BBD
UNIVERSITY of CALIFOKNU
it, ;'/;LES
ELEMENTS
THEOBY OF THE NEWTONIAN POTENTIAL
FUNCTION,
BY
B. O. PEIRCE, PH.D.,
ASSISTANT PROFESSOR OF MATHEMATICS AND PHYSICS
IX HARVARD UNIVERSITY.
BOSTON:
PUBLISHED BY GIXX & COMPANY.
1886.
105460
Entered, according to Act of Congress, in the year 1886, by
GINN & COMPANY,
in the Office of the Librarian of Congress, at Washington.
J. 8. CUSHINO & Co., PRINTERS, BOSTON.
T 3
PREFACE.
THIS book is almost entirely made up of lecture-notes
which from time to time during the last four years I
have written out for the use of students who have begun
I with me the study of what I have ventured to call, after
,4 Neumann, the Newtonian Potential Function.
The notes were intended for readers somewhat familiar with
the principles of the Differential and Integral Calculus, but
unacquainted with many <tf the methods commonly used in
N applying Mathematics to the study of physical problems.
These students, I learned, found it difficult to get from any
single book in English a treatment of the subject at once
elementary enough to be within their easy comprehension,
%. and at the same time suited to the purposes of such of them
^L as intended eventually to pursue the subject farther, or
wished, without necessarily making a . -\:'./?alty of Mathe-
matical Physics, to prepare themselves to study Experi-
pV mental Physics thoroughly and understandingly. "What is
here printed seems to have been of use to some of those
who have read it in manuscript, and it is hoped that it may
now be helpful to a larger number of students.
Since these notes are professedly elementary in character,
I feel that no apology is needed for what may seem to be
the rather prolix way in which some of the subjects are
treated, or for an arrangement of matter which would be
IV PREFACE.
unsuitable in a book intended for a different class of readers.
I have not hesitated to use a long proof whenever this has
seemed to me more easily comprehensible than a short and
mathematically neater one, and I have often given more than
one demonstration of a single theorem in order to illustrate
different methods of work. Although I have used freely
the notation * of the Calculus, I have assumed on the part
of the reader only an elementary knowledge of its principles.
The short treatment of Electrostatics in Chapter v. is in-
troduced to show how the theorems of the preceding chapters
may be used in solving physical problems ; but it is hoped that
a person who has mastered even the little here given will be
able to understand, with the aid of some good treatise on
Experimental Ph}"sics, most of the phenomena of Electro-
statics. It is also hoped that those readers who mean to study
the subject of Electricity from the mathematical point of
* In this book the change made in a function u by giving to the
independent variable x the arbitrary, finite increment AT, and keeping the
other independent variables, if there are any, unchanged, is denoted by
A x u. Similarly, A y u and z u express the increments of u due to changes
respectively in y alone and in z alone. The total change in u due to
simultaneous changes in all the independent variables is sometimes
denoted by Au; so that if u=f(x, y, z),
A v u
, . * ,
Ay -\ ---- Az + e,
AT Ay Az
where e is an infinitesimal of an order higher than the first.
The partial derivatives of u with respect to x, y, and z are denoted by
D x u, D y u, and D z n, and the sign = placed between a variable and a con-
stant is used to show that the former is to be made to approach the
latter as its limit. In those cases where it is desirable to draw attention
to the fact that a certain derivative is total, the differential notation
dn .
is used.
dx
PREFACE. V
view will find what they have learned here useful when
they take up standard works on the subject.
My sincere thanks are due to H. N. Wheeler, A.M., who
has read much of the manuscript of the following pages and
all of the proof, and to Dr. E. H. Hall, who has examined
parts of Chapters iv. and v. and helped me with various
suggestions. I am indebted to other friends also, and among
them to Mr. "W. A. Stone for the use of some of his problems.
The reader who wishes to get a thorough knowledge of
the properties of the Newtonian Potential Function and of
its applications to problems in Electricity is referred to the
following works, which, with others, I have consulted and used
in writing these notes.
