
the second positive. Let now
so that d/dj; is ultimately equal to a/ir, that is to say when g is
first made equal to 1, and then the limits of f, and therefore those
of f, are made to coalesce. Let now G, L be respectively the
greatest and least values of (1  A) a~ l dg/di; {/(#) + R] within
the limits of integration. Then if we observe that
W + *
were tan" 1 denotes an angle lying between ?r/2 and ?r/2, putting
THE SUMS OF PERIODIC SERIES. (SECT. I. SERIES.) 245
 f p 2 for the limits of , we shall see that the value of the inte
gral (8) lies between
G (tan 1 fjh + tan 2 //t) and L (tan 1 gjh + tan" 1 gjh) :
but in the limit, that is to say, when we first suppose h to vanish
and then and f g , 6r and Z become equal to each other and to
ir~ l f(x) t and tan" 1 f x /& + tan" 1 f 2 //i becomes equal to TT. Hence,
/(#) is the limit of (7).
Next, suppose that the value of x which we are considering
lies between and a, and that as x passes through it /(# ) alters
suddenly from M to N. Then the reasoning will be exactly as
before, except that we must integrate separately for positive and
negative values of f , replacing f(x) + R by M + R in the latter
case, and by N + R in the former. Hence, the limit of (7) will be
Lastly, if we are considering the extreme values as = and
sc = a, it follows at once from the form of (7) that its limiting value
is zero.
Hence the limit to which the sum of the convergent series (5)
tends as g tends to 1 as its limit is f(x) for values of x lying
between and a, for which f(x) alters continuously, it is J (M+N)
for values of as for which f (x) alters suddenly from M to N, and it
is zero for the extreme values and a.
5. Of course the limiting value of the series (5) is f(fty and
not zero, if we suppose that g first becomes 1 and then x passes
from a positive value to zero. In the same way, if/(#) alters
abruptly from M to N as x increases through x lt the limiting value
of (5) will be M if we suppose that g first becomes 1 and then x
increases to x^ and it will be N if we suppose that g first becomes
1 and then x decreases to x lt It would be futile to argue that the
limiting value of (5) for # = is zero rather than /(O), or /(O)
rather than zero, since that entirely depends on the sense in which
we employ the expression limiting value. Whichever sense we
please to adopt, no error can possibly result, provided we are only
consistent, and do not in the course of the same investigation
change the meaning of our words.
It is a principle of great importance in these investigations,
that a function of two independent variables which becomes
246 ON THE CRITICAL VALUES OF
indeterminate for particular values of the variables may have
different limiting values according to the order in which we
suppose the variables to assume their particular values, or ac
cording to the nature of the arbitrary relation which we conceive
imposed on them as they approach those values together.
I would here make one remark on the subject of consistency.
We may speak of the sum of an infinite series which is not con
vergent, if we define it to mean the limit of the sum of a con
vergent series of which the first n terms become in the limit
the same as those of the divergent series. According to this
definition, it appears quite conceivable that the same divergent
series should have a different sum according as it is regarded
as the limit of one convergent series or of another. If however
we are careful in the same investigation always to regard the
same divergent series, and the series derived from it, as the
limits of the same convergent series and the series derived from
it, it does not appear possible to fall into error, assuming of course
that we always reason correctly. For example, we may employ
the series (3), and the series derived from it by differentiation, &c.,
without fear, provided we always regard these series when di
vergent, or only accidentally convergent, as the limits of the
particular convergent series formed by multiplying their n th terms
6. We may now consider the convergency of the series (3), in
order to find whether we may employ it directly, or whether we
must regard it as the limit of (5).
By integrating by parts in the ri b term of (3), we have
mrx 7 , 2 , , N nirx
sin dx = f (x ) cos
a WIT a
Suppose that / (x) does not necessarily vanish at the limits
x = and x = a, and that it alters discontinuously any finite
number of times between those limits, passing abruptly from
M^ to N t when x increases through a lt from M 2 to JV 2 when x
increases through 2 , and so on. Then,. if we put 8 for the sign of
summation referring to the discontinuous values of /(a? ), on
taking the integrals in (9) from # = to x = a, we shall get for
THE SUMS OF PERIODIC SERIES. (SECT. I. SERIES.) 247
the part of the integral corresponding to the first term at the
righthand side of the equation
^ {/(O) ()"/() + S(NM) cos  ........... (10).
It is easily seen that the last two terms in (9) will give a
part of the integral taken from to a, which is numerically
inferior to L/n*, where L is a constant properly chosen. As far
as regards these terms therefore the series (3) will be essentially
convergent, and its sum will therefore be the limit of the sum
of (5).
Hence, in examining the convergency or divergency of the
series (3), we have only got to consider the part of the coefficient
of sin mrx/a of which (10) is the expression. The terms /(O),
/(a) in this expression may be included under the sign 8 if we
put for the first a = 0, M 0, ZV=/(0), and for the second a = a,
M=f(a), N=Q. We have thus got a set of series to consider
of which the type is
2 ., r ,,, ^ 1 mrx . niroc ..,,.
 (N M)2,  cos  sm  ............ (11).
TT n a a
If we replace the product of the sine and cosine in this ex
pression by the sum of two sines, by means of the ordinary
formula, and omit unnecessary constants, we shall have for the
series to consider
Let now
2sinw* (12).
n ^ }
u sin^ + Jsin 2z ... f  sin nz (13),
Tl
then
du sin (n 4 ] z
T = cos z f cos 2z . . . + cos nz = ,\ V
dz 2 sm z
and since u vanishes with z, in which case m \ l ^ z J s finite,
sm \z
we shall have, supposing z to lie between 2?r and + 2?r, so
that the quantity under the integral sign does not become infinite
within the limits of integration,
*H;=EH^*i W>
248 ON THE CRITICAL VALUES OF
and we have to find whether the integral contained in this
equation approaches a finite limit as n increases beyond all limit,
and if so what that limit is. Since u changes sign with z, we
need not consider the negative values of z.
First suppose the superior limit z to lie between and 2?r ;
and to simplify the integral write 2z for z, n for 2n + 1, so that
the superior limit of the new integral lies between and TT ;
then the integral
[ z sinnz , f sOnnz z 7 f^sinnz,., ^ . 7
= . dz =  . dz \ (1 + Ez\ dz,
J o sin z Jo z sin z J o z
where R = ; , a quantity which does not become infinite
z sin z
within the limits of integration. Hence, as is known, the limit of
sin nz . Rdz when n increases beyond all limit is zero. Hence,
o
if / be the limit of the integral,
,.,.. [*smnz 7 r ., f C
I = limit of dz limit of
JQ Z JO
b
Now, z being given, the limit of nz is oo , and therefore
7T
2
Secondly, suppose z in (14) to be equal to 0. Then it follows
directly from this equation, or in fact at once from (13), that
u = 0, and consequently the limit of u = 0.
The value of u in all other cases, if required, may be at once
obtained from the consideration that the values of u recur when z
is increased or diminished by 2?r,
Hence, the series (12) is in all cases convergent, and has for
its sum when z = 0, and \ (TT  z} when z lies between
and 27T.
Now, if in the theorem of Article 4, we write z for x, and
put a = TT, f (z) =  (TT  z), we find, for values of z lying between
and TT, and for z = TT,
limit of 2,  g n sin nz = J (TT z) ;
and evidently
limit of 2  # M sin n^ = 0, when a = 0,
THE SUMS OF PERIODIC SERIES. (SECT, I. SERIES.) 249
that is of course supposing z first to vanish and then gio become 1.
Also the limit of ^n~ l g n sin nz changes sign with z, and recurs
when z is increased or diminished by 2?r. Hence, the series (12),
which has been proved to be convergent, is in all cases the limit
to which the sum of the convergent series %n~ l g n sin nz tends
as g tends to 1 as its limit. Now the series (11) may be de
composed into two series of the form just discussed, whence it
follows that the series (3) is always convergent, and its sum for
all values of x, critical as well as general, is the limit of the
sum of the series (5), when g becomes equal to 1.
The examination of the convergency of the series (3) in the
only doubtful case, that is to say, the case in which / (x) is dis
continuous, or does not vanish for x = and for x = a, is more
curious than important. For in the analytical applications of
the series (3) it would be sufficient to regard it as the limit of
the series (5) ; and in the case in which (3) is only accidentally
convergent, we should hardly think of employing it in the
numerical computation of f(x) if we could possibly help it, and
it will immediately appear that in all the cases which are most
important to consider we can get rid of the troublesome terms
without knowing the sum of the series.
The proof of the convergency of the series (3) which has just
been given, though in some respects I believe new, is certainly
rather circuitous, and it has the disadvantage of not applying
to the case in which f (x) is infinite*, an objection which does
not apply to the proof given by M. Dirichletf. It has been
supposed moreover that f"(x) is not infinite. The latter re
striction however may easily be removed, as in the end of the
next article.
7. Let f(x) be a function of x which is expanded between
the limits x = and x a in the series (3). Let the series be
A . TTOC A . 7rx . . mrx , .
^sin  h^4 2 sm  ... + A H sin  K.., ..,.(15),
CL a a
and suppose that we have given the coefficients A } , A 2 ... , but
do not know the sum of the series f(x). We may for all that
find the values of /(O) and /(a), and likewise the values of x
* This restriction may however be dispensed with by what is proved in Art. 20.
t Crelle s Journal, Tom. iv. p. 157.
250 ON THE CRITICAL VALUES OF
for which f(x) is discontinuous, and the quantity by which f(x) is
increased as x increases through each of these critical values.
For from (9) and (10)
E being a quantity which does not become infinite with n. If
then we use the term limit in an extended sense, so as to include
quantities of the form (7 cos ny, [of course C ( l) n is a particular
case,] or the sum of any finite number of such quantities, we
shall have for n = oo ,
limit of ^=
Let then the limit of nA n be found. It will appear under the
form
y ...... . ........... (17).
Comparing this expression with (16), we shall have
and for each term of the series denoted by 8 we shall have
In particular, if f(x) is continuous, and if the limit of nA n is
L Q or L e according as n is odd or even, we shall have
+/()}, A = = !/(0) /());
whence
If f(x) were discontinuous for an infinite number of values
of x lying between and a, it is conceivable that the infinite
series coming under the sign 8 might be divergent, or if con
vergent might have a sum from which n might wholly or partially
disappear, in which case the limit of nA n might not come out
under the form (17). It was for this reason among others, that
in Art. 1, I excluded such functions from consideration.
If f(x) be expressible algebraically between the limits x = Q
and x = a, or if it admit of different algebraical expressions within
THE SUMS OF PEEIODIC SERIES. (SECT. I. SERIES.) 251
different portions into which that interval may be divided, A n
will be an algebraical function of n, and the limit of nA n may
be found by the ordinary methods. Under the term algebraical
function, I here include transcendental functions, using the term
algebraical function in opposition to what has been sometimes
called an empirical function, or a general function, that is, a function
in the sense in which the ordinate of a curve traced liberd manu
is a function of the abscissa. Of course, in applying the theorem
in this article to general functions, it must be taken as a postulate
that the limit of nA n can be found, and put under the form (17).
The theorem in question has been proved by means of equation
(9), in which it is supposed that f"(sc] does not become infinite
within the limits of integration. The theorem is however true
independently of this restriction. To prove it we have only got to
integrate by parts once instead of twice, and we thus get for
T>
the quantity which replaces the integral
n
9 /*
I/.
nirx , ,
cos   ax ,
which by the principle of fluctuation* vanishes when n becomes
infinite. There is however this difference between the two cases.
When the series (15) has been cleared of the part for which the
limit of n A n is finite, by the method which will be explained in
the next article, the part which remains will be at least as con
vergent in the former case as the series ^2 + ^... + ^+...,
L .Z n
whereas we cannot affirm this to be true, and in fact it may
be proved that it is not true, in the case in which f"(x) becomes
infinite. Observing that the same remark will apply when we
come to consider the critical values of the differential coefficients
* I borrow this term from a paper by Sir William E. Hamilton On Fluctuating
Functions, (Transactions of the Royal Irish Academy, Vol. xix. p. 264.) Had I
been earlier acquainted with this paper, and that of M. Dirichlet already referred
to, I would probably have adopted the second of the methods mentioned in the
introduction for establishing equation (2) for any function, or rather, would have
begun with Art. 7, taking that equation as established. I have retained Arts (2)
(6), first, because I thought the reader would enter more readily into the spirit of
the paper if these articles w r ere retained, and secondly, because I thought that
Section in, which is adapted to this mode of viewing the subject, might be found
useful.
252 ON THE CRITICAL VALUES OF
of f(x\ I shall suppose the functions and derived functions
employed in each investigation not to become infinite, according
to what has already been stated in Art. 1.
8. After having found the several values of a, and the cor
responding values of N M, we may subtract the expression (10)
from A n , provided we subtract from the sum of the series (15) the
sums of the several series such as (11). Now if X be the sum of
the series (11),
X = (NM) s S in +2 gm... (19).
JT \ n a n a )
But it has been already shewn that 2  sin nz = \ (TT z) when z
n
lies between and 2?r, = when z = 0, and = J (IT + z) when z
lies between and 2?r. Now when x lies between and a,
TT (x + a) /a lies between and 2?r, and TT (x a) /a lies between
2?r and ; and when x lies between a. and a, TT (x + a) /a still lies
between and 2?r, and TT (a? a) /a now lies between the same
limits. Hence
x
X = (N M)  , when x lies between and a
...(20).
/ _ /
(N M)  , when x lies between a and a
a
We need not trouble ourselves with the singular values of the
sum of the series (15), since we have seen that a singular value is
always the arithmetic mean of the values of the sum for values
of x immediately above and below the critical Value. This rule
will apply to the extreme cases in which x = and x = a, if we
consider the sum of the series for values of x lying beyond those
limits. The rule applies to the series in (19), which is only a
particular case of (15), and consequently will apply to any combi
nation of series having this property, formed by way of addition
or subtraction; since, when we increase or diminish any two.
quantities M Q) N by any other two M, N respectively, we increase
or diminish the arithmetic mean of the two former by the arith
metic mean of the two latter.
It has been already stated that we may, with a certain conven
tion, include quantities referring to the limits x = and x = a
under the sign of summation S. If we do so, and put H for the
THE SUMS OF PERIODIC SERIES. (SECT. I. SERIES.) 253
sum of the series (15), and B n for the remainder arising from sub
tracting the expression (10) from A n , we shall have
i n TT ^ r, .
s, SX = zB n sm 
CL
and the sum of the series forming the righthand side of this equa
tion will be a continuous function of x. As to SX, the value of
each series contained in it is given by equation (20).
To illustrate this, suppose H the ordinate of a curve of which
x is the abscissa. Let OGr be the axis of x\ OA, MB, ND, Ob
right lines perpendicular to it, and let OG = a. Let the curve of
Q
which % is the ordinate be the discontinuous curve AB, CD,
EFG. Take Gb equal to BC, and on the positive or negative
side of the axis of x according as the ordinate decreases or increases
as x increases through OM, and from measure an equal length
Oc on the opposite side of the axis. Take Gd, Oe, each equal to
DE, and draw the right lines AG, Ob b, cc G, Od d, ee G. Then
it will be easily seen that if X v X lt X z be the values of X cor
responding to the critical values of x, x = 0, x OM, x = ON,
respectively, X will be represented by the right line AG ; X l by
254 ON THE CRITICAL VALUES OF
the discontinuous right line 06 , c G\ and X^ by the discontinuous
right line Od , e G. Take MP equal to the sum of the ordinates
of the points in which the right lines lying between OA and c B
cut the latter line; M Q equal to the sum of the ordinates of the
points in which the right lines lying between c B and d E cut the
former, and so on, the ordinates being taken with their proper
signs. Let P, Q, R, S be the points thus found, and join AP,
QR, SO. Then SX will be represented by the discontinuous right
line AP, QR, 8G. Let the ordinates of the discontinuous curve
be diminished by those of the discontinuous right line last men
tioned, and let the dotted curve be the result. Then H SX will
be represented by the continuous, dotted curve. It will be
observed that the two portions of the dotted curve which meet in
each of the ordinates MB, NE may meet at a finite angle. If
there should be a point in one of the continuous portions, such as
AB, of the discontinuous curve where two tangents meet at a
finite angle, there will of course be a corresponding point in the
dotted curve.
If we choose to take account of the conjugate points of
the curve of which SX is the ordinate, it will be observed that
they are situated at 0, and midway between P and Q, and between
R and S.
9. There are a great many series, similar to (3), in which f (x)
may be expanded within certain limits of x. I shall consider one
other, which as well as (3) is of great use, observing that almost
exactly the same methods and the same reasoning will apply in
other cases.
The limit of the sum of the series
1 f%/ N J 2 V n f%/ N nrjTX J U7rX /M \
 f (x) dx +  2.g H I /(a?) cos dx .cos  ...(21),
d J o & J M &
when g from having been less than 1 becomes 1, is f (#), x being
supposed not to lie beyond the limits and a. For values, how
ever, of x for which f (x) alters discontinuously, the limit of the
sum is the arithmetic mean of the values of f (x) for values of x
immediately above and below the critical value. I assume this as
being well known, observing that it may be demonstrated jusfc as a
similar theorem has been demonstrated in Art. 4.
THE SUMS OF PERIODIC SERIES. (SECT. I. SERIES.) 255
10. Let us now consider the series
1 [ a /./ ,x 7 , 2 , [ a , . , N nirx , , mrx /c .<*\
 f(x) dx + 2, /UP) cos das . cos .... (22).
Q>JO a Jo a a
We have by integration by parts
mrx 7 ,
cos dx
and now, taking the limits properly, and employing the letters
M, N, a. and S in the same sense as before, we have
2 [ a
jj(x)a
mr a n 2 "
E being a quantity which does not become infinite with n. It
follows from (23), that the series (22) is in all cases convergent, and
its sum for all values of x } critical as well as general, is the limit of
the sum of (21).
It will be observed that if f (x) is a continuous function the
series (22) is at least as convergent as the series 2 l/n*. This is
not the case with the series (3), unless /(O) =/() = 0.
If the constant term and the coefficient of cos mrx fa in the
general term of (22) are given, f (x) itself not being known,
except by its developement, we may as before find the values of x
for which /(a?) is discontinuous, and the quantity by which f (x)
is suddenly increased as x increases through each critical value.
We may also, if we please, clear the series (22) of the slowly con
vergent part corresponding to the discontinuous values of f (x}.
11. Since the series (3) is convergent, if we have occasion to
integrate /(a?) we may, instead of first summing the series and
then integrating, first integrate the general term and then sum.
More generally, if < (x) be any function of x which does not
become infinite between the limits x = and x = a, we shall
have
f * ^ / N i / \ 7 2 _^ [ a . . ,. . UTTX , , f* , . U7TX ,
f(x] <l>(x)dx = 2 f(x ) am   dot. <#> (x) sm   dx,
Jo ft J o &Jo a
the superior limit x of the integrals being supposed not to lie
beyond the limits and a ; and the series at the second side of the
250 ON THE CRITICAL VALUES OF
above equation will be convergent. In fact, even in the case in
which f (x) is discontinuous the series will be as convergent as the
series 2 l//i 2 . A second integration would give a series still
more rapidly convergent, and so on. Hence, the resulting series
may be employed directly, and not merely when regarded as limits
of converging series. The same remarks apply in all respects to
the series (22) as to the series (3).
12. But the series resulting from differentiating (3) or (22)
once, twice, or any number of times would not in general be conver
gent, and could not be employed directly, but only as limits of the
convergent series which would be formed by inserting the factor y n
in the general term. This mode of treating the subject however
appears very inconvenient, except in the case in which the series
are only temporarily divergent, being rendered convergent again
by new integrations ; and even then it requires great caution.
The series in question may however be rendered convergent by
means of transformations to which I now proceed, and which,
with their applications, form the principal object of this paper.
The most important case to consider is that in which / (a?) and
its derivatives are continuous, so that the divergency arises from
what takes place at the limits and a. I shall suppose then, for
the present, that f (x) and its derivatives of the orders considered
are continuous, except the last, which will only appear under the
sign of integration, and which may be discontinuous.
Consider first the series of sines. Suppose that / (x) is not
given in finite terms, but only by its developement
where A n is supposed to be given, and where the developement of
f(x) is supposed to be that which would result from the formula
(3). I shall call the expansions of / (.*) which are obtained, or
which are to be looked on as obtained from the formulae (3) and
(22) direct expansions, as distinguished from other expansions
which may be obtained by differentiation, and which, being diver
gent, cannot be directly employed. Let us consider first the even
differential coefficients of/(0), and let A\ A* ... be the coeffi
cients of smmrx/a in the direct expansions of /" (x), f 4 (x) ...
The coefficient of sin mrx/a in the series which would be obtained
THE SUMS OF PERIODIC SERIES. (SECT. I. SERIES.) 257
by differentiating twice the several terms in the series in (24)
would be  nV/a* . A n . Now
2 r a mrx ,
and we have by integrating by parts
mrx , 2w7r ,, , x mrx 2 ,,, ,. . mrx
d*=^f(*)<f(*)* a
Taking now the limits, remembering the expression for A^ , and
transposing, we get
.
Any even differential coefficient may be treated in the same
way. We thus get, /ut, being even,
13. In the applications of these equations which I have
principally in view, /(O), f(a), /"(O)... are given, and A lt A 2 ,
A 3 ... are indeterminate coefficients. If however A lt A z ... A n ...
are given, and /(O), f (a) ... unknown, we must first find /(O),
f(a)..., and then we shall be able to substitute in (25) and (26).
This may be effected in the following manner.
We get by integrating by parts
r ,. , . mrx , ">/ * n7rsc>
/ sm dx =f(x) cos 
a m
. O /
mrx
+ f(cc } sm + if (x cos
Multiplying now both sides by 2/a, and taking the limits of the
integrals, we get
s. 17
258 ON THE CRITICAL VALUES OF
Hence, if n be always odd or always even, A n can be expanded,
at least to a certain number of terms, in a series according to
descending powers of n, the powers being odd, and the first of
them 1. The number of terms to which the expansion in this
form is possible will depend on the number of differential coef
ficients of f(x) which remain finite and continuous between the
limits # = and x = a. Let the expansion be performed, and
let the result be
A n = + a l+0 4 l,+ ... when n is odd, 1
; I i [.(28).
A n E  + E 2 3 4 E 4 5 4 . . . when n is even
W Yl 111 J
Comparing (27) and (28), we shall have
\.(0.+E& /(a) = I (0 fl
7T 3 ,
..(29),
and so on. The first two of these equations agree with (18).
If we conceive the value of A n given by (27) substituted
in (26), we shall arrive at a very simple rule for finding the direct
expansion off*(x). It will only be necessary to expand A n as
far as l/w 4 " 1 , admitting ( l) n into the expansion as if it were
a constant coefficient, and then, subtracting from A n the sum
of the terms thus found, employ the series which would be ob
tained by differentiating the equation (24) /z, times. It will be
necessary to assure ourselves that the term in I/nf vanishes in
the expansion of A n , since otherwise f*(x) might be infinite,
or/**" 1 ^) discontinuous without our being aware of it. It will
be seen however presently (Art. 20) that the former circumstance
would not vitiate the result, nor introduce a term involving 1/n*.
THE SUMS OF PERIODIC SERIES. (SECT. I. SERIES.) 259
Should A n already appear under such a form as
l +c r; (lyJ + nV, &c.,
/ It
where c 2 < 1, it will be sufficient to differentiate equation (24)
//, times, and leave out the part of the series which becomes
divergent. For it will be observed that the terms c n , nV, in
the examples chosen, decrease with I/n faster than any inverse
power of n.
14. Let us now consider the odd differential coefficients
of /(#), supposing f(x) to be expanded in a series of cosines, so
that
/(*) =1*0 + 21?. cos .................. (30).
\AJ
Let A f n , A " n ... be the coefficients of smnirx/a in the direct
expansions of f (x\ / "(#)... in series of sines. If we were to
differentiate (30) once we should have mr/a . B n for the coef
ficient of sin n7rx/a. Now
nir 2 f  , ,. mrx , .
 .  f(x) cos  dx
a aj j ^ a
2 ., ,. . UTTX 2 /",,,. ,. . H7TX , ,
= fix ) sm  f  / (x ) sm  dx ;
u j a a] J a
and taking the limits of the integrals, and introducing B n and
A n , we get
A n=*n ....................... (31).
Hence, the series arising from differentiating (30) once gives
the direct expansion of/ (x) in a series of sines.
The coefficient of sin n7rx/a in the series which would be
obtained by differentiating (30) p times, /u. being odd, would
be ( 1)0* +1 )/ 2 (nw/aYB n . By proceeding just as in the last article
we obtain
~!/" 2 W(i)"/" 2 ()!.
172
260 ON THE CRITICAL VALUES OF
When / (O), f (a), &c., are known, this series enables us to develope
f*(x) in a direct series of sines, the direct developement of f(x)
in a series of cosines being given.
15. If we treat the expression for B n by integration by
parts, just as the expression for A n was treated, going on till
we arrive at the integral which gives Af, and observe that the
very same process is used in deducing the value of Af from
that of B n as in expanding the latter according to inverse powers
of n, and that the index of n in the coefficient of Af is yu,
and that A n vanishes when n becomes infinite, we shall see that
in order to obtain the direct expansion of f^(x) we have only
got to expand B n as far as l/n M , (the coefficient of 1/n* will
vanish,) and subtract from B n these terms of the expansion, and
then differentiate (30) //< times.
The expansion of B n , at least to a certain number of terms,
will proceed according to even powers of 1/w, beginning with 1/Vi 2 .
If we suppose that
B. = ^ + 8 i + U . . . when n is odd,
I \ \
B E. 2 + E 3  4 + E. 3 + . . . when n is even I
1 ri* 3 n* 5 n" J
and compare these expansions with that given by integration by
parts, we shall have
...(34),
and so on, the signs of the coefficients being alternately + and ,
and the index of vr/a increasing by 2 each time.
16. The values of / (O) and /**(a) when / M (#) is expanded
in a series of sines and p is odd, or when f(x) is expanded in a
series of cosines and JJL is even, will be expressed by infinite series.
To find these values we should first have to obtain the direct ex
pansion of /^(.T), which would be got by differentiating the equa
tion (24) or (30) p times, expanding A n or B n according to powers
of 1/w, and rejecting the terms which would render the series
contained in the /A th derived equation divergent. The reason of
this is the same as before.
THE SUMS OF PERIODIC SERIES. (SECT. I. SERIES.) 261
17. The direct expansions of the derivatives of f(x) may
be obtained in a similar manner in the cases in which f(x} itself,
or any one of its derivatives is discontinuous. In what follows,
cc will be taken to denote a value of x for which f(x) or any one
of its derivatives of the orders considered is discontinuous; Q,
Qi> Qi*. will denote the quantities by which f(x], f (x), ...f*(x)
are suddenly increased as x increases through a; S will be used
for the sign of summation relative to the different values of a,
and will be supposed to include the extreme values and a, under
the convention already mentioned in Art. 6. Of course f(x) may
be discontinuous for a particular value of x while f*(x) is con
tinuous, and vice versa. In this case one of the two Q, Q^ will
be zero while the other is finite.
The method of proceeding is precisely the same as before,
except that each term such as f(x) cos mrx/a in the indefinite
integral arising from the integration by parts will give rise to a
series such as SQ cos mra/a in the integral taken between limits.
We thus get in the case of the even derivatives of f(x), when f(x)
is expanded in a series of sines,
*\~ l SQco8 *
/ a
nira.
2 f^Y~ 2 S Q sin !^ + 2 (^Y~ 2 S Q :
a \a J a a \ a J
In the case of the odd derivatives of f(x), when f(x) is ex
panded in a series of cosines, we get
UTTOL
a
When the several values of a, Q, Q^... are given, these equa
tions enable us to find the direct expansion of f* (x). The series
corresponding to the odd derivatives in the first case and the even
in the second might easily be found.
If we wish to find the direct expansion of f^(x) in the case
in which A n or B n is given, we have only to expand A n or B n in a
series according to descending powers of n, regarding cos ny or
262 .ON THE CRITICAL VALUES OF
sin ny, as well as ( 1)", as constant coefficients, and then reject from
the series obtained by the immediate differentiation of (24) or (30)
those terms which would render it divergent. This readily follows
as in Art. 15, from the consideration of the mode in which A n *
is obtained from A n or B n .
The equations (85) and (36) contain as particular cases (26)
and (32) respectively. It was convenient however to have the
latter equations, on account of their utility, expressed in a form
which requires no transformation.
18. If we transform A n and B n by integration by parts, we
get
A 2
A= 
n7r" a
(37),
n 2 ~ . mra 2a ar .
B n =  SQ sin  a g SQ. cos
mr a nV a
2a? ,y^ . UTTO. /oox
+ n ,^SQ tS m +..., ............. (38),
where the law of the series is evident, if we only observe that
two signs of the same kind are always followed by two of the
opposite kind. The equations (37), (38) may be at once obtained
from (35), (36). The former equations give the true expansions
of A n and B n according to powers of 1/n; because when we stop
after any number of integrations by parts the last integral with
its proper coefficient always vanishes compared with the coefficient
of the preceding term.
Hence A n and B n admit of expansion according to powers
of 1/n, if we regard 003717 or sin ny as a constant coefficient in
the expansion. Moreover quantities such as cos ny, smny will
occur alternately in each expansion, the one kind going along
with odd powers of 1/n and the other along with even. If we
suppose the value of A n or B n , as the case may be, given, and
the expansion performed, so that
y. + SFt sin ny., + SF, cos W7 . 8 + ..., ...(39),
B H = SG sin ny .  + 0, cos ny . \ + #. sin ny .  3 + . , . , ... (40),
THE SUMS OF PERIODIC SERIES. (SECT. I. SERIES.) 263
and compare these expansions with (37) or (38), we shall get the
several values of a, and the corresponding values of Q, Q v Q 2 ...
We may thus, without being able to sum the series in equation
(24) or (30), find the values of x for which f(x) itself or any one
of its derivatives is discontinuous, and likewise the quantity by
which the function or derivative is suddenly increased. This
remark will apply to the extreme values and a of x if we con
tinue to denote the sum of the series by f(x) when x is outside
of the limits and a.
19. Having found the values of a, Q, Q^..., we may if we
please clear the series in (24) or (30) of the terms which render
f(ss) itself, or any one of its derivatives, discontinuous. If we
wish the function which remains expressed by an infinite series
and its first //, derivatives to be continuous, we have only to sub
tract from A n or B n the terms at the commencement of its ex
pansion, ending with the term containing I/nf +1 , and from f(x)
itself the sums of the series corresponding to the terms subtracted
from A n or B n . These sums will be obtained by transforming
products of sines and cosines into sums or differences, and then
employing known formulae such as
COS Z COS 3.3 7T 2 7T ~ _ , . _ .
Br + T + = ^jT> from 2; = to z = TT (41),
_L O o "r
which are obtained by integrating several times the equation
sin z f J sin 2z f l sin 3.Z + . . . =  (IT z\ from z = to z ZTT,
or the equation deduced from it by writing TT z for z, and taking
the semisum of the results. It will be observed that in the
several series to be summed we shall always have sines coming
with odd powers of n and cosines with even. Of course, by clear
ing the series in (24) or (30) in the way just mentioned we shall
increase the convergency of the infinite series in which a part of
f(x) still remains developed.
When A n or B n decreases faster than any inverse power of n
as n increases, (as is the case for instance when it is the w th term of
a geometric series with a ratio less than 1,) all the terms of its
expansion in a series according to inverse powers of n vanish. In
this case, then, f(x) and its derivatives of all orders are con
tinuous.
204 ON THE CRITICAL VALUES OF
20. In establishing the several theorems contained in this
Section, it has been supposed that none of the derivatives of / (x)
which enter into the investigation are infinite. It should be
observed, however, that if/ 4 (x) is the last derivative employed,
which only appears under the sign of integration, it is allowable to
suppose that /> (so) becomes infinite any finite number of times
within the limits of integration. To shew this, we have only got
to prove that
ra ra
/> (x) sin vx dx or /> (#) cos vx dx
Jo Jo
approaches zero as its limit as v increases beyond all limit. Let us
consider the former of these integrals, and suppose that /** (x)
becomes infinite only once, namely, when x = a, within the limits of
integration. Let the interval from to a be divided into these
four intervals to a f, a  f to a, a to a + f , a + f to a, where
f and f are supposed to be taken sufficiently small to exclude all
values of x lying between the limits a f and a + f for which
/ /x ~ 1 (#) alters discontinuously, or for which f*(x) changes sign,
unless it be the value a. For the first and fourth intervals /** (x)
is not infinite, and therefore, as it is known, the corresponding parts
of the integral vanish for v = oo . Since sin vx cannot lie beyond
the limits + 1 and 1, and is only equal to either limit for parti
cular values of x, it is evident that the second and third portions
of the integral are together numerically inferior to /, where
the symbol A ~B denoting the arithmetical difference of A and B,
and e being an infinitely small quantity, so that /(a e), f(a + e)
denote the limits to which f(x) tends as x tends to the limit a by
increasing and decreasing respectively. Hence the limit of the
integral first considered, for v = oo , must be less than I. But /may
be made as small as we please by diminishing f and f , and there
fore the limit required is zero.
The same proof applies to the integral containing cos vx, and
there is no difficulty in extending it to the case in which f* (x) is
infinite more than once within the limits of integration, or at one
of the limits.
21. It has hitherto been supposed that the function expanded
in the series (3) or (22) does not become infinite ; but the expan
sions will still be correct even if / (x} becomes infinite any finite
THE SUMS OF PERIODIC SERIES. (SECT. I. SERIES.) 265
number of times, provided that jf(x)dx be essentially convergent.
Suppose that f (x) becomes infinite only when x = a. Then it is
evident that we may find a function of x, F (x), which shall be
equal to f(x) except when x lies between the limits a ? and
a + f, which shall remain finite from x a. % to a? = a + f , and
/+? /+
which shall be such that F(x)dx=\ f(x}dx. Suppose
J a J af
that we are considering the series (3). Then, if G n be the coeffi
cient of sin mrx/a in the expansion of F (x) in a series of the
form (3), it is evident that C n will approach the finite limit A n
2 C a mrx
when t and ? vanish, where A  I /(a?) sin  cZar. But so
n
long as f and f differ from zero the series 2) (7 n sin mrx I a is conver
gent, and has jP T (a?) for its sum, and F (x) becomes equal to / (as)
when f and f vanish, for any value of # except a. We might
therefore be disposed to conclude at once that the series (3) is
convergent, and has f(x) for its sum, unless it be for the particular
value x = a. but this point will require examination, since we
might conceive that the series (3) became divergent, or if it
remained convergent that it had a sum different from f (x), when
f and f were made to vanish before the summation was performed.
If we agree not to consider the series (3) directly, but only the limit
of the series (5) when g becomes 1, it follows at once from (7) that
for values of x different from a that limit is the same as in Art. 4.
For x = a the limit required is that of \ {/(a e) +/ (a + e)} when
evanishes. IS. f(x) does not change sign as x passes through a
the limit required is therefore positive or negative infinity, accord
ing as f(x) is positive or negative; but if f (x) changes sign in
passing through oo the limit required may be zero, a finite quantity,
or infinity. The expression just given for the limit may be
proved without difficulty. In fact, according to the method of
Art. 4, we are led to examine an integral of the form
where f is a constant quantity which may be taken as small as we
please, and supposed to vanish after h. Now by a known property
of integrals the above integral is equal to
\. {/() +/( + &} , where f, lies between and
2G6 OX THE CRITICAL VALUES OF
But JTT% > which is equal to tan" 1  t } becomes equal to ^ when
J o M ~r 5 ft 2t
h vanishes, and the limit of , when h vanishes must be zero, since
it cannot be greater than f, and may be made to vanish
after h.
22. The same thing may be proved by the method which con
sists in summing the series S sin nirx/a . sin mrx /a to n terms.
If we adopt this method, then so long as we are considering a value
of x different from a it will be found that the only peculiarity in
the investigation is, that the quantity under the integral sign in
the integrals we have to consider becomes infinite for one value of
the variable ; and it may be proved just as in Art. 20, that this
circumstance has no effect on the result. If we are considering the
value x = a, it will be found that the integral we shall have to con
sider will be
where v is first to be made infinite, and then f may be supposed to
vanish. If /(a + e) +/(a e) approaches a finite limit, or zero,
when e vanishes, as may be the case if f(x) changes sign in passing
through oo, it may be proved, just as in the case in which f (x)
does not become infinite, that the above integral approaches the
same limit as \ {/( + e) + /( e)]. In all cases however in
which / (x) does not change sign in passing through oo , and in
some cases in which it does change sign, f(a + e) +/(% e) becomes
infinite when e vanishes. f
In such cases put for shortness
and let the numerical values of the integral I "*""* dj: taken from
to TT/V, from ir/v to ZTT/V ... or which is the same those of
1 S1 di; taken from to TT, from TT to 2?r . . . be denoted by / t , / 2 . ..
Then evidently J 1 >/ 2 >/3... Also, if f be sufficiently small,
.F (f ) will decrease from = to f = f, if we suppose, as we may,
F(%) to be positive. Hence the integral (42), which is equal to
...} (43),
THE SUMS OF PERIODIC SERIES. (SECT. I. SERIES.) 267
where , 2 ... are quantities lying between and TT/V, TT/V and
27T/z/ ... is greater than
if we neglect the incomplete pair of terms which may occur at the
end of the series (43), and which need not be considered, since they
vanish when v = oo . Hence, the integral (42) is a fortiori
>T~ 1 (/ 1 / 8 )^(? 1 ). But vanishes and F(%^ becomes infinite
when v becomes infinite ; and therefore for the particular value
x = a the sum of the first n terms of the series (3) increases
indefinitely with n.
If a coincides with one of the extreme values and a of x, the
sum of the series (3) vanishes for x = a. This comes under the
formula given above if we consider the sum of the series for values
of x lying beyond the limits and a. The same proof as that
given in the present and last article will evidently apply if / (x)
become infinite for several values of x, or if the series considered
be (22) instead of (3). In this case, the sum of the series becomes
infinite for x = a when a = or = a.
23. Hence it appears that f(x] may be expanded in a series
of the form (3) or (22), provided only If (as) dx be continuous. It
should be observed however that functions like (sin c/x)~% , which
become infinite or discontinuous an infinite number of times within
the limits of the variable within which they are considered, have
been excluded from the previous reasoning.
Hence, we may employ the formulae such as (26), (35), &c., to
obtain the direct developement of/** (x), without enquiring whether
it becomes infinite or not within the limits of the variable for
which it is considered. All that is necessary is that f(x) and its
derivatives up to the (/t l) th inclusive should not be infinite within
those limits, although they may be discontinuous.
24. In obtaining the formulae of Arts. 7 and 13, and generally
the formulae which apply to the case in which A n or B n is given,
and f(x) is unknown, it has hitherto been supposed that we knew
a priori that f (x) was a function of the class proposed in Art. 1 for
consideration, or at least of that class with the extension mentioned
ON THE CRITICAL VALUES OF
in the preceding article. Suppose now that we have simply pre
sented to us the series (3) or (22), namely
, . . UTTX ^ mrx
zA n sm  , or B Q + 2<B n cos  ,
Oj d
where A n or B n is supposed given and want to know, first, whether
the series is convergent, secondly, whether if it be convergent it is
the direct developement of its sum f(x), and thirdly, whether we
may directly employ the formulae already obtained, trusting to the
formulae themselves to give notice of the cases to which they do
not apply by leading to processes which cannot be effected.
25. If the series 24 n or ^B n is essentially convergent, it is
evident a fortiori that the series (3) or (22) is convergent.
If A n = S  cos ny + G n , or if B n = 8  sin ny + C n , where 2(7
11 n
is essentially convergent, the given series will be convergent, as is
proved in Art. 6.
In either of these cases let / (as) be the sum of the given series.
Suppose that it is the series of sines which we are considering.
Let E n be the coefficient of sin mrx/a in the direct developement of
f(x). Then we have
* / x ~ A . niTX ^ r, . H7TX
f(x) = 2A n sm = ^E n sm  
U/ (Jj
and since both series are convergent, if we multiply by any finite
function of vc, <f> (x), and integrate, we may first integrate each
term, and then sum, instead of first summing and then integrating.
Taking < (x) = sin nirx/a, and integrating from,, a = to x = a, we
get E n =.A n , so that the given series is the direct developement
of its sum f(x). The proof is the same for the series of cosines.
26. Consider now the more general case in which the series
2 I/n . A n is essentially convergent. The reasoning which is about
to be offered can hardly be regarded as absolutely rigorous ; never*
theless the proposition which it is endeavoured to establish seems
worthy of attention. Let u n be the sum of the first n terms of the
given series, and F (n, x) the sum of the first n terms of the series
2 a/mr . A n cos mrx I a. Then we have
5( u n+ m ~ u n) dx = F (n + ra, x)  F (n, x} = ^r (n, x), suppose. . .(44).
Now by hypothesis the series S I/n . A H is essentially convergent,
and therefore a fortiori the series 2 a/HTr . A n cos mrx/a is con
THE SUMS OF PERIODIC SERIES. (SECT. I. SERIES.) 269
vergent, and therefore ^ (co , x) = 0, whatever be the value of m.
Let the limits of x in (44) be x and x + A#, and divide by A#, and
we get
and as we have seen the limit of the second side of this equation
when we suppose n first to become infinite and then Ace to vanish
is zero. But for general values of x the limit will remain the same
if we first suppose A# to vanish and then n to become infinite ; and
on this supposition we have
limit of (w n+m u n ) = 0, for n = oo ;
so that for general values of x the series considered is convergent.
To illustrate the assumption here made that for general values
of x the order in which n and A# assume their limiting values is
immaterial, let ty (y, x) be a continuous function of x which
becomes equal to ^r (n, x) when y is a positive integer ; and con
sider the surface whose equation is z = ty (y, x). Since
^ (x , x) =
for integral values of y, the surface approaches indefinitely to the
plane xy when y becomes infinite ; or rather, among the infinite
number of admissible forms of ^r (y, x) we may evidently choose
an infinite number for which that is the case. Now the assertion
made comes to this ; that if we cut the surface by a plane parallel
to the plane xz, and at a distance n from it, the tangent at the
point of the section corresponding to any given value of x will
ultimately lie in the plane xy when n becomes infinite, except in
the case of singular, isolated values of x, whose number is finite
between x = and x = a. For such values the sum / (x) of the
infinite series may become infinite, while $f(x)dx remains finite.
The assumption just made appears evident unless A n be a function
of n whose complexity increases indefinitely with its rank, i. e. with
the value of n.
Since the integral of /(a?) is continuous, f(x] may be expanded
by the formula in a series of sines. Let E n be the coefficient
of smn7rx/a in its direct expansion ; so that,
*i \ ^ A 
/= S^sm 
(45),
i
270 ON THE CRITICAL VALUES OF
where both series are convergent, except it be for isolated values
of x. Consequently, we have, in a series which is convergent,
at least for general values of x,
(46).
The series (45) may become divergent for isolated values of x y
and are in fact divergent for values of x which render f (x] infinite.
But the first side of (46) being constantly zero, and the series
at the second side being convergent for general values of x, it
does not seem that it can become divergent for isolated values.
Hence according to the preceding article the second side of the
equation is the direct developement of the first side, i.e. of zero;
and therefore E n A n , or the given series is the direct develope
ment of its sum, which is what it was required to prove. The
same reasoning applies to the series of cosines.
It may be observed that the well known series,
J + cos x + cos 2x + cos 3# ........ , ........ (47),
forms no exception to the preceding observation. This series
is in fact divergent for general values of x, that is to say not
convergent, and in that respect it totally differs from the series
in (46). When it is asserted that the sum of the series (47)
is zero except for x = or any multiple of 2?r, when it is infinite,
all that is meant is that the limit to which the sum of the
convergent series \ + %g n cos nx approaches when g becomes 1
is zero, except for x or any multiple of 2?r, in which case it is
infinity.
27. It follows from the preceding article that even without
knowing a priori the nature of the function f(x) we may employ
the formulae such as (35), provided that if n~* be the highest
power of I/n required by the formula, and n~>* C n the remainder in
the expansion of A n) the series ^n~ l C n be essentially convergent.
For let G n be the sum of the terms as far as that containing n~* in
the expansion of A n , those terms having the form assigned by (35),
that is to say cosines like cos ny coming along with odd powers
of 1/n, and sines along with even powers. Then
Let 2 sin
THE SUMS OF PERIODIC SERIES. (SECT. II. INTEGRALS.) 271
then f(x)F(x) = 2w* C n sin  ............ (48).
Now if <(#) = Sw n , where the series 2w n , %du n /dx are both
convergent, we may find < (a?) by differentiating under the sign of
summation. This is evident, since by the theorem referred to
in Art. 2 (note), we may find \t~ dx by integrating under the
sign of summation. Consequently we have from (48)
..... (49);
and since the series 2 ri~ l C n is essentially convergent, the con
vergency of the series forming the righthand side of (49) cannot
become infinitely slow (see Sect. III.), and therefore, the n ih term
being a continuous function of x, the sum is also a continuous
function of x, and therefore /**(#) F ti (x} is a function which
by Art. 23 can be expanded in a series of sines or cosines. But
Fp (x) is also such a function, being in fact a constant, and
therefore /**(#) is a function of the kind considered in Art. 23,
which is what is assumed in obtaining the formula (35).
It may be observed that these results do not require the
assumptions of Art. 26 in the case in which the series 2<7 n is
essentially convergent, or composed of an essentially convergent
series and of a series of the form S/Stan" 1 sin wy or "ZScrT 1 cosny,
according as C n is the coefficient of a cosine or of a sine.
SECTION II.
Mode of ascertaining the nature of the discontinuity of the
integrals which are analogous to the series considered in Section I.,
and of obtaining the developements of the derivatives of the expanded
functions.
28. Let us consider the following integral, which is analogous
to the series in (1),
I 00S)sin#eZ (50),
J o
where <f>(j3) =  f" f(x) sinfa dx (51).
272 ON THE CEITICAL VALUES OF
Although the integral (50) may be written as a double in
tegral,
2 f 00 [ a
I f(x)$mjBxsmBxdBc!x (52),
7T J Jo
the integration with respect to x must be performed first, because,
the integral of sin fix sin fix df3 not being convergent at the
,00
limit oo , I sin /3x sin fBx d/3 would have no meaning. Suppose,
Jo
however, that instead of (52) we consider the integral,
2 f C a
7TJO Jo
where h is a positive constant, and e is the base of the Napierian
logarithms. It is easy to see that at least in the case in which the
integral (50) is essentially convergent its value is also the limit to
which the integral (53) tends when h tends to zero as its limit.
It is well known that the limit of (53) when h vanishes is in
general / (x) ; but when x = the limit is zero ; when x a the
limit is J/(a); and when f(x) is discontinuous it is the arith
metic mean of the values of f (x) for two values of x infinitely
little greater and less respectively than the critical value. When
x > a it is zero, and in all cases it is the same, except as to sign, for
negative as for positive values of x.
We may always speak of (53), but we cannot speak of the
integral (50) till we assure ourselves that it is convergent. Now we
get by integration by parts, ,
/(# ) sin fix dx  ~~ o/ ( x } cos fa
~T ry>j (x ) sin px ~ a I j (x j sin px dx \^^V*
When this integral is taken between limits, the first term will
furnish a set of terms of the form G/@ . cos fix, where a may be
zero, and the last two terms will give a result numerically less than
Z//3 2 , where L is a constant properly chosen. Now whether a. be
zero or not, /cos /3a sin /3x . /3" 1 d/3 is convergent at the limit oo , and
moreover its value taken from any finite value of /3 to /3 = oo is
the limit to which the integral deduced from it by inserting the
factor e~ 7 ^ tends when h vanishes. The remaining part of the
integral (50) is essentially convergent at the limit oo . Hence the
THE SUMS OF PERIODIC SERIES. (SECT. II. INTEGRALS.) 273
integral (50) is convergent, and its value for all values of x, both
critical and general, is the limit to which the value of the integral
(53) tends when h vanishes.
29. Suppose that we want to find f" (x), knowing nothing
about f(x), at least for general values of x, except that it is the
value of the integral (50), and that it is not a function of the class
excluded from consideration in Art. 1. We cannot differentiate
under the integral sign, because the resulting integral would,
usually at least, be divergent at the limit oo , We may however
find /" (x) provided we know the values of x for which f(x) and
/ (x) are discontinuous, and the quantities by which f(x) and / (a;)
are suddenly increased as x increases through each critical value,
supposing the extreme values included among those for which
f(x) or f (x) is discontinuous, under the same convention as in
Art. 6. Let a be any one of the critical values of x ; Q, Q t the
quantities by which f(x), f (x) are suddenly increased as x
increases through a ; $ the sign of summation referring to the
critical values of #; fa(ft) the coefficient of sin fix in the direct
developement of /" (x) in a definite integral of the form (50). Then
taking the integrals in (54) between limits, and applying the
formula (51) to/" (x), we get
2 9
(j) (fi) =  /3* 6(P) +  /3SQ cos fa   SO sin 82.
7T 7T
We may find fa(/3) in a similar manner. We get thus when
yu, is even
 2 2
( I) 2 fa ()  0ty(/9)   0*~ 1 8Q cos fiy. +  0**8Q. sin fa + ...
7T 7T
where sines and cosines occur alternately, and two signs of the
same kind are always followed by two of the opposite. The expres
sion for fr (@). when //, is odd might be found in a similar manner.
These formulae enable us to express f^ (x) when <f> (/3) is an
arbitrary function which has to be determined, and /(O), &c. are
given.
30. If however <p ((3) should be given, and /(O), &c. be
unknown, <f> (fi) will admit of expansion according to powers of pi 1 ,
beginning with the first, provided we treat sin/3a or cos fa as if it
s. 18
274 ON THE CRITICAL VALUES OF
were a constant coefficient ; and sin fix, cos /3a will occur with even
and odd powers of ft respectively. The possibility of the expan
sion of </> (ft) in this form depends of course on the circumstance
that (f> (x) is a function of the class which it is proposed in Art. 1
to consider, or at least with the extension mentioned in Art. 23.
It appears from (55) that in order to express /** (x) as a definite
integral of the form (50) we have only got to expand c/> (ft), to
differentiate (50) p times with respect to x, differentiating under
the integral sign, and to reject those terms which appear under the
integral sign with positive powers of ft or with the power 0. The
same rule applies whether //, be odd or even.
31. If we have given < (a), but are not able to evaluate the
integral (50), we may notwithstanding that find the values of x
which render f(x) or any of its derivatives discontinuous, and the
quantities by which the function considered is suddenly increased.
For this purpose it is only necessary to compare the expansion of
</> (ft) with the expansion
jftBQmfrp8Q l mnl* ......... (5G),
given by (55), just as in the case of series.
We may easily if we please clear the function <f> (ft) of the part
for which f(x) or any one of its derivatives is discontinuous, or
does not vanish for x = and x = a. For this purpose it will be
sufficient to take any function F (x) at pleasure, which as well as
its derivatives of the orders considered has got the same discon
tinuity as f(x) and its derivatives, to develope F (x) in a definite
f
integral of the form <J> (ft) sin fix dft by the formula (51), and to
Jo
subtract F (x) from f(x) and 4> (ft) from $ (ft). It will be
convenient to choose such simple functions as I + mx f nx 2 ;
I sin x + m cos x ; le~ x + we **, &c. for the algebraical expressions of
F(x) for the several intervals throughout which it is continuous,
the functions chosen being such as admit of easy integration
when multiplied by sin fixdx, and which furnish a sufficient number
of indeterminate coefficients to allow of the requisite conditions
as to discontinuity being satisfied. These conditions are that
the several values of Q, Q lt &c. shall be the same for F(x) as for
/<*)
THE SUMS OF PERIODIC SERIES. (SECT. II. INTEGRALS.) 275
32. Whenever / f(x) dx is essentially convergent, we may at
once put a = oo in the preceding formulae. For, first, it may be
easily proved that in this case, (though not in this case only,) the
limit of (53) when h vanishes is f(x) ; secondly, the limit of (53)
is also the value of (52); and, lastly, all the derivatives of f(x)
have their integrals, (which are the preceding derivatives,) essentially
convergent, and therefore oo may be put for a in the developements
of the derivatives in definite integrals.
When f(x} tends to zero as its limit as x becomes infinite, arid
moreover after a finite value of x does not change from decreasing
to increasing nor from increasing to decreasing,
f e** f (x) sin flat dx
J o
/.oo
will be more convergent than I f(x)sm$x dx t and the latter
Jo
integral will be convergent, and its convergency will remain finite*
when j3 vanishes. In this case also we may put a = oo .
Thus if /(a?) = sin Ix (b* + a 2 ) 1 , we may put a = oo because f(x)
has its integral essentially convergent : if / (x) = (b 4 x)~*, we may
put o = co because f(x) is always decreasing to zero as its limit.
But if f(x) = smlx (b + x)~*, the preceding rules will not apply,
because f(x), though it has zero for its limit, is sometimes increas
ing and sometimes decreasing. And in fact in this case the
integral in equation (51) will be divergent when/3 = /, and $(J3)
will become infinite for that value of @. It is true that/(#) is still
the limit to which the integral (53) tends when h vanishes ; but I
do not intend to enter into the consideration of such cases in
this paper.
33. When oo may be put for a, and/(#) is continuous, we get
from (55)
+ (l)/3/> 2 (0) ............ (57).
In this case $ (fi) will admit of expansion, at least to a certain
* See next Section.
182
276 ON THE CRITICAL VALUES OF
number of terms, according to odd negative powers of (3. If
we suppose < (/3) known, and the expansion performed, so that
and compare the result (49), we shall get
/(0)=//o; /"(0)=tf 2 ; / 4 (0) = # 4 ; &c ......... (58).
34. The integral
[ t(/3)cos/3^/3 ......................... (59),
Jo
where
(^>os/3^ ............ (60),
which is analogous to the series (22), is another in which it is some
times useful to develope a function or conceive it developed. For
positive values of x the value of (59) is the same as that of (50).
When x = the value is /(O); and for negative values of x it is
the same as for positive. It is supposed here that the integral (59)
is convergent, which it may be proved to be in the same manner
as the integral (50) was proved to be convergent.
Suppose that we wish to find, in terms of ^r (/3), the develope
ment of /> (x) in a definite integral of the form (50) or (59),
according as //, is odd or even. We cannot differentiate under the
integral sign, because the resulting integral would be divergent.
We may however obtain the required developemepit by transform
ing the expression ^ (@) by integration by parts, just as before. We
thus get for the case in which //, is odd
Mi 9 9
(1) fa(&)
*
* ............ (61),
7T
where </y (/3) is the value of </> (/3) in the direct developement of
/* (x) in the integral (50). In the same way we may get the value
of.^(/3) when //, is even, ^(/3) being the value of ir (/3) in the
direct developement of/ 1 (a?) by the formulae (59), (60).
The equation (61) is applicable to the case in which ^(/3) is an
arbitrary function, and a, Q, &c., are given. If however
THE SUMS OF PERIODIC SERIES. (SECT. II. INTEGRALS.) 277
should be given, we may find ^(/5) or ^(/3) by the same rule as
before.
In the case in which ^ (/3) is given, we may find the values of
a, Q, &c., without being able to evaluate the integral (59). For
this purpose it is sufficient to expand ty (/3) according to negative
powers of /3, and compare the expansion with that furnished by
equation (61).
35. The same remarks as to the cases in which we are at
liberty to put oo for a apply to (60) as to (51), with one exception.
In the case in which f(x) approaches zero as its limit, and is at
last always decreasing numerically, or at least never increasing, as
x increases, while $f(x) dx is divergent at the limit oo , it has been
observed that </>(#) remains finite when /5 vanishes. This however
is not the case with ^(/8), at least in general. I say in general,
rx
because, although I f(x] dx increases indefinitely with its superior
limit, we are not entitled at once to conclude from thence that
I cos @xf(x) dx becomes infinite when /3 vanishes, as will appear
in Section III. It may be shewn from the known value of
)
x~ n cos jSxdx, where 1 > n > 0, that if f(x) = F(x) + Cx~ n , where
F(x) is such that fF(x) dx is convergent at the limit oo , ty (ft) be
comes infinite when /3 vanishes; and the same would be true if
there were any finite number of terms of the form Cx~ n . There is
no occasion however to enquire whether ^(/3) always becomes in
finite : the point to consider is whether the integral (59) is always
convergent at the limit zero.
In considering this question, we may evidently begin the inte
gration relative to x at any value X Q that we please. Suppose first
\ve integrate from x =x to x = X, and let r($) be the result
so that
2 t x
*r(/9) =  fix ) cos fix dx.
TJ J *
Let ix i (/3) be the indefinite integral of or (/3) dp : then, c being a
positive quantity, we get from the above equation
w / (8) C 7 / (c) =  I f(x } {sin fix sin cx \ ^ .
T . jr e
278 ON THE CRITICAL VALUES OF
Now put JT=oo. Then since I /(a/) Sm , dx is a convergent
J Xo &
integral, and its convergency remains finite (Art. 39) when @
vanishes, as may be proved without much difficulty, its value can
not become infinite, and therefore vr,(/3) does not become infinite
when @ vanishes. Now
/() cos fad/3 = w,(/8) cos fa + xfa, (0) sin pxdfi ......... (62),
when x is positive ; and when x = 0,
hence in either case JW (/3) cos fa d/3 is convergent at the limit
zero. Now the quantity by which vr({3) differs from ty(@) evi
dently cannot render (59) divergent, and therefore in the case con
sidered the integral (59) is convergent at the limit zero.
00
By treating I sy(/3) e~*0cos (3%d{3 in the manner in which
J o
JW (/S) cos /3x d/3 is treated in (62), it may be shewn that the con
vergency of the former integral remains finite when h vanishes.
Hence, not only is the integral (59) convergent, but its value is
the limit to which the integral similar to (53) tends when h
vanishes.
When f(x) is continuous, and oo may be put for a, we have
from (61), /// being odd,
9 9
/"(O)
7T " * *T ^ V
...... (63).
If v/r(yS) be given we can find the values of/ (0), / "(O) ... just
as before.
36. The integral
cos/3(x x)f(x )d/3dx .............. (64),
in which the integration with respect to x is supposed to be per
formed before that with respect to fi, so that the integral has the
form
*eo ,.
X(P)co*fad/3+l tr(P)&afad0 ......... (65),
Jo Jo
THE SUMS OF PERIODIC SERIES. (SECT. III. DISCONTINUITY.) 279
may be treated just as the integral (59); and it may be shewn that
in the same circumstances we may replace the limits a t and a by
oo , + oo respectively. If we suppose %(/3) and cr(/3) known, we
may find as before the values of x for which f(x), f (x) ... are
discontinuous, and the quantities by which those functions are sud
denly increased. We may also find the direct developement of
f (x),f"(x) ... in two integrals of the form (65); and we may if we
please clear the integrals (65) of the part which renders f(x), f (x)...
discontinuous.
37. In the developement of f(x) in an integral of the form (50)
or (59), or in two integrals of the form (65), it has hitherto been
supposed that f(x) is not infinite. It may be observed however
that it is allowable to suppose f(x) to become infinite any finite
number of times, provided ff(x) dx be essentially convergent about
the values of x which render f(x) infinite. This may be shewn
just as in the case of series. Hence, the formulae such as (55) which
give the developement off t (x) are true even when/** (a?) is infinite,
f*~ l (x) being finite.
SECTION III.
On the discontinuity of the sums of infinite series, and of the
values of integrals taken between infinite limits.
38. LET
u l + u a ... + u n + (66),
be a convergent infinite series having U for its sum. Let
Wj+v,... +v n + (67),
be another infinite series of which the general term v n is a function
of the positive variable h, and becomes equal to u n when h vanishes.
Suppose that for a sufficiently small value of h and all inferior
values the series (67) is convergent, and has V for its sum. It
might at first sight be supposed that the limit of V for h = was
necessarily equal to U. This however is not true. For let the sum
to n terms of the series (67) be denoted by f(n, h) : then the limit
of V is the limit of f(n, h) when n first becomes infinite and then
h vanishes, whereas Uis the limit of f(n, h) when h first vanishes
280 ON THE CRITICAL VALUES OF
and then n becomes infinite, and these limits may be different.
Whenever a discontinuous function is developed in a periodic series
like (15) or (30) we have an instance of this ; but it is easy to form
two series, having nothing to to with periodic series, in which
the same happens. For this purpose it is only requisite to take for
f(n, h) U n , (U n being the sum of the first n terms of (66),) a
quantity which has different limiting values according to the order
in which n and h are supposed to assume their limiting values, and
which has for its finite difference a quantity which vanishes when n
becomes infinite, whether h be a positive quantity sufficiently small
or be actually zero.
For example, let
(68),
which vanishes when n = 0. Then
A {/(, h)  U.} = V M  u, M = ^ ny *L__ I5 .
Assume
U =1  1  r , sothatw =Af/ H =  r ,
n + 1 " 1 n (n + 1)
and we get the series
5&
2 (1 + h) " n (n + 1) {(w  1; h + Ij (nh + 1)
which are both convergent, and of which the general terms become
the same when h vanishes. Yet the sum of the first is 1, whereas
the sum of the second is 3.
If the numerator of the fraction on the righthand side of (68)
had been pnh instead of 2nh, the sum of the series (70) would have
been p + 1, and therefore the limit to which the sum approaches
when h vanishes would have beenp + 1. Hence we can form as
many series as we please like (67) having different quantities for
the limits of their sums when h vanishes, and yet all having their
n th terms becoming equal to u n when h vanishes. This is equally
true whether the series (66) be convergent or divergent, the series
like (67) of course being always supposed to be convergent for all
positive values of h however small.
THE SUMS OF PERIODIC SERIES. (SECT. III. DISCONTINUITY.) 281
39. It is important for the purposes of the present paper to
have a ready mode of ascertaining in what cases we may replace
the limit of (67) by (66). Now it follows from the following
theorem that this substitution may at once be made in an extensive
class of cases.
THEOREM. The limit of Fcan never differ from U unless the
convergency of the series (67) becomes infinitely slow when h
vanishes.
The convergency of the series is here said to become infinitely
slow when, if n be the number of terms which must be taken in
order to render the sum of the neglected terms numerically less
than a given quantity e which may be as small as we please, n
increases beyond all limit as h decreases beyond all limit.
DEMONSTRATION. If the convergency do not become infinitely
slow, it will be possible to find a number n^ so great that for the
value of h we begin with and for all inferior values greater than
zero the sum of the neglected terms shall be numerically less than
e. Now the limit of the sum of the first n^ terms of (67), when h
vanishes is the sum of the first n^ terms of (66). Hence if e be the
numerical value of the sum of the terms after the n^ of the series
(66), Z7and the limit of V cannot differ by a quantity so great as
e + e. But e and e may be made smaller than any assignable
quantities, and therefore U is equal to the limit of V.
COR. 1. If the series (66) is essentially convergent, and if,
either from the very beginning, or after a certain term whose rank
does not depend upon h, the terms of (67) are numerically less than
the corresponding terms of (66), the limit of Fis equal to U.
For in this case the series (67) is more rapidly convergent than
(66), and therefore its convergency remains finite.
COR. 2. If the series (66) is essentially convergent, and if the
terms of (67) are derived from those of (66) by multiplying them
by the ascending powers of a quantity g which is less than 1, and
which becomes 1 in the limit, the limit of V is equal to U.
It may be observed that when the convergency of (67) does
not become infinitely slow when h vanishes there is no occasion to
prove the convergency of (66), since it follows from that of (67).
In fact, let V H be the sum of the first n terms of (67), U n the
same for (66), F the value of V for h = 0. Then by hypothesis
282 ON THE CRITICAL VALUES OF
we may find a finite value of n such that V V n shall be numeri
cally less than e, however small h may be ; so that
V V n + a quantity always numerically less than e.
Now let h vanish : then V becomes F and V n becomes U n . Also
e may be made as small as we please by taking n sufficiently great.
Hence U n approaches a finite limit when n becomes infinite, and
that limit is F .
Conversely, if (66) is convergent, and if 7 = F , the convergency
of the series (67) cannot become infinitely slow when h vanishes.
For if U n , V n represent the sums of the terms after the n th in
the series (66), (67) respectively, we have
F=F n +F n , U U n +U u i
whence FJ = F  U  ( V n  U n ) + tT.
Now F U, V n U n vanish with h, and U n r vanishes when n
becomes infinite. Hence for a sufficiently small value of h and all
inferior values, together with a value of n sufficiently large, and
independent of h, the value of F n may be made numerically less
than any given quantity e however small; and therefore, by defini
tion, the convergency of the series (67) does not become infinitely
slow when h vanishes.
On the whole, then, when the convergency of the series (67)
does not become infinitely slow when h vanishes, the series (66) is
necessarily convergent, and has F for its sum : but in the contrary
case there must necessarily be a discontinuity of some kind. Either
F must become infinite when h vanishes, or the, series (66) must
be divergent, or, if (66) is convergent as well as (67), U must be
different from F .
When a finite function of x, f (#), which passes suddenly from
M to N as x increases through a, where a > a > 0, is expanded in
the series (15) or (30), we have seen that the series is always con
vergent, and its sum for all values of x except critical values is
f(x), and for x a its sum is J (M+ N). Hence the convergency
of the series necessarily becomes infinitely slow when a x
vanishes. In applying the preceding reasoning to this case it will
be observed that h is a  x, F is M, and U is \ (M + N), if we
are considering values of a? a little less than a ; but h is x a.
and F is N, if we are considering values of x a little greater
than a.
THE SUMS OF PERIODIC SERIES, (SECT. III. DISCONTINUITY.) 283
When the series (66) is convergent as well as (67), it may be
easily proved that in all cases
where L is the limit of V lt when h is first made to vanish and then
n to become infinite.
40. Reasoning exactly similar to that contained in the preced
ing article may be applied to integrals, and the same definitions
,00
may be used. Thus if F (x, h) dx is a convergent integral, we
J a
may say that the convergency becomes infinitely slow when h
vanishes, when, if X be the superior limit to which we must inte
grate in order that the neglected part of the integral, or
F(x } h)dx,
may be numerically less than a given constant e which may be as
small as we please, X increases beyond all limit when h vanishes.
The reasoning of the preceding article leads to the following
theorems.
If F= [ F (x, h) dx, if F be the limit of F when h = 0, and if
J a
F (x, 0) =/ (x) ; then, if the convergency of the integral F do not
become infinitely slow when h vanishes, / (x) dx must be con
J a
vergent, and its value must be F . But in the contrary case either
F must become infinite when h vanishes, or the integral
J
must be divergent, or if it be convergent its value must differ
from F .
When the integral I f(x)dx is convergent, if we denote its
J a
value by U, we shall have in all cases
when L is the limit to which \ F (x, h) dx approaches when h is
J x
first made to vanish and then X to become infinite.
284 ON THE CRITICAL VALUES OF
The same remarks which have been made with reference to the
convergency of series such as (15) or (30) for values of x near
critical values will apply to the convergency of integrals such as
(50), (59) or (65).
The question of the convergency or divergency of an integral
might arise, not from one of the limits of integration being oo , but
from the circumstance that the quantity under the integral sign
becomes infinite within the limits of integration. The reasoning
of the preceding article will apply, with no material alteration, to
this case also.
41. It may not be uninteresting to consider the bearing of the
reasoning contained in this Section on a method frequently given
of determining the values of two definite integrals, more especially
as the values assigned to the integrals have recently been called
into question, on account of their discontinuity.
Consider first the integral
[ x sin ax ,
u= dx 71),
JQ x
where a is supposed positive. Consider also the integral
f 30 , sina# 7
v= ~ hx   dx.
Jo
It is easy to prove that the integral v is convergent, and that its
convergency does not become infinitely slow when h vanishes.
Consequently the integral u is also convergent, (as might also be
proved directly in the same way as in the case of v,) and its value
is the limit of u for h = 0. But we have ,,<
dv^_r
dh Jo
e hx sm axdx =
9
+ h
whence v = G tan J  ;
a
and since v evidently vanishes when h = oo , we have (7=7r/2>
whence
7T _i h 7T
= ^ tan , tt = 2
Also u = when a = 0, and u Tr/2 when a is negative, since u
changes sign with a. By the value of u for a = 0, which is
C si o air
asserted to be 0, is of course meant the limit of  dx when
Jo x
a is first made to vanish and then X made infinite.
THE SUMS OF PERIODIC SERIES. (SECT. III. DISCONTINUITY.) 285
It is easily proved that the convergency of the integral u
becomes infinitely slow when a vanishes. In fact if
, f sin ax ,
u = dx,
Jx %
we get by changing the independent variable
, [ sinx 7
u = ax :
JaX ^
but for any given value of X, however great, the value of u
becomes when a vanishes I dx, an integral which might have
Jo &
been very easily proved to be greater than zero even had we been
unable to find its value. It readily follows from the above that if
u has to be less than e the value of X increases indefinitely as a
approaches to zero.
42. Consider next the integrals
00 cos ax dx [ , cos ax dx
y I ^flX
It is easily proved that the convergency of the integral v does not
become infinitely slow when h vanishes, whatever be the value of a.
Consequently u is in all cases the limit of v for h = 0. Now v
satisfies the equation
tfv
da*
f ii
v = e hx cos a&dx = rj*  z (73).
Jo ti + a
It is not however necessary to find the general value of v ; for if
we put h = we see that u satisfies the equation
so long as a is kept always positive or always negative : but we
cannot pass from the value of u found for positive values of a to
the value which belongs to negative values of a by merely writing
a for a in the algebraical expression obtained. For although u
is a continuous function of a, it readily follows from (73) that
r is discontinuous. In fact, we have from this equation
da
(dv\ (dv\ f x ^ o+_i x
h
fdv\ fdv\ f x ,
I p 1 ~" ( j~ J vaa ~ * ^ an
286 ON THE CRITICAL VALUES OF
Now let h first vanish and then X. Then v becomes u, and
vda vanishes, since v does not become infinite for a = 0, whether
h be finite or be zero. Therefore du/da is suddenly decreased by
TT as a increases through zero, as might have been easily proved
from the expression for u by means of the known integral (71),
even had we been unable to find the value of u in (72), The equa
tion (74) gives, a being supposed positive,
But u evidently does not increase indefinitely with a, and
[ dx TT ,
u = I g = when a ;
J Q JL ftG L
whence C = 0, G Tr/2,^ u = Tr/2 . e~ a . Also, since the numerical
value u is unaltered when the sign of a is changed, we have
u = 7T/2 . e a when a is negative.
It may be observed that if the form of the integral u had been
such that we could not have inferred its value for a negative from
its value for a positive, nor even known that u is not infinite
for a = oc , we might yet have found its value for a negative by
means of the known continuity of u and discontinuity of du/da
when a vanishes. For it follows from (74) that u = C7 X e a + C g e~ a
for a negative ; and knowing already that u = Tr/2 . t~ for a positive,
we have
177 _ n n 7r _r< rt
whence G l = Tr/2, <7 2 = 0, u = Tr/2 . e a , for a ne gative.
Of course the easiest way of verifying the result u = TT/% . e~ a for
a positive is to develope e~ x for x positive in a definite integral of
the form (59), by means of the formula (60).
SECTION IV.
Examples of the application of the formulce proved in the preceding
Sections.
43. Before proceeding with the consideration of particular
examples, it will be convenient to write down the formulae which
THE SUMS OF PERIODIC SERIES. (SECT. IV. APPLICATIONS.) 287
will have to be employed. Some of these formulae have been proved,
and others only alluded to, in the preceding Sections.
In the following formulae, when series are considered, f(x) is
supposed to be a function of x which, as well as each of its deriva
tives up to the (fj, l) th order inclusive, is continuous between
the limits x = and x = a, and which is expanded between those
limits in a series either of sines or of cosines of irx/a and its multi
ples. A n denotes the coefficient of smmrx/a when the series is
one of sines, B n the coefficient of cos mrx/a when the series is one
of cosines, Af or B* the coefficient of sin mrx/a or cos mrx/a in
the expansion of the /t th derivative. When integrals are considered
f (x) and its first //, 1 derivatives are supposed to be functions of
the same nature as before, which are considered between the limits
x = and # = oo ; and it is moreover supposed that / (x) decreases
as x increases to oo , sufficiently fast to allow ff(x)dx to be
essentially convergent at the limit oo , or else that f(x) vanishes
when x oo , and after a finite value of x never changes from
increasing to decreasing nor from decreasing to increasing. < (fi)
or \/r(/3) denotes the coefficient of sin fix or cos fix in the develope
/oo
ment of f(x) in a definite integral of the form (f>(fi) siufixdxor
J o
cosfixdx, $ (fi) or ^ (J3) denotes the coefficient of sin 8x
,
J
o
or cos fix in the developement of the /Lt th derivative of /(#). The
formulas are
^7r
) ( /() (DV(<)} + 0* even) ..... (5),
288 ON THE CRITICAL VALUES OF
+ ~ f^TV (O)  (D"/) ..0* even) .....
except when n = 0, in which case we have always
JB being the constant term in the expansion of / M (V) in a series of
cosines. In the formulae (4), (B), (C), (D) we must stop when
we have written the term containing the power 1 or 0, (as the
case may be,) of n7r/a.
The formulae for integrals are
( I/** f , 08) 0* * (0)  0 1 / (0)
+ Vs/"(0) ...(/, odd) ............. (a),
08) =
 * 8 /"(0)   fa even ) .............. (&)
o
/^ 2 / (0)
 /^ 4 / "(0) + ... (P odd) ........... (c),
7T t
 />*/ " (0) + ... (/t even)
7T
where we must stop with the last term involving a positive power
of ft or the power zero.
44. As a first example of the application of the principles
contained in Sections I. and II. suppose that we have to determine
the value of < for values of x lying between and a, and b
respectively, from the equation
79 I
(75),
THE SUMS OF PERIODIC SERIES. (SECT. IV. APPLICATIONS.) 289
with the particular conditions
T? = a>(x %a), when y = or =b .......... (76),
^ = co(y%b), when x = or =a ......... (77).
This is the problem in pure analysis to which we are led in
seeking to determine the motion of a liquid within a closed
rectangular box which is made to oscillate.
For a given value of y, the value of <f> can be expanded in
a convergent series of cosines of irx/a and its multiples; for
another value of y, < can be expanded in a similar series with
different coefficients, and so on. Hence, in general, <f> can be
expanded in a convergent series of the form
...... (78),
where Y n is a certain function of y, which has to be determined.
In the first place the value of < given by (78) must satisfy (75).
Now the direct developement of dty/dy* in a series of cosines will
be obtained from (78) by differentiating under the sign of sum
mation ; the direct developement of cPfydar will be given by the
formula (D). We thus get
~d*Y nV 2o>
and the lefthand member of this equation being the result of
directly developing the righthand member in a series of cosines,
we have
* 17 ,
^b) or =0,
2
o
dy* a* a v<y
according as n is odd or even. This equation is easily integrated,
and the integral contains two arbitrary constants, C n , D n , suppose.
It only remains to satisfy (76). Now the direct developement
of dYJdy will be obtained by differentiating under the sign of
summation, and the direct developement of &&gt; (x J a) is easily
found to be 2 4&)a/7rV . cos mrxja, the sign S denoting that
odd values only of n are to be taken. We have then, both for
s. 19
290 ON THE CKITICAL VALUES OF
y = and for y = &,
dY n
7 =  j, or =0,
dy irn
according as n is odd or even, which determines G n and D n .
It is unnecessary to write down the result, because I have
already given it in a former paper*, where it is obtained by
considerations applicable to this particular problem. The result
is contained in equation (4) of that paper. The only step of the
process which I have just indicated which requires notice is,
that the term containing (x J a) (y J 6) at first appears as an
infinite series, which may be summed by the formula (41). The
present example is a good one for shewing the utility of the
methods contained in the present paper, inasmuch as in the
Supplement referred to I have pointed out the advantage of the
formula contained in equation (6), with respect to facility of nu
merical calculation, over one which I had previously arrived at
by using developements, in series of cosines, of functions whose
derivatives vanish for the limiting values of the variable.
45. Let it be required to determine the permanent state of
temperature in a rectangle which has two of its opposite edges
kept up to given temperatures, varying from point to point, while
the other edges radiate into a space at a temperature zero. The
rectangle is understood to be a section of a rectangular bar of
infinite length, which has all the points situated in the same line
parallel to the axis at the same temperature, so that the pro
pagation of heat takes place in two dimensions. ,
Let the rectangle be referred to the rectangular axes of x, y,
the axis of y coinciding with one of the edges whose temperature
is given, and the origin being in the middle point of the edge.
Let the unit of length be so chosen that the length of either
edge parallel to the axis of x shall be TT, and let 2/3 be the length
of each of the other edges. Let u be the temperature at the
point (as, y), h the ratio of the exterior, to the interior conductivity.
Then we have
hu = 0, when 2/ = /3 ............... (80),
u
* Supplement to a Memoir On some Cases of Fluid Motion, p. 409 of the
present Volume [Ante, p. 188].
THE SUMS OF PERIODIC SERIES. (SECT. IV. APPLICATIONS.) 291
^ + tt = 0, when y= .................. (81),
u=f(y) ) when # = ................... (82),
u =F(y} ) when x = a ..................... (83),
/(?/), JP(y) being the given temperatures of two of the edges.
According to the method by which Fourier has solved a similar
problem, we should first take a particular function Fe^, where Y
is a function of y, and restrict it to satisfy (79). This gives
Y = A cos \y + B sin \y, A and B being arbitrary constants. We
may of course take, still satisfying (79), the sum of any number
of such functions. It will be convenient to take together the
functions belonging to two values of X which differ only in sign.
We may therefore take, by altering the arbitrary constants,
u = 2 [A (^(*$  *(*tf) + B (<**  e**)} cos Xy,
+ 2{C(^^e^^)+D(e^ e * x )}sm\y ..... (84),
in which expression it will be sufficient to take only one of two
values of X which differ only by sign, so that X, if real, may be
taken positive. Substituting now in (80) and (81) the value of
u given by (84), we get either (7=0, D = 0, and
X/3.tanX/3 = A/3 ......................... (85),
or else A = 0, B = 0, and
X/3.cotX/3 = fy3 ..................... (86).
It is easy to prove that the equation (85), in which X/3 is
regarded as the unknown quantity, has an infinite number of
real positive roots lying between each even multiple of ?r/2, in
cluding zero, and the next odd multiple. The equation (86)
has also an infinite number of real positive roots lying between
each odd multiple of Tr/2 and the next even multiple. The
negative roots of (85) and (86) need not be considered, since the
several negative roots have their numerical values equal to those
of the positive roots; and it may be proved that the equations
do not admit of imaginary roots. The values of X in (84) must
now be restricted to be those given by (85) for the first line, and
those given by (86) for the second. It remains to satisfy (82)
and (83). Now let
 y} = 2^(3,), F(y) F(y} =
192
292 ON THE CRITICAL VALUES OF
then we must have for all values of y from to @, and therefore
for all values from /3 to 0,
F 1 (y) .......... (87),
F % (y) .......... (88),
where L = e Xv  e~ A7r , M = e* n  e~^ n }
JJL denoting one of the roots of the equation
(89),
and the two signs S extending to all the positive roots of the
equations (85), (89), respectively. To determine A and B, multiply
both sides of each of the equations (87) by cos\ ydy, X being
any root of (85), and integrate from y = to y = /3. The integral
at the first side will vanish, by virtue of (85), except when X = X,
in which case it will become l/4\ . (2X/3 + sin 2X/3), whence A
and B will be known. G and D may be determined in a similar
manner by multiplying both sides of each of the equations (88)
by smpydy, fi being any root of (89), integrating from 2/ =
to y /3, and employing (89). We shall thus have finally
u = 42X (2X + sin 2X/3)
cos \y dy + (e Xa5 "**)) F^ (y) cos X^ dy] cos \y,
J o
 sin 2/*) 1 (d"  e^)" 1 {(e^~^ ^ (
sin ^^ + (e^  e^) ( F t (y) sin pydy} bin M y. . .(90).
JO
46. Such is the solution obtained by a method similar to that
employed by Fourier. A solution very different in appearance
may be obtained by expanding u in a series 2 I^sin nx, and em
ploying the formula (B). We thus get from the equation (79)
which gives
r= Ae" + Bf~  \f(y )  ( l)"F(,j)} (e^  ."("I) dy ;
THE SUMS OF PERIODIC SERIES. (SECT. IV. APPLICATIONS.) 293
whence, dujdy 2 Y sin nx, where
The values of A and B are to be determined by (80) and (81),
which require that
dy
We thus get
(n + h) * A(h)"#B^ l\f (y 1 )  ( 1)" F (y)}
{(n + h) <*&& +(nh) e~ n ^^} dy = 0,
and the equation derived from this by changing the signs of h and
/3 ; whence the values of A and B may be found. We get
finally
u = 2Ysmnx ........................ (91),
where
Y=  {(n + h) e n ?  (n  h) e ^} 1 (<* + e ^) I {(n + h) e(0>
7T J
+ (n  h) e W1} {/, (y}  ( 1)" F, (y )} dy
+  {(n + A) e 1 ^ + (w  A) e *J ^e""  e "") [ {( + h)
"" Jo
+ (n  A) 6tf V>} {/ 2 (y f )  ( 1)"
47. The two expressions for w given, one by (90), and the
other by (91) and (92), are necessarily equal for values of x and y
lying between the limits and TT, /3 and /3 respectively. They
are also equal for the limiting values y = [B and y = ft, but not
for the limiting values x = and x = TT, since for these values (91)
fails ; that is to say, in order to find from this series the value of u
for x = or x = TT, we should have first to sum the series, and then
put x = or x = TT.
The comparison of these expressions leads to two remarkable
formulae. In the first place it will be observed that the first and
294 ON THE CRITICAL VALUES OF
second portions of the righthand side of (92) are unchanged when y
changes sign, while the third and fourth portions change sign with y.
This is obvious with respect to the first and third portions, and may
be easily proved with respect to the second and fourth by taking y
instead of y for the variable with respect to which the integration
is performed, and remembering thatj^ (y), F^ (y) are unchanged, and
/ 2 (y)t F z (y) change sign, when y changes sign. Consequently the
part of u corresponding to the first two portions of (92) is equal to the
part expressed by the first two lines in (90), and the part correspond
ing to the last two portions of (92) equal to the part expressed by the
last two lines in (90). Hence the equation obtained by equating the
two expressions for u splits into two ; and each of the new equa
tions will again split into two in consequence of the independence
of the functions f, F, which are arbitrary from y = to y = /3.
As far however as anything peculiar in the transformations is con
cerned, it is evident that we may suppress one of the functions
f, F, suppose F, and consider only an element of the integral by
which f is developed, or, which is the same, suppose f t (y f ) or f 2 (y)
to be zero except for values of the variable infinitely close
to a particular value y, and divide both sides of the equa
tion by
//,(/) dy or J/ 2 (/)<% .
We get thus from the first two lines of (90) and the first two
portions of (92), supposing y and y positive, and y the greater
of the two,
e
e^
where the first 2 refers to the positive roots of (85), and the second
to positive integral values of n from 1 to GO .
Of course, if y become greater than y, y and y will have to
change places in the second side of (93). This is in accordance
with the formula (92), since now the second line does not vanish;
and it will easily be found that the first and second lines together
give the same result as if we had at once made y and y change
places. Although y has been supposed positive in (93), it is easily
seen that it may be supposed negative, provided it be numerically
less than y.
THE SUMS OF PERIODIC SERIES. (SECT. IV. APPLICATIONS.) 295
The other formula above alluded to is obtained in a manner
exactly similar by comparing the last two portions of (92) with
the last two lines in (90). It is
sn
(n+fceP^ + (w + A)eP^
 ~
where the first 2) refers to the positive roots of (89), the second to
positive integral values of n, and where x is supposed to lie
between and TT, y between and /3, y between and y, or, it
may be, between y and y. Although x has been supposed less
than TT, it may be observed that the formulae (93), (94) hold good
so long as x t being positive, is less than 2?r.
48. Let it be required to determine the permanent state of
temperature in a homogeneous rectangular parallelepiped, suppos
ing the surface kept up to a given temperature, which varies from
point to point.
Let the origin be in one corner of the parallelepiped, and let
the adjacent edges be taken for the axes of x, y, z. Let a, b, c be
the lengths of the edges ; /j (y, z), F l (y, z), the given temperatures
of the faces for which x and x = a respectively ; f 2 (z, x),
F 9 (z, x) the same for the faces perpendicular to the axis of y ;
fs ( x > 2/)> ^3 ( x > y) * ne same f r those perpendicular to the axis of z.
Then if we put for shortness v to denote the operation otherwise
denoted by
as will be done in the rest of this paper, and write only the charac
teristics of the functions, we shall have,, to determine the tempera
ture u, the general equation ^u = with the particular condi
tions
u= fi> wnen # = 0; u = F l , when x a ...... (95);
u= fz> when # = 0; u = F 2 , when y = b ...... (96);
u=f 9 , when = 0; u~F 3 , when zc ...... (97);
It is evident that u is the sum of three temperatures u lt u 2 , u 3 ,
where u : satisfies the conditions (95), and vanishes at the four
remaining faces, and w 2 , u s are related to the axes of y, z as w t is
296 ON THE CRITICAL VALUES OF
related to that of x y each of the quantities u lt u. 2 , u 3 representing
a possible permanent temperature. Now u 3 may be expanded in
a double series 2S Z mn sin mTrx/a . sin njry/b, where Z mn is a function
of z which has to be determined. Let for shortness
rmr nir PTT
then the substitution of the above value of u 3 in the equation
V^ 3 = leads to the equation
tfa^o.
where (f $ + z> 2 , which gives Z mn = A mn e qz + B mn e~ 9 " ; and the con
stants A mn , B^ are easily determined by the condition (97). We
may find u t and w 2 in a similar manner, and the sum of the results
gives u. It is thus that such problems are usually solved.
We may, however, expand u in a series of the form
on px sn vy,
even though it does not vanish for x and x = a, and for y =
and y = b } and the formula proved in Section I. enable us to
make use of this expansion.
Let then u = SS^sin fix sin vy,
the suffixes of Z being omitted for the sake of simplicity. We
have by the formula (B)
vy + * [/,_( 1JJJ j. sin ^. .
/i (2/j *0 ~~ ("" l) m ^i(3/j s ) ^ e expanded in the series 2Qsini>y
by the formula (3), so that will be a known function of z, m,
and w. Then
Q h sin /Lta? sin i/?
The value of d*u/dy* may be expressed in a similar manner, and
that of d?u/dz 2 is found by direct differentiation. We have thus,
for the direct developement of yw, the double series
SS j^jf  (^ 2 + ^ ! ) ^+ y P + J Q } sm ^ sin v y,
THE SUMS OF PERIODIC SERIES. (SECT. IV. APPLICATIONS.) 297
where P is for x what Q is for y. The above series being the
direct developement of yw, and v^ being equal to zero, each co
efficient must be equal to zero, which gives
d*Z 2v 2a
j T q i Zt r P+iQ = Q m .. (98)
dz b a \J)>
where q means the same as before. The integral of the equation
(98) is
Q > o f[ J o
2T denoting the sum of the last two terms of (98). It only re
mains to satisfy (97). If the known functions f B (x, y\ F 3 (x, y) be
developed in the double series 22(7 sin px sin.z/y, 22//sin (JLX sin vy y
we shall have from (97)
  e c F <T*Tdz +  e^ c l\*Tdz = H.
2 Jo q. Jo
A and B may be easily found from these equations, and we shall
have finally
~ e~ (c  z) ) l\e qz  ") T dz
Jo
T being the value of T when z =  z. It will be observed that the
letters Z, P, Q, T, A, B, G, H ought properly to be affected with
the double suffix inn. It would be useless to write down the
expression for u in terms of the known quantities /j (y, z), &o.
It will be observed that u might equally have been expressed by
means of the double series SSZ^sin vysm >&z, or 22 F^sm^sinnr^,
where p is any integer. We should thus have threeTdifferent ex
pressions for the same quantity u within the limits x = and x = a,
y = and y = &, * = and z = c. The comparison of these three
expressions when particular values are assigned to the known
functions / x (y, s) & c . would lead to remarkable transformations.
The expressions differ however in one respect which deserves
notice. Their numerical values are the same for values of the
298 ON THE CRITICAL VALUES OF
variables lying within the limits and a, and b, and c. The
first expression holds good for the extreme values of z, but fails for
those of x and y : in other words, in order to find from the series
the value of u for the face considered, instead of first giving x or y
its extreme value and then summing, which would lead to a result
zero, we should first have to sum with respect to m or n, or con
ceive the summation performed, and then give x or y its extreme
value. The same remarks apply, mutatis mutandis, to the second
and third expressions ; so that the three expressions are not equi
valent if we take in the extreme values of the variables.
49. Many other remarkable transformations might be obtained
from those already referred to by differentiation and integration.
We might for instance compare the three expressions which would
ra rb re
be obtained for I udxdydz. and we should thus have three
JoJoJo
different expressions for the same function of the three independent
variables a, b, c, which are supposed to be positive, but may be of
any magnitudes. Some examples of the results of transformations
of this kind may be seen by comparing the formulae obtained in
the Supplement alluded to in Art. 44 with the corresponding for
mulae contained in the Memoir itself to which the Supplement has
been added. Such transformations, however, when separated from
physical problems, are more curious than useful. Nevertheless, it
may be worth while to exhibit in its simplest shape the formula
from which they all flow, so long as we restrict ourselves to a func
tion u satisfying the equation VM = 0, and expanded between the
limits x = and x a, &c. in a double series of sines.
The functions /^y, z] &c., which are supposed known, are arbi
trary, and enter into the expression for u under the sign of double
integration. Consequently we shall not lose generality, so far as
anything peculiar in the transformations is concerned, by consider
ing only one element of the integrals by which one of the functions
is developed. Let then all the functions be zero except / 3 ; and
since in the process / 3 has to be developed in the double series
4r52 If f a (, y } s i n P sm vy dx dy . sin JJLX sin vy,
cio Jo Jo
consider only the element / 8 (# , y ) sin pat sin vy dx dy of the double
integral, omit the dx dy, and put/ 8 (# , y ) = 1 for the sake of sim
THE SUMS OF PERIODIC SERIES. (SECT. IV. APPLICATIONS.) 299
plicity. If we adopt the first expansion of u, and put <f for fi + z/ 2 ,
we shall have
Z= A (e 9(c  0)  e 9(c " 2) ), (e 9C  <T qc ) A=~ b sin i*x sin vy ;
whence
4 ?(C~3) _ 6 ?(c2)
u = r SS   sin fjix sin i/y sin /z# sin vy. . . (99).
O/U ~~
By expanding u in the double series 2S Fsin //,# sin nrz we should
get
2 __> cr (*  ew) (e*( 6 ^  e^tf) .
w = 2,2,   ;  ^   sm LLX sin ua; sin txz
ac s e sb ~ s
...... (100),
where s 2 = ^ + &*, and y is the greater of the two y, y. The
third expansion would be derived from the second by inter
changing the requisite quantities. In these formulsB z may have
any positive value less than 2c.
We should get in a similar manner in the case of two variables
x,y
2 ^ e v(ax] _ e v(ax) ^ f m
^ sm v sm
aoi)f
where x is supposed to lie between and a, y between and 6,
and y between and y . This formula is however true so long
as x lies between and 2a, and y between y and y .
rb rb ra
If we compare the two expressions for I I udy dy dx
Jo Jo Jo
obtained from (101), taking 2 for the sign of summation corre
sponding to odd values of n from 1 to oo, putting a = rb, and
replacing S 1/n 2 by its value 7r 2 /8, we shall get the formula
which is true for all positive values of r, and likewise for all
negative values, since the lefthand side of (102) is not changed
when  r is put for r. In integrating the second side of (101),
supposing that we integrate for y before integrating for y , we
must integrate separately from # = to y = y, and from y = y
300 ON THE CRITICAL VALUES OF
to y = b, since the algebraical expression of the quantity to be
integrated changes when y passes the value y.
It would be useless to go on with these transformations, which
may be multiplied to any extent, and which cease to be useful
when they are separated from physical problems to which they
relate, and of which we wish to obtain solutions.
It may be observed that instead of supposing, in the case of
the parallelepiped, the value of u known for all points of the
surface, we might have supposed the value of the flux known,
subject of course to the condition that the total flux shall be
zero. This would correspond to the following problem in fluid
motion, u taking the place of the quantity usually denoted by </>,
"To determine the initial motion at any point of a homogeneous
incompressible fluid contained in a closed vessel of the form of
a rectangular parallelepiped, which it completely fills, supposing
the several points of the surface of the vessel suddenly moved
in any manner consistent with the condition that the volume be
not changed." In this case we should expand u in a series of
cosines instead of sines, and employ the formula (D) instead of (B).
We might, again, suppose the value of u known for the faces
perpendicular to one or two of the axes, and the value of the
flux known for the remaining faces. In this case we should
employ sines involving the coordinates perpendicular to the first
set of faces, and cosines involving the others.
The formulae would also be modified by supposing some one or
more of the faces to move otf to an infinite distance. In this
case some of the series would be replaced by integrals. Thus, in
the case in which the value of u at the surface is known, if we
supposed a to become infinite we should employ the integral (50)
instead of the series (3), as far as relates to the variable x, and
the formula (b) instead of (B). If we were considering a rect
angular bar infinitely extended both ways we should employ the
integral (65). Of course, if we had already obtained the result for
the case of the parallelepiped, the shortest way would be thence
to deduce the result for the case of the bar infinite in one or in
both directions, but if we began with considering the bar it would
be best to start with the integrals (50) or (65).
50. To give one example of transformations of this kind,
let us suppose b to become infinite in (101). Ol serving that
THE SUMS OF PERIODIC SEMES. (SECT. IV. APPLICATIONS.) 301
v = n7r/b, Av = 7T/6, we get on passing to the limit
2
= 2 (e^e^e^ sin/^ ..... (103).
Multiply both sides of this equation by dx dy, and integrate from
x = to # = a, and from y = to y = oo . With respect to the
integration of the second side, it is only necessary to remark that
when y becomes greater than y r , y and y must be made to change
places in the expression written^ down in (103). As to the in
tegration of the first side, if we first integrate from y = to y = Y,
we get, putting f(v t x) for the fraction involving x,
I f(v, x) sin vy (1 cos v Y) .
J o v
Now let Y become infinite; then the term involving cos^Fmay
be omitted, not because cosz^F vanishes when F becomes infinite,
which is not true, but because, as may be rigorously proved, the
integral in which it occurs vanishes when F becomes infinite.
If we write 1 for a, as we may without loss of generality, we
get finally
"?=;![ 2,4(16*) (104).
51. Hitherto in satisfying the general equation v^ = 0, to
gether with the particular conditions at the surface, the value
of u has been expanded in a double series involving two of the
variables, and the functions of the third variable which enter as
coefficients into the double series have been determined by an
ordinary differential equation such as (98). We might however
expand u in a triple series, and thus satisfy at the same time
the equation v^ = and the conditions at the surface, without
using an ordinary differential equation at all, but simply by means
of the terms introduced into the series by differentiation, which
are given by the formulae at the beginning of this Section ; and
then by summing the triple series once, which may be done in
any one of three ways, we should arrive at the same results as
if we had employed in succession three double series, involving
circular functions of x and y, y and z, z and x respectively, and
the corresponding ordinary differential equations. I am indebted
302 ON THE CRITICAL VALUES OF
for this method to my friend Prof. William Thomson, to whom
I shewed the method given in Art. 48.
Let us take the case of the permanent state of temperature
in a rectangular parallelepiped, supposing the temperature at the
several points of the surface known. For more simplicity suppose
the temperature zero at the surface, except infinitely close to
the point (x, y ) in the face for which = 0, so that all the
functions f, &c. are zero, except f a (x, y), and / 3 (x, y) itself zero
except for values of x, y infinitely close to x t y respectively ; and
let fff a (x, y) docdy = 1, provided the limits of integration include
the values x = x, y = y . Let u be expanded in the triple series
sin ixz .............. (105),
where yu,, v, is mean the same as in Art. 48. Then
a? = ^ p \
sn ^ sn
/ 3 0, y) sin
Now the expansion of f s (x, y) in a double series is
4/a6 . 22 sin fix sin vy sin px sin vy,
that is to say with this understanding, that the result is to be
substituted in (106); for it would be absurd to speak, except
by way of abbreviation, of a quantity which is zero except for
particular values of x and y y for which it is infinite. The values
of tfuldx* and d^u/dy* will be obtained by direct differentiation.
We have therefore for the direct developement of TJU in a triple
series
O 2 + i? + *r 2 ) A mnp
SOT , ,}
+ j sin IJLX sin vy \ sin px sin vy sin isz.
But \7w being equal to zero, each coefficient will be equal to
zero, from whence we get A mnp , and then
u j 222 3   i sin fj,x f sin vy sin px sin vy sin ixz ..... (107).
One of the three summations, whichever we please, may be
performed by means of the known formulae
THE SUMS OF PERIODIC SERIES. (SECT. IV. APPLICATIONS.) 303
c e k ^^ e k ^~^ .
t ^v t c
M + S 1P1V = 2~6 >^^>^>Q ...... (109),
which may be obtained by developing the second members be
tween the limits z and z = c, y / = and y t b by the formulae
(2), (22), and observing that the expansions hold good within
the limits written after the formulae, since 6* (c ~* } e~* (c ~*> has
the same magnitude and opposite signs for values of z equidistant
from c, and e*^J +e*(&tU has the same magnitude and sign
for values of y t equidistant from 6. If in equation (107) we
perform the summation with respect to p, by means of the formula
(108), we get the equation (99) : if we perform the summation
with respect to n, by means of the formula (109), we get the
equation (100).
52. The following problem will illustrate some of the ideas
contained in this paper, although, in the mode of solution which
will be adopted, the formulas given at the beginning of this Section
will not be required.
A hollow conducting rectangular parallelepiped is in com
munication with the ground : required to express the potential,
at any point in the interior, due to a given interior electrical
point and to the electricity induced on the surface.
Let the axes be taken as in Art. 48. Let of, y, z be the
coordinates of the electrical point, m the electrical mass, v the
required potential. Then v is determined Jirst by satisfying the
equation y v = 0, secondly by being equal to zero at the surface,
thirdly by being equal to m/r infinitely close to the electrical
point, r being the distance of the points (x, y, z}, (x, y , z \ and
"by being finite and continuous at all other points within the
parallelepiped.
Let v = m/r + v l , so that v^ is the potential due to the elec
tricity induced on the surface. Then v^ is finite and continuous
within the parallelepiped, and is determined by satisfying the
general equation yt^ = 0, and by being equal to m/r at the
surface. Consequently v l can be determined precisely as u in
Art. 48 or 51. This separation however of v into two parts
seems to introduce a degree of complexity not inherent in the
304 ON THE CRITICAL VALUES OF
problem ; for v itself vanishes at the surface ; and it is when
the function expanded vanishes at the limits that the application
of the series (2) involves least complexity. On the other hand
we cannot immediately expand v in a triple series of the form
(105), on account of its becoming infinite at the point (x , y\ z }.
Suppose, therefore, for the present that the electricity is
diffused over a finite space ; then it is evident that we may
suppose the electrical density, p, to change so gradually, and pass
so gradually into zero, that the derivatives of v, of as many orders
as we please, shall be continuous functions. We may now suppose
v expanded in a triple series, so that
v = nMy, sn fuff sn vy sn vrz :
and we shall have
yy = SSS (yu, 2 + v* + or 2 ) A mnp sin fix sin vy sin iaz.
But we have also, by a wellknown theorem, y v = 4firp ; and
p = S2SKnnp sin fix sin vy sin wz,
where
g ra rb re
~r~ P sm I* 31 ? sin v y s i n **% dx dy dz,
abcJo Jo Jo
p being the same function of # , y\ z that p is of x, y, z. W
get therefore by comparing the two expansions of yv
whence the value of v is known. We may now,,if we like, suppose
the electricity condensed into a point, which gives
8m
Rp = 7 sm px sm vy sin TSZ ,
sin fix sin vy sin vrz sin fix sin vy sin r^. . . (110).
One of the summations may be performed just as before. We
thus get, by summing with respect to p,
8Trm
ab q e* c 6<i c
sin/Aic sin i/y sin fix sin z/y ....... (HI);
THE SUMS OF PERIODIC SERIES. (SECT. IV. APPLICATIONS.) 305
where # 2 = //, 2 + v z , and z is supposed to be the smaller of the
two z t z. If z be greater than z, we have only to make z and z
change places in (111).
53. The equation (110) shows that the potential at the
point (x, y, z) due to a unit of electricity at the point (sc t y\ z)
and to the electricity induced on the surface of the parallelepiped
is equal to the potential at the point (x t y, z) due to a unit of
electricity at the point (x, y, z) and to the electricity induced
on the surface. This however is only a particular case of a general
theorem proved by Green*.
Of course the parallelepiped includes as particular cases two
parallel infinite planes, two parallel infinite planes cut at right
angles by a third infinite plane, &c. The value of v being known
the density of the induced electricity at any point of the surface
is at once obtained, by means of a known theorem.
If we suppose a ballpendulum to oscillate within a rectangular
case, the value of < belonging to the motion of the fluid which
is due to the direct motion of the ball and to the motion reflected
from the case can be found in nearly the same manner. The
expression reflected motion is here used in the sense explained
in Art. 6 of my paper, "On some Cases of Fluid Motionf." In
the present instance we should expand in a triple series of
cosines.
54. Let a hollow cylinder, containing one or more plane
partitions reaching from the axis to the curved surface, be filled
with homogeneous incompressible fluid, and made to oscillate
about its axis, both ends being closed : required to determine
the effect of the inertia of the fluid on the motion of the cylinder.
If there be more than one partition, it will evidently be suffi
cient to consider one of the sectors into which the cylinder is
divided, since the solution obtained may be applied to the others.
In the present case the motion is such that udx + vdy + wdz (ac
cording to the usual notation) is an exact differential d(f>. The
motion considered is in two dimensions, taking place in planes
perpendicular to the axis of the cylinder. Let the fluid be referred
to polar coordinates r y 6 in a plane perpendicular to the axis, r
being measured from the axis, and 6 from one of the boundin^
* Essay on Electricity, p. 19.
t See p. Ill of the present Volume. [Ante, p. 28.]
20
306 ON THE CKITICAL VALUES OF
partitions of the sector considered, being reckoned positive when
measured inwards. Let the radius of the cylinder be taken for the
unit of length, and let a. be the angle of the sector, and o> the
angular velocity of the cylinder at the instant considered. It will
be observed that a = 2?r corresponds to the case of a single partition.
Then to determine </> we have the general equation
dr* r dr r 2 d6 2
with the conditions
i g= cor, when = or = a (113),
. T^=0, whenr = l (114),
and, that <> shall not become infinite when r vanishes.
Let r = e~ A , and take 0, X for the independent variables ; then
(112), (113), (114) become
(115),
= , when ^0 or a ....... ..... (116),
du
^ = 0, whenX=0 ........................ (117).
aA
Let $ be expanded between the limits 6 = and 9 = a in a series
of cosines, so that
(118),
A , A rt being functions of X. Then we have by the formula
(D) and the condition (116) applied to the general equation (115)
dK*
whence
A. = 4.6
THE SUMS OF PERIODIC SERIES. (SECT. IV. APPLICATIONS.) 307
Since <> is not to be infinite when r vanishes, that is when X
becomes infinite, we have in the first place A Q = 0, A n 0. We
have then by the condition (117)
H7T
when n is odd, and B n = Q when n is even. If then we omit
which is useless, and put for X its value, we get
C S nv
The series multiplied by r 2 may be summed. For if we
expand sin 2(6 Ja) between the limits 6 = 0, 6 a in a series of
cosines, we get
. . Q ^ 82 cos a
sin (20  a) =  S cos
whence
W7r/a cos mr#a co
2
mr (nV 4a 2 ) 2 cos a
. /C . A . ,t^\
r 2 sm(20a)...(120).
In determining the motion of the cylinder, the only quantity
we care to know is the moment of the fluid pressures about the
axis. Now if the motion be so small that we may omit the square
of the velocity we shall have, putting (f> = u>f(r t 6),
where p is the pressure, >r () a function of the time t, whose
value is not required, and where the density is supposed to be 1,
and the pressure due to gravity is omitted, since it may be taken
account of separately. The moment of the pressure on the curved
surface is zero, since the direction of the pressure at any point
passes through the axis. The expression (119) or (120) shows
that the moments on the plane faces of the sector are equal, and
act in the same direction ; so that it will be sufficient to find the
moment on one of these faces and double the result. If we con
sider a portion of the face for which 6 = whose length in the
direction of the axis is unity, we shall have for the pressure on an
element dr of the surface dw/dt . f (r, 0) dr ; and if we denote the
whole moment of the pressures by G dco/dt, reckoned positive
202
308 ON THE CRITICAL VALUES OF
when it tends to make the cylinder move in the direction of
positive, we shall have
(7 = 2 P/(n 0)r<Zr.
.0
Taking now the value of /(/ , 0) from (120), and performing the
integration, we shall have
(7=jtana16a 3 2 0/  N 1 ,  ^~ 2 ......... (121).
4 (mr 2y) mr (mr + 2a/
The mass of the portion of fluid considered is Ja; and if we
put
C = i*fc",
and write S7T/2 for a, so that s may have any value from to 4, we
shall have
k 9  tan  2
K>  Uclll o ^W
o Q . ,
STT 2 TT (n s)n(n + s)
When s is an odd integer, the expression for 7/ 2 takes the
form oo oo , and we shall easily find
2
where all odd values of n except s are to be taken.
The quantity k may be called the radius of gyration of the
fluid about the axis. It would be easy to prove from general
dynamical principles, without calculation, that if k be the corre
sponding quantity for a parallel axis passing through the centre of
gravity of the fluid, h the distance of the axes,
k" 2 = k*+h* ....................... (124),
in fact, in considering the motion of the cylinder, which is sup
posed to take place in two dimensions, the fluid may be replaced
by a solid having the same mass and centre of gravity as the
fluid, but a moment of inertia about an axis passing through the
centre of gravity and parallel to the axis of the cylinder different
from the moment of inertia of the fluid supposed to be solidified.
If K , K be the radii of gyration of the solidified fluid about the
axis of the cylinder and a parallel axis passing through the centre
of gravity respectively, we shall have
.__, T ^ 4 sin J a 8 . STT /,K\
J fiT 2 = i = A 2 + A 2 , h =  ^ = smr .... (125).
3 a 3.97T 4 N
THE SUMS OF PERIODIC SERIES. (SECT. IV. APPLICATIONS.) 309
If we had restricted the application of the series and the
integrals involving cosines to those cases in which the derivative
of the expanded function vanishes at the limits, we should have
.00
expanded <f> in the definite integral I (0, 0) cos ft\d/3, and
Jo
the equation (115) would have given
f, ^ denoting arbitrary functions, which must be determined by the
conditions (116). We should have obtained in this manner
*
_ 6
cos
B log d/3... (126),
= s2 r i  6p*
Try. Jo l + e0*
(127).
It will be seen at once that & 2 is expressed in a much better form
for numerical computation by the series in (122) than by the
integral in (127). Although the nature of the problem restricts
a. to be at most equal to 2?r, it will be observed that there is no
such restriction in the analytical proof of the equivalence of the
two expressions for $, or for k 2 .
In the following table the first column gives the angle of the
cylindrical sector, the second the square of the radius of gyration
of the fluid about the axis of the cylinder, the radius of the
cylinder being taken for the unit of length, the third the square of
the radius of gyration of the fluid about a parallel axis passing
through the centre of gravity, the fourth and fifth the ratios of
the quantities in the second and third to the corresponding quanti
ties for the solidified fluid.
a
2
k 2
fc 2
K*
fc 2
K*
50000
05556
10000
10000
45
45385
03179
9077
4079
90
39518
03492
7904
2499
135
34775
07442
6955
3283
180
31057
13044
6211
4078
225
28101
18261
5620
4547
270
25703
21700
5141
4718
315
23720
22858
4744
4652
360
22051
22051
4410
4410
310 ON THE CRITICAL VALUES OF
55. When a. is greater than TT, it will be observed that the
expression for the velocity which is obtained from (119) becomes
infinite when r vanishes. Of course the velocity cannot really
become infinite, but the expression (119) fails for points very near
the axis. In fact, in obtaining this expression it has been assumed
that the motion of the fluid is continuous, and that a fluid particle
at the axis may be considered to belong to either of the plane
faces indifferently, so that its velocity in a direction normal to
either of the faces is zero. The velocity obtained from (119)
satisfies this latter condition so long as a is not greater than TT.
For when a < TT the velocity vanishes with r, and when a. = TT the
velocity is finite when r vanishes, and is directed along the single
plane face which is made up of the two plane faces before con
sidered.
But when a is greater than TT the motion which takes place
appears to be as follows. Let OABC be a section of the cylindrical
sector made by a plane perpendicular to the axis, and cutting it in
0. Suppose the cylinder to be turning round in the direction
indicated by the arrow at B. Then the fluid in contact with OA
and near will be flowing, relatively to OA, towards 0, as indi
cated by the arrow a. When it gets to it will shoot past the
face OC ; so that there will be formed a surface of discontinuity
Oe extending some way into the fluid, the fluid to the left of Oe
and near flowing in the direction AO, while the fluid to the
THE SUMS OF PEEIODIC SERIES. (SECT. IV. APPLICATIONS.) 311
right is nearly at rest. Of course, in the case of fluids such as
they exist in nature, friction would prevent the velocity in a direc
tion tangential to Oe from altering abruptly as we pass from a
particle on one side of Oe to a particle on the other ; but I have
all along been going on the supposition that the fluid is perfectly
smooth, as is usually supposed in Hydrodynamics*. The extent of
[* It may be said that the motion of a perfect fluid which is at first at rest,
and is then set in motion by the action of solid bodies of finite curvature in contact
with it, is unique and continuous, so that no surface of discontinuity can be
formed ; and that that being always true will be true in the limit, when we suppose
the curvature at a certain point or along a certain line to become infinite, in such
a manner as to pass in the limit to a salient conical point or edge intruding into
the fluid, and therefore even in this case no surface of discontinuity can be formed.
This may be perfectly true in one sense and yet not in another. A perfect fluid
is an ideal abstraction, representing something which does not exist in nature. All
actual fluids are more or less viscous, and we arrive at the conception of a perfect
fluid by starting with fluids such as we find them, and then in imagination making
abstraction of the viscosity. Similarly any edge that we can mechanically form is
more or less rounded off ; but we have no difficulty in conceiving of an edge
perfectly sharp. The motion that belongs to a perfect fluid and perfectly sharp
edge may be regarded as the limit, if unique limit there be, of the motion which
belongs to a slightly viscous fluid interrupted by a solid presenting a salient,
slightly rounded edge, when both the viscosity of the fluid and the radius of cur
vature of the edge are supposed to vanish. For the sake of clear ideas we may
suppose that the mass of fluid we are dealing with is contained in a vessel differing
from that mentioned in the text in being bounded on the side towards the centre by
^cylindrical surface of very small radius a, coaxial with the outer cylinder. Then
we may represent the motion, in a sense which the reader will readily apprehend,
by f(fj,, a), where fj, denotes the coefficient or index of internal friction. We pro
posed to contemplate the limit of /(,u, a) when /* and a vanish. But for anything
that appears to the contrary there may be no such unique limit, but the limit
lim.ao lim.p=o/(Ai, a) may be one thing, and the limit lim.^o lim. a =o /(/*, a) a
totally different thing ; and I am strongly disposed to believe that such is actually
the case. When a is finite and yu = 0, we pass to a case of motion of a perfect fluid
similar to that in the text, and capable of being attacked by a similar analysis, but
in which the motion nowhere becomes infinite ; and the limit to which this tends as
a vanishes does not present a surface of discontinuity, but the velocity near the
centre increases indefinitely as a decreases indefinitely. But when on the other
hand fjt. though small remains finite, and a diminishes without limit, the motion
which would be investigated by an analysis resembling that in the text would be
such that near the centre there would be an enormous gliding, which would call
into play a great tangential force, in which work would be consumed, that is,
converted into heat. This I believe would be an unstable condition, and what
would actually take place would be that so large a local consumption of work would
be avoided by the fluid rushing past the corner, somewhat as represented in the
figure, carrying with it by adhesion a narrow stratum in which there would be very
great molecular rotation, inasmuch as the fluid of which the stratum consists had
previously been pent up between the radial wall of the vessel on one side of it,
312 ON THE CRITICAL VALUES OF THE SUMS OF PERIODIC SERIES.
the surface of discontinuity Oe will be the less the smaller be the
motion of the cylinder; and although the expression (119) fails
for points very near 0, that does not prevent it from being sensibly
correct for the remainder of the fluid, so that we may calculate k *
from (122) without committing a sensible error. In fact, if 7 be
the angle through which the cylinder oscillates, since the extent of
the surface of discontinuity depends upon the first power of 7, the
error we should commit would depend upon 7*. I expect, there
fore, that the moment of inertia of the fluid which would be
determined by experiment would agree with theory nearly, if not
quite, as well when a > TT as when a < TT, care being taken that
the oscillations of the cylinder be very small.
As an instance of the employment of analytical expressions
which give infinite values for physical quantities, I may allude to
the distribution of electricity on the surfaces of conducting bodies
which have sharp edges.
56. The preceding examples will be sufficient to show the
utility of the methods contained in this paper. It may be observed
that in all cases in which an arbitrary function is expanded
between certain limits in a series of quantities whose form is
determined by certain conditions to be satisfied at the limits, the
expansion can be performed whether the conditions at the limits
be satisfied or not, since the expanded function is supposed per
fectly arbitrary. Analogy would lead us to conclude that the
derivatives of the expanded functions could not be found by direct
differentiation, but would have to be obtained from formulae
answering to those at the beginning of this Section. If such
expansions should be found useful, the requisite formula? would
probably be obtained without difficulty by integration by parts.
This is in fact the case with the only expansion of the kind which
I have tried, which is that employed in Art. 45. By means of
this expansion and the corresponding formulae we might determine
in a double series the permanent temperature in a homogeneous
rectangular parallelepiped which radiates into a medium whose
which had no radial motion and but little in a perpendicular direction, and the
rapidly rushing fluid on the other side. The smaller /* is made, the narrower will
this stratum be, but not, so far as I can see, the shorter; and a very narrow
stratum in which there is intense molecular rotation passes, or may pass, in the
limit to a surface of discontinuity.
The above is what was referred to by anticipation in the footnote at p. 99.]
THE SUMS OF PERIODIC SERIES. (SECT. IV. APPLICATIONS.) 313
temperature varies in any given manner from point to point ; or
we might determine in a triple series the variable temperature in
such a solid, supposing the temperature of the medium to vary in
a given manner with the time as well as with the coordinates,
and supposing the initial temperature of the parallelepiped given
as a function of the coordinates. This problem, made a little
more general by supposing the exterior conductivity different for
the six faces, has been solved in another manner by M. Duhamel
in the Fourteenth Volume of the Journal de lEcole Poly technique.
Of course such a problem is interesting only as an exercise of
analysis.
ADDITIONAL NOTE.
If the series by which r 2 is multiplied in (119) had been left
without summation, the series which would have been obtained for
A/ 2 would have been rather simpler in form than the series in (122),
although more slowly convergent. One of these series may of
course be obtained from the other by means of the development
of tan x in a harmonic series. When s is an integer, k 2 can be
expressed in finite terms. The result is
& 2 = 8s 1 7T 2 log e 2 + 8s 1 7T 2 {2 1 + 4" 1 . . .
+ (s  I) 1 } + 47T 2 {2 2 + 4 2 . . . + (s  I)" 2 )  J, (s odd,)
F = 8s 1 7T 2 {r 1 + 3 1 . . . + (s  1) 1 } + 47T 2 {r 2 + 3~ 2 . . .
+ (s  1)" 2 )  J. (s even.)
Moreover when 2s is an odd integer, or when a = 45, or = 135, &c.,
k * can be expressed in finite terms if the sum of the series
1 2 + 5~* f 9~ 2 + ... be calculated, and then be regarded as a known
transcendental quantity.
[Not before published. (See page 229.)]
SUPPLEMENT TO A PAPER ON THE THEORY or OSCILLATORY
WAVES.
THE labour of the approximation in proceeding to a high order,
when conducted according to the method of the former paper
whether we employ the function </> or ty, depends in great measure
upon the circumstance that the two equations which have to be
satisfied simultaneously at the free surface are both composed in a
rather complicated manner of the independent variables, and in the
elimination of y the length of the process is still further increased
by the necessity of expanding the exponentials in y according to
series of powers, giving for each exponential a whole set of terms.
This depends upon the circumstance that of the limits of y belong
ing to the boundaries of the fluid, one instead of being a constant
is a function of x, and that too a function which is only known
from the solution of the problem.
If we convert the wave motion into steady motion, and refer
the fluid to two independent variables of which one is the para
meter of the stream lines or a function of the parameter, and the
other is x or a quantity which extends with x from co to + oo ,
we shall ensure constancy of each independent variable at both its
limits, but in general the equations obtained will be of great com
plexity. It occurred to me however that if from among the infinite
number of systems of independent variables possessing the above
character we were to take the functions </>, i/r, where
<f> =f(ud% f vdy), ty = $(udy vdx),
simplicity might be gained in consequence of the immediate rela
tion of these functions to the problem.
SUPPLEMENT TO A PAPER ON OSCILLATORY WAVES. 315
We know that <, ^ are conjugate solutions of the equation
satisfying the equations
dcf> d^r d(f> _ dty , .
dx dy dy dx "
so that if the form of either be assigned, satisfying of course the
equation (1), the other may be deemed known, since it can be
obtained by the integration of a perfect differential. If now we
take (/>, fy for the independent variables, of which x and y are re
garded as functions, we get by changing the independent variables
in differentiation
d<}> _ 1 dy d</> _ 1 dx d^r _ 1 dy d^r _ldx , ,
dx^Sd^ d^ = ~Sd$ ~dx~~Sd^ ~dj~"
a dx dy dx dy
where ^ TT TT ~~ TT j >
d^jr d(f>
whence from (2)
dx _ dy dx _ _ dy ...
~d$~d^ ~d^~~d$"
so that x, y are conjugate solutions of the equation
tfx d*x .
We have also from (4)
We get from (3), (4) and (6)
\ 2 (dty\* 1 (fda;\* f dx
+(/) =^\ ,, +
j \dxj 8 [\d<t>J \
whence
(8),

where C is an arbitrary constant.
The mode of proceeding is the same in principle whether the
depth of the fluid be finite or infinite ; but as the formulae are
simpler in the latter case, it may be well to consider it separately
in the first instance.
316 SUPPLEMENT TO A PAPER ON THE
If be the velocity of propagation, c will be the horizontal
velocity at a great depth when the wave motion is converted into
steady motion. The difference between </> and ex will be a
periodic function of x or of (/>. We may therefore assume in ac
cordance with equation (5)
x = ^ + ^(ApWI* + B i eWl*)*\nim<l>lc ......... (9).
c
No cosines are inserted in this equation because if we take, as we
may, the origins of x and of <f> at a trough or a crest (suppose a
trough), x will be an odd function of </>, in accordance with what
has already been shown at page 212. Corresponding to the above
value of x we have
y =  + (A.e im ^ c  Re im ^ c ) cos im^/c ..... (10),
c
the arbitrary constant being omitted, as may be done provided we
leave open the origin of y.
The origin of ty being arbitrary, we may take, as it will be
convenient to do, ^ = at the free surface. We see from (10) that
ty increases negatively downwards ; and therefore of the two ex
ponentials that with imjr/c for index is the one which must be
omitted, as expressing a disturbance that increases indefinitely in
descending.
We may without loss of generality shorten the formulas during
a rather long approximation by writing 1 for any two of the con
stants which depend differently on the units of space and time.
These constants can easily be reintroduced in the end by rendering
the equations homogeneous. We may accordingly put m = 1 and
c = 1. The expressions for x and y as thus shortened become, on
retaining only the exponential which decreases downwards,
l> .................. (11),
l> .................. (12).
At the free surface i/r = 0, and we must therefore have for ty
which gives
(C + 2 A. cos fy) (1  22t4 4 cos ty + 2*% f + VtijAA> cos [(t  ;
THEORY OF OSCILLATORY WAVES. 317
where in the last term within parentheses each different combina
tion of unequal integers i, j is to be taken once.
On account of the complicated form of this equation, we can
proceed further only by adopting some system of approximation.
The most obvious is that adopted in the former paper, namely to
proceed according to powers of the coefficient of the term of the
first order. If we multiply out in equation (13), and replace pro
ducts of cosines by cosines of sums and differences, we may arrange
the equation in the form
jB + S l cos $ + 2 cos 2< + . . . = 0,
where the several It s are series of terms involving the coefficients
A. And as the equation has to be satisfied independently of <,
we must have separately
= 0, ^=0, 2 = 0, &c.
A slight examination of the process will show that A. is of the
order ", and that consequently the product of any number of the
A s is of the order marked by the sum of the suffixes, and that B.
is of the order i. In proceeding therefore to any desired order we
can see at once what terms need not be written down, as being of
a superior order.
Thus in proceeding to the fifth order we must take the six
equations B Q = 0, B l = 0, . . . B 6 = 0, which when written at length
are
C (1 + A* + 4^1 2 2 )  A* + 2A*A 2  2A*  J g* = 0,
+ 3^4, 544, = 0,
C ( 4J 2 + 644) + A 2  A? + 34 2 J 2  444 = 0,
(64+ 844) + J 3  344 + 44M a + 2A l A? 5^J 4 = 0,
c (
These equations may be looked on as giving, the first, the
arbitrary constant C, the second, the velocity of propagation, and
the succeeding ones taken in order the values of the constants
A 2 , A 3 , A 4 , A 5 , respectively. I say " may be looked on as giving",
for it is only when we restrict ourselves to the terms of the lowest
order in each equation that those quantities are actually given in
succession ; the equations contain also terms of higher orders ; and
318 SUPPLEMENT TO A PAPER ON THE
to get the complete values of the quantities true to the order to
which we are working, we must use the method of successive sub
stitutions. As to the second equation, if we take the terms of
lowest order in the first two we get C = J^" 1 , and then by substi
tution in the second equation 1 = g, the constant A^ dividing out.
The equation 1 =g becomes on generalizing the units of space and
time c 2 = g/m, and accordingly gives the velocity of propagation to
the lowest order of approximation.
On eliminating the arbitrary constant in the above equations,
and writing b for A i9 the results become
(14),
x =  + 6e*sin  (b 2 + PV / sin 20 + (f b 5 + if 6 5 )^sin 30
............ (15),
f 6V* cos 40 + 1^ 6V* cos 50 ............ (16).
The equation (14) gives to the fifth order the square of the
velocity of propagation in the wave motion; and (15), (16) give
the point where the parameters 0, >r have given values, and also,
by the aid of the formulae previously given, the components of the
velocity, and the pressure, in the steady motion. These same
equations (15), (16), if we suppose ^jr constant give implicitly the
equation of the corresponding stream line, or if we suppose
constant the equation of one of the orthogonal trajectories.
To find implicitly the equation of the surface, we have only to
put ir= in (15), (16), which gives
as =  + b sin  (V +  6 4 ) sin 20  (f tf + if b 5 ) sin 30
f & 4 sin 40+ ^& B sin5 ............... (17),
y = b cos  (6 2 + i 6 4 ) cos 20  (f b* + if I 5 } cos 30
6 4 cos40} % 5 6 5 cos50 ............... (18).
It is not necessary to form the explicit equation, but we can do so
if we please, most conveniently by the aid of Lagrange s theorem.
The result, carried to the fourth order only, which will suffice for
the object more immediately in view, is
6 3 ) cos x (i& 2 + V & 4 ) cos 2#
+ f b 3 cos 3a?  J6 4 cos 4^... (19).
THEORY OF OSCILLATORY WAVES. 319
If we put b + b 3 = a, we have to the fourth order
b = a   a 3 ,
and substituting in (19) we get
2/41 a 2 i a 4 = a cos x (J a 2 + J a 4 ) cos 2% + f a 3 cos 3#
 J a 4 cos4# (20).
The expression (14) for the square of the velocity of propagation,
and the equation of the surface (20), agree with the results pre
viously obtained by the former method (see p. 221) to the degree
of approximation to which the latter were carried, as will be seen
when we remember that the origins of y are not the same in the
two cases ; but it would have been much more laborious to obtain
the approximation true to the fifth order by the old method.
It has already been remarked (p. 211) that the equation of the
profile in deep water agrees with a trochoid to the third order,
which jsas. far as the approximation there proceeded. This is no
longer, true when we proceed to the fourth order. On shifting
the origin of y so as to get rid of the constant term, the equation
(20) of the profile becomes
y a cos x ( J a 2 + JJ a 4 ) cos Zx + f a 3 cos 3# J a 4 cos 4#. . . (21).
On the other hand, the equation of a trochoid is given impli
citly by the pair of equations
x = a6 + j3 sin 0, y = j3cos0 + y.
In order that x may have the same period in the trochoid as in
the profile of the wave, we must have a = 1. We get then by
development to the fourth order, choosing 7 so as to make the
constant term vanish,
y = (p f 3 ) cos x  (4 /3 J 4 ) cos 2^ + /3 3 cos 3x  J/3 4 cos 4a?,
and putting
/3t/3 ! = ,
we get to the fourth order
y = aco$x (i a 2 + V a*) cos Zx + f a 3 cos 3#  J a 4 cos 4^.. .(22).
Hence if y w , y t denote the ordinates for the wave and trochoid
respectively, we have to the fourth order
Hence the wave lies a little above the trochoid at the trough and I/
crest, and a little bejow it in the shoulders.
320 SUPPLEMENT TO A PAPER ON THE
This result agrees well witli what might have been expected.
It has been shown (p. 227) that the limiting form for a series of
uniformly propagated irrotational waves is one presenting edges of
120, and that the inclination in this limiting form is in all proba
bility restricted to 30, whereas in the trochoidal waves investigated
by Gerstner and Kankine the limiting form is the cycloid, presenting
accordingly cusps, and an inclination increasing to 90. Hence the
limiting form must be reached with a much smaller value of the
parameter a in the former case than in the latter. Hence when
a is just large enough to make the difference of form of the irrota
tional and trochoidal waves begin to tell, since the limiting form
is more nearly approached in the former case than in the latter, we
should expect the curvature at the summit to be greater, while at
the same time as the general inclination is probably rather less,
and the inclination begins by increasing more rapidly as we recede
from the summit, the troughs must be shallower and flatter for an
equal mean height of wave.
Let us proceed now to the case of a finite depth. As before
we may choose the units of space and time so that c and m
shall each be 1, and we may choose for the value of the para
meter ^/r at the surface. Let & be its value at the bottom.
Then since dfyjdy = at the bottom we have from (3) and (4)
dy/d<p = when ty + k = 0, and consequently
whence writing A.e ik for A. we have
o * i
iDMty ........... (23),
Putting for shortness
e * + e* = S t , e ik e ik =D it
we have by the condition at the free surface
(C + ^,A t D t cos ty) {1  2&A t iS t cos i</> + $iAj3 t cos /</>/
As the expressions are longer than in the case of an infinite
depth, and the problem itself of rather less interest, I shall content
THEORY OF OSCILLATORY WAVES. 321
myself with proceeding to the third order. We have to this
order from (25), on taking account of the relations
(C + AJ)^ cos + A 2 D 2 cos 2</> + A 3 D 3 cos 30)
x  1  2A& cos cf)  4A& cos 20  6A 3 S 3 cos 30 1
{+ A*S Z + 4AtAj8 9 cos f 2^ 2 cos 20 + ^^cos 30j
**
]\Iultiplying out, retaining terms up to the third order only,
arranging the terms according to cosines of multiples of 0, and
equating to zero the coefficients of the cosines of the same
multiple, we get the four equations
j
3
C ( 4,A& + 2A*)  AS t D t + A,D, = 0,
C ( QA S
A slight examination of the process of approximation will
show that, whatever be the order to which we proceed, C, and
the coefficients A z , A^ ... with even suffixes, will contain only
even powers, and the coefficients A 3 , A 5) ... with odd suffixes
only odd powers, of the first coefficient A lt Writing b for A lt we
may therefore assume, in proceeding to the third order only,
Substituting in the last three equations of the preceding group,
which after the substitution may be divided by b, b 2 , b* respect
ively, arranging, and equating coefficients of like powers of 6,
we get
 2^a f D, = 0,
) 7  2^/3 + S& 4 A = 0,
(D,  66 8=0.
21
322 SUPPLEMENT TO A PAPER ON THE
The substitution for C and the coefficients A 2 , A a , ... of series
according to even or odd powers of b with indeterminate coeffi
cients was hardly worth making in proceeding to the third order
only, but seems advantageous when we want to proceed to a
rather high order. In proceeding to the n ih order it is to be noted
that the coefficients of C in the group of n + 1 equations got by
equating to zero the coefficients of cosines of multiples of </>
(including the zero multiple, or constant term), are of the orders
0, 1, 2, ... n in b, so that C being determined only to b n ~ l in the
equations after the first, the terms of the order n in the first
equation (which could only occur when n is even) are not required,
but this first equation need only be carried as far as to n 1.
In fact, in proceeding to the orders 1, 2, 3, 4, 5, 6, ..., the velocity
of propagation is given to an order not higher than 0, 1, 2, 3,
4, 5, ... in b, and therefore actually to 0, 0, 2, 2, 4, 4, ... since
it involves only even powers of b.
The last equations give in succession
(26),
(28),
(29),
and then by substituting in the first equation of the group on
the middle of p. 321, we get
We get now from (23), (24), after rendering the equations ho
mogeneous,
x = 
c
...... (31),
cos m$/c + &c ...... (32),
THEORY OF OSCILLATORY WAVES. 323
the expression for y after the first term differing from that for x
only in having a minus sign before the second exponential in each
term, and cosines in place of sines. We have also from (30)
= + A (S t +aS, + 12) V ............. (33),
9 t ^1^1
which gives the velocity of propagation according to one of its
possible definitions (see Art. 3, p. 202). In these expressions
it is to be observed that
imk/c J) e imk/c _ e 
We might of course in the numerators of the coefficients have
used expressions proceeding according to powers of 8 t instead of
according to the functions 8 l) S 2 , S 3 ...
Let h be the value of y at the bottom, which is a stream line
for which ty= k, then we have from (24) generalized as to
units
k = ch ............................... (34),
so that it remains only to specify the origin of y and the meaning
of c. To the first order of small quantities we have
C
and at the surface
/c ......... (35),
< ) cos m^/c ......... (36),
lc ................................. (37),
c
y = bD t cos ra</>/c ......................................... (38).
Since y in (38) is a small quantity of the first order, we may
replace <f>/c in its expression by #, in accordance with (37), which
gives for the equation of the surface
y = bD t cos mx,
so that to this order of approximation the origin is in the plane
of mean level, and therefore h denotes the mean depth of the fluid.
Also since u = dfy/dx we have to the first order from (3), (4), (6)
u = Fjjj] = j i + ^ ( e (*+A:)/c + e m(*+*)/c) cos m^
= cc.mb (e h ~ri + e  m(h ^) cos mx,
324 SUPPLEMENT TO A PAPER ON THE
and consists therefore of two parts, one representing a uniform
flow in the negative direction with a velocity c, and the other
a motion of periodic oscillation. To this order therefore there
can be no question that c should be the horizontal velocity in
a positive direction which we must superpose on the whole mass
of fluid in order to pass to pure wave motion without current.
In passing to the higher orders it will be convenient still to regard
this constant as the velocity of propagation, and accordingly as
representing the velocity which we must superpose, in the positive
direction, on the steady motion in order to arrive at the wave
motion ; but what, in accordance with this definition, may be
the mean horizontal velocity of the whole mass of fluid in the
residual wave motion, or what may be the mean horizontal velocity
at the bottom, &c., or again what is the distance of the origin from
the plane of mean level, are questions which we could only answer
by working out the approximation, and which it would be of
very little interest to answer, as we may just as well suppose
the constant h defined by (34) given as suppose the mean depth
given, and similarly as regards the flow.
Putting \/r = in (31) and (32), we have implicitly for the
equation of the surface the pair of equations
x   + SJ) sin m^/c  ^ ($ 2 + 1) S z mb* sin 2m<j)/c
c U
4) SX&" sin
y = Dp cos infyjc   ($ 8 + l) D 2 mb 2 cos 2mc/>/c
+ Q n 4 (3$ 4 + 4$ 2 + 4) D 3 m 2 & 3 cos
The ratios of the coefficients of the successive cosines in y or
sines in x to what they would have been for an infinite depth,
supposing that of cos md>/c the same in the two cases, are
i i
multiplied respectively by
#
THEORY OF OSCILLATORY WAVES. 325
for the cosines in y, and by
8. 8, 8,
n> n , . n a
1 "!
for the sines in #. Expressed in terms of D t , the first three ratios
become
1, l + W~\ I
and increase therefore as the depth diminishes, and consequently
D 1 diminishes. The same is the case with the multipliers
D 2 /A 2 > DJD?, SJD 1} &c, and on both accounts therefore the
series converge more slowly as the depth diminishes. Thus for
1)^ = 3 the first three ratios are 1, 2, 3$. 1^ = 3 corresponds to
A/X = 125, nearly, so that the average depth is about the one
eighth of the length of a wave.
The disadvantage of the approximation for the case of a finite
as compared with that of an infinite depth is not however quite so
great as might at first sight appear. There can be little doubt
that in both cases alike the series cease to be convergent when
the limiting wave, presenting an edge of 120, is reached. In the
case of an infinite depth, the limit is reached for some determinate
ratio of the height of a wave to the length, but clearly the same
proportion could not be preserved when the depth is much
diminished. In fact, high oscillatory waves in shallow water tend
to assume the character of a series of disconnected solitary waves,
and the greatest possible height depends mainly on the depth of
the fluid, being but little influenced by the length of the waves,
that is, the distance from crest to crest. To make the comparison
fair therefore between the convergency of the series in the cases of
a finite and of an infinite depth, we must not suppose the co
efficient of cos m<p/c the same in the two cases for the same length
of wave, but take it decidedly smaller in the case of the finite
depth, such for example as to bear the same proportion to the
greatest possible value in the two cases.
But with all due allowance to this consideration, it must be
confessed that the approximation is slower in the case of a finite
depth. That it must be so is seen by considering the character of
the developments, in the two cases, of the ordinate of the profile
in a harmonic series in terms of the abscissa, or of a quantity
having the same period and the same mean value as the abscissa.
The flowing outline of the profile in deep water lends itself readily
326 THEOKY OF OSCILLATORY WAVES.
to expansion in such a series. But the approximately isolated
and widely separated elevations that represent the profile in very
shallow water would require a comparatively large number of
terms in their expression in harmonic series in order that the
form should be represented with sufficient accuracy. In extreme
cases the fact of the waves being in series at all has little to
do with the character of the motion in the neighbourhood of the
elevations, where alone the motion is considerable, and it is not
therefore to be wondered at if an analysis essentially involving the
length of a wave should prove inconvenient.
INDEX TO VOL. I.
Aberration of light, 134, 153; Fresnel s
theory respecting, 141
Airy, Sir G. B., on tides and waves, 163,
165, 169, 171
Angular velocities of a fluid, 81, 112
Axes of extension, 82
Babinet s result as to noninfluence of
earth s motion on interference ex
plained, 142
Ball pendulum, resistance to, 180, 186;
resistance to a, within a concentric
spherical case, 41, 181; in presence of
a distant plane, 43 ; within rectangu
lar box, 111, 305
Box, motion of fluid within a closed, of
the form of a parallelepiped, 60, 66,
194, 288 ; equilateral triangular prism,
65 ; elliptic cylinder, 65 ; sector of
cylinder, 305
Cauchy s proof of a fundamental pro
position in hydrodynamics, 107, 160
Challis, Prof., aberration, 138; hydro
dynamical theorem, 160; ball pendu
lum, 180
Convergency, essential and accidental,
241 ; infinitely slow, 281
Current, superficial, in water an accom
paniment of waves, 208
Cylinder, motion of a piston and of the
air within a, 69
Cylindrical surfaces, (circular) motion
of perfect fluid between, 30 ; (elliptic)
approximate motion within or outside,
54 ; (circular) motion of viscous fluid
between, 102
Determinateness of problems in fluid
motion, 21
Discharge of air through small orifices,
paradox relating to, 176
Discontinuity, determination of, in a
function expressed by series or inte
grals of periodic functions, 239, 271 ;
propositions respecting, in the sums of
infinite series, &c., 279 ; of motion in
a fluid, 310
Doubly refracting crystals, formula for
determining the principal indices of,
148
Earnshaw, S., on solitary waves, 169
Eddies, production of resistance by, 53,
99 ; production of, 311
Elastic solids, isotropic, equations of
equilibrium &c. of, 113; necessity of
two arbitrary constants in the equa
tions, 120
Fresnel s theory of noninfluence of
earth s motion on the reflection and
refraction of light, 141
Friction, internal, of fluids, theory of, 75,
182; production of eddies by, 99, 311
Gerstner s investigation of a special case
of possible waves, 219
Green, notice of his papers on waves,
162 ; on sound, 178 ; on the motion of
fluid about an ellipsoid, 54, 179
Hydrodynamics, report on, 157
Impulsive motion of fluids, 23
328
INDEX TO VOL. I.
Instability of motion, 53, 311
Integrating factor of homogeneous dif
ferential equations, 130
Kelland, Prof., long waves in canal of
any form, 163 ; oscillatory waves in tri
angular canal, 165
Lee way of a ship, effect of waves on,
208
Lines of motion (see Stream lines)
Luminiferous ether, equations of mo
tion of, 124; constitution of, 153
Motion of fluids, some cases of, 17; sup
plement, 188
Newton s solution of velocity in a vortex,
correction in, 103
Parallelepiped, rectangular, motion of
fluid within, 60, 66, 288; experiments
as to the motion, 194; different ex
pressions for permanent temperature
in, 295, 302; expression for the poten
tial in a hollow conducting, due to an
interior electric point, 303
Periodic series, critical values of the
sums of, 236
Pipe, linear motion of fluid in a, 105;
production of eddies in a, 99
Poisson s theory of elastic solids, 116;
of viscous fluids, 118, 182; reduction
of his two arbitrary constants in the
latter case to one, 119, 184
Poisson s solution of the problem of a
ball pendulum, correction in, 42, 49
Eankine s investigation of a special case
of possible rotational waves, 219; of
the limiting form of irrotational waves,
225
Eectangle, different expressions for the
permanent temperature in, 290
Eeflection, principle of, as applied to
the motion of liquids, 28
Eesistance referable to instability of
motion and eddies, 52, 99
Saint Venant and Wantzel, discharge of
air through small orifices, 176
SaintVenant, equations of motion of a
viscous fluid, 183
Sound, intensity and velocity of, theo
retical effect of viscosity of air on, 100
Spheres, motion of fluid between two
concentric, 38; nonconcentric, 230
Steady motion of incompressible fluids, 1
Stream lines, determination whether a
given family of curves can be a set of,
in two dimensions, 5, 9; for motion
symmetrical about an axis, 15
Thomson, F. D. , demonstration of a the
orem due to him, 7
Thomson, Sir W., hydrodynamical ap
plication of his method of images, 230;
expression suggested by, for perma
nent temperature in rectangular pa
rallelepiped, 301
Tides (see Waves)
Triangular prism, (equilateral) motion of
fluid within, 8, 65
udx + vdy+wdz an exact differential,
proposition relating to, 1, 20, 106,
158
Uniqueness of xpf ession for p, q, or 0,
23
Waves and tides, report respecting, 161
Waves, theory of oscillatory, of finite
height, 197, 314 ; of small, at the com
mon surface of two liquids, 212;
greatest height of, at the surface of a
liquid, 225 ; Gerstner and Eankine s
investigation of a special possible case
of, 219; solitary, 168, 325
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With a Metrical Translation, Notes and Introduction, by E. H.
PALMER, M.A., BarristeratLaw of the Middle Temple, Lord
Almoner s Professor of Arabic and Fellow of St John s College
in the University of Cambridge. 3 vols. Crown Quarto.
Vol. I. The ARABIC TEXT. ios. 6d.\ Cloth extra, 15.?.
Vol. II. ENGLISH TRANSLATION, ios. 6d. y Cloth extra, i$s.
" Professor Palmer s activity in advancing
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lege, Cambridge. He has now produced an
admirable text, which illustrates in a remark
able manner the flexibility and graces of the
language he loves so well, and of which he
seems to be perfect master.... The Syndicate
of Cambridge University must not pass with
out the recognition of their liberality in
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an Arabic text. It is not the first time that
Oriental scholarship has thus been wisely
subsidised by Cambridge." Indian Mail.
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of Cambridge. It may be pronounced one of
the most beautiful Oriental books that have
ever been printed in Europe : and the learning
of the Editor worthily rivals the technical
getup of the creations of the soul of one of
the most tasteful poets of Isl^m, the study
of which will contribute not a little to save
honour of the poetry of the Arabs."
MYTHOLOGY AMONG THE HEBREWS (Engl.
Transl.}, p. 194.
"For ease and facility, for variety of
metre, for imitation, either designed or un
conscious, of the style of several of our own
poets, these versions deserve high praise
We have no hesitation in saying that in both
Prof. Palmer has made an addition to Ori
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grateful ; and that, while his knowledge of
Arabic is a sufficient guarantee for his mas
tery of the original, his English compositions
are distinguished by versatility, command of
language, rhythmical cadence, and, as we
have remarked, by not unskilful imitations of
the styles of several of our own favourite
poets, living and dead." Saturday Review.
" This sumptuous edition of the poems of
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accessible to readers who are not Oriental
ists. ... In all there is that exquisite finish of
which Arabic poetry is susceptible in so rare
a degree. The form is almost always beau
tiful, be the thought what it may. But this,
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lation. It is excellently well done. Mr
Palmer has tried to imitate the fall of the
original in his selection of the English metre
for the various pieces, and thus contrives to
convey a faint idea of the graceful flow of
the Arabic Altogether the inside of the
book is worthy of the beautiful arabesque
binding that rejoices the eye of the lover of
Arab art." Academy,
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NALOPAKHYANAM, OR, THE TALE OF NALA ;
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student. We must be contented to thank is one which no classical master should be
Professor Kennedy for his admirable execu without." Examiner.
THE THE^TETUS OF PLATO by the same Author.
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THE NEMEAN AND ISTHMIAN ODES.
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PLATO S PH^DO,
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A SELECTION OF THE STATE TRIALS.
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first volume reduced to a commodious torm. Howell s bulky series of volumes has been
The Examiner. done with much judgment, merely curious
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THE FRAGMENTS OF THE PERPETUAL
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LUCIANI SOMNIUM CHARON PISCATOR ET DE
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