Full text of "Volume and surface integrals used in physics"

See other formats

i 0 A
825
IL4
PHYS

^^^^^K

^HR

^^?^^^^^

UC-NHLP

B H b4T MDl

Cambridge Tracts in Mathematics

and Mathematical Pl^|pg5|lt 9? -- '^'^^^■

VOLUME AND SURFACE
IKTEGRTALS USED IN PHYSICS

J. g.|li

LEATHEM, Sc.D.

Cambridge University Press

C. F. Clay, Manager

London : Fetter Lane, E.G. 4

1922

Cambridge Tracts in Mathematics
and Mathematical Physics

No. I

Volume and Surface Integrals
used in Physics

CAMBRIDGE UNIVERSITY PRESS

C. F. CLAY, Manager

LONDON : FETTER LANE, E.G. 4

NEW YORK : THE MACMILLAN CO.

BOMBAY ^

CALCUTTA I MACMILLAN AND CO., Ltd.

MADRAS J

TORONTO : THE MACMILLAN CO. OF
CANADA, Ltd.

TOKYO : MARUZEN-KABUSHIKI-KAISHA

VOLUME AND SURFACE
INTEGRALS USED IN PHYSICS

by
J. G. LEATHEM, Sc.D.

Fellow and Bursar of St John's College

Cambridge

at the University Press

1922

PHYSICS UBR^

First Edition, 1905

Second Edition, 1913

Reprinted 1922

PHYSICS
LIBRARY

PREFACE TO THE SECOND EDITION

THE present edition of this Tract differs from the first edition only
by the inckision of two additional Sections. One of these deals
with Gauss's theorem of the surface integral of normal force in the
Theory of Attractions. The other discusses some theorems in Hydro-
dynamics, and includes a short account of the theory of ' suction '
between solid bodies moving in liquid.

The author's arrow notation for passage to a limit, since its publi-
cation in the first edition of this work in 1905, has been adopted by
many writers on Pure Mathematics, and may be regarded as -now well
established. Its application has rightly been confined to continuous
passages to limit, and there is evidently room for some corresponding
symbol to indicate saltatory approach to a limit value. A dotted
arrow might perhaps appropriately serve this purpose ; it would present
no difficulty to the printer, but it is just doubtful whether it would be
convenient in manuscript work.

The author desires again to express his thanks to Dr T. J. I'A.
Bromwich for help in the preparation of the first edition of this work,
more particularly for valuable suggestions with reference to the dis-
cussion of tests of convergence in § 13 and to the restriction upon/' in
the theorem of § 38.

J. G. L.

St John's College, Cambridge.
October, 1912.

M687645

CONTENTS

II.

Introduction

On the validity of volume -integral expressions for the
potential and the components of attraction of a
body of discontinuous structure ....

Potentials and attractions of accurately continuous
bodies ........

ART.

PAGE
1

III. Volume integrals

IV. Theorems connecting volume and surface integrals

V. The difiierentiation of volume integrals

VI. Applications to Potential Theory

VII. Applications to Theory of Magnetism .

VIII. Surface integrals

IX. Volume integrals through regions that extend to
infinity ........

X. Gaus-s's theorem in the Theory of Attractions

XL Some Hydrodynamical Theorems

VOLUME AND SURFACE INTEGRALS
USED IN PHYSICS

Introduction

1. The student of Electricity, and of the theory of attractions in
general, is constantly meeting with and using volume integrals and
surface integrals ; such integrals are the theme of the present tract.
It is proposed, in the first instance, to examine how far it is justifiable
to represent by such integrals the potential and other physical quantities
associated with a body which is supposed to be of molecular structure ;
and, in the second place, to give proofs of certain mathematical
properties of these integrals which there is a temptation to assume
though they are not by any means as obvious as the assumption of
them would imply. Illustrations will be taken for the most part from
the theory of the Newtonian potential, and from Electricity and
Magnetism ; and attention will be directed, not to all the peculiarities of
integrals which can be imagined by a pure mathematiciaUj but only to
those difiiculties which constantly present themselves in the usual
physical applications.

I. On the validity of volume-integral expressions for the
potential and the components of attraction of a body
of discontinuous structure.

2. The generally accepted formulation of the Newtonian law of
gravitation is that two elements of mass, m and m', at a distance r
apart attract one another with a force mrn'r'^ in the line joining them*.
This statement of the law may be regarded as a generalisation founded

* In order to shorten the formulae the constant of gravitation is omitted here
and elsewhere ; its presence or absence in no way affects the questions to be
discussed.

L. 1

2 POTENTIAL OF BODIES OF DISCONTINUOUS STRUCTURE [l

on the observed motions of heavenly bodies, and its simplicity commends
its adoption as the startinj? point of mathematical discussions of
^'ravitation problems. But the fact must not be ignored that the
statement is really lacking in precision ; for in the first place the phrase
'element of mass' is somewhat vague (even when the term 'mass'
i.s sufticiently understood), and must be taken to mean simply a very
small body or portion of a body, so small, namely, that its linear
dimensions are very small compared with r ; and in the second place r
itself is rather indefinite, meaning the distance between some point
of the first element and some point of the second.

The principle of superposition, that is to say the assumption
that the force exerted on one element of mass by two others is that
obtained by compounding according to the parallelogram law the forces
that would be exerted on it by each of the other elements alone, is part
of the fundamental hypothesis of the Newtonian law; and the principle
is commonly used to evaluate the attraction of a body which is not
extremely small by compounding the attractions of the small component
elements of mass of which it may be regarded as built up. In works
which deal witii the mathematics of the gravitation potential and
attractions, the values of these quantities for a body of definite size are
invariably obtained in the form of volume (or surface) integrals taken
through the space occupied by the body, the element of mass being
represented by the product of the element of volume and a function of
position called the * density.' But it ought to be clearly understood
that this procedure virtually involves either using the 'element of
mass ' of tlie Newtonian law as an element of integration, and thereby
attributing to it properties which are directly contrary to accepted
views as to the constitution of matter, or else using the word 'density'
in a special sense which is by no means simple or precise. For there
is no limit to the fineness of the subdivision of a region into volume
elements for purposes of integration, and the process must get endlessly
near to a limit represented by vanishing of the volume elements ; if
this extreme subdivision cannot be applied equally to mass, there
come8 a stage in the process when a volume element becomes too
small to contain a mass element, and so the average density in the
element, mass divided by volume, ceases to have a meaning, and the
mathematical passage to limit which constitutes the usual definition of
the density at a point is now impossible.

If the body considered is of mathematically continuous structure,
so that the portion of it occupying any space however minute has the

2] POTENTIAL OF BODIES OF DISCONTINUOUS STRUCTURE 3

gravitation propert)^ then density is a term having a precise meaning ;
but if the distribution of the gravitation property through the space
occupied by the body has not this mathematical continuity, we cannot
attach any meaning to the volume integrals till we have first invented
a suitable new meaning for the term 'density'; and the inevitable
vagueness that will arise in the new definition will preserve in the final
results that slight lack of precision present in the terms of statement
of the gravitation law, which might at first sight appear to have
dropped out of results represented by such precise mathematical
expressions as volume integrals.

It is practically certain that no substance can be subdivided without
limit into small portions each of which possesses the gravitation
property. There must be a stage of subdivision beyond which the
component portions cease to have the properties of larger portions of
the substance, and we may speak of the smallest portion of a substance
that has the gravitation property as a 'particle' for purposes of the present
discussion. What the order of magnitude of a particle may be it is
difficult to guess, but the kind of generalisation from large bodies to
small bodies which led to the conception of an element of mass suggests
the possibility that the process of subdivision without loss of the
gravitation property might be continued till we arrive at the molecule
of Chemistry or Gas Theory. There is no experimental evidence to
prohibit, and possibly some to justify our carrying the generalisation
so far (provided we set some limit to the smallness of the distance at
which the attraction of two particles is supposed to obey the law of the
inverse square), and the great simplicity of the law thus obtained
makes it an interesting one to study.

If a body has not got mathematical continuity in the distribution
of the gravitation property throughout its volume, and so is to be
regarded as made up of particles, we may speak of it as having
'discontinuous' structure. And if it be supposed that the particles
may be as small as molecules, we must form a mental picture of the
structure in which there appears no trace of material continuity, the
substance being represented by discrete molecules or systems occupying
spaces with somewhat indefinite boundaries, separated by more or less
empty regions which may be called intermolecular space; (if the
molecules move it may be supposed that their motions do not affect
the properties under consideration). The problem of finding the
potential or attraction of such a body at any point, if formulated on a
greatly magnified and coarser scale, would in some respects resemble

1—2

4 POTENTIAL OF BODIES OF DISCONTINUOUS STRUCTURE [l

the problem of evaluating,' the potential or attraction of a mass of sand
or other granular matter. We want to see how volume integrals present
themselves as approximate solutions of such problems.

It is unnecessary to dwell here upon the familiar definitions of the
intensity of force at a point, or the potential at a point, due to a
gravitating body. But, for future reference, we may emphasize the
fact that, among the mathematical properties of the potential at a
point outside the body, that which may be taken as the fundamental
physical definition is the fact that its space-gradient at any point is
vectorially equal to the intensity of force there. If a point is so
situated that a physical definition of intensity of force there is
in)possible, this physical definition of potential breaks down, and we
are at liberty to substitute some convenient purely mathematical
definition for which it may be possible afterwards to find a physical
interpretation.

The potential of a body of discontinuous structiire at an external
point P is the sum of the potentials at P due to the particles that
compose the body, i.e. 1mr~^, where m is the mass of a particle and
r its distance from P. Here, in accordance with what has been said
above, r is not precisely defined, and a corresponding lack of precision
must be present in 2w?r~\ By assuming P to be not too close to any
particle of the body we can ensure that each r shall always be great
compared with the linear dimensions of the corresponding particle.

When we endeavour to compare the values of this expression for
the potential at difierent points, we recognise that the sum of a finite
but extremely great number of extremely small terms is a most trouble-
some function to work with, and so there naturally suggests itself the
device of getting a probably very approximate equivalent function by
replacing the sum by the limit to which it woidd tend if the number
of terms could be increased indefinitely while each separate term
decreased corresi)ondingly ; this process would give us the potential in
tiie form (.f the definite integral fpr-^dr, where dr is an element of
volume and pdr the corresponding mass.

Jiut, as has been suggested already, the transition from a sum of
terms to a definite integral would imply the possibility of endless
subdivision of the material mass into elements each possessing the
gravitation property, whereas it is practically certain that matter
cannot be so endlessly subdivided. In fact the use of the definite
integral form implies a regarding of matter as continuously extended
through the space which it effectively occupies, and attributes to the

2-3] POTENTIAL OF BODIES OF DISCONTINUOUS STRUCTURE 5

density p at any point the value obtained by passing to a mathematical
limit in the usual fashion, that is to say the Hmit of the quotient of
mass by volume for a region surrounding the point as the dimensions
of the region tend to zero. The molecular view, however, requires us
to cease subdividing matter beyond a certain stage, and so prevents
our ever arriving at the kind of limit which is known as an integral.

3. Nevertheless the potential at the point P of an assemblage of
discrete particles in a finite region may be equal in value to a volume
integral taken throughout the region if the integral be supposed to
refer to a hypothetical continuous medium occupying the same region
and having a suitably chosen density at each point. It is only
necessary to choose the law of density properly, and to this end there
suggests itself the device of taking for each point A some sort of
average density, based on a consideration of all the masses within
a very small but finite region surrounding A. The dimensions of this
small region might be settled by convention, but we need only consider
the order of magnitude of these dimensions.

The kind of smallness that we want in this connexion is what
we may call physical smallness, as distinguished from mathematical
smallness to which there is no limit. Physical smallness implies
smallness which appears extreme to the human senses, but it must not
be a smallness so extreme as to necessitate passage from molar physics
to molecular phj-sics; it must leave us at liberty, for example, to
attribute to matter occupying a physically small space the properties
of matter in bulk if these should be different from the properties of
isolated molecules. In fact a physically small region, though extremely
small, must still be large enough to contain a very great number of
molecules. It is estimated that a gas, at normal temperature and
pressure, has about 4 x 10^'' molecules per cubic centimetre; thus a cube
whose edge is 6 x 10~^ centimetres (roughly the wave length of sodium
light) contains more than 8,000,000 molecules ; if we regard a million
as a large number, the wave length of sodium light is (for other than
optical purposes) physically small, and it is known that very much
greater lengths than this appear to our senses extremely small. We
are therefore in a position to speak of lengths which, though extremely
small, are very great compared with other physically small lengths.

Now it is not suggested that the gravitation property is a molar
property of matter, not possessed by a single molecule, for we have
adopted just the opposite hypothesis ; and so it might be thought
justifiable to make the region round A, used for calculating p,

6 POTENTIAL OF BODIES OF DISCONTINUOUS STRUCTURE [l

smaller than merely physically small. This point will be referred to
again, hut at present it suffices to remark that the p generally used in
potential theory i^< a continuous function whose value is not subject to
very rapid tluctuations as A moves from one position to another. To
get such I'cntinuity and smoothness in the suggested average, and to
avoid the dillirulty tiiat would arise if the number of molecules inter-
sected by the boundary of the region so as to be neither obviously
inside nor obviously outside were not very small in comparison with
tlie total number inside, we must take account of a large number of
molecules at a time, and so we have to assume that the region
surrounding the point A and u.sed in getting a value for p is only
jdiysically small.

As regards the system of averaging, it is clear that in the case
under discussion we get the best agreement between the potential of
the actual and that of the hypothetical system if we give to each
mok^cule an importance proportional to the product of its mass and the
reciprocal of its distance from F ; but it would be unfortunate to be
obliged to average in this fashion, as we should thereby get a value of
P at A which would not be independent of the position of P. And we
should get quite a different law of density if we were dealing with some
other integral, say an attraction integral, instead of that representing
potential.

So long, however, as P is at a distance from A great compared
with the linear dimensions of the physically small region used for the
purpose of averaging, the values of r~^ for the molecules in this region
are very nearly e([ual, and so there is very little error if in taking the
average we give to each molecule an importance simply proportional to
its mass. We thus get for the density p of the hypotlietical continuous
medium the (juotient of mass by volume for the jihysically small region
considered. This value of p has the advantage that it is independent
of the i)osition of P, and of the particular physical quantity whose
integral expression is being investigated. But its great advantage,
and the real reason why we adopt it, is that it is that density of a
substance whicii we actually arrive at by practical methods of measure-
ment ; for ordinary laboratory measurings and weighings are applied
to jHjrtions of a substance which are far from the limits of physical
snuiUness, and so give us, not the sizes and masses of individual mole-
cules, but only the total mass and the total space effectively occupied.

4. So far we have .considered oidy the case in which the point P
at wiiicii the jjotential (or otiier function of position) is to be estimated

3-4] POTENTIAL OF BODIES OF DISCONTINUOUS STRUCTURE 7

is at a distance from the nearest portion of the gravitating mass great
compared with the linear dimensions of what we have called a physically
small region. Such a distance, which, as we have seen, may be extremely
small compared with the smallest distance we can measure directly,
would seem to mark the limit of nearness of P to the gravitating body
if the integral taken for the hypothetical continuous medium is to serve
as equivalent to the true potential. But further consideration may
enable us to push this limit still closer to the body. For the inac-
curacies whose importance is magnified by decreasing distance do not,
for a given position of P, occur in the case of each molecule of the
body ; they arise only in connexion with the molecules that are near
P. JN^ow such molecules, though perhaps absolutely numerous, are
generally few in comparison with the remaining molecules of the body,
and it is possible that their numerical inferiority may prevail over the
advantage of their position in such a way as to render the total
inaccuracy corresponding to them a negligibly small fraction of the
whole potential.

Instead of the function r~' which occurs in expressions for the
potential, let us consider some other function / of position relative to
P, which tends, as r becomes smaller, to become infinite of the same
order as r"*^, so that, for small values of r, f is of the form h'~>^ where
^ is a finite function of relative angular position. Taking, in the first
instance, a single physically small element of matter, say of volume c*,
it is clear that the difference between the sum 2w/ and the integral
^pfdr through the element will in general be of the same order of magni-
tude as either quantity separately so long as r for all points of the
element is of the same order of magnitude as e, but that the difference
will diminish to a quantity smaller in a ratio comparable with er"^
when r becomes great compared with c Hence, for purposes of
estimating order of magnitude, it is fair to represent the difference
between '%mf and /p/(/t by the expression jAer^^p/dr where A is
a finite number.

To include all elements of the body near to P, we suppose the least
value of r for a molecule near to P to be rj, and take the integral
representing inaccuracy through all space between the concentric
spheres /• = •>? and r = a, where a is large compared with e. If p is the
greatest value of p in this space, the order of magnitude of the inac-
curacy is the same as or less than that of

p'k'A fer-'r-'^iTrr-dr,

8 POTENTIAL OF BODIES OF DISCONTINUOUS STRUCTURE [l

where /•' is a finite constant replacing /•, and 4n)"dr, the volume
between the spheres of radii r and /• + dr, takes the place of dr. This

is eciual tu

Anpk'Ae{a-''-r-'']/i2-H-),

of which, when r; is of the same order as c, the first term is small of
order « and therefore negligible always, while the second term is small
of the same or higher order provided /a < 2 ; the second term would be
small, but not of so high an order of smallness, if fx. were between 2 and 3.
The case of /t* = 2 would turn on the^ order of magnitude of t log v or
€ lof c, which is very small though not as small as c. Sometimes the
.special form of the function /•, taken in connexion with probable sym-
metry in the average distribution of molecules round P, increases still
further the order of smallness.

It follows, therefore, that if /a<2 the inaccuracy is certainly as
negligible as eb~\ and that if /^ < 2^ the inaccuracy is certainly as
negligible as Jelr\ where b is some length which is physically not small,
e.g. a centimetre. The case of /a = 1 corresponds to potential ; for
attraction components /t = 2.

5. Thus it appears that the representation of potentials and
attractions by means of integrals extended through the hypothetical
continuous medium which replaces the actual gravitating body is valid
without sensible error not only for points well outside the body, but
also for points whose distance from the nearest portion of the body is
small of the order of the physically small length e. This includes the
case when P is so clo.se to the apparent outer surface of the body as to
be sensibly just not in contiict with it, and also the case when P is in
a small but not imperceptibly small cavity cut in the body, that is
a cavity of such a size that the piece excavated would have the pro-
perties of matter in bulk rather than the properties of a few molecules.

As might be expected, any attempt to ju&tify the use of the same
integral expressions for the potential and attractions at a point P
which is at a distance from the nearest molecule of a higher order
of smallness than e, results in failure. For now a simjile molecule
contributes to the potential (for example) a term mr~^ which, in spite
of the smallness of m, may become very great as r diminishes ; how
small r may become we cannot say, its least possible value must
depend on the extent to which the 'impenetrability' of matter is
true of isolated molecules, for, since potential is only physically
interpretable as the negative potential energy per unit mass of a particle
(at least one molecule) at P, the least value of ;• is the least possible

4-7] POTENTIAL OF BODIES OF DISCONTINUOUS STRUCTURE 9

distance between the centres of two molecules. While not knowing
this least value, we cannot but admit the possibility that a few terms
of the type mr'^ might easily become so important as to make the
potential quite different from the value of jpr'^dr to which, as we
shall see later, the parts of the hypothetical continuous distribution
near P contribute only a negligible amount. But there is in any case,
from the point of view of physics, no motive for pursuing the enquiry
to such small values of r, for there are reasons for supposing that the
Newtonian law of attraction does not hold good at such distances. In
proving that the ordinary integTal representations of potential and
attractions are valid for distances of P from the attracting body which
are indefinitely small from the point of view of molar physics, we have
done all that would be required to justify their ordinary use in the
theory of gravitation.

6. One point requires emphasis. By the attraction at a point P
inside a body we mean the attraction (per unit mass) on a molecule or
particle at P, situated in a cavity of dimensions which are only
physically small. Hence if we describe a closed geometrical surface >S',
however small, in a body, we cannot calculate the force exerted on the
part of the body inside S by the rest of the body by use of the attrac-
tion integrals. This can only be done when there is a gap, not
smaller than physically small, between the attracting and the attracted
matter, such a gap as might be made by cutting the body along the
surface S and keeping the fissure open so that the opposite sides of it
are nowhere in contact. In the absence of such a physical separation,
account must be taken of unknown forces between molecules that are
very near together. When the volume enclosed by ;S' is not small
beyond the physical limit of smallness, such molecules will all lie
relatively near the surface S, and the forces between them will appear
as surface forces between the geometrically separated portions of the
body. In fluids the extra force is the fluid pressure, in solids it is less
simple.

In the case of an absolutely continuous body there is nothing
corresponding to the limit of physical smallness, and if the Newtonian
law were supposed to hold for all distances however small there would
be no surface forces of the kind described. The assumption of surface
forces in the ideal case of continuity is really a tacit assumption that
the Newtonian law breaks down ultimately as r diminishes.

7. We might, of course, devise other definite integrals than those
above considered, in the hope of representing the same physical

10 POTENTIAL OF BODIES OF CONTINUOUS STRUCTURE [H

quantities with possibly graiter accuracy. For example p might be
obtiiimHl by averaging through a region smaller than physically small,
so small that the number of molecules in it might be sometimes two or
one or even zero : in this case fractions of molecules would become
important, and the question would arise how a molecule ought to be
regarded when it is neither altogether inside nor altogether outside the
region. Again we might reduce the region of averaging to the limit
of mathematical smallness and so get a p which is absolutely zero in
intermolecular .space, and presumably finite and continuous in the
spaces occupied by tlie various molecules. Against such integrals it is to
be urged firstly that one important factor of the function to be integrated,
namely p, is inaccessible to experimental measurement, secondly that
even if p were knowai the integrals would probably be more difficult to
evaluate than the sum 1mr~\ and thirdly that the gi-eater accuracy
which they seem to possess would be entirely vitiated by the probable
failure of the Newtonian law for short distances. Moreover a method
which involves integrating through the volumes of individual molecules,
if it has any physical significance at all, implies the view that a
molecule is of the nature of a small continuous mass w'hose smallest
parts have the same kind of properties as the whole ; this view is
directly contrary to modern views of the constitution of matter, and
the mathematical method corresponding to it, so far from being the
best possible representation of the facts, must share all the defects of
the method of summation for the various molecules.

II. Potentials and Attractions of accurately continuous

bodies.

8. The potential and the attraction components of a finite body
of accurately continuous substance, at an external point P, are
represented by volume integrals which, for ordinary laws of density,
give rise to no mathematical difficulties. The subjects of integration
are finite at all points of the region of integration, and the integrals them-
selves are finite and differentiable with respect to the coordinates of P
by the method known as 'differentiation under the sign of integration.'
Thus the i)oten(ial integral, defined as jpr'^dr, justifies its right to the
name 'potential' by possessing the property that its differential
coefficients with respect to the coordinates (^, t/, C) of P are the attrac-
tion integrals of the type jp {.v - c) r~'dT.

7-9] POTENTIAL OF BODIES OF CONTINUOUS STRUCTURE 11

But it is quite another thing when we come to consider the potential
and attractions at a point inside the gravitating body. For now, for
example, if we define the potential as jpr'^dr taken throughout the
whole body, the subject of integration pr~^ becomes infinite at the
point P, a point in the volume of integration, and it becomes a question
whether the integral symbol represents a finite quantity at all, and, if
so, whether it is differentiable and what are its differential coefficients.

These troublesome questions might be avoided by introducing, as
in the investigation for a body of molecular structure, a cavity within
which P must be situated. And, indeed, this still seems to be
demanded by the physical interpretation, since potential and attraction
are physically defined as work function and force per unit mass for a
hypothetical small mass or particle at the i^oint P; such particle
cannot be supposed to occupy space already occupied by other matter,
and hence must be situated in a cavity made for it. But whereas, in
the case of molecular structure, there was suggested from physical
considerations a limit to the order of smallness of the cavity contem-
plated, corresponding in fact to the order of smallness of the necessarily
present inaccuracy in the mathematical representation adopted, no
such limit suggests itself in the case of continuous bodies. The
retention of a cavity, of any definite though arbitrarily chosen order
of smallness, is not demanded when there is no limit to the possible
smallness of a portion of matter, and w^ould moreover involve a want
of precision or at least a restriction on the meaning of the mathematical
symbols employed which would considerably discount their utility.
Whereas it is only for the sake of mathematical precision that the
hypothetical continuous bodies are generally made the subject of study
in preference to the actual molecular bodies of which they are
approximate representations.

9. We obtain the definiteness we desire, and, as will be seen,
conform at the same time to the conventions and definitions of Integral
Calculus, by framing new definitions of the potential and the attraction
components at a point P ($, -q, t), inside a continuous body. We first
suppose the point P to be in a cavity, we then make the cavity smaller
and smaller, and define the limits (if such exist) to which the potential
and the attraction components at P tend with the vanishing of the
cavity as the potential and the attraction components respectively at
P when no cavity exists. It must be recognised that this passage to
the limit entirely destroys the physical meaning which the quantities
considered possess at any stage short of the limit, but on the other

12 VOLUME INTEGRALS [ill

hand it gives us extremely convenient standard approximations to
these quantities in cases of physical interest ; the very definition of
the term limit implies that the approximation can be made as close as
we please by taking the cavity sufficiently small.

It is also to be noticed that a relation such as X=-^ (where X

is a force component and V the potential), which holds inside a cavity
of finite size however small, might not persist after passage to the
limit. That is to say, though of necessity

Lim A' = Lim ^7-,

it is not equally inevitable that

Lim X= 57. Lim V.

In fact if, as is customary, we drop the phrase 'limit' from our

notation, though keeping the idea in mind, we have to face the fact

dV . . . .

that the formula X--^, valid for free space, requires examination

before we can l)e sure that it is true at a point in the substance
of the body. And if it be objected that the formula is known to be
true in all cases of physical interest, and that no such interest attaches
to its validity or otherwise in the case which has avowedly no pliysical
significance, an answer is that if we decide to use a certain kind of
mathematical functions as approximate representations of physical
quantities, we must become acquainted with the meanings and pro-
perties of these functions before we can make intelligent use of them.

Hence it is natural for the student of the theory of attractions
to turn his attention to that part of pure mathematics which has to
do with the definition and properties of volume and surface integrals.

III. Volume Integrals.

10. Let / be a function of position, and let a finite vohime T be
divided into a great number of elements At, of small linear dimensions;
let Ji be a quantity associated with an element of volume At, chosen
according to some law, so that it is either the value of/ at some point
of the element, or at any rate not greater than the greatest or less than
the least value of / for points in the element. If / is finite at all
points in the volume T, the sum 2/, At extended to all elements of T
is finite, and will remain so no matter how small and correspondingly
numerous are the elements At. If this sum tends to a limit as the

9-11] VOLUME INTEGRALS 13

number of elements tends to infinity, and the linear dimensions of each
tend to zero, and if this limit is independent of the law specifying J\
and of the manner of subdivision into elements, the limit is called the
volume integral of/ through the volume T, and is denoted by //c?r.
This definition is only valid on the supposition that / is finite at all
points in T.

Whether the limit here spoken of does or does not exist depends on
the nature of the function /; we shall assume that it does exist for all
the forms of/ which we meet with in potential theory.

11. Next consider the case in which / is a function which becomes
infinite at a point P within the volume 7^; clearly we need a new
definition, and that which has been generally adopted is as follows.
Surround the point P by a small closed surface t, and take the volume
integral through the whole of the volume T except the part included
by # ; we thus exclude P from the range of integration, and so get a
finite integral. Now let the surface t become smaller and smaller,
whilst always surrounding P ; if the volume integral tends to a finite
limit as the space enclosed by t tends to vanishing, and if the limit is
independent of the shape of t, then this limit is defined to be the
integral of/ throughout the whole volume T. The definition may be
expressed symbolically thus :

rT rT

\ /c?T = Lim fdr,

•where the symbol -* is used to denote such phrases as ' tending
towards ' or ' tends towards,' so that t ^0 reads ' as t tends towards
zero.' Here and elsewhere the subscript to the integral specifies the
inner boundary of the region of integration.

If we call the space inside the vanishing surface t a 'cavity' in
the volume of integration, we see at once the parallelism between the
definition of this kind of volume integral and that of the so-called
potential and attractions at a point in the substance of a continuous
body.

The volume integral (if it exists) through a region within which /
becomes infinite at some point is seen, by the above definition, to be a
mathematical conception of a different character from the integral for
a region in which / is everywhere finite. In a sense one might say
that the latter is a true volume integral while the former is the limit
of a true volume integral. The latter bears to the former the kind of
relation that a single limit bears to a double limit, or that a finite
series bears to the so-called sum of an infinite series.

14 VOLUME INTEGRALS [ill

12. Analogously with tlie terminology of series, we speak of the
volume integral as convergent if it tends to a finite limit with the
vanishing of the cavity, divergent if it tends to become infinitely great,
and semi-convergent if, as sometimes happens, there is a finite limit
whose value is not independent of the shape or mode of vanishing of
the cavity. Divergent integrals are, for ordinary purposes, as meaning-
less as divergent infinite series, and so we must satisfy ourselves that
the integrals in use in gravitation problems are convergent either ab-
solutely or in the conditional manner corresponding to semi-convergence.

To decide whether, for a given form of/, the integral is convergent
or not, we have the following rule, depending on the order of the
infinity of/ at P in terms of r, the distance from P to the point at
which / is estimated. If / becomes infinite at P of an order lower
than r~" the volume integral is convergent, if of an order higher than
r'^ the integral is divergent; if of the order r~^ exactly the integral
may be divergent, semi-convergent, or convergent, according to the
way in which / in the neighbourhood of P depends on the angular
position of r. This rule is not stated with sufficient accuracy to rank
as a theorem, and one can easily think of exceptions to it ; for example
the case of /= r"*cos^ (in the notation of spherical polar coordinates),
which is obviously convergent iov any cavity symmetrical about the
plane B = ^ir though otherwise likely to be divergent, shews that some-
thing like semi-convergence may be associated with infinities of order
greater than 3*; but the rule is a convenient approximation to the
facts.

13. With a view to justifying the rule here given, it will be
convenient to re-state, with slight modification of form, the definition
of convergence. The integral of / through the volume T, which
includes a point of infinity P, is convergent if, corresponding to any
arliitrarily chosen small quantity o-, there can always be found a closed
surface 6 surrounding P such that all closed surfaces t surrounding P
and lying wholly inside 0 have the property that

; . . . I /•'■'*

That this is essentially the same as the definition of § 11 appears at
once when we think of the ordinary definition of a limit ; for if the
integral through 7' has a limit A, we can choose 6 so that

\\ fdr-A < la, fdr-A

* See Article 70.

12-13]

VOLUME INTEGRALS

nnd therefore

/>iHP''^-r

fd.

it is, of course, to be understood that P must not lie on the surface 0.
Thus the property constituting the definition of convergence of the
present Article is a consequence of the property laid down as a
definition in § 11.

Conversely, possession by an integral of the property specified in
the present Article involves as a necessary consequence the existence of
a limit A, though giving no indication of its actual value. For by
taking B sufficiently small we can keep the fluctuation of the value of
the integral for different cavities within Q as small as we please, that is
small without limit ; and infinitely restricted fluctuation is the same as
infinite approximation to some definite (and therefore finite) value.

Part of the rule of § 12 may be formulated in the following theorem.
If within a sphere of finite radius (a), having P as centre, f is everywhere
less in absolute value than Mr~f^, where M is a definite constant and
/x<3, the integral is convergent. To prove this let us take for the
surface 0 the sphere r = -q, where y]<a, and let us denote by e the
distance from P to the nearest point of the surface t of the cavity ; the
cavity is of course entirely inside 6, but is otherwise unrestricted as to
shape. Since the modulus of a sum is not greater than the sum of the
moduli, and since an integral is the limit of a sum.

f/'^

^ \f\dr
Jt

< \f\dT

<3I r-f'd^

where the subscript c means that the inner boundary of the integrals
is the sphere /* = € ; the second inequality holds because \f\ is positive
and € is completely inside 0.

In dealing with a function of ;• only, we may combine all elements
djT that lie between spheres of radii r and r + dr in the single expression
Airr-dr, so that

< IttJ/ / r^-^dr

AttM , q ON

3-fi
47rif

3-/*'

,3-/i

16 VOLUME INTEGRALS [ill

it being noted that f < »? and that 3 - /* is positive, so that the last

expression obtained is positive. Now if o- be any arbitrarily chosen

small quantity, we have only to take rj less than {(3 - /x) o-/47ril[/}3-'^
in order to get a surface 6 such that

whatever shape t may have provided only it lies inside 0, Thus the
convergence of the integral of/ is established.

It will be noticed that the convergence of the integral of / in
accordance with this theorem involves also, as the proof indicates, the
convergence of the integral of \f\.

It need hardly be pointed out that the position and shape of the
outer boundary T of the region of integration do not, in general, affect
the question of convergence ; whatever the outer boundary may be,
provided it does not include other points of infinity, it is only the
part of tlie volume just round P that is in danger of making the
integral very great, and so only that part need be studied with a view
to detecting divergence.

It is clear that the theorem holds equally well for cases in which
the point P where the infinity occurs is not inside but just on the
boundary of the region of integration.

14. The corresponding theorem for divergence is as follows. If
within a sphere of finite radius (ci), having P as centre, f is everywhere
algebraically greater than mr~>^, where m is a constant greater than zerOy
and /x ^ 3, the integral is divergent. To prove this, we take as outer
boundary the sphere r - a, and as inner boundary a surface t, and we
denote by e the distance from P to the furthest point of the surface f,
so that the sphere r = e completely surrounds the cavity. Then

/•ft ra ra

I fdr > ni I r~>^dr > m \ r~>^dT ;

Ji Jt Je

as before, we collect all the elements dr between r and r + dr into the
expression Airr-dr, and so get

ra ra

I fdT>ATrmj i^'i^dr

> — - {e-<'*-3)-a-('^-3)i or Attih {log a - \og e} ,

according as /a is greater than or equal to 3. In either case the
expression obtained tends to infinity for « -*- 0 ; and so the integral,
being greater than a quantity which tends to become endlessly great,
is divergent.

13-16] SEMI-CONVERGENCE 17

15. The case of semi-convergence need only be illustrated by an
example. Suppose that / is r-' cos 6, and consider the values of the
integral in the regions having a common outer boundary r = a, and
having for inner boundaries in the first instance the sphere r = e, and
in the second instance the sphere )•"- - re cos 6 = 2e^ these being two
small spheres of which the latter touches and completely surrounds the
former.

The difference between the integrals over these two regions is the
integral through the space between the two small spheres, which is
certainly not zero so long as € is different from zero, since, if we consider
the volumes of the region cut off by a cone of small solid angle having
the origin as vertex, the positive contribution to the integral from the
frustum where cos 0 is positive is greater in absolute value than
the negative contribution from the frustum where cos 0 is negative.
Further, the magnitude of the integral is independent of c, since if we
multiply € by ^ we can get the new region of integration by multiplying
all radii vectores from F by k, and thus each element of volume dr is
multiplied by P; the subject of integration r~^cos 0 is correspondingly
multiplied by k'"', and so the integral is unaltered. Thus the integral
over the space between the small spheres, being finite when e is not
zero, has the same finite value as € tends to vanishing ; in other words
there is a finite difference between the values of the original integral
corresponding to the different cavities. It is clear, from the symmetry
of/ about the plane 6 = ^tt, that the integral is zero for the cavity whose
centre is P, and therefore not zero for the cavity whose centre is not
at P ; but in neither case is it infinite. Thus the semi-convergence of
the integral is demonstrated.

This example suggests the remark that two cavities are to be
regarded as of different shapes even if they are similar, if they are not
similarly situated with respect to P. The cavities considered in the
example are both spheres, but since one has P at its centre while in
the other P trisects a diameter, the cavities are regarded as of different
shapes for purposes of the present discussion.

IV. Theorems connecting volume and surface integrals.

16. There is a well-known theorem connecting volume integrals
with surface integrals taken over the boundary of the region of
volume-integration. If the region be finite, if I, m, n denote the
direction cosines of the normal drawn outwards from the region at

L. 2

18 THEOREMS CONNECTING VOLUME [iV

a point of the boundary B, and if f, -q, 4 be functions having finite
space differential coefficients at all points in the region,

/(«.„„.»0<f..=/(|.|.|)<^. (1),

where dS represents an element of area of the boundary, the surface
integral is taken over the complete boundary, and the volume integral
through the complete volume. A proof is given in Williamson's
Integral Cahulus, Chapter XL

It is worth while to enquire whether this theorem can be extended
to the case in which there is a point P in the volume where the
subject of volume-integration becomes infinite. The course which
suggests itself is to surround the point P by a small closed surface o-,
and to apply the original theorem to the region bounded internally
by o- and externally by the surface B, The complete boundary
consists of both B and a-, and so we get

where the suffixes to the surface integrals indicate the surfaces over
which they are taken. Now if the subject of integration of the
volume integral is such as to make it convergent with respect to the
infinity at P, the left-hand side of the equality tends to a definite
limit as the dimensions of a- decrease towards zero. Consequently we
get as the limiting form of the equality,

J \dx dy dzj

= I (l$+ mrj + nQ dS + Lim ( (1$ + m-q + nl) dS ... (2).

JB <r-*0 J<T

When the volume integral is convergent, the left side of (2) is a
perfectly definite finite quantity, and hence the limit indicated on the
right-hand side must exist and be independent of the shape of o- ; it
will exist, but have a value dependent on the shape of a-, if the volume
integral is semi-convergent. In the former case it is frequently con-
venient to determine the value of the limit by taking some specially
simple shape for a-, sucli as a sphere with centre at P. Generally, if
the subject of volume-integration is, in the neighbourhood of P, of
order r"** (/x < 3), where r denotes distance from J\ the subject of the
surface integral under the limit sign is of order r~i^+\ where r equals
the radius € of the si)here o- ; also dS is «Vw, where dw is an element H
of solid angle, and so the surface integral is of order e^-*^ and tends to

16-17] AND SURFACE INTEGRALS 19

the limit zero for € -^ 0. If /x = 3, the case of possible semi-convergence,
the surface integral is of order e", so far as its dependence on c is
concerned, and therefore may have for limit a value different from
zero.

17. A generalisation, and at the same time a particular case, of
the fundamental ' surface and volume integral theorem ' is (^ot by
putting <l>$, <l>rj, <f)^, instead of $, rj, ^, where <f> is another function of
position which has finite space differential coefficients at all points in
the region. The volume integral then becomes

/{ai(«"*a^(*')^-.(«)}*

SO that the theorem takes the form

f^.(l(.„.r,.„OdS-f^.(l.py£)dr

i'^^'-'^^g)* w-

Now <^ is continuous and therefore finite throughout the whole

region of integration, but let us suppose that some or all of the

r^^ p'M f^y

functions i, v, C, ^t ^ > ^ become infinite at a point P in the
da; CI/' dz

region. If the infinities are such that both the volume integrals are

convergent, it is clear that by introducing the cavity o- and making it

tend to zero dimensions we set the relation

I </) . (/^ + mr] + ni) dS + Lim / <t>.(l^ + mr] + nC)

JB o-»-0 Ja-

^ros-^^D* ■■■■•; «.

and that when the convergence is due to $, yj, I being infinite of lower
order than r~^ the limit of the surface integral is zero. When the
volume integrals are semi-convergent, or when ^, -q, ^ are infinite of
the same order as r~^, the limit of the surface integral may be
different from zero, possibly depending on the shape of the cavity ;
it would usually be convenient to take it in the ultimately equivalent
form

(f>p . Lim I (li + mr] + nC) dS (5),

where <t>p signifies the value of ^ at the point P.

2—2

20 green's theorem [iv

18. Green's Theorem is got from the ' surface and volume integral
theorem ' by putting

^^^a^' ''^^^j' ^=^8F'

where U, V are functions of position ; if we use the notation — for
differentiation along the outward normal, and A for Laplace's operator

(T' a^ 32

whence

= jv~diSl~jv^UdT (6)

by symmetry. These two equalities constitute Green's Theorem. The
statement of the theorem requires modification when the region of
integration is multiply connected and either ZZ or Fis many- valued ;
this modification is discussed in Maxwell's Electricity and in Lamb's
Hydrodynamics, and need not be entered upon here.

If one of the functions, say U, together with all of its differential
coefficients which occur in the formula (G), is finite throughout the
region, and if V becomes infinite at a point P in the region, we
isolate P in a cavity or and make the dimensions of o- tend to zero. If I
the volume integrals converge, we get a pair of equalities precisely
similar to the above, save that to the first member we must add

• -V T r /■ 3 TT

Lim I U -:r-dS, and to the third member Lim I V-^r-dS.

<7-..0 J<r O" o-*0 J<T ^v

Generally the convergence of all the volume integrals would involve;
that V should become infinite at P of an order lower than r~^, since
when F" is a function of position relative to P, as is frequently the
case, a space differentiation adds one to the order of the infinity so
tliat A ]' is of an order higher by r~- than V. In this case both thei
integrals over the surface o- are of the order of a positive power of thei
small length r, and their limits are zero.

Tlie case of F=r~' is one of special interest in physical applicar
tions ; it is also interesting mathematically because it is just the*
case in which semi-convergence is to be looked for. There is noi
semi-convergence however, for /6^A (/•-') c?t is absolutely zero, sincai

18-19] THE DIFFERENTIATION OF VOLUME INTEGRALS 21

A (?-~^) = 0 at all points between o- and the outer boundary, and the other
volume integrals are convergent because the subjects of integration are
infinite of lower order than r-^ ; and if all the terms but one of an
equality are definite or convergent, that one cannot be semi-convergent.
Clearly, since dS is comparable with t'^dw, where dm is an element of

solid angle, / i'~^y~ ^'^ ^^^ ^ ^^^^ \\ni\t ; but I Z7— (r-^)c?>S^(when o-

is a sphere of radius e, which, in the absence of semi-convergence,
may be assumed without loss of generality) has the same limit as

Up \J^{r-')r'dm or - ^^^£g^(Or^^u>,

that is Up j dot or 4frrUp.

Thus Green's Theorem in this case gives the equalities

= jr-^^-§dS-jr-'i^[rdr (7).

Almost identical reasoning applies to the case in which V satisfies
AF=0 at all points of the region other than P, and becomes infinite
at P in such a way that Lim (rV) = 3I, where iHf is a definite constant.

The theorem becomes

ju'-^dS.,.MU^--f.(-^'^)dr

. = jV^-^dS-jv^Udr.

Another case of Green's Theorem much used in Physics is that in
which U= V. The two equalities reduce to the single one

fv>-IdS-fr^Vdr = f^(^^Jdr (8);

no particular interest attaches to an examination of possible modifica-
tions in this formula when V has an infinity at a point in the region
of integration.

V. The differentiation of volume integrals.

19. "We shall next discuss the possibility of differentiating a
volume integral with respect to a parameter which occurs in the subject
of integration but does not affect the boundary of the region of

22 THE DIFFERENTIATION OF VOLUME INTEGRALS [V

integration. Tlie only parameters that need be considered here are
the coordinates of a point P at which the subject of integration / has
an infinity. We shall call these coordinates ^, -q, i, keeping .r, y, z to
denote the coordinates of the element dr of integration. The question
to be settled is whether the integral has a differential coefficient with
respect to ^, and if so whether that differential coefficient is e<iual to
the integral of ri/jd^. Integration means passage to a limit, and so also
does differentiation ; we have to find out whether alteration of the
order of the two passages to limit alters the value of the final result.
When P is in the region of integration there is in each case the
additional passage to limit corresponding to the closing of the cavity,
and the question to be settled is whether by first integrating, second
closing the cavities, third making a ^-»-0, (where a ^ is an increment
of ^), we get the same result as by first making a ^ ^ 0, second in-
tegrating, third closing the cavity.

20. First we shall consider the case in which P is outside the
region of integration, and shew that if /has at all points of the region
and for all contemplated values of ^ a differential coefficient with
respect to ^, which differential coefficient is a uniformly continuous*
function of ^ throughout the region, the differential coefficient of the
integral is the same as the integral of the differential coefficient.

The incremental ratio of the integral is

which, by the theorem of mean value,

wliere « =/' (^ + ^ a ^) -/' {i\ and 1 > ^ > 0.

Now the uniform continuity of/' {t) implies that for an arbitrarily
chosen small quantity o- we can always find a quantity w such that for
all values of a ^ less than w,

l/'(^+ At')-/'(^)|,

and consequently also | « i, is less than a, for all points in the region T of
integration. Thus by choosing a ^ less than w we can ensure that | jtdr \
shall be less than (tT, which, in virtue of the finiteness of T and the
arbitrariness of tr, is arbitrarily small. In fact the difference between the

• It can be proved that if a function is continuous at all points in a region it is |
uniformly continuous throughout the region.

19-21] THE DIFFERENTIATION OF VOLUME INTEGRALS 23

integral of /' (<) and the incremental ratio of the integral of f{t) can
be made arbitrarily small. Hence the limit of the incremental ratio, i.e.

^ l/(^) dr, is equal to //' (^) dr, as we set out to prove.

21. Next we consider the case in which P is within the region of
integration. The rough rule for this case is that if the original integral
is convergent, and if the integral obtained by differentiating under the
sign of integration is convergent, the latter is the differential coefficient
of the former.

It will serve our purpose to prove this proposition for a particular
case, namely that in which the subject of integration /is the product
of two factors, each subject to special restrictions. One of these, which
we denote by p (^, y, z) or briefly by p, is supposed to be a function of
absolute position, not involving ^, 17, t, at all ; it is assumed to be finite,
not to vanish at P, and to have space differential coefficients which are
uniformly continuous throughout the region of integration. The other
factor is supposed to be a function of position relative to P, and to
become infinite at P; we may denote it by fj^ix- ^, y-rj, z-C), or
sometimes for brevity by ^ (i, x) or {i). It has obviously the
property that ^ (^ + a ^, ,r) ^ (f> (|, .v - a t), so that

dcf> _ d(l>

The integral to be differentiated is I p.<f>. dr, or, written in full,

Lim I p.t^.dr, where € is a cavity surrounding the point P. The
incremental ratio is

-—. Lim / p (a\ y,z).4>{i+ a ^) ^t - Lim I p {x, y,z).4> (^) dr ,

A^Le'^OJe' e-*0./€ -J

where «' is a cavity surrounding the point P' (^ + a ^, 17, C), which may
be taken to be in all respects similar to c To get the differential
coefficient of the integral it is necessary, in the incremental ratio, first
to make e and «' tend to vanishing, and afterwards to make a ^ -^ 0.

Before we pass to either limit, however, we are at liberty to simplify
the form of the incremental ratio in any manner that seems desirable.

In the integral 1 p (x, y,z) .<^{i + ^C> dr imagine the boundaries of

the region of integration, and every volume element, shifted a distance
A ^ in the negative direction of the axis of x. An element of volume
originally at a point K' is merely shifted to a point ^ whose position

24 THE DIFFERENTIATION' OF VOLUME INTEGRALS [v

relative to the point ($, v, 0 is the same as that of K' relative to
(^ + A ^, rj, 0 ; so that if K is (x, ij, c), the value of <f> belonging to the
shifted volume element, originally presenting itself as fj>{x+ Ai,$+ a^),
is equally well represented by the form <f> (.r, i) ; but the value of p
for the element now brought to K is that appropriate to K', i.e.
p{x + Ai,ij, z). The inner boundary e of the integral is brought by
the shifting into coincidence with e, but the outer boundary is changed
to a surface T which is simply T displaced without change of form.
In fact

"T fT'

/ p(.r).«^(^+A^,.r)(/T = j p(.r + A^).<^(|, ^)c?T,

80 that the incremental ratio of the integral with as yet unclosed
cavity is equal to

i^[[%(^+AO.<^(0^r-^%(.r).<^(^)^r],

or to

where the latter integral is extended to the region between T and 7",
being taken positively where the boundary of J" lies outside T,
negatively where the reverse is the case. By the theorem of mean
value the above expression equals

j%J{a^ + eA$,y,z).<t>(^)dr^-^-J%{:i'+Ai).($)dr

rT fT I rT

= I Px {X, y,z). (i) dr + 1^ o)<^(|) ^T + 7| jy p (.r + A ^) . (/) {i) dr,

where o» ^p^ {x + 6 a ^, y, z) - pj (.r, i/, z), and 1 > ^ > 0.

Now as P is not on the boundary of 7^, a ^ can always be taken
small enough to prevent P from being in the region between Tand T' ;
hence the integral for this region has no dependence on" the cavity e.
The assumed uniform continuity (which includes finiteness) of pj, and
the assumed convergence of the original integral, are sufficient guaran-
tees of the convergence of the integrals of pJ <{> and w^. Hence we
may proceed to the limits for €-*0 and c'^- 0, and so get the relation :

Incremental ratio of / p<f)dT

fT fT I fT'

= j Px<f>d-r+j tocfidT+ ^ j p{d'-i-A$).if,(i)dT...{9),

and the cavities «, c' are now closed up and finished with.

21] THE DIFFERENTIATION OF VOLUME INTEGRALS 25

As A $ becomes smaller it is clear that the volume between T and
T' approximates to a very thin shell over the surface of T whose normal
thickness (outwards from T) is — a$. I, where / is the x cosine of the
outward normal. Thus the corresponding volume integral approximates
to and has the same limit as the surface integral

And in virtue of the uniform continuity of pj, corresponding to any
arbitrary small quantity o-, we can always find a quantity k such that
for all values of a ^ less than k, and for all points of the region of inte-
gration, I 0) I < cr, and so

I a)^C?T < O" / \<l>\dT.
This last expression is cr multiplied by a finite quantity, for, since p is
finite and does not vanish at P, the convergence of / | <^ | c?t is implied

in the assumed convergence of I p<l>dT, at least if the latter be

convergence of the kind discussed in § 13. Hence I w^o?t can, by

suitable choice of a $, be made smaller than any arbitrary small
quantity, and so tends to the limit zero for a ^ -* 0. Proceeding now
to the limit a ^-^ 0 in relation (9), we get

^j\dT=j\j<t>dr-jjpcl.dS (10).

The theorem of § 17, formula (4), enables us to transform the right-
hand side of (10), giving

^j p(f>dT = -j pxdT

= f^{p)dr (11),

provided the last integral is convergent.

If the function p is of such a simple character near P that the
nature of the integral depends entirely on the form of ^, it is clear
that the case of possible semi-convergence is covered by the above
reasoning, certainly as far as formula (10); but in formula (11) the
use of formula (4) may introduce an extra term on the right-hand

26 APPLICATIONS TO POTENTIAL THEORY [vi

side, namely the limit of the surface integral of Ipcf) over a new cavity
round F. One can imagine that peculiarities in the form of p might
invalidate some of the steps of the argument, but such peculiarities are
not to be expected in physical applications.

21 a. The more general problem of the differentiation of a volume
integral with respect to a parameter 0 which affects the boundary of
the region of integration as well as the subject of integration may be
mentioned here. The formula for the differentiation is

where the region of volume integration is bounded by the surface
J^(.r, y, z, 0)-0, dS is an element of area on this surface, and/' is
the first derived function of/. When it is assumed that the stibjects
of both integrations on the right-hand side are uniformly continuous
in their respective regions of integration the proof presents no difficulty
and may be left to the reader. Infinities of/ would require special
investigation and might introduce exceptions to the formula.

VI. Applications to Potential Theory.

22. The potential at a point P {$, rj, Q of a finite mass of
continuous matter, whose density at a point {x, y, z) is p, a function
of X, y, z but not of ^, >/, ^, is the volume integral V = jpr'^dr, where
r = Vs (x — $y^ ; the attraction component parallel to the axis of .v is Jl
where JC= J (-a* - i) r'-^dr ; both integrals are taken through the whole
space occupied by the body.

When (i, r), ^) is outside the body, and p is finite and subject to
such restrictions as are required for the validity of the theorems proved
in the preceding Articles, there is no infinity of the subjects of inte-
gration in the region, and so

. , ^V '^v 8-F /■ /a^ a^ a^N, ,, ^ ^

21-23] APPLICATIONS TO POTENTIAL THEORY 27

23. When the point P is inside the body, the potential and the
attraction components have no longer the simple physical interpreta-
tion suggested by their names, but are defined as the limits of these
physical quantities in a vanishing cavity. And this, as we saw in
§11, implies their equivalence to the integrals represented by the
same sjTnbols as in § 22, but referring to a region including the point
of infinity P of the subjects of integration, and therefore only
intelligible when the integrals are convergent.

The subject of the potential integral, being infinite at P of the
order of r~\ the integral is convergent; and the attraction integrals,
having subjects of integration that are infinite of the order of r~-, are
also convergent. Hence we have, by § 21,

xJ-y

But if we differentiate A" with respect to i under the sign of
integration, we get an integral whose subject of integration is of the
order of r~*, so that there is a possibility of semi-convergence or
divergence. Instead, therefore, of merely quoting a simple differen-
tiation rule in this case, we must proceed with care.

It is clear that for the integral X = jp {x — C) i'~'^ dT the argument
of ^ 21 holds as far as the formula (10), which in this case becomes

%- \'^%^^-^)r-'dr- ^^k{x-i)r-^dH (12),

"the volume integrals involved being convergent and all cavities being

closed up. This formula shews that -^ has a definite value.

Now surround P by a small surface o- and use formula (4) of § 17,
putting p for the quantity there called ^, and {x - ^) r"^ for the
quantity there called L Thus we get

f Ip {x - k) r-'dS + pp Lim { l{x- $) r-?dS

whence

^f = PpUm (l(x-^)r-'dS-Um f % ^ {(.r-^)r-^}o?r ...(13).

The sum of the limits here indicated is perfectly definite and inde-
pendent of the shape of o-, but either limit taken separately has a value
which depends on the shape of o- ; this can be seen readily by studying

28 APPLICATIONS TO POTENTIAL THEORY [VI

the surface integral first wlieii o- is a sphere and second when o- is a
very flat cylinder with plane ends parallel to the plane of .t:

Since formulae corresponding to (13) hold for Y and Z, we get
by addition

dX dY dZ

tT-*-0j<r

oi drj dC, '^a-*QJ<r

Lim I p2 T- {{x - $) 7-'^] dr

Now here the subject of volume-integration is identically zero at all
points outside the cavity, and so the integral is zero whatever the
shape of tlie cavity, and its limit is zero. Hence the value of the
surface integral is independent of the shape of the cavity, and may be
calculated on the assumption that cr is the sphere r = c ; in this case
I- -(a'- i) €~\ so that 2/ (.r — ^) = - e, and dS = rdai, where dm is an
element of solid angle at P ; thus the integral becomes - jdw, which
equals - 47r. Thus our equality becomes

dX dY dZ , ,,„ ,

'W'^'B^^yr'^''^' ^^^''^'

S'V d'V dPV

which is Poisson's equation.

24. The theorems proved above for the differential coefficients of
V are perfectly intelligible for a body of the hypothetical continuous
structure which we have postulated. But when applied to a body of

molecular structure such a symbol as -^7- requires qualification. It has

been seen that for a continuous body a $ was only made to tend to zero
after we had first closed the cavities c and c' corresponding to V and
V+ A V ; in fact a ^, though small, was always large compared with
the dimensions of the cavities, but this fact did not interfere with our
making a ^ as near to zero as we pleased.

But for a body of molecular structure the cavities must always be
large enough to be capable of containing a great number of molecules,
and so we can never close them entirely ; hence a ^, so far from ever
vanishing, must be actually large compared with the smallest length
which can be regarded as only physically small ; nevertheless a $
may be, to our senses, extremely small. Hence instead of the true

differential coefficient -57- we have the incremental ratio — x- , where

K A^ '

A ^ though very small is still definitely prevented from attaining the

23-25] APPLICATIONS TO THEORY OF MAGNETISM 29

higher orders of smallness which lie on the way to the limit zero

dV
Thus it is clear that the symbol -^ , just as V itself, is inexact and

stands for something not precisely defined; but the inaccuracy or
deviation from a precise value is no greater than the inaccuracy which
regards matter as continuous, and is in fact an inaccuracy so small as
to be inappreciable to our senses. Accordingly the relations

Jl = ^T- and 2 ^r = - 'k-rrp

, have a sufficiently precise meaning when applied to bodies of molecular
structure.

VII. Applications to Theory of Magnetism.

25. If a body is magnetised so that the components of the intensity
of magnetisation at a point (ar, y, z) are A, B, C, the magnetic potential
at an external point P, (^, rj , ^), is given by

''=/(^3V^r''^.)('-)* (1*).

and the x component of magnetic force is a where

—\IMI-^^4,*'^1>-)^^ a^).

the body being regarded as of continuous structure ; so long as P is
outside the body it is clear that

«=-f ('8)-

Outside the body the induction (a, b, c) is the same as the force
(a, /?, y), and therefore remembering that A, B, C are functions of
Xy y, z but not of ^, -q, ^ while r depends only on x~i, y -r], z— C, we
see that

J\ djf dz^ dxdy dxdzj

= f-| ('')>

30 APPLICATIONS TO THEORY OF MAGNETISM [VII

where

^=/(^a^-^ el) (••"■)* (•^>-

F, G, H are the components of the vector potential at P ; the relation
between induction and vector potential is frequently written

{a, b, c) = cut\{F, G, H) (19).

26. When P is inside the magnetised body the integral of
formula (14) is convergent, and so the formula may stand as the
definition of the potential at P. Tlie properties of V, thus defined,
are most easily deduced from another expression, obtained by making
a cavity round P and applying the theorem of § 17, formula (4), taking
^ to be ?•-' and writing A, B, C for ^, 17, t,. The surface integral over
the cavity has a zero limit, and so we get

r=/^(M .»5.«C),-rf^-/'('^ .'|.|),-- A ...(20).

where the surface integral refers only to the boundary T of the body,
and the cavity is now closed and finished with. This form of the
magnetic potential exhibits it as equivalent to the gravitation potential
of a volume distribution of density

^_dA _^_B_dC

da: dy dz '
combined with a distribution of surface density lA +mB + 7iC spread
over the boundary of the body. (Of course formula (20) is equally true
when P is outside the body.)

If we suppose that P is right inside the body, i.e. not on the
boundary, there is no infinit)'^ in the subject of surface-integration;
the volume-integral part of V has the properties of the gravitation
potential studied in Section VI. Thus V has definite space difterential
coefiicients obtained by difierentiating under tlie sign of integration
in formula (20) (not in formula (14)*) ; accordingly, since

a^('-) = -|('-).

-s('I,M'^*"'"*'"^)Kc'-'- *<^*-./ U;- * si, * Was ('■"■>*■

* The theorem of § 21 does not apply to the integral of formula (14), since the
infinity at P is of the order r~2.

25-26] APPLICATIONS TO THEORY OF MAGNETISM 31

the volume integral having a perfectly definite value. Now we cut
a cavity o- round P, and apply the theorem of § 17, formula (4), and
we get

-%^=-Um \ {IA+ mB + nC) ^ {r~') dS

These two limits combined give a value which is independent of the
shape of a-, but the value of each limit taken separately depends on the
shape of cr ; the volume integral we see, by comparison with (15), to be
the a; component of the magnetic force in the cavity due to all the matter

dV
outside the cavity. So in general - -^ is not the limit of the com-
ponent of force in the cavity, but differs from it by an amount repre-
sented by the limit of the surface integral. If, however, we can choose
a shape for the cavity which shall make the limit of the surface integral
zero, the limit of the force component in the cavity will be accurately

represented by - . ^ ; and this is effected by making the cavity a

cylinder whose generators are parallel to the direction of the vector
(A, B, C) at the point F, with flat ends perpendicular to the
generators, all the linear dimensions of the cylinder tending to zero
in such fashion that the linear dimensions of the ends tend to become
vanishingly small compared with the length ; this may, for brevity, be
called a 'long' cylinder. The direction chosen for the generators
ensures that the integral of lA + mB + nC for the curved portion of
the surface tends to zero, and the relative smallness of the flat ends
makes the integral over these tend also to zero. The definition of the
magnetic force (a, p, y) at a point P in the body is ' the limit of the
force in a cavity in the form of a long cylinder with generators parallel
to the resultant intensity of magnetisation ' ; and this definition, in
connexion with the present argument, justifies the statement that

The definition of 4;he induction (a, b, c) at a point P in the body is
* the limit of the force in a cavity in the form of a very flat circular
cylinder with generators parallel to the resultant intensity of magnetisa-
tion,' where by a very flat cylinder is meant one whose linear dimensions
tend to zero in such a way that the length tends to become vanishingly
small in comparison with the linear dimensions of the plane ends. For

32 APPLICATIONS TO THEORY OF MAGNETISM [VII

such a cavity I A +mB + nC tends to zero on the curved part of the

surface, to - / over one of the plane ends, and to + / over the other,

/being the resultant intensity of magnetisation ; and each of these ends

ultimately subtends a solid angle 27r at F. Thus the three surface

integrals of which that in (21) is a type have for limits the components

of force at a point between two infinite circular* parallel planes, the

one covered with a uniform surface density /, the other with a uniform

surface density - /, of matter that attracts according to the Newtonian

law ; this force is known to be AttI perpendicular to the planes, and so

its components are 4:TrA, 4:irB, A-tC. So, for the flat cavity, (21) yields

the equality

a=-47r^ +a (22).

27. The integrals of formulae (18) representing vector potential
are convergent for a point inside the body, and may therefore stand
as the definition of the vector potential at such a point. If in the
formula (4) of § 17 we put 0 for $, -C for rj, B for t,, and r~^ for
<^, we get

F^\^{nB-mC)r-^dS-f(^f^-f^)r-^dT (23),

the cavity of formula (4) being closed and finished with, and the surface
integral over the cavity having a zero limit. This result exhibits jPas
the gravitation potential of a finite volume distribution combined with
a surface distribution ; it shews, therefore, that F has definite differ-
ential coefficients with respect to the coordinates of P.
From the two formulae analogous to (23),

Cut a cavity o- round P and apply the theorem of formula (4), § 17,
and the result is readily seen to be

* Tho word 'circular' is introduced in order to exclude cases in which the
resultant force at a point between the parallel planes is not normal to them. The
circles are supposed to have a common axis, passing through P,

26-28] SURFACE INTEGRALS 33

wherein the limits on the right-hand side together give a value
independent of the shape of a-, though the value of each separately
depends on the shape of a-. The volume integral is the same as

and accordingly represents the x component of force due to all the
magnetisation outside the cavity ; the surface integral, together with
the corresponding surface integrals in the two other formulae analogous
to (25), will tend to zero if A/l = B/m= C/n over practically the whole
surface of the cavity, and this is ultimately the case when the cavity
is the flat cylinder used in defining the induction. For this shape of
cavity (25) is equivalent to

'^-Tr"" ^''^'

which, with the two other equalities of the same type, constitutes the
vector relation

(a, h, c) = curl {F, G, H),

true now for points inside as well as for points outside the magnetised
body.

It should be noticed that the definition of vector potential used
in the present discussion is not that which is regarded as fundamental
in the physical theory, though equivalent to it. The usual definition
is contained in the relation (19) coupled with the relation

dF dG dH ^

It is easy to verify, on the lines of the present Article, that the
vector defined by the relation (18), whether for internal or for external
points, satisfies this further condition.

VIII. Surface Integrals.

28. When gravitating matter is distributed in a very thin layer,
or when the surface of a body is charged with electricity, the corre-
sponding potential and attraction at a point P are represented by surface
• integrals, a surface density o- taking the place of a volume density.
The integrals are of the type JafdS where a- is usually free from such
mathematical pecuHarities as might raise doubts concerning the
existence of the integrals, and / is a function having an infinity at the
point P ($, t], Q.

34 SURFACE INTEGRALS [VIII

So long as P is not actually in the region of integration the

integrals do not present any difficulties, and the formulae X=-^,

A F = 0, are clearly valid.

When the point P is in the surface distribution we must cut
a cavity round it of dimensions that tend to zero, and the question
of convergence necessarily arises. We consider first the case in which
the integration takes place over a portion oi o, plane surface.

The chief test of convergence is now as follows. If within a circle

of finite radius (a), having the point P as centre, the subject <fi of

integration is always less in absolute value than Mr~i^, where /x<2 and

M is a definite constant, the integral j^dS is convergent. To prove

this we shall shew that, corresponding to any arbitrarily chosen small

quantity or, there can always be found a closed curve 0 surrounding P

such tliat all closed curves t surrounding P and lying wholly inside

B have the property that

/"*
I <^dS

Take for the curve 0 the circle r - -q, where ■q<a, and denote by c the
distance from P to the nearest point of the boundary t of the cavity ;
the cavity is of course entirely inside 6, but is otherwise unrestricted
as to shape. Since the modulus of a sum is not greater than the sum

of the moduli,

re re

<l>dS ^ \<}>\dS,

^ f\<l>\diS, <MJ\-t^dS,

< 27rM r r'->^dr, < ?^ (r'>^ - e^-'^), < ^^ rf'i^.
Jt 2-/X ^ ' 2— /u,

Hence by choosing -q less than {(2-/u.)o-/27rJ/}--'* we get a curve

6 satisfying the specified condition ; the integral is accordingly

convergent.

When the order of the infinity of ^ is the same as that of r'-,
semi-convergence may appear.

29. Passing to the case in which the region of integration is a
portion of a curved surface, we shall assume P to be a point at which
there is a definite tangent plane and such that at all points of the
region within a finite distance of P the principal curvatures are both
finite. We need consider only the integral taken through a finite

28-30] SURFACE INTEGRALS

region not extending far from P, and in virtue of the finiteness of the

curvatures at and near P we can always choose this region so that,

if Q is any point of it and 6 the inclination of the tangent plane at Q

to the tangent plane at P, for all positions of Q in the region 6<a,

where a is a definite acute angle. Let the projection of Q on the tangent

plane at P be $„, let r, u represent PQ, PQ, respectively, dS an

element of area round Q, dSo the projection of dS on the tangent plane

at P, B the boundary of the area of integration, and B^ its projection

on the tangent plane at P.

Since d8 = dSo sec 0,

'^ ... f^o

<j>secOdSo,

/ d8=\

the second integral being taken in the tangent plane at P.
If within the region of integration

where M is a constant and 2 > /* > 0, then
I ^ sec ^ I < Mr'!^ sec 0

< Mt'o-i^ (ro/t-y sec 6,
or, since r,, < r, and sec 0 < sec a,

I </) sec ^ I < 31 sec aro'f^,
where Msec a is finite since a is acute.

Hence I <ji sec OdS^ is convergent, and therefore so also is

ffidS ; thus the test of convergence is the same whether the surface

of integration be plane or curved provided the curvatures be finite.
The existence of a definite tangent plane at P is not a necessary
feature in the proof, the essential thing is that there shall be a finite
region round P for which 0 <a<^Tr, 6 being inclination to some fixed
plane through P ; for example the surface might be a cone and P its
vertex. (Compare Poincard, Potentiel Newtonien, § 33.)

30. Applying the test of the preceding Articles we see that at
a point in a surface distribution of gravitating matter or electricity the
potential is represented by a convergent integral, but the attraction
components in the tangent plane are represented by integrals whose
order renders semi-convergence possible. It is not difficult to shew,
by a particular example,, that semi-convergence does occur; for the
attraction of a uniform plane elliptic disc (of eccentricity e and surface
density a) at a focus is 27ro-(l - J I - e')/e if the cavity is circular, but

3—2

36 SURFACE INTEGRALS [VIII

is zero if the cavity is an ellipse similar and similarly situated to the
edge of the disc, with the focns for centre of similitude ; the verification
of these statements, by using polar coordinates and integrating, is
quite easy.

The component of attraction at P normal to the surface is
represented by a convergent integral, but this quantity is the attraction
in a cavity, though a vanishing one, and must be distinguished from
the normal component of attraction at a point very close to the
unbroken surface but not in it ; it is, in electrical applications, the
•mechanical force per unit charge,' the quantity denoted by B.2 in
Prof. Sir J. J. Thomson's Elements of Electricity and Magnetism, § 37,
whereas the normal attraction at a point just not in the surface is the
quantity there denoted by R.

31. The distinction drawn above, between the attraction at a
point in the surface and that at a point just not in the surface, brings
us to a question of a kind which, for lack of a fourth dimension,
does not arise geometrically in the case of volume integrals, the
question, namely, whether an integral jdS tends to a definite limit
if the point P, where <^ has an infinity, is not originally in the surface,
but approaches a point of the surface as a limiting position.

Let 0 be the point of the surface to which P gets continually
nearer ; it will be convenient to take 0 as origin of coordinates and the
tangent plane at 0 as plane oi z; we shall suppose that there is a
limiting position of the line PO, as P moves up to coincidence with 0,
which makes with the plane of ;2 a definite angle different from zero
and so has definite direction cosines Iq, m^, Wo, of which the last is
numerically greater than zero. The length PO will be denoted by
K, and the coordinates of P by (I, r}, Q or (- Ik, - niK, - hk), while
a, y, z represent the coordinates of a variable point Q on the surface ;
the subject of integration, <f>(a; y, z, i, rj, 0, may for brevity be
represented by <^, while <f> (a; y, z, 0, 0, 0), the value of <f> when P
is coincident with 0, will be represented by ^o. If integration be
extended to a finite part of the surface round 0, bounded by a closed
curve B, the quantities to whose different meanings and possibly
different values it is desired to draw attention are respectively

I ft>ud>S and Lim I cfydS.

J K^OJ

The first thing to notice is that, while the integral of (f) recjuires no
cavity so long as k is different from zero, which is the case at all stages

30-31] SURFACE INTEGRALS 37

in the passage to limit denoted by k ^ 0, the integral of <^o is only
intelligible in terms of a cavity e round the point 0, though this cavity
of course tends to vanishing. If, therefore, we set out to find the
algebraic difference between the two quantities which form the subject
of discussion (which may conveniently be denoted by D) we have

i> = Lim f (j>dS- I <f>odS

rB rB

= Lim I (jidS-lAm (f^dS.

Since the term involving the limit for k -* 0 in no way depends upon
c, and the term involving the Hmit for c -* 0 in no way depends upon
K, it is a matter of indifference in what order we suppose the passages
to limit to be made ; accordingly we are at liberty, if we please, to
make first the passage to the limit for k -* 0, so that at any stage short
of the limits we shall think of*: as extremely small compared with the
linear dimensions of the cavity e. The difference, then, before passage
to either limit, may be put in the form

j' cf>dS+ j c{>dS- j <l>,dS.

Now if we proceed first to the limit for k -* 0, the points F and 0 at
all stages of this passage are quite outside the region of integration of
the last two integrals, and the functions <^ and ^o are kept definitely
removed from their infinite values ; hence in the absence of peculiarities
of <f> other than that infinity at P which is the special subject of our

investigation, we get the same limit for I (f>dS whether we first

integrate and then make k -^ 0 or first make k -* 0 and then integrate.
In fact

whence

.B fB rB

Lim I <l>dS= I Lim <f)dS= I <t>odS ;

K-*-0.'e Je ic-*.0 /e

i> = LimrLim r<f>dS + Lim f 4>dS-\ ods\

= LimLim 1 <l>dS (27),

e-»-0 K^oJ

the notation implying that e is kept constant while k ^ 0, thus yielding
a limit which is a function of c, and that afterwards the limit of this
function of e is taken for e ^ 0.

38 SURFACE INTEGRALS [VIII

Let us suppose ^ to be of the form (z - CY r~i^ where r denotes PQ
and A and /x are positive, and let us proceed to make a closer examina-
tion of D for this particular case. We shall assume that the principal
curvatures of the surface are finite at all points in a finite region round
0, and that the cavity e is determined by the intersection of the surface
with the narrow cylinder x^ -^y^ = ^\ and we picture to ourselves a
small piece of the surface, which we may call the 'cap,' bounded by
this curve whose projection on the plane of z is a circle of radius c and
centre 0, and a point P at a distance k from 0 which is extremely small
compared with €. To begin with, we observe that there is a finite
constant a such that for all points Q in the cap \z\< as^, where s stands
for Ja^ + y-, the distance of Q from the axis of z ; for, since the surface
has finite curvature at all points of the cap, | z \/^ tends to a finite limit
for any given azimuth as Q approaches 0, and so is finite at all points
of the cap ; and the various values, being finite, have a finite superior
limit a which is of the same order of magnitude as the greatest
curvature of a normal section through 0 ; thus the inequality is
proved.

Consider now the curve of intersection of the surface with the
cylinder or + y- ~ k- ; this divides the cap into two regions, an inner
region whose linear dimensions are of the order of k and therefore
small in comparison with those of the cap, and an outer region
comprising most of the cap, in which there is no point whose distance
from 0 is of a higher order of smallness than k. The integral whose
limit is D can be regarded as the sum of two integrals, one over the
inner region, one over the outer region, and these will be considered
separately.

Taking first the outer region, and remembering the assumption
Wo + 0, we see that, while there may be points in this region for which
r<s, there are no points in it for which s/r becomes infinite, so that
there is a finite superior limit /3 for the values of s/r in the region ;
the finiteness of the curvature is a guarantee that there is a finite
superior limit c (differing from unity by a quantity of the order of e^)
to the secant of the inclination to the axis of z of the normal to the
surface at Q, i.e. the ratio of f/>S' to its projection dS^, on the plane of z ;
and \i\Js is finite at all points of the region and has therefore a finite
stiperior limit y', so that | C| < y's ; as | c | < cur, it follows that

\z — ^\s~^ <y' + as<y
where y is a finite quantity.

31] SURFACE INTEGRALS 39

Thus in the outer region

\ ^\ = \(z — (Yr'f^l < y'^s^ files' f^,
dS <cdSo,

and so

UdS\<y^/3'^GJs^-'^d8„

< 27ryA /Si^C I S^ - f" + 1 ds,

J K

< 27r//3'^C (X - /x + 2)-i [e^-M+2_KA-M+2],

The Kmit of this for k ^^ 0 and c -^ 0 is zero provided )u, — X < 2.

Taking next the inner region, we know that in it | c | < ar < aK-,
while C is of the same order of smalhiess as k, so that \z — t,\ is of the
same order as k, and there must be an inequality ! « - C| <gK where g is
a definite constant. Clearly there is a superior limit to the secant of
the angle between the normal at Q and the line PQ, so that there is an
inequality dS <h)"dui, where doi represents an element of solid angle at
P ; and r always bears to k a finite ratio, so that there is a double
inequality pK > r > (jk, where p and q are constants. Hence in the
inner region

Us- 0^ r-i'dS < g^q-i^pViK^->^+'^ ( dto,

and the limit of this, for k -^ 0, is zero provided /x — X < 2.

Thus I) = 0 for ^ = (0 - Cy r~>^ subject to /u, - X < 2, and it is an
immediate inference that i) = 0 for (l) = a-(z- 4)^ r-i^ subject to the
same condition, if a- is a function of a', y, z which is finite throughout
the region of integration. For this type of integral the condition for the
vanishing of D is not the same as the condition for the convergence of
the integral of </)o, for, in the neighbourhood of 0, z becomes small of
the order of r^^, so that the condition of convergence of ^0 is

/>t-2X<2.

Powers of .r - ^, y-r} might appear in ^ ; they would count as the
corresponding powers of r in applying the test just proved, though of
course they would not be equivalent to powers of r if one were
examining a case where something analogous to semi-convergence
seemed probable.

For the potential integral X = 0, /a = 1, and therefore D = 0. Thus
the potential at 0 is the same as the limit of the potential at P as
P approaches 0, so that V is not discontinuous at points on the
surface.

40 SURFACE INTEGRALS [VIII

32. For the attraction components /x - X - 2, and the test of the
previous Article is not applicable so that a special investigation is
required. Let us consider first a tangential component, say that whose
subject of integration is <r (.r - $) ;■"''.

The corresponding integi-al of <^o is semi-convergent, and it may
appear useless to investigate the difference Z> between the unknown
limit of the integral of <^ and the quantity of uncertain value which is
the integral of </>,,. But the integral of <t> is quite independent of the
shape of the cavity, in fact it does not require any cavity, so we are as
free as in the case of absolute convergence to give to the cavity any
shape we please, so long as we attach to the integral of ^o the value
associated with that particular shape ; thus the semi-convergence does
not introduce any uncertainty into the meaning of Z>.

Let us take the same cavity as in the preceding Article, using the
same notation and applying whatever parts of the reasoning remain
valid for the changed form of (ft. And let us consider what error is
introduced into the subject of integration if we replace o- by o-q, its
value at 0, dS by its projection dS,> on the plane of z, and r by r' the
distance from F to the projection of Q on the plane of z. The error
due to the change in o- corresponds to the omission of a factor which
(lifters from unity b)'" a quantity of the order of smallness of s, if we
assume the function o- to have no troublesome peculiarities* at 0 ; the
error due to the change in dS corresponds to the omission of a factor
differing from unity by a quantity of the order of sr ; and, since
\r — r'\<\z\< as", the error due to the change in r corresponds to
the omission of a factor differing from unity by a quantity of the
order sV~\ So the most important terms representing error in the
integral over the cap are of the order of the integrals over the cap of
(x — $) ?•"*« and (x — $) r~*s'' ; for these error integrals X = 0, /x = 1, s in
the numerator counting as equivalent to r since there is a finite
superior limit to sr~^, and so, by the previous Article, the error has a
zero limit for k-^0 and « ^^ 0.

Hence, for the .r component of attraction,

D - Lim Lim I o-„ (.r - $) 7-'~^dSo,

where it is clear that the integral is now taken over a circular area of

* If further precision be desired, we may assume j o- - o-q ] < i^/s'", where 3[ is
finite and in positive. The error corresponding to this is less than the integral of
Mn'" (.r - f) r-'', for which X = 0, ^ = 2 - wi, and the limit of the error is zero. In the
text m is taken to be unity.

32-33] SURFACE INTEGRALS 41

radius € in the plane of z, and represents the component of attraction
at P of a circular disc of uniform surface density o-„. Now, before
passage to the limit, P is not in such a position of symmetry that the
a; attraction component must vanish, but if we describe the reflexion
with respect to the plane x = ^ of the lesser of the two arcs into which
this plane divides the circumference of the disc, we obtain a division
of the disc into two areas the greater of which, on account of the
symmetry of the position of P with respect to it, contributes nothing
to the attraction component. The component is therefore that due to
the crescent- shaped smaller area, whose mass is very nearly 4^€o-o and
whose nearest point is at a distance from P comparable with c ; thus the
component of attraction is of the same order of magnitude as o-„|e~',
which has a zero limit if we make « ^- 0 (i.e. ^ -* 0) before e ^- 0. Thus
D is zero, so that the x component of attraction of the whole surface
at P tends to a limit, as P approaches 0, equal to the corresponding
component of attraction at 0 reckoned for a vanishing circular cavity
with 0 as centre; the limit is the same fi"om whichever side of the
surface P moves up to 0.

33. In the case of the normal attraction component Z, the subject
of integration is <t{z — t,) '*~^ and the corresponding component at 0
is represented by a convergent integral. Thus

D - Lim Lim f o- {z - V) i-' dS.

The most important terms of the error introduced into the integral
of this formula by putting a-^ for a, r for r, and dS^, for dS, are of the
same order of magnitude as

\,{z-Osr-'dS* or C a, (z - Q ^7-' dS ;

for both of these A. = l, /a = 2, and therefore, by § 31, the error has
a zero limit. Accordingly D is the limit of

j^cr,zr-'dSo-j^cT,tr'-'d.%;

and in the former of these we notice that z is of the order of smallness
of s", and therefore the integral of the same order as 1 o-gS-r'-^dS^, which
has A. = 0, /A= 1, and therefore the limit zero. Thus

D = ~ Lim Lim o-^ I ^r'-^dS^,

* If we make the same assumption with regard to a as in the footnote of § 32
the index of s will be m in this integral.

42 SURFACE INTEGRALS [VIII

the integral being now taken over the plane circular area bounded by
r - €, c = 0. Now if c/w represent the solid angle subtended by dH^^ at P,
\t\r'~^d8^-d(ii, and the integral is ±/c?w, the sign being positive if t
is positive, negative if ^ is negative. If we make /f-^0 before t-^0,
clearly the limit of /c?w is 27r, and so we get

the upper sign corresponding to t positive. So the limit of Z diifers
from the value of Z a\ 0 by 2-0-^,, the excess of the former over the
latter corresponding to an attraction 2Tr(Tg towards the surface; the
difference between the limits of ^ as P approaches 0 from different
sides of the surfiice is ina^,, and the arithmetic mean of these limits is
the value of Z at 0.

34. The potential at F of a double sheet, or normally magnetised
shell, of strength /j.' at the point Q, is given by

V = jix'r~^cos\f/dS,

where ip is the angle between QP and the normal at Q drawn in the
sense for which /x' is reckoned positive. The error introduced into the
subject of integration by taking \f/ at points near 0 to mean the angle
between QP and the axis of z, and so replacing cos ij/ by —(z-C) r~^,
corresponds to dropping a factor which differs from unity by a quantity
of the order of s^, and the integral of this error taken over the cap is
one for which, in the notation of§31,X=l,/A=l, and therefore has
a zero limit. Hence the potential of a double sheet has, for purpose
of finding D, the same form as the integral investigated in the
preceding Article ; and the limit of V as P approaches 0 from the
I)ositive side of the sheet exceeds by iTr/x^' the limit as P approaches
0 from the negative side.

35. The potential of a surface distribution of gravitating matter,
whose surface density is free from such peculiarities as would render
invalid the properties already established, has in a certain sense a
space differential coefficient in any direction at any point 0 of the
surface ; this is not a differential coefficient as generally defined, since
it is a limit which has different values according as the consecutive
point /-' approaches 0 from one side of the surface or from the other.
It is to be noticed that the existence of a differential coefhcient cannot
be inferred from the physical property that force equals gradient of
potential, since 0 is a point not in free space, but in the gravitating
matter.

33-35] SURFACE INTEGRALS 43

The theorem is that

Lim {{Vo- Vp)/OP} = Lim Fp ,

OP-s-O OP-*0

where Fp is the component of the force at P resolved along the tangent
to the path by which P approaches 0.

To prove this we must shew that if rj be any arbitrary small
quantity we can always choose a point K on the curve by which P
approaches 0 such that for all positions of P on the curve between K
and 0

Vo-Vp

-Fo

<v,

where Fo represents Lim Fp .

OP-*0

Let us regard r] as the sum of three arbitrary parts >7i, r}^, and r/j.
We take a point / on the curve between P and 0, and notice that

Vo-Vp „ Vo-Vj , Vj-Vp JP_j^ .
~Qp~-J^o- Qp + jp OP ""'

and, remembering that

Vj-Vp^j^ds,

where F is the tangential force and ds an element of the curve, we
apply the first theorem of mean value and so get

Vj-Vp = JP.Fq,

where Q is some point on the curve between / and P.

Thus

Vo-Vp „ Vo-Vj , JP „ J,
-QP -^^=—OP~^OP^^''~^'■

^mcQ F is definite at all points between 0 and P, and has a
definite limit for a point tending to coincidence with 0, there is a
definite superior limit to the absolute value of F for the points of the
curve lying between 0 and any definite point K, we call this superior
limit M.

Now we choose K so near to 0 that, for all points P between K
and 0,\Fp- Fo\<y]i and therefore also \Fq- Fo\<->ii> this we can do
because Fp has the limit Fo •

"We next take P anywhere on OK, and P having been chosen, we
can choose a point L^ so that for all points / between Zo and 0,
\yo-Vj\<OP .-q.,, this being possible because Vj has the limit Vo-

44 VOLUME INTEGRALS THROUGH REGIONS [iX

And a point L^ can be chosen so that for all points J between Z;;

and 0

JP ,

where | i\<r)JM. We now take J to be between 0 and the nearer of
the points Z.., L,,.
Thus

Vq-Vj , JP j,y ET ^o-Vj , . j^ „

= [^']^[^V^<-].[.3/.f

where we notice that \Fq\<3L The modulus of the first expression
in square brackets is less than ■q.y, that of the second is less than rji,
that of the third is less than r/j; hence the modulus of the sum of the
three expressions is less than r]i + rj.y + r/3 or rj. Thus we have been able
to choose A" so that for all points P on the curve between 0 and K

which establishes the theorem.

Vo-Vp

<V,

IX. Volume Integrals through regions that extend
to infinity.

36. The integrals to which we have so far been devoting most
attention are those whose peculiarity consists in the subject of
integration becoming infinite at a point in the range. Another kind
of integral requiring special study occurs frequently in mathematical
physics, namely, a volume integral taken through a region which
extends to infinity.

By the integral J/dr taken through all space outside certain finite
closed surfaces Si, S^, etc. is meant the limit of the integral taken
through a region bounded internally by Si, S.,, etc., and externally by
a surface B, as the linear dimensions of B and the distances of all its
points from the inner boundaries become indefinitely gi-eat, provided such
a limit exists and is indei)endent of the shape of B. When the limit
exists and is independent of the shape of B, the integral is said to be
convergent ; if the limit has a finite value which is not independent of
tlie shape of B, the integral is said to be semi-convergent.

35-37] THAT EXTEND TO INFINITY 45

The following is the chief test of convergence. If we measure r
from some fixed origin, and if f is such that, fw all values of r greater
than a definite length a, f is less in absolute value than Mr->^, where M
is a constant and fi>3, the integral is convergent. We shall prove this
by shewing that, corresponding to any arbitrarily chosen small quantity
0-, there can always be found a closed surface 6 surrounding 0 and all
the surfaces ^*i, /So, etc., such that all closed surfaces t surrounding 0
have the property that

I/:

t I

fdr < 0-.

Take for the surface 0 a sphere r^r) large enough to surround the
sphere r = a and all the inner boundaries of the region; and let w be'
the distance from 0 to the furthest point of the outer boundary t.
Then

ffdr ^f\f\dr, ^Hfldr,

JB JQ JO

the upper limit in the last integral being the sphere r = u). Thus

f fdr < M rr-i^dT, <iirM ri^'-'^dr,
Je Je Jt,

AttM

< (r] f*^-^) - w-('^"^)), a positive quantity,

/A O

1
Hence by choosing rj greater than {47rJ//(/x-3)o-}'^-^ we get a surface
6 satisfying the specified condition; the integral is accordingly con-
vergent. It will be noticed that there is no restriction on the shape of
the outer boundary t.

Generally speaking, if/ is zero at infinity of an order higher than
r~^, the integral is convergent; if the zero is just of the order r~^, the
integral may be semi-convergent or divergent.

37. When Green's theorem and allied theorems are applied to
volume integrals of this type, the outer boundary which tends to
become infinitely large must not be left out of account, and so we
have limits of surface integrals which are spoken of as integrals over
the surface infinity. If the subject i/' of integration is, for values of
r greater than a finite length a, less in absolute value than J//-~^
where M is finite and /x>2, then |/i/^c?>S^| taken over the sphere r = u>
is less than J/w--'^ // sin 6 dO c?</), which has the limit zero for w -* x .

46 VOLUME INTEGRALS THROUGH REGIONS [iX

Thus the surface integral vanishes if i/^ is zero at infinity of a higher
order than ;•--'. If i/' is zero of tlie order r'- the limit of the surface
■ integral may be different for different shapes of -6; if this is the case
there is of course corresponding semi-convergence of one of the volume
integrals, and special investigation is required.

38. The differentiation with respect to a parameter of a volume
integral through a region extending to infinity, involving as it does
two distinct passages to limits, requires special consideration. Let
us consider the case in which the parameter ^ affects the subject of
integration, but does not affect the specification of the inner boundaries
^S*!, /Si, etc. Let us suppose that y/3| (or/') exists and is uniformly
continuous through all finite portions of the region of integration for
all values of ^ considered, and that the integral of / is convergent ;
and further that, for all values of I considered and for all values of r
greater than a, there is an inequality \f' \<Mr~^, where /'t>3, and
M, fi, and a are constants whose values do not depend on the value
of i*. Take the outer boundary to be the sphere r = w ; then

= Lim Lim f-^. r{/(^+ a$) -f($)}dT ^ fy (^)dr]...{2S),
and this, by the theorem of mean value,

= Lim Lim / edr,

Af-*-0 (a-*-<X: J

where e =/'(! + ^ a ^) -/'(^) and 1>^>0, and the notation implies
that first to ^- Qo and afterwards a f ^- 0. If we can shew that the
subject of this double limit can, by first making <d -»- oo , and afterwards
taking a $ sufficiently small, be made less than any arbitrarily assigned
small quantity <r, clearly the double limit will be zero.

Now since/' satisfies the conditions of the theorem of § 36, and
moreover in such a way that 3f, /x, and a are independent of $, it is
clear by the reasoning of that Article that, for all values of $ considered
and therefore in particular for all possible values of ^ + 6 a ^, we can
choose a definite length 77 such that for all values of o> greater than r?

//'(^ + ^A^)rfT and f7'(^)

* We cau get greater generality by simply requiring that the integral of/' shall
be uniformly convergent for the contemplated range of values of f ; but it seems
better not to introduce into the text the idea of uniform convergence, especially as
there are additional difficulties in the proof.

37-40] THAT EXTEND TO INFINITY 47

are both less than any assigned small quantity, which we shall take to
be io". Hence

Lim

r <icr

<^o-.

and Lim | I f'(i)dr

to -*■ X .' 7J

And V being chosen and therefore finite, however large, the uniform
continuity of /' ensures our being able to choose a value of a | such
that for it and for all smaller values 1 € | is less than an arbitrary small
quantity; this small quantity we choose to be l(rT~^, where T is the

<V. Thus

finite volume I dr. This makes 1 I cdi

fc^rULiml rf\^+eAi)dr-fy'(^)dT+(\dr

Lim

<o-.

Hence the double limit on the right-hand side of (28) is zero, and
therefore the differentiation of the integral of/ is effected by the rule
of differentiating under the sign of integration.

Differentiation with respect to a parameter which affects only the
specification of one of the inner boundaries, say >S'i, clearly gives rise
merely to a surface integi-al over Si.

39. Volume integrals through regions extending to infinity occur in
electrical theory as expressions for electrostatic and for electrodynamic
energy, and in other ways. They occur in the theory of gravitation
and electrostatic potential in proofs of the important 'theorems of
uniqueness.' They occur in Hydrodynamics as representing kinetic
energy, and 'impluse.' Differentiation of such integrals is employed
in the dynamical theory of solid bodies moving through an infinitely
extended liquid. In every such application of these integrals it is
necessary to make sure that there is such convergence as will render
the formulae valid.

X. Gauss's Theorem in the Theory of Attractions*.

40. In a memoir on attractions and repulsions according to the
law of the inverse square! Gauss enunciates the theorem that the
surface integral of normal force taken over a closed surface is equal to

* This section is a reprint, with sHght modifications, of a paper published in the
Proceedings of the London Mathematical Society, Series 2, Vol. viii. p. 200.
+ C. F. Gauss, Ges. Werke, Bd. v., s. 224.

48 gauss's theorem in the theory of attractions [x

47rJ/+ 27rJ/', Avliere M is the total mass of all the matter which is
surrounded by the surface and M' the total mass of all the matter
which lies as a surface distribution in the surface considered.

One way of proving this theorem is to deduce it from Poisson's
equation (>^ 23) by a volume integration, but a direct proof from first
principles is preferable.

The proof given by Gauss and reproduced in most text-books im-
plicitly involves the view that the attracting or repelling substance
(whether matter or electricity) is made up of discrete particles of
dimensions which are either absolutely zero or negligibly small in com-
parison with other distances in the contemplated configuration. It is
shewn that the contribution of a particle of mass m to the surface
integral of normal force is zero if the particle is definitely outside the
space enclosed by the surface, and Airm if the particle is definitely
inside that space.

A particle of absolutely zero dimensions is necessarily in the
enclosed region, outside the enclosed region, or in the surface which
constitutes the boundary. In the last case it is easy to shew that the
contribution of the particle to the surface integral of normal force is
generally 27r;«, but may have some other value if the particle is at a
singular point of the surface. There is therefore no difficulty in veri-
fying, for point particles, the complete theorem as stated originally by
Gauss, namely so as to include particles of no dimensions situated in
the boundary.

When the particles considered have size, the ordinary argument
applies, with as much precision as there is in our knowledge of the
truth of the Newtonian Law, to such as are at a distance from the
boundary great in comparison with their own linear dimensions. But
in the case of particles, some of whose points are at a distance from the
boundary which is not great in comparison with their linear dimensions,
two difficulties arise. In the first place the law of the inverse square
may fail adequately to represent the field of force at such close
proximity to the particle ; and, in the second place, in the absence of
information as to the size and structure of a particle, it may be uncertain
whether the particle is or is not wholly on one side of the boundary
surface, and, if it is cut by the surface, how its contribution to the
integral is to be reckoned.

The mathematical theory of attractions, however it may appear
fundamentally and originally to have treated of particles, has by modern
convention been in great measure transferred to another field of

40-41] gauss's theorem in the theory of attractions 49

investigation; and the most familiar propositions of the theory are
enunciated as applying to an ideal attracting substance which is not
made up of discrete particles, i.e. is not of molecular structure, but is
a continuum. The transition to such a substance from the particles
originally discussed is made by dividing space into volume elements,
and treating the continuous matter in a volume element as a particle.
Clearly this is justifiable so long as it is recognized that the matter in
a volume element constitutes a particle whose dimensions are not zero,
and which therefore only comes under the elementary reasoning which
leads to the Gauss theorem when it is at a distance from the boundary
great compared with its linear dimensions.

Thus the transition, usually assumed without discussion, from the
Gauss theorem for particles to the corresponding theorem for continuous
matter is quite safe provided the surface S over which the integral is
taken does not cut through the continuous matter. But the transition
is not obviously safe, and requires special proof, for a surface that
intersects the matter ; for in this case some of the volume elements
which are treated as particles must necessarily be actually in contact
with the surface of integration.

41. In order to see what additional proof is required in this case,
let us draw two surfaces parallel to
the surface >S at a small distance «
from it, the one Si inside it, the
other Sq outside it.

The normal force N at any point
of the surface >S^ may be regarded as
made up of three parts, namely :
(i) Ni, the part due to all the matter
inside Si, (ii) No, the part due to all
the matter outside So, (iii) iV', the
part due to all the matter in the space between aS'i and So.

For any selected value of c, the volume elements which take the
place of particles can always be chosen so small that their linear
dimensions are as small as we please in comparison with €. Hence the
ordinary elementary reasoning is valid for all the matter Mi inside Si,
and for all the matter 3Io outside S^. Hence, for the surface >S',

JNidS = 47r J/i , j^odS = 0.

Thus JNdS = JNidS + JNJS + JN'dS = 4.^3Ii + JN'dS

50 gauss's theorem in the theory of attractions [x

If now we pass to the limit, for t -*■ 0, it is obvious that Lim Mi = M,
where M is the mass of all the matter inside S*. Hence

JNdS = AttM + Lim JN'dS ;

so the Gauss theorem is true if, and only if,

Lim JN'dS^O.

e-»0 J

42. A class of cases in which this limit is zero can easily be

specified. For if, for any selected value of c, there is a maximum

value or superior limit to the values of | N' \ at points on *S', say n,
then

JN'ds\<nS,

where S is the complete area of the closed surface. And if, further,

Lim n = 0,

e-».0

then clearly Lim jN'dS=0.

We shall see that these conditions are satisfied in ordinary cases of
matter so distributed that the volume-density is everywhere finite.

Volume- Density. If the distribution of continuous matter between
Si and ^0 l^a.s everywhere finite volume-density, we know (§ 23) that the
force-intensity at every point in that region is definite, and so N' is
definite at every point of S. Consequently the superior limit or maxi-
mum value n exists. It remains to ascertain whether w ^0 as e ^0.
The first step towards this is to shew that in all ordinary cases A^' -* 0
as £-»-0. When this has been established, if /* is a maximum value of
\N'\, the convergence of all values of N' to zero necessarily involves
also the convergence of n to zero. But if n is a superior limit to | N' \
without being a maximum value, we can be sure of the convergence of
n to zero only if the convergence of A^' to zero is uniform for the values
of N' corresponding to all points of the surface S.

43. If we assume that the part of the material distribution con-
tained between ^i and S^ consists only of finite volume-density which,

* M does not include matter distributed with finite surface-density in the surface S.

41-44] gauss's theorem in the theory of attractions 51

though not necessarily continuous, is a function of position free from

other analytical peculiarities, then singularities in the function N' can

only arise from peculiarities in the shape of the surface S and in the

direction of the normal along which N' is the component of force. At

an ordinary conical point, or at a point on the intersection of two sheets

of the surface S, the direction of the normal is indeterminate and so

also is the value of N'; nevertheless there are definite directions to

I which the normal tends as to a limit, and corresponding values which are

limits of the function N' without being values of the function. One of

Ithe limits of A""' at such a point might be the superior limit n and yet

jijot be a maximum value. This sort of case must not be omitted from

tne present discussion, for one of the closed surfaces most frequently

employed in applications of the Gauss theorem is a cylinder with flat

ends, i.e. a surface of three sheets with two nodal lines.

The study of the influence of peculiarities of the surface S can be
I avoided if we express our test in terms of jP', the resultant force-intensity
I due to the matter between S„ and >S^i, instead of N' its normal com-
ponent. For N' ^ F', and, if/ be the maximum value or superior limit
oiF',

N'dS </S,

and the Gauss theorem holds provided/ -^0 as e-*-0.

For the simple kinds of material distribution contemplated, F' is
definite and continuous everywhere between Si and So, so there is no
possibility of /being a superior limit without being a maximum value.
And therefore, if we can shew that F' -*■ 0 for every point on >S', we are
thereby assured that /-^ 0. The convergence of F' to zero is proved
by shewing that the component G' oi F' in any arbitrarily selected
direction tends to zero.

44. We aim, then, at shewing that the component in any direction
of the intensity of force at a point, due to a distribution of given
volume-density contained between two parallel surfaces, tends to zero as
the surfaces tend to coincidence. This result would be obvious if the
point were definitely outside the attracting distribution. But it is not
obvious in the present instance since the point is inside the dis-
tribution, nor would it be obvious if the point, though outside the
distribution, were at a distance from its boundary which tended to
vanishing.

4—2

GAUSS S THEOREM IN THE THEORY OF ATTRACTIONS

Let P be the point at which F' is estimated. With P as centre,
describe a sphere of radius 6.
Then (r' at P is the sum of two
terms, G',' due to that part of the
matter between ^S"; and S^ which
is exterior to the sphere, and G^
due to that part of the matter
between /S', and S^ wliich is in-
terior to the sphere.

Now in consequence of the
convergence of the integral repre-
senting the component of force
in any direction due to a volume
distribution, it is always possible
to choose 6 so small that G.2 shall be less than any assigned small
quantity ^w. And when 0 has been selected we notice that P is
definitely outside the matter which gives rise to G^, so that G^ tends
to zero as the quantity of matter tends to zero. Thus we can always
choose c so small that Gi shall be less than \ w. Consequently we have
been able to choose c so small that

G^ + G:^G'<o^.
Thus Lim^' = 0.

This is all we require for a proof of the Gauss theorem in the case
in which there is no distribution of matter between >So and ^S*! save such
as has finite volume-density. We have seen that, since (r'-^- 0,

/^O and

JN'dS^O.

So we conclude that when the surface S cuts only through matter of
finite volume-density the Gauss theorem is valid.

45. Surface- Density. Let us now consider the case in which the
distribution of matter between ^S', and S^^ includes a surface distribution,
say on a surface 12, of finite surface-density. In general the surface
S2 will cut the surface S in a curve. A particular case, requiring
special consideration, is that in which the surfaces aS and O coincide
over a definite area.

At the outset it is important to remember that the force-intensity
at a i)oint of a surface distribution such as fi is represented by a
seiui-convergcnt integral (§ 30) and is therefore not completely defined.

44-47] gauss's theorem in the theory of attractions 53

By a suitable convention the definition may be completed, and in such a
way that the force-intensity is finite. Adopting some such convention, we
have N' finite at points on the curve of intersection of O and S ; we
have already seen that K' is finite at all other points of S, so that
there is a finite superior limit or maximum value n of the values of
J N' I for all the points of >S^.

46. It is to be remarked that the particular convention adopted in
order to give definiteness to iV does not generally affect the value of
JN'dS. If the surface fl cuts the surface S in a curve, then the
elements of area associated with dubious values of X' in the surface
integral form a narrow strip on >S' along the curve of intersection of the
two surfaces, and in the process of integration the area of this strip
tends to zero, making its contribution to the surface integral also zero.
On the other hand if the surface 12 coincides, over an area A, with the
surface S, N' at points of J. is the component of force-intensity normal to
the former surface as well as to the latter. Now all the indeterminateness
or semi-convergence of force due to a surface distribution is in the
tangential component ; the normal component is definite, and so there
is no indeterminateness in the value of jN'dS.

Incidentally we recall the fact (.§ 33) that the normal force-intensity
at a point in a surface containing surface-density o- differs from the
limit of the normal force for a point approaching the surface but not in
it by 27ro-. Consequently the surface integral jii'dS is less when the
surface of integration coincides with the surface on which the surface-
density resides than if the former surface were just outside the latter
by the amount 2Tr J a dS taken over the common area A.

47. In the more general case, in which fi intersects >S^, having seen
that jN'dS is definite we must examine what is the limit of its value
as €-^0. This we do by considering the component G' in any direc-
tion of the force-intensity at a point P in the curve of intersection of
fi and S, due to that part of the surface O which is intercepted between
>S'i and /So, and by investigating whether, as e^^O, G' tends to infinity
or to a finite or zero limit.

The problem is strictly analogous to that already discussed for
volume-density, being, namely, that of examining the limit of a com-
ponent of attraction at any point in a distribution of finite surface
density, as the distribution is diminished to zero in one of its dimensions
without change of surface-density. The result is, however, quite different
from that obtained for volume-density.

54 gauss's theorem in the theory of attractions [x

Let us begin by attempting to apply the method already used for
volume-density, with such changes as are demanded by the changed
circumstances. The attempt will be unsuccessful, but a study of the
reason for its failure will help to make clear the nature of the difficulty
which has to be faced. jP is a point in the strip of breadth 2t cosec x
cut by So and *S', on the surface il, where x is the angle of intersection
of S and 12. Suppose the ambiguity arising out of the semi-convergence
of the surface integral representing G' to have been removed by selecting
some special shape for the vanishing cavity round P. Describe round
P, in the surface fi, a curve 6 of the special shape selected. The com-
ponent of force at P due to the whole strip of O contained between Si
and So consists of two parts, namely, Gi due to the part of the strip
outside 0, and G^' due to the part of the strip inside 6. On account
of the convergence of the integral representing force due to a surface
distribution, it is possible to choose the dimensions of 6 so small that
I G2 I is less than any selected small quantity h w. And when 6 is chosen
and fixed, P is definitely outside the distribution that gives rise to Gi,
so clearly (r/ tends to zero as the distribution that gives rise to it tends
to vanishing. Accordingly we can choose € so small that \Gi'\<^u>.
Hence we have been able to choose e so small that Gi + G2' = G' <o).
Hence, apparently,

UmG' = 0.

48. This reasoning, however, is not sound, and the result obtained
is false. Semi-convergence is equivalent to convergence associated with
a vanishing cavity about P of a definite shape, and implies that the
limit of the attraction component (defined by means of such a vanishing
cavity) of a portion of the distribution enclosed by a curve 0 of the
same shape about P is zero, as the linear dimensions of 6 tend to zero.
Now when the dimensions of 0 have been selected and fixed in the
above argument, the subsequent selection of c may give us a strip (as
in the diagram of § 44) which is narrower than 6, so that the whole of
6 is not occupied by matter. Thus the actual matter within 0 is
bounded, not by 6, but by a curve of different shape, and the reasons
fur regarding the corresponding attraction at P as less than |to cease
to be applicable.

If we had absolute convergence instead of semi-convergence it would
be quite another matter. For then we should not be tied to any par-
ticular shape of cavity or of boundary, and the fact of the boundary's
becoming something else instead of 0 would not invalidate the assumed

47-49] gauss's theorem in the theory of attractions 55

inequality. We should, in fact, be entitled then to say that the attraction
at P due to any area surrounding P and lying entirely within 6 is less
than io). Now the component of force normal to Q, is represented by
an absolute convergent integral, and so its limit is zero as the breadth
of the strip tends to vanishing. But it is not so with the component in
any other direction.

Consider, for example, a tangential component of force at P. For
different shapes of vanishing cavity the values of this force are different.
The difference between the values for two selected shapes of cavity is
due to the area bounded internally by one cavity and externally by the
other, or rather to the limit of this area as it vanishes. Hence we infer
the important fact that the difference of the values of the force corre-
sponding to different cavities is independent of the size and shape of
the outer boundary of the surface distribution. It is simply a function
of the shapes of the two cavities, proportional of course to the surface-
density at P. Now since these values of a tangential component
corresponding to different cavities have definite differences independent
of the outer boundary of the distribution, their limits, as the outer
boundary tends to any limiting form, must have the same definite dif-
ferences ; one may vanish, but certainly not all. [In the case of a plane
surface of uniform density the one which vanishes would correspond to
a cavity similar and similarly situated to the limiting form of the outer
boundary, with P for centre of similitude.] Thus the result apparently
obtained in § 47 could not be true.

49. At this stage it is natural to note the probability that the
exclusion of infinitely great differences between the values of the force
for different cavities implies a restriction on the nature of the contem-
plated cavities and their mode of vanishing. And we therefore consider
for a moment the question of what kinds of passage to limit make the
semi-convergent force-integrals converge to definite values.

An answer, though not a complete one, may be founded upon a well-
known theorem already made use of without explicit quotation at the
end of § 48. The theorem is that an annular plane lamina of uniform
surface-density, whose inner and outer boundaries are similar and
similarly situated curves, exerts no tangential attraction at the centre
of similitude. Applying this result to the part of a uniform disc con-
tained between two successive positions of a contour which is diminishing
in size without change of shape round a fixed centre of similitude P,
we find that the shrinkage of the contour makes no difference in the

56 gauss's theorem in the. theory of attractions [x

value of the tangential force at P due to the part of the disc outside
the contour ; accordingly the limit of this force is definite for vanishing
of the cavity. If we have to do with a surface-density which is not
uniform, residing in a curved surface, the errors involved in neglecting
the variability of the surface-density and the curvature of the surface
can be rendered as small as we please by taking a sufficiently small
contour, provided the curvature of the surface is definite and the
surface-density continuous at P. Hence the reasoning for a plane
uniform surface can be rendered applicable to the more general case,
and we arrive at the following result : — The semi-convergent integral
representing tangential force at any point P of a surface distribution
is rendered convergent by selecting a cavity of definite shape, and
diminishing to zero, without change of shape, the scale of the geometrical
configuration consisting of the cavity and the point P.

50. A wider range of cases of convergence might be obtained by
study of the form of the integral representing the component of attrac-
tion in the direction of the axis of x, at an origin situated in a plane
disc of uniform surface-density o-, occupying part of the plane c = 0.
The integral is, in the notation of polar coordinates,

jjar-'coiedrdS,

and if the external boundary and the boundary of the cavity are respec-
tively r = F{B) and r=/(6), this reduces to

a- r'{\og F (6) - log/(e)} cos edO.
Jo

The danger of an infinity here arises out of the tendency oi/(6) to
zero, as the inner boundary closes in round the origin. But if the
form of /(B) is such that we can write it r](t> (O'), where r; is independent
of 0 and tends to zero with the vanishing of the cavity, while <l> (0) is
neither zero nor infinite for any value of 6, we may put

\og/ie) = \ogv + \og(e),

and the only term tlireatening an infinity is now

/•2ir

(r log ri I cosddd,
Jo

which vanishes for all values of rj different from zero, and therefore has
the limit zero for 77 -»- 0.

If 4> (6) is independent of rj we have the case discussed otherwise

49-52] gauss's theorem in the theory of attractions 57

iu ,^ 49. If ^ {6) involves r/, but in such a way that its order of
magnitude is not determined by that of t/, we have a type of con-
vergence more general than that of ^ 49, but not of a seriously different
character.

We note here that the disappearance of the term which threatened
infinity is bound up with one special restriction on the cavity con-
templated, namely that the contour of the cavity, i.e. the nearer edge
of the matter, must completely surround the point at which the attrac-
tion is estimated, otherwise the integrals of cos 6 and sin 0 would not
vanish. This explains the fact that at a point on the edge of a disc the
attra<?tion is infinite.

Another feature of the cavities now under discussion is that the
distances from the origin of the various points of the edge of the cavit}'
all become small of the same order as the cavity closes in.

51. The class of cavity just described (which it' will be convenient to
call class a) does not, of course, represent the only mode of making the
force integrals converge to definite values. It is to be expected that there
are other classes of vanishing cavity capable of producing convergence,
and of these an important example is that which comprises such as are
symmetrical about two axes at right angles through the point P. For
the contours of such cavities, both

[log/ (6) cos Ode and |log/(^) sin 6 d6

vanish, so that both components of tangential force are definite. In
this case the value of eaeh force-component is independent of the
shape of the cavity, so that the vanishing of the ca^^ty need not be
effected by keeping the shape unaltered and gradually diminishing the
scale ; for it is permissible, while the linear dimensions are diminished,
to keep changing the shape, so long as the s>Tnmetry conditions are
never infringed.

For example, the cavity might be a rectangular slit whose length,
though tending to zero, tends to be infinitely great in comparison with
the breadth ; here the distances of the various points of the edge of the
cavity do not aU become small of the same order.

The class of symmetrical cavities we may call class ;S.

52. In the present application, in order to be able to employ the
foregoing principles of semi-convergence, we shall suppose that the
surface-density in Q is a continuous function at the points where fi

58 gauss's theorem in the theory of attractions [x

meets S, which inchides the supposition that the material distribution
in 17 does n<»t terminate on the surface >S', but passes through it. We
postulate, further, that the force shall always be reckoned for a cavity
of the class leading to finite values. And now we see that, since the
ditferences of the values of the force for different cavities of this class
are definite and independent of c, one value of i'^' may tend to zero as
€ -»• 0, but certiiinly not all ; but, if one has a definite limit for e -* 0,
then also all the others have definite limits.

53. Now let us return to the argument in § 47 by which the attempt
was made to prove that a force-component G', specified by a cavity 0
of assigned shape, tends to zero with e. The reasoning broke down
because the boundary of the portion whose attraction was denoted by 6rV
ceased to be the curve 0. In the construction by which it was proposed
to make \G' \ less than an arbitrarily assigned small quantity w, the
dimensions of 6 were first selected, and afterwards a value of c was
chosen such that for it and all smaller values \Gi'\< h w. This order of
choice of the dimensions of 6 and c implies that the efiective contour of
tlie portion whose attraction is G.2 is a quadrilateral on the surface O
bounded by parallel curves on the surfaces Si and So and by two
opposite portions of the curve 6 ; and the breadth of the quadrilateral
between the former curves may be taken as tending to infinite smallness
in comparison with the length from one to the other of the opposite
portions of the curve 6.

The limit form of this quadrilateral contour is simply the kind of
rectangle quoted above as an example of a cavity of the class /?. True
it is not plane, not rectilineal, and not right-angled. But, given
simple analytical circumstances, small errors may always be considered
separately, and therefore when the curves and surfaces concerned are
free from geometrical singularity the errors due to neglect of their
curvatures may always be rendered as small as we please by making e
and the dimensions of 6 sufficiently small. And when the <iuadrilateral
may be as narrow as we please in comparison with its length, the con-
tributions of its short and relatively distant ends to the contour integi-al
may be made as small (relatively to the whole) as we please. Thus the
possible obliquity of the ends is unimportant, and the contour may be
regarded as tending ultimately to the narrow rectangular form.

Thus the reasoning wliich failed to prove that G' tends to zero with
€, when calculated for a cavity of the shape 6, does actually prove that
G' tends to zero with « when calculated for a particular cavity of the
class p, and therefore also for all cavities of the class /8.

52-55] gauss's theorem in the theory of attractions 59

Now some cavities of the class ;8 belong also to the class a ; there-
fore G' tends to zero with c when calculated for some cavities of the
class a. Therefore G' tends to a definite limit, as e -* 0, when calculated
for any selected cavity of the class a.

54. This is just the result which we require in order to demonstrate
the vanishing of Lim jN'dS, when there is present a surface dis-

e-*0 J

tribution of finite surface-density. For N' tends to zero at all points
of S except those on a certain curve, the intersection of S and O ; and
non-zero values, provided they are definite, confined merely to a curve
on the surface of integration, do not contribute to the value of the
integral. Now we have shewn that, at points on the curve, N' calcu-
lated for any cavity of the class already discussed has a finite limit for
c ->■ 0. Hence

Lim (N'dS=0.

6-*0 J

55. The case in which the angle x of intersection of the surfaces
S and O vanishes at a point P is not covered by the preceding reasoning,
and it is desirable to give a brief outline of the manner in which it may
be discussed. In this case the surfaces O and >S^ touch at the point P,
and it is desired to prove that at P

Lim N' = 0.

The portion of CI contained between ^i and 6',, is not now a long
narrow strip, and we must examine the shape of that part of its boundary
which is nearest to P. Let us suppose (noting that the supposition
excludes geometrical singularities in fi and S) that with P as origin
and suitably chosen axes the part of /S' near P is approximately

2z = aaf + 2ha:y + bif,
and the equation to the part of CI near P is approximately

2z = a'a^ + 2Kxy + h'lf' ;
then the equations of the parts of ^S'l and S^ near to P are (to a sufficient
approximation for the present purpose)

2z = + 2£ + aa? + 2hxy -i- %".
Hence the projections on the tangent plane at P of the parts of the
curves of intersection of CI with >S'i and >So which are near to P are given
approximately by the equations

± 26 = («' - a) x" + 2 ill - h) xy + {b! - b) /,

60 gauss's theorem in the theory of attractions [xi

and are, in fact, curves analogous to the indicatrix of fl and no less
definite in character. These may be called 'quasi-indicatrices.'

If the surface fi does not cross the surface S at P, i.e. if
(a - a) {b' -b)> {h' - h)-,
one of the quasi-indicatrices is imaginary, the other a real ellipse with
P as centre. This ellipse is an approximation to the boundary of the
part of Q contained between S^ and S^^, and tends to the same limit form
closing in round P for e -»- 0. The convergence of N (absolute in this
case) is a guarantee that the limit, for e-^0, of the corresponding N'
is zero.

If (a' - a) {U -h)< (k' - Jif, so that the surfaces S and fi cross one
another at P, the two quasi-indicatrices are conjugate hyperbolas with
P for centre, their vertices tending to coincidence with P as « ^0.
These hyperbolas are an approximation to the part of the boundary near
P of the part of O contained between >Si and S^. Now describe round
P, in the surface fi, a curve 6 which (since N' is absolutely convergent
at P) may be taken to be a circle. The diagram would be an oblique
curvilinear cross inside a circle. N' may be split up into N^ due to
the i)art of O between ^ and S^ but outside 6, and i\V due to the part
of fi between S-^ and /% and inside 6, namely the cross-shaped area.
Having selected any small quantity w we can, on account of the
absolute convergence of iV, choose 0 so small that i\V<|w; and then,
since when 6 is once chosen P is definitely outside the distribution that
gives rise to iV/, we can choose c so small that i\^i'<ia); hence we have
been able to choose c so small that A"'<w. Thus Lim N' = 0.

Therefore the contact of fi and >S' at P does not disturb the value of

Lim {n'

dS.

56. Thus, both for distributions of finite volume-density and for
those which include finite surface-density, the validit)'' of the Gauss
theorem has been established. •

XI. Some Hydrodynamical Theorems.

57. Uniqueness Theorems. A well-known application of
Green's Theorem (Article 18, formula 6) is to prove that the problem
of finding a potential function </> for a given definite region, which shall
satisfy certain conditions at the surface or surfaces which constitute the
boundary of the region, cannot have two essentially different solutions.

55-58] SOME HYDRODYNAMICAL THEOREMS 61

The nature of the boundary conditions depends on the particular
physical application which is contemplated. Thus if <t> is to be an
electrostatic potential it will be required to have a constant value over
each isolated portion of the boundary and to be such that the surface
integral of its normal gradient over each such isolated portion has a
prescribed value. If (f> is to be the hydrodynamical velocity potential
of a liquid in motion the normal gradient of (ft at each point of the
boundary is prescribed, being required to be equal to the normal com-
ponent of the velocity of the moving boundary.

In the simpler cases A^ = 0, and the proof is got by assuming two
different <^'s, <^i and 4>2, which satisfy the prescribed conditions, and
by substituting ^1-^2 for V in formula (8) of Article 18. This gives

/(*.-*=) C^'-t) <*«=/= {^.(*.- 4' <^'.

and the surface conditions are such as to ensure the vanishing of the
surface integral ; hence the volume integral must vanish, and as the
subject of integration cannot be negative anywhere it must vanish
everywhere in the region of integration.

Less simple cases are the electrical applications to conductors
situated in a heterogeneous dielectric, or in a region in which there are
interfaces between different dielectrics. Both the enunciations and the
proofs of the theorems require care but present no serious difficulty.

58. As has been already suggested in Article 39, the application
of a theorem of this type to regions which have no outer boundary and
so ' extend to infinity ' is not merely the taking of a particular case of
a general theorem, but involves an additional step of passage to limit
which requires special justification. No doubt the applied mathema-
tician who is not disposed to give time to refinements of logic will take
many such passages to limit on trust, without a qualm, for he has
convictions based on physical considerations as weighty as any reasoning
by pure mathematics. But, in approaching the usual hydrodynamical ap-
plications of such theorems, the more one thinks of an infinitely extended
absolutely incompressible Hquid, a system which instantaneously trans-
mits force and energy to unlimited distance, the more one realises that
it has no proper place in physics, and that (however useful it may be as
an approximation to physical circumstances) it is a conception of the pure
mathematician and must be studied by purely mathematical methods.

It is therefore within the scope of the present Tract to enquire
whether for a region occupied by liquid, bounded internally by the

62 SOME HYDRODYNAMICAL THEOREMS [XI

surfaces of solids but without external boundar)^, there can be established
any theorem corresponding to the fundamental ' uniqueness theorem '
for hydrodynamical velocity potential, namely that, for a region bounded
internally by closed surfaces which it surrounds and externally by a
containing surface, two solutions of Laplace's equation having a pre-
scribed normal gradient at each point of the complete boundary can
differ only by a constant.

To take the known theorem and press it to a limit by endless
extension of the containing boundary surface would be a task of some
difficulty. For the functions dealt with are dependent in form upon the
form of the boundary and must change as it changes, so that the volume
integral and one of the surface integrals which appear in the proof
would have not only changing regions of integration but also changing
subjects of integration. It is best therefore to begin with a region
which is externally unbounded, and to consider functions <^ which
satisfy the equation A<^ = 0 at all points in this region.

59. Let us denote by dS an element of area of the closed surface
or surfaces which constitute the inner boundary of such an infinite
region, and by dv an element of the outward-drawn normal at a point
of such a surface. Let us also take another closed surface, whose
element of area we may denote by da-, which surrounds all the surfaces
S. We begin by applying to the region which is bounded internally by
S and externally by o- the theorem of Article 18, formula (8). If F is a
function which satisfies the equation A F=0 at all points in the infinite
region bounded internally by S, it does so at every point of the part of
this region which lies within o- ; hence the volume integral on the left-
hand side of the equality vanishes, and we have

I r'Jd.-f vfdS^r%CJ-Xdr (29).

Now suppose that the surface o- expands without limit, so that the
distance of every part of it from some definite origin 0 tends to infinite
greatness ; then clearly the volume integral and the first surface integral
either both do or both do not converge to definite limit values, and if
the former alternative obtains the two convergences are either both
dependent on or both independent of the manner or form in which
the surface o- tends to infinity.

On applying the theorem of Article 36 it appears that the volume
integral converges absolutely, i.e. independently of the manner or form

58-61] SOME HYDRODYNAMICAL THEOREMS 63

ill which o- tends to infinity, provided 2 f — j , or (as we may call it

for brevity) (f, is such that for all values of the distance r measured
from 0 greater than a definite length a

(f<Mr-v- (30),

where J/ is a constant and /A > 3.

Under these circumstances the surface integral over o- also tends to
a definite limit whose value may be calculated for any special form of
<r which is convenient. If o- be taken to be a sphere with 0 as centre
and ;• as radius, da- — r-dtn where d(M is an element of solid angle, and
9 Vjdv is the same as 9 Vjdr ; as q is, for great values of r, of the
order of greatness of v'^i^ at most, 9 Vjdr is at most of the same order
of greatness, and V at most of the order r'sf^+i. Hence the surface
integral is at most of the order of greatness of

-''doy

and therefore tends to the limit zero for r -^ oc .

Thus we have, as the result of passage to limit, the theorem'

-ij'^oH^^P^ (-)■

valid for functions V which satisfy the above specified conditions.

60. In the hydrodynamical application, where V is a velocity
potential, q is the resultant velocity, and the inequalities

<f<Mr-i-, fJL>S (32)

take the place of the common but only imperfectly intelligible statement
that the velocity 'vanishes at infinity.' To the physicist, however, the
interpretation of this restriction on V which appears most significant
is that which is expressible in terms of the kinetic energy, namely that
the motion is one having a definite amount of kinetic energy. If q^
were of the order of r~^ or of a greater order of magnitude than r~^ the
integral representing the total kinetic energy would almost certainly
tend to indefinite greatness.

When the kinetic energy of the motion is definite formula (31) gives
an expression of its value as a surface integral over the surface S.

61. It may be remarked here that the inequalities (32) allow of
fractional values of /x provided only that jti>3. It is known however

64 SOME HYDRODYNAMICAL THEOREMS [XI

from the general theory of the sohition.s of Laplace's equation* that, in
problems dealing with regions of the same general character as the region
surrounding a closed surface -S', fractional powers of r do not occur. So
we may think of i/A- 1, the negative power of r associated with V, as
integral, and of /i. as an even integer.

62. A uniqueness theorem for the infinite region under considera-
tion is obtained as follows. Let <^i and <^2 be two functions which
satisfy Laplace's equation at all points of the region extending from S
to infinity, which have equal normal gradients at all points of the
surfaces S, and to each of which corresponds a liquid motion having a
definite (i.e. finite but not prescribed) amount of kinetic energy. In
equation (31) substitute 4>i~^-2 for Fand we get

Now the left-hand side vanishes because ^ = ^ at S, and consequently

ov ov

the volume integral must vanish ; on account of the positive character
of the subject of integration this requires that at every point

Hence ^i and ^o cannot differ except by a constant.

63. Theorems concerning Kinetic Energy. It is well known
that, for liquid in a given region whose boundaries are moving in a pre-
scribed manner, the irrotational motion has less kinetic energy than
any possible rotational motion. The following theorem, which is in a
certain sense a particular case of the former, is given here because of
the important dynamical principles which can be deduced from it.

In any region bounded by given surfaces moving in given manners^
consider alternatively two possible liquid motions, (a) coritinuous irrota-
tional motion, or (/3) sevei-al continuous irrotational motions in various
sub-regions separated from one another by surfaces at which there is
continuity of nwmal but not of tangential velocity. TJie. kinetic energy
of the motion (/?) is greater than that of the motion (a) by an amount
equal to the kinetic energy of such motion as tvould have to be super-
posed on (a) in order to produce (ft).

" Cf. Thomson and Tait, Natural Philosophy, Edition of 1890, Vol. i. p. 181.

61-63] THEOREMS ON KINETIC ENERGY 65

The proof, which consists of a simple application of the theorems
of Article 18, varies slightly in detail according to the nature of the
region and the surfaces dealt with. Let us consider a region bounded
internally by a surface S and externally by a containing surface a-, and
let each of these surfaces be moving (not necessarily rigidly) with
velocity whose normal component at any point is typified by V and v
respectively. Let dv represent an element of normal drawn outwards
from any closed surface, and let the density of the liquid be taken as
unity.

Consider (a) a continuous motion in the region between S and a,
having a velocity potential </> ; (/3) a motion having a discontinuity of
tangential flow over a closed surface >S" which does not surround S, the
motion having a velocity potential <^ + x inside S' and a velocity potential
<l> + if/ outside S', and the normal velocity at >S" being typified by V.

Then ^, x and ij/ all satisfy Laplace's equation, and in addition the
following surface conditions are satisfied: —

at a-, • d(fi/dv = V, dr^jdv = 0 ;

OV OV CV ov

Denoting the kinetic energies by T with appropriate sufiix, we know
from Article 18, equation (8), that

Hence, remembering the surface conditions, we get

66 SOME HYDRODYNAMICAL THEOREMS [xi

But by Green's Theorem (Article 18, formula 6), applied to the region
outside S'

and by the same theorem applied to the region inside S'
so that

the final expression is essentially positive, being the kinetic energy of
the motion (represented by x and i/') which if superposed on (a) would
yield (/8).

The internal boundary S is not necessary, but gives generality to
the theorem ; clearly it may consist of one or of several distinct closed
surfaces. S' also may be regarded as typical of several closed surfaces
exterior to one another, or some surrounding others. The theorem also
holds good if S' cuts any of the surfaces S and a-, being bounded by the
curves of section, or if S' surrounds some or all of the surfaces ^S* ; for
such cases the proof would require slight and fairly obvious modifica-
tions which need not be set out in detail. Enough has been said to
establish the truth of the theorem as enunciated.

64. Let us now pass to the consideration of some special cases of
the general theorem.

(\) Suppose S' to surround *S^ but to be wholly inside o-, and let
both tS' and cr be at rest; and let the motion of 6* be any motion which
is compatible with rigidity of each of the surfaces S. Then in the (3
motion the complete boundary of the region between aS" and o- is at rest,
and so there is no motion there, i.e. x - ~ ^- ^^^ ^i^^y think of S as
made up of the surfaces of solid bodies moving in the liquid, and S'
in the /3 motion as a new fixed outer boundary substituted for the fixed
outer boundary a of the a motion. Hence the theorem : Ani/ number
of solid bodies are moving with given linear and angular velocities in
fiomoffeneous lit/uld which is bounded by a fixed outer boundary. If for
t/ti.s outer boundary there icere substituted another fixed boundary lying

63-64] THEOREMS ON KINETIC ENERGY 67

completely inside thejormer one, the kinetic energy of the liquid motion
would be increased by an amount equal to the kinetic energy of the
motion ivhich would have to be superposed on the f(yrmer motion in order
to produce the latter.

Of course the outer boundary of the original motion may be at
infinity, provided the motion has definite kinetic energy. In this case
the new boundary *S" might be a plane or any open surface extending
to infinity.

The solid bodies and the liquid constitute a dynamical system
whose motion is determined by the motion of the solids, so that it
has six times as many coordinates as there are movable solids. The
inertia coefficients depend on the configuration, including the shape
and position of the boundary. An increase of kinetic energy for given
velocities of the solids means an increase of the inertia coefficients.
Hence our theorem tells us that a closing in of the fixed boundary
involves increase of inertia.

Thus it might be expected, for example, that a submarine vessel
would be more difficult to propel or to steer when near to the bottom
of the sea, or to the shore, than when out in the open deep sea.

(ii) As a second special case suppose /S" to be wholly inside o-
but not to surround /S', and let both S' and o- be at rest, while B moves
in any manner compatible with the rigidity of each of the surfaces
typified by 8. As before we think of 8 and 8' as rigid material
boundaries, and note that in the /? motion there is no motion inside 8' .
Hence the theorem : Any number of solid bodies are moving with given
linear and angular velocities in homogeneous liquid having a fixed outer
boundary. If another fixed solid ivere present the kinetic energy of the
liquid motion would be greater than it actually is by an amount equal to
the kinetic energy of the motion ivhich would have to be superposed on the
first motion in order to pi'oduce the second.

This theorem indicates that the eff"ect of the presence of a fixed
solid is to increase the eff"ective inertia coefficients ot movable solids in
its neighbourhood.

(iii) As a third special case suppose 8' to be wholly inside o- but
not to surround >S', let o- be at rest, and let both 8' and 8 be moving in
any manner compatible with the rigidity of each separate surface. As
before we think of 8 as made up of the surfaces of moving solid bodies,
and in the first instance we think of 8' as a rigid massless material
shell. There is then, in the ^ system, motion inside as well as outside
8\ and we may conveniently split T^ into two parts T^ for the motion

68 SOME HYDRODYNAMICAL THEOREMS [Xl

outside S' and Tp" for the motion inside S'. The general theorem now
takes the form

Tp' + Tp"-Ta=T(^)^T(x) (34),

the meaning of the symbols on the right-hand side being obvious.

Now suppose the liquid inside S' to be replaced by solid matter,
whose kinetic energy in the given motion of S' is t. If t^ Tp" our
equality leads to the ine(iuaHty

Tp+t>Ta+T(^) + T{x)>Ta (35).

Hence the theorem : Anj/ number of solid bodies are moving ivith given
linear and angular velocities in homogeneous liquid having a fixed outer
boundary. If in addition there were present another solid body moving
in any manner the kinetic energy of the motion oj the liquid and the new
solid ivould be together greater than the kinetic energy of the original
fluid motion, provided the new solid has for its given motion not less
kinetic energy than that of the irrotational motion of liquid occupying a
boundary similar to the boundary of the solid and moving in a similar
maimer.

It may be remarked that the motion of a liquid as if solid, when
not a motion of mere translation, is rotational, and so has greater
kinetic energy than the irrotational motion having the same boundary.
Hence the condition t ^ J/s' is certainly satisfied if the solid body *S" is
homogeneous and of the same specific gravity as the liquid. And as
the moving of matter to the boundary of a solid, without change of
total mass, increases the moments of inertia, a hollow solid having the
same mass as the liquid it displaces would have not less kinetic energy.
Hence the condition t ^ Tp is likely to be satisfied for many solids
which are not lighter than the liquid they displace.

This theorem accordingly indicates that the effect of the presence of
a movable solid wliich is not lighter than the liquid it displaces is
generally to increase the kinetic energy of the total motion, and there-
fore to increase the effective inertia coefficients of movable solid bodies
in its neighbourhood.

It is readily seen that, when the boundary of the additional solid
body is given, its total mass and the distribution of its mass within
its boundary can afiFect only those coefficients which multiply the
squares and products of the fluxes of the six coordinates of the body
itself in tlie expression for the total kinetic energy. All the other
coefficients may be affected by tlie geometrical boundary-configuration

64-65] HYDRODYNAMICAL SUCTION 69

but not by the mass-configuration of the new solid. Hence the
theorem, as regards the inertia coefficients of the original solids, is
perfectly general.

65. 'Suction.' Let T be the kinetic energy and U the work
function of the acting forces for the dynamical system consisting of one
or more solid bodies moving in liquid, and let 0 be typical of the
generalised coordinates of the system. In the Lagrangian equation of
motion

dt\d$ )~ dd "^ dd

the term dTIdd represents an inertia eflfect which can in a certain sense
be regarded as equivalent to a force tending so to modify the configura-
tion as to increase T, just as dUjdd is a force tending so to modify the
configuration as to increase U. Thus when the kinetic energy of a
system is a function not only of the time-fluxes of the coordinates but
also of the coordinates themselves there are apparent forces, which are
really inertia effects, making for increase of the kinetic energy.

Now from the three dynamical theorems stated above, and from
others on similar lines which it would be easy to formulate, it is fairly
clear that generally the approach of a movable solid to a fixed boundary
or to another solid which is held fixed, or even to another movable solid,
so changes the configuration as to increase the kinetic energy. In
some cases this is capable of complete logical proof, as for example when
a single solid moves in liquid which has no boundary except a single
infinite plane. In other cases it is difficult to distinguish exhaustively
between changes of configuration which tend to increase the kinetic
energy and those which tend to decrease it, but various kinds of change
can be assigned to one class or the other with such a high degree of
probability as is equivalent to certainty for practical purposes. Gener-
ally the question at issue is whether what has been called T{\f) is
increased or decreased by the contemplated change of configuration, and
one feels justified in stating (though the term used is not precise) that
T{}1/) increases with the proximity of two bodies, or of one body and
a fixed boundary.

Hence the inertia term dTjdO usually manifests itself as a force
making for increase of proximity, as it were an attraction between the
bodies or the body and the boundary. This is what is called ' suction.'
It is an additional effect to the increase of inertia previously discussed.
If, for example, a submarine were passing near another vessel the

5—3

70 SEMI-CONVERGENT VOLUME INTEGRALS [XI

theory points not only to abnormal heaviness in steering and propelling,
but also to the i)(>ssil)ility of the steering being utterly vitiated by
forces and couples due to suction.

66. Semi=convergent Volume Integrals to Infinity. The

theorem of Article HG suggests the rough rule that a volume integral to
infinity whose subject of integration /at great distance r from a definite
origin 0 tends to smallness of the order r->^ is convergent if /x>3,
semi-convergent or divergent if /* = 3, divergent if /a < 3. This is, how-
ever, by no means an accurate statement, for the divergence theorem
analogous to that of Article 1 4 is as follows : If at all points uutdde a
.sphere, having 0 an centre and a definite radius a, f is alyebraicall y
greater than mr~i^, where m is a constant greater than zero and /x <$ 3,
the integral jfdr, taken through a region ivhose outer boundary tends to
infinite remoteness from 0 in all directions, is divergent. This theorem
indicates that for /x < 3 there is no possibiUty of convergence if/ is a
function which has (outside the sphere a) everywhere the same sign and
is such that r'^' is everywhere dehuitely different from zero, but it by
no means shuts the door on semi-convergence, i.e. convergence associated
with some special mode of infinite widening of the outer boundary, if/
changes sign from place to place.

67. A criterion for the existence of special modes leading to con-
vergence, and a complete specification of them (when they exist) for a
general subject of integration, would probably be extremely difficult to
obtain. But there are two particular types of subject of integration for
which certain modes lea-ling to convergence can be readily recognised.

(i) Using spherical polar coordinates r, 6, ^, let us first suppose
f to be of the form

r-^(^, <^) + r/,

where i// is a finite single-valued function of angular position which
satisfies the condition

f d6 rd^smdil^iO, <^)-0,

.'() .'0

and ^f is a function of position which tends to smallness of a higher
order than r""^ The volume integral of g in general converges abso-
lutely. As regards the first term of/ let us consider its integral through
the volume contained between the two similar and similarly situated
boundaries, having the origin for centre of similitude, whose e((uations
are

r~-ftF{B,<f>j.,

65-68] SEMI-CONVERGENT VOLUME INTEGRALS 71

where F is a function which is always definitely greater than zero, and
a and /3 are positive parameters of which a is the greater. The volume
integral is

wdiich becomes, on integration with respect to r,

log(a/^) jjif^iO, cf>)Hin6ded4>,
which is zero by liypothesis.

Passing now to the volume integral of r~^ \p (0, ^) through a region
whose outer boundary is r = f3F(6, <^), we see from the above that the
value is the same as if we had integrated through the wider region
wdiose outer boundary is r= aF(e, cf)), no matter how great a may be.
Thus the integral out to the a surface has a definite constant value, and
tlierefore a definite limit value, while a becomes great without limit ;
the value of course depends on the form of the function F(0, </>). In
other words our semi-convergent integral is rendered convergent by
selecting an outer boundary of arbitrary but definite shape, and a definite
origin 0 within it, and by increasing indefinitely without change of
shape the scale of the geometrical configuration consisting of the outer
boundary and the point 0.

It will be noticed that the property of if/ (6, (^) which leads to this
convergence is a property of Laplace's functions of integral order other
than zero. Hence t// may be the sum of any number of such complete
surface harmonics*. A particular case of importance is that in which
/ is a finite single-valued solution of Laplace's equation, for then ip is
a complete surface harmonic of order 2. »

68. The proof suggests a certain slight and perhaps unimportant
extension of the theorem. Instead of r = aF(0, 0) we might take the
outer boundary to be

r = a^('^'^^F{6, <f),

where x is a function which is positive (so that the outer and inner
boundaries may not intersect) and free from infinities ; the volume
integral between this and the surface r = F{0, </>) is

log a jjxio, (f>) xi,{e, <f>)smeded<p.

■■■• Any single- valued function of angular position, provided it be of limited
variation, can be expanded in a uniformly convergent series of Laplace's functions.
(Jordan, Cours d' Analyse, t. ii, § 244.) If the Laplace's function of order zero is
absent from the expansion of xp, the condition for convergence of the volume
integral is satisfied.

72 SEMI-CONVERGENT VOLUME INTEGRALS [xi

If X and tj/ be sucli as to make this vanish there is convergence for the
mode of expansion of tlie outer boundary corresponding to a -*- co . For
example x niight be a constant plus a sum of surface harmonics of
integral orders dift'erent from any which occur in ij/. It may be noted
that X may involve a without invalidating the argument*.

69. (ii) In the second place let us suppose/ to be of the form

where yj/ satisfies the same criterion as in Article 67, and X(r) is any
function of r which does not become infinite for any definite value of r
which occurs in any contemplated region of integration. Consider the
volume integral of this /for the volume between the concentric spheres
r = a and r = /?, where a > ^. The integral is

jjjx (r) il^ {$, </>) r"" sin ed6ddr,

which becomes, on integration with respect to r,

r r'X (r) dr iU {$, <A) sin Oded<t>,

which is zero in virtue of the hypothesis with regard to i/'.

From this it follows, by reasoning similar to that employed above,
that the volume integral for this type of / is rendered convergent by
taking as outer boundary a spliere whose centre is 0 and by increasing
the radius of the sphere without limit.

Probably both this theorem and the preceding one could be gene-
ralised by taking other curvilinear coordinates instead of r, 6, <f>.

70. It is clear that theorems analogous to those just established
hold for the semi-convergence of certain integrals of the kind discussed
in Section III, it being a question of the mode of closing in of a cavity
instead of the mode of expansion of an outer boundary.

71. T/u) Integral of Linear Momenturn in Hydrodynamics. A
familiar example of a semi-convergent volume integral to infinity is the

* It may be possible, even in cases where ^ does not comply with the hypo-
thesis of Article 67, to secure convergence by giving a suitable form to X' But in
such cases the expansions of both \p and x contain the Laplace's function of zero
order (i.e. a constant), and therefore the surface integral of their product over the
unit-sphere contains at least one non-vanishing term. The vanishing of the whole
is secured by providing for one or more other non-vanishing terms, each resulting
from the integration of the product of two surface harmonica of the same order,
with constants adjusted to give a zero sum, if this can be done without sacrificing
the positive character of x-

68-72] SEMI-CONVERGENT VOLUME INTEGRALS 73

integral representing the total linear momentum, resolved in any direc-
tion, of the motion of unbounded liquid due to the motion of a solid
body through it. When the velocity potential ^ tends to smallness of
the order r~- a velocity component u generally tends to smallness of the
order r~^. And since <^ satisfies Laplace's equation so also does d^jdx
or u. Hence the momentum integral judr is semi-convergent and
converges if the outer boundary of the region of integration tends to
infinity in any of the manners specified in Articles 67, 68 and 69.

72. The Integral of Angular Momentum in Hydrodynamics. In
the integrals of the type

/(■

yvz-'ry)^'

which represent components of the moment of momentum of a liquid
motion, when 4> is of the order r~^ the subjects of integration are
generally also of the order r~-. But since ^ satisfies Laplace's equation
so also does yd<f>/dz — zd(fi/dy, and likewise the two similar expressions.
Hence the volume integrals are of the class whose subjects of integration
are finite single-valued solutions of Laplace's equation. The expansion
of the subject of integration in a series of solid spherical harmonics
gives a series of integrals of the type discussed in Article 69, which can
be rendered convergent by always taking for outer boundary a sphere
whose centre is the origin. Thus the angular momentum integrals are
semi-convergent.

PRINTED IN ENGLAND BY J. B. PEACE, M.A.
AT THE CAMBRIDGE UNIVERSITY PRESS

Cambridge Tracts in Mathematics
and Mathematical Physics

No.

1. VOLUME AND SURFACE INTEGRALS USED IN PHYSICS,

by J. G. Leathem, Sc.D. 3rd edition. 45. net.

2. THE INTEGRATION OF FUNCTIONS OF A SINGLE

VARIABLE, by G. H. Hardy, M.A., F.R.S. 2nd edition.
4^. net.

3. QUADRATIC FORMS AND THEIR CLASSIFICATION

BY MEANS OF INVARIANT FACTORS, by T. J. I'A.

Bromwich, Sc.D., F.R.S. Out of print.
THE AXIOMS OF PROJECTIVE GEOMETRY, by A. N.

Whitehead, Sc.D., F.R.S. 3.^. net.
THE AXIOMS OF DESCRIPTIVE GEOMETRY, by A. N.

Whitehead. 3^. fiet.
ALGEBRAIC EQUATIONS, by G. B. Mathews, ^^$l^^R.S.

2nd edition. 4^. ()d. fiet. ^ V " >/

THE THEORY OF OPTICAL INSTRUME^TTSf by E. T.

Whittaker, M.A., F.R.S. 2nd edition. ^S.^ ^u. net.
THE ELEMENTARY THEORY OF THEJ^YMMETRICAL

OPTICAL INSTRUMENT, by J. G. Leathem. 3^. net.
INVARIANTS OF QUADRATIC DIFFERENTIAL FORMS,

by J. E. Wright, M.A. is. net.
AN INTRODUCTION TO THE STUDY OF .INTEGRAL

EQUATIONS, by Maxime B^chep, B.A., Ph.D.- standi edition.

3^. net. •->«'■» -.

11. THE FUNDAMENTAL THEOREMS OF THE DIFFER-

ENTIAL CALCULUS, by W. H. Young, Sc.D., F.R.S.
35. net.

12. ORDERS OF INFINITY: The 'Infinitarcalcul' of Paul

Du Bois-Reymond, by G. H. Hardy. 3^. 7iet.

13. THE TWENTY-SEVEN LINES UPON THE CUBIC SUR-

FACE, by A. Henderson, M.A., Ph.D. With table and 13

plates. %s. 6d. net.
THE TWISTED CUBIC, by P. W. Wood, M.A. With 10

diagrams. 3^. net.
COMPLEX INTEGRATION AND CAUCHY'S THEOREM,

by G. N. Watson, M.A., F.R.S. 3^. dd. net.
LINEAR ALGEBRAS, by L. E. Dickson, Ph.D. 7,s. 6d. net.
THE PROPAGATION OF DISTURBANCES IN DISPER-
SIVE MEDIA, by T. H. Havelock, D.Sc, F.R.S. 4J. 6d. net.
THE GENERAL THEORY OF DIRICHLET'S SERIES, by

G. H. Hardy and Marcel Riesz, Dr. Phil. 45. dd. net. .
THE ALGEBRAIC THEORY OF MODULAR SYSTEMS,

by F. S. Macaulay, M.A., D.Sc. 55. dd. Tiet.
THE ELEMENTARY DIFFERENTIAL GEOMETRY OF

PLANE CURVES, by R. H. Fowler, M.A. 75. net.

In preparation

THE INTEGRALS OF ALGEBRAIC FUNCTIONS, by
Prof. H. F. Baker, Sc.D., F.R.S.

RETURN PHYSICS LIBRARY

TO— ^ 351 LeConte Hall 642-3122

LOAN PERIOD 1
" QUARTER

ALL BOOKS MAY BE RECALLED AFTER 7 DAYS
Overdue books ore subject to replacement bills

DUE AS STAMPED BE

LOW

UNIVERSITY OF CALIFORNIA, BERKELEY
FORMNO.DD25, 2.5m, 477 BERKELEY, CA <;>47 20 ^^

CD3E7DS3E(!

-J

Internet Archive Audio

Featured

Top

Images

Featured

Top

Software

Featured

Top

Books

Featured

Top

Video

Featured

Top

Mobile Apps

Browser Extensions

Archive-It Subscription

Save Page Now

Full text of "Volume and surface integrals used in physics"

See other formats