Betti : Teorica delle Forze Newtoniane e sue Application! all'
Elettrostatica e al Magnetismo.
Clausius : Die Potentialf unction und das Potential.
Gumming : An Introduction to the Theory of Electricity.
Chrystal : The article " Electricity " in the Ninth Edition of the
Encyclopaedia Britannica.
Dirichlet: Yorlesungeu iiber die im umgekehrteu Verhaltniss des
Quadrats der Entfernung \virkenden Kriifte.
Gauss : Allgemeine Lehrsiitze in Beziehung auf die im verkehrten
Verhaltnisse des Quadrates der Entfernung -wirkenden Anzieh-
ungs- und Abstossungskriifte. Also other papers to be found
in Volume Y. of his Gesammelte Werke.
Green : An Essay on the Application of Mathematical Analysis to
the Theories of Electricity and Magnetism.*
Mascart: Traite d'Electricite Statique. Also Wallentin's translation
of the same work into German, with additions.
*A copy of the original edition of this paper is to be found in the
Library of Harvard University, Gore Hall, Cambridge. The paper has
boon reprinted by Ferrers in " The Mathematical Papers of George Green,"
and by Thomson in CrelU-'s Journal.
vi PREFACE.
Mascart et Joubert :. Lemons sur 1'Electricite et le Magnetisme.
Also Atkinson's translation of the same work into English, with
additions.
Mathieu: Theorie du Potential et ses Applications a 1'Electro-
statique et au Magnetisme.
Maxwell: An Elementary Treatise on Electricity. A Treatise on
Electricity and Magnetism.
Minchin : A Treatise on Statics.
C. Neumann: Untersuchuugen iiber das Logarithmische und New-
ton'sche Potential.
Riemann: Schwere, Electricitat und Magnetismus, edited by Hatten-
dorff.
Schell: Theorie der Bewegung und der Krafte.
Thomson : Eeprint of Papers on Electrostatics and Magnetism.
Thomson and Tait: A Treatise 011 Natural Philosophy.
Todhunter : A Treatise on Analytical Statics.
Watson and Burbury: The Mathematical Theory of Electricity
and Magnetism.
Wiedemann : Die Lehre von der Electricitat.
TABLE OF CONTENTS.
CHAPTER I.
THE ATTRACTION OP GRAVITATION.
SECTION. PAGE
1. The law of gravitation 1
2. The attraction at a point 1
3. The unit of force 2
4. The attraction due to discrete particles .... 2
5. The attraction of a straight wire at a point in its axis . 3
6. The attraction at any point due to a straight wire . . 4
7. The attraction at a point in its axis due to a cylinder of
revolution ......... 7
8. The attraction at the vertex of a cone of revolution due
to the whole cone and to different frusta ... 8
9. The attraction due to a homogeneous spherical shell; to
a solid sphere ........ 11
10. The attraction due to a homogeneous hemisphere . . 13
11. Apparent anomalies in the latitudes of places near the
foot of a hemispherical hill 15
12. The attraction due to any ellipsoidal homo?oid is zero at
all points within the cavity enclosed by the shell . .1(5
13. The attraction due to a spherical shell whose density at
any point depends upon the distance of the point from
the centre IS
14. The attraction at any point due to any given mass . .19
15. The component in any direction of the attraction at a point
P due to a given mass is always finite . . . .21
viii TABLE OF CONTENTS.
SECTION PAGE
16. The attraction between two straight wires . . . .22
17. The attraction between two spheres 23
18. The attraction between any two rigid bodies . . .24
CHAPTER II.
THE NEWTONIAN POTENTIAL FUNCTION IN THE CASE OF
GRAVITATION.
19. Definition of the potential function 29
20. The derivatives of the potential function relative to the
space coordinates are functions of these coordinates
which represent the components parallel to the coordi-
nate axes of the attraction at the point (x, y, z) . 30
21. Extension of the statement of the last section . . .31
22. The potential function due to a given attracting mass is
everywhere finite, and the statements of the two pre-
ceding sections hold good for points within the attract-
ing mass 32
23. The potential function due to a straight wire . . .34
24. The potential function due to a spherical shell . . .35
25. Equipotential surfaces'and their properties . . . .37
26. The potential function is zero at infinity . . . .40
27. The potential function as a measure of work . . .40
28. Laplace's Equation 42
29. The second derivatives of the potential function are finite
at points within the attracting mass . . . . 43
30. The first derivatives of the potential function change con-
tinuously as the point, (x, y, z) moves through the
boundaries of an attracting mass 48
31. Theorem due to Gauss. The potential function can have
no maxima or minima at points of empty space . . 50
32. Tubes of force and their properties 53
33. Spherical distributions of matter and their attractions . 54
34. Cylindrical distributions of matter and their attractions . 58
TABLE OF CONTENTS. IX
SECTION PAGE
35. Poisson's Equation obtained by the application of Gauss's
Theorem to volume elements 59
36. Poisson's Equation in the integral form . . . .62
37. The average value of the potential function on a spherical
surface 64
38. The equilibrium of fluids at rest under the action of
given forces 66
CHAPTER III.
THE NEWTONIAN POTENTIAL FUNCTION IN THE CASE OF
REPULSION.
39. Repulsion according to the " Law of Nature "... 72
40. The force at any point due to a given distribution of repel-
ling matter ......... 73
41. The potential function due to repelling matter as a measure
of work x . . . . . . . . .75
42. Gauss's Theorem in the case of repelling matter . . .75
43. Poisson's Equation in the case of repelling matter . . 76
44. The coexistence of two kinds of active matter . . .77
CHAPTER IV.
THE PROPERTIES OF SURFACE DISTRIBUTIONS. GREEN'S
THEOREM.
45. The force due to a closed shell of repelling matter . . 80
46. The potential function is finite at points in a surface distri-
bution of matter .82
47. The normal force at any point of a surface distribution . 85
48. Green's Theorem ......... 87
49. Special cases under Green's Theorem . . . . .01
50. Surface distributions which are equivalent to certain vol-
ume distributions ..... .05
51. Those characteristics of the potential function which are
sufficient to determine the function . .96
52. Thomson's Theorem. Dirichlet's Principle . . .98
X TABLE OF CONTENTS.
CHAPTER V.
ELECTROSTATICS.
SECTION PAGE
53. Introductory 103
54. The charges on conductors are superficial .... 104
55. General principles which follow directly from the theory
of the Newtonian potential function . . . .106
56. Tubes of force and their properties 108
57. Hollow conductors 110
58. The charge induced on a conductor which is put to earth . 114
59. Coefficients of induction and capacity .... 115
60. The distribution of electricity on a spherical conductor . 117
61. The distribution of a given charge on an ellipsoidal con-
ductor .......... 118
62. Spherical condensers ........ 119
63. Condensers made of two parallel conducting plates . . 122
64. The capacity of a long cylinder surrounded by a concentric
cylindrical shell .124
65. Specific inductive capacity ....... 125
66. The charge induced on a conducting .sphere by a charge
at an' outside point . . . . . . .130
67. The energy of charged conductors ..... f34
THE
NEWTONIAN POTENTIAL FUNCTION,
CHAPTER I.
THE ATTEACTION OF GEAVITATIOU.
1. The Law of Gravitation. Every body in the universe
attracts every other body with a force which depends for mag-
nitude and direction upon the masses of the two bodies and
upon their relative positions.
An approximate value of the attraction between any two rigid
bodies may be obtained by imagining the bodies to be divided
into small particles, and assuming that every particle of the one
body attracts every particle of the other with a force directly
proportional to the product of the masses of the two particles,
and inversely proportional to the square of the distance between
their centres or other corresponding points. The true value of
the attraction is the limit approached by this approximate value
as the particles into which the bodies are supposed to be divided
are made smaller and smaller.
2. The Attraction at a Point By "the attraction at any
point P in space, due to one or more attracting masses," is
meant the limit which would be approached by the value of the
attraction on a sphere of unit mass centred at P if the radius of
the sphere were made continually smaller and smaller while its
mass remained unchanged. The attraction at 7* is, then, the
attraction on a unit mass supposed to be concentrated at P.
2 THE ATTRACTION OF GRAVITATION.
If the attraction at every point throughout a certain region
has a value other than zero, the region is called ' ' a field of
force " ; and the attraction at any point P in the region is called
" the strength of the field" at that point.
3. The Unit of Force. It will presently appear that all spheres
made of homogeneous material attract bodies outside of them-
selves as if the masses of the spheres were concentrated at their
middle points. If, then, Jc be the force of attraction between
two unit masses concentrated at points at the unit distance
apart, the attraction at a point P due to a homogeneous sphere
A 3
of radius a and of density p is &'~- g ? where r is the dis-
tance of P from the centre of the sphere. In all that follows,
however, we shall take as our unit of force the force of attrac-
tion between two unit masses concentrated at points at the unit
distance apart. Using these units, k in the expression given
above becomes 1, and the attraction between two particles of
mass mi and m 2 concentrated at points r units apart is i-^-
r
4. Attraction due to Discrete Particles. The attraction at a
point P, due to particles concentrated at different points in the
same plane with P, may be expressed in terms of two com-
ponents at right angles to each other.
FIG. 1.
Let the straight lines joining P with the different particles be
denoted by r n r 2 , r 3 , , and the angles which these lines make
with some fixed line Px by a l5 a 2 , a 3 , . If, then, the masses
THE ATTRACTION OF GRAVITATION. 3
of the several particles are respectively wij, m 2 , m 3 , , the
components of the attraction at P are
m, COSa 2 . rin
^ -- 1 = / - [I]
2
in the direction P#, and
v _ ?ft! sin a! _j_ 7?i 2 sn a 2
~
rf ?y
in the direction Py, perpendicular to Px.
The resultant force at P is
_,_ _ ro -,
' ' ' * 7 5 ' L^ J
2
F 2 , [3]
and its line of action makes with Px the angle whose tangent
. T
is
X
If the particles do not all lie in the same plane with P, we
may draw through P three mutually perpendicular axes, and call
the angles which the lines joining P with the different particles
make with the first axis a n a 2 , a 3 , ; with the second axis,
/?n /?2i &, ; and with the third axis, y a , y,, y 3 , .... The three
components in the directions of these axes of the attraction at
P due to all the particles are then
Wl COS a T r_X^W COS /? . _X^Wl COSy
- ^ ' / ~> - ~
** r
The resultant attraction is
. _ r .-,
~~ / - ~> '
Y 2 +Z*, [5]
and its line of action makes with the axes angles whose cosines
are respectively
1 1' and I ra
5. Attraction at a Point in the Produced Axis of a Straight
Wire. Let ft be the mass of the unit of length of a uniform
straight wire AB of length /, and of cross section so small that
4 THE ATTRACTION OF GRAVITATION.
we may suppose the mass of the wire concentrated in its axis
(see Fig. 2), and let P be a point in the line AB produced at a
I HI Hr
FIG. 2.
distance a from ^1. Divide the wire into elements of length
Aa;. The attraction at P due to one of these elements, Jf, whose
nearest point is at a distance x from P, is less than ^ and
ar
greater than
(z + Aa;) 2
The attraction at P due to the whole wire lies between
Z^ and 7 ; but these quantities approach the
ar L^(x -f- Aa;)-
same limit as Aa; is made to approach zero, so that the attrac-
tion at P is
limit
Ax = (
If the coordinates of P, .4, and B are respectively (a;, 0, 0) ,
(a?!, 0, 0), and (ccj + Z, 0, 0), this result may be put into the form
_j 11
x l x x t . x + Ij
[8]
6. Attraction at any Point, due to a Straight Wire. Let P
(Fig. 3) be any point in the perpendicular drawn to the straight
wire AB at A, and let PA = c, AB = Z, AM = #, and the angle
ABP= 8. Let MN be one of the equal elements of mass (/^Aa 1 )
into which the wire is divided, and call PM, r. The attraction
II y\ /y
at P due to this element is approximately equal to ' , and
r-
acts in some direction lying between PM and PN. This attrac-
tion can be resolved into two components whose approximate
values are ^ ' in the direction PA. and ^ x ' - in the
2 (c 2 +ar z )i
THE ATTRACTION OF GRAVITATION.
direction PL. The true values of the components in these
directions of the attraction at P, due to the whole wire, are,
then, respectively :
p t,cdx tf x 1 /
Jo ((r'-f-ar)tJ c [ Vc^+^Jo c
and
[9]
[10]
FIG. 3.
The resultant attraction is equal to the square root of the sum
of the squares of these components, or
and its line of action makes with PA an angle whose tangent is
= tan ^ APB.
cos 8
sin APB 2 sin | APB cos ^ APE
That is, the resultant attraction at P acts in the direction of
the bisector of the angle APB.
From these results we can easily obtain the value of the
attraction at any point P, due to a uniform straight wire B'B
(Fig. 4) . Drop a perpendicular PA from P upon the axis of
the wire. Let AB = Z, AB' = /', PA = c, ABP = 8, AB'P = 3',
BPB'=0. The component in the direction PA of the attrac-
tion at P is [9]
~ (cos 8+ cos 8'),
6 THE ATTRACTION OF GRAVITATION..
and that in the direction PL is
-(sin 8' sin 8),
so that the resultant attraction is
FIG. 4.
The line of action PKof R makes with PA an angle < such
that
sin 8' sin S
cos 8 + cos 8
, ,<-,, x
= tanks'- S);
1-10-1
[13]
and
It is to be noticed that PK bisects the angle 0, and does not
in general pass through the centre of gravity or any other fixed
point of the wire. Indeed, the path of a particle moving from
rest under the attraction of a straight wire is generally curved ;
for if the particle should start at a point Q and move a short
distance on the bisector of the angle BQB' to Q', the attraction
of the wire would now urge the particle in the direction of the
bisector of the angle BQ'B', and this is usually not coincident
with the bisector of BQB'.
THE ATTRACTION OF GRAVITATION. 7
If q is the area of the cross section of the wire, and p the
mass of the unit volume of the substance of which the wire is
made, we may substitute for p. in the formulas of this section
its value qp.
If instead of a very thin wire we had a body in the shape of
a prism or cylinder of considerable cross section, we might
divide this up into a large number of slender prisms and use the
equations just obtained to find the limit of the sum of the attrac-
tions at any point due to all these elementary prisms. This
would be the attraction due to the given body.
7. Attraction at a Point in the Produced Axis of a Cylinder
of Revolution. In order to find the attraction due to a homo-
geneous cylinder of revolution at any point P (Fig. 5) in the
axis of the cylinder produced, it will be convenient to imagine
the cylinder cut up into discs of constant thickness Ac, by
means of planes perpendicular to the axis.
Let p be the mass of the unit of volume of the cylinder, and
a the radius of its base. Consider a disc whose nearer face is
at a distance c from P, and divide it into elements by means of
B B
FIG. 5.
radial planes drawn at angular intervals of A0 and concentric
cylindrical surfaces at radial intervals of Ar.
The mass of any element J/ whose inner radius is r is equal
to pAc-A0[rAr + (Ar) 2 ], antl tne whole attraction at P due to
M is approximately p - [ (Ar) ] iu a Une j omiug p
with some point of J/. The component of this attraction in
the direction PC is found by multiplying tbe expression just
8 THE ATTRACTION OF GRAVITATION.
/>
given by , the cosine of the angle CP/S, so that the
/~2 i - 2
V c" + r
attraction at P in the direction P(7, due to the whole disc, is
approximately
Ac -
/27T x<
= Ac I dO \
Jo Jo
pcrdr
1 - . c
L v?^
If the bases of the cylinder are at distances c and c + h
from P, the true value of the attraction at P in the direction
PC, due to the cylinder QQ', is
limit X.^o A C r> C c + h f-> C \ 7
2wpAc 1 - = 2 ?rp I [1 )(ic
[15]
This is evidently the whole attraction at P due to the cylin-
der, for considerations of symmetry show us that the resultant
attraction at P has no component perpendicular to PC.
[14] gives the attraction due to the elementary disc ABA'JB',
on the assumption that the whole matter of the disc is concen-
trated at the fape ABC. The actual attraction at P due to
this disc may be found by putting c = c and h = Ac in [15].
If a, the radius of the cylinder, is very large compared with
h and c , the expression [15] for the attraction at P due to the
cylinder approaches the value
8. Attraction at the Vertex of a Cone. The attraction due to
a homogeneous cone of revolution, at a point at the vertex of
the cone, may be found by the aid of [14].
If Fig. 6 represents a plane section of the cone taken through
the axis, and if PM= c, MM' = Ac, and MB = ?, the attraction
at P due to the disc ABCD is approximately
27rpAc(l COS a),
THE ATTRACTION OF GRAVITATION.
and the attraction due to the whole cone is
- COSa) AC = 2,rp(l - COSa)
= 27rp(l-COSa).PZ. [16]
The attraction at P due to the frustum ABKN is found by
subtracting the value of the attraction due to the cone ABP
from the expression given in [1C], The result is
[17]
and it is easy to see from this that discs of equal thickness cut
out of a; cone of revolution at different distances from the vertex
by planes perpendicular to the axis exert equal attractions at
the vertex of the cone.
FIG. 6.
It follows almost directly that the portions cut out of two
concentric spherical shells of equal uniform density and equal
thickness, by any conical surface having its vertex at the
common centre P of the shells, exert equal attraction at this
centre ; but we may prove this proposition otherwise ^ as fol-
lows :
Divide the inner surface of the portion cut out of one of the
shells by the given cone into elements, and make the perimeter
of each of these surface elements the directrix of a conical
surface having its vertex at P. Divide the given shells into
elementary shells of thickness Arby means of concentric spheri-
cal surfaces drawn about P. In this way the attracting masses
will be cut up into volume elements.
Let ML 1 (Fig. 7) represent one of these elements, whose
inner surface has a radius equal to r ; then, if the elementary
10 THE ATTRACTION OF GEAVITAT1ON.
cone APB intercept an element of area Aw from a spherical sur-
face of radius unity drawn around P, the area of the surface
element at MM' is i & Aw, and that at LL' is (r + Ar) 2 Aw. The
attraction at P in the direction PJf, due to the element ML', is
approximately
P
and the component of this in any direction Px, making an
angle a with PM, is approximately p Aw Ar cos a. The attraction
at P in the direction Px, due to the whole shell EDFG, is,
then, ^-v
X = lim ^ p Ar Aw cos a,
where the sum is to include all the volume elements which go to
make up the shell. If PF= r c , PG = r 1} PF'= r ', PG' = r/,
and . = FG =
X I c^r j cos adu pn | cosadw.
The attraction at P in the same direction, due to the shell
E'D'F'G', is
X 1 = p I l dr I cosadw = pp ( cos adw.
But the limits of integration with regard to w are the same in
both cases ; .-. X= X', which was to be proved.
If the shells are of different thicknesses, it is evident that
they will exert attractions at P proportional to these thick-
nesses.
THE ATTRACTION OF GRAVITATION.
11
The area of the portion which a conical surface cuts out of a
spherical surface of unit radius drawn about the vertex of the
cone is called " the solid angle " of the conical surface.
9. Attraction of a Spherical Shell. In order to find the
attraction at P, any point in space, due. to a homogeneous
spherical shell of radii ?' and 1\, it will be best to begin by
dividing up the shell into a large number of concentric shells
of thickness Ar, and to consider first the attraction of one of
these thin shells, whose inside radius shall be r.
Let p be the density of the given shell, that is, the mass of
the unit of volume of the material of which the shell is com-
posed. Join P (Fig. 8) with by a straight line cutting the
inner surface of the thin shell at J\ T , and pass a plane through
PO cutting this inner surface in a great circle NLSL', which
FIG. 8.
will serve as a prime meridian. Using N as a pole, .describe
upon the inner surface of the thin shell a number of parallels of
latitude so as to cut off equal arcs on NLSL'. Denote by A0
the angle which each one of these arcs subtends at 0. Through
PO pass a number of planes so as to cut up each parallel of
latitude into equal arcs. Denote by A< the angle between any
two contiguous planes of this series. By this moans the inner
surface of the elementary shell will be divided into small quad-
rilaterals, each of which will have two sides formed of meridian
arcs, of length r-A'J, and two sides formed of arcs of parallels
of latitude, of length rsin0-A< and r sin(#-f- A0)- A<, where
12 THE ATTRACTION OF GRAVITATION.
G is the angle which the radius drawn to the parallel of higher
latitude makes with ON. The area of one of these quadri-
laterals is approximately r^sinO- A0- A<, and the thickness of
the shell is Ar, so that the element of volume is approxi-
mately ? <2 sin0- Ar- A0 A<. Let PMy, then the attrac-
tion at P, due to an element of mass which has a corner at
nr <- i P^ 2 sin0ArA0A<f> . ,, ,. ,.
M, is approximately f - , in the direction PM.
This force may be resolved into three components : one in the
direction PO, the others in directions perpendicular to PO
and to each other ; but it is evident from considerations of
symmetry that in finding the attraction at P due to the whole
shell, we shall need only that component which acts in PO. This
, , or 2 sin 6 Ar M A< cos KPM
is approximately ; or, if PO = c,
y
The attraction at P due to the whole elementary shell is, then,
approximately (truly on the assumption that the whole mass of
the shell is concentrated at its inner surface),
Ar f C
J J
pr * Sin e ( c ~ r cos
y 3
and the true value at P of the attraction due to the iven shell is
f ri Xdr
Jr
[20]
If in the expression for X we substitute for its value in
terms of y, we have, since
y 2 = (r + y 2 2 crcos0,
and hence . 2ydy = 2crsinOd9,
X- Cc
J
[21]
THE ATTRACTION OF GRAVITATION. 13
In order to find the limits of the integration with regard to y,
we must distinguish between two cases :
I. If P is a point in the cavity enclosed by the given shell,
= r c and l = r + c ;
2 "] 0? ^
..
c 2 r + c r c
and f ri Xtfr = 0; [23]
so that a homogeneous spherical shell exerts no attraction at
points in the cavity which it encloses.
II. If P is a point without the given shell,
y =c r and y l = c -f- r ;
and
From this it follows that the attraction due to a spherical
shell of uniform density is the same, at a point without the shell,
as the attraction due to a mass equal to that of the shell con-
centrated at the shell's centre.
If in [25] we make r = 0, we have the attraction, due to a
solid sphere of radius i\ and density p, at a point outside the
sphere at a distance c from the centre. This is
4 irp
,3
[2G]
oir
10. Attraction due to a Hemisphere. At any point P in the
plane of the base of a homogeneous hemisphere, the attraction
of the hemisphere gives rise to two components, one directed
toward the centre of the base, the other perpendicular to the
plane of the base. AVe will compute the values of these com-
ponents for the particular case where P lies on the rim of the
hemisphere's base, and for this purpose we will take the origin
14
THE ATTRACTION OF GRAVITATION.
of our sj'stem of polar coordinates at P, because by so doing
we shall escape having to deal with a quantity which becomes
infinite at one of the limits of integration. Denote the coordi-
nates of any point L in the hemisphere by r, 0, <, where (Fig. 9)
XPN= </>, IPL = 0, and PL = r.
If r l be the radius of the hemisphere,
PT = PNcos NPT = PX cos XPN- cos NPT = 2 r^ sin 6 cos <.
V r> T IK IK
cosXPL = - = =
PL r
ODT IL sin d> . a . ,
cos SPL = = - = 2- = sm# sin d>.
PL r r
The mass of a polar element of volume whose corner is at
L is approximately p IL &$ PL &6 Ar or /3r 2 sin#ArA0A<,
and this divided by r 2 is the attraction at P in the direction PL
of the element, supposed concentrated at L. The components
of this attraction in the direction PX and PI" are respectively
psinflArAtfA^cosXPZ, and P s\u0^rM^cosSPL.
The component in the direction Py of the attraction at P due
to the whole hemisphere is, then,
J f* 1t /2r, sinS co8<Ji
2d(J> I dO I p shrO sincfrdr =
[27]
THE ATTRACTION OF GRAVITATION. 15
and the component in the direction Px is
xr /"TT x2r! sinO cos<J>
I 2 cZ(| cZ0 I p sin- cos <(?>= fTrpTV [28]
/0 /0 c/0
This last expression might have been obtained from [26] by
making c equal to r and halving the result.
11. Attraction of a Hemispherical Hill. If at a point on the
earth at the southern extremity of a homogeneous hemispheri-
cal hill of density p and radius i\ the force of gravity due to the
earth, supposed spherical, is gr,- the attraction due to the earth
and the hill will give rise to two components, g ^pi\ down-
wards, and f Trp^ northwards. The resultant attraction does
not therefore act in the direction of the centre of the earth, but
makes with this direction an angle whose tangent is
FIG. 10.
Let < (Fig. 10) be the true latitude of the place and (< )
the apparent latitude, as obtained by measuring the angle wh'mh
the plumb-line at the place makes with the plane of the equator.
Let a be the radius of the earth and <r its average densitv. Then
tana =
[29]
16 THE ATTRACTION OF GRAVITATION.
The radius of the earth is very large compared with the
radius of the hill, and a is a small angle, so that approximately
a = -- i -, and the apparent latitude of the place is < -- -i
2 acr 2 acr
If < x is the true latitude of a place just north of the same hill,
its apparent latitude will be ^ -f- -^^ , and the apparent differ-
2 acr
ence of latitude between the two places, one just north of the
hill and the other just south of it, will be the true difference
plus -J. If there were a hemispherical cavity between the two
acr
places instead of a hemispherical hill, the apparent difference of
latitude would be less than the true difference.
12. Ellipsoidal Homceoids. A shell, thick or thin, bounded
by two ellipsoidal surfaces, concentric, similar, and similarly
placed, shall be called an ellipsoidal homoeoid.
FIG. 11.
It is a property of every such shell that if any straight line
cut its outer surface at the points $, S' (Fig. 11) and its inner
surface at Q, Q', so that these four points lie in the order
SQQ'S 1 , the length SQ will be equal to the length Q'S'.
We will prove that the attraction of a homogeneous closed
THE ATTRACTION OF GRAVITATION. 17
ellipsoidal bomoeoid, at any point P in the cavity which it shuts
in, is zero.
Make P the vertex of a slender conical surface of two
nappes, A and B, and suppose the plane of the paper to be
so chosen that PQ is the shortest and PM the longest length
cut from any element of the nappe A by the inner surface of
the homceoid. Draw about P spherical surfaces of radii PQ,
PM, PS, and PO, and imagine the space between the inner-
most and outermost of these surfaces filled with matter of the
same density as the homceoid. The nappe A cuts out a portion
from this spherical shell whose trace on the plane of the
paper is QLOT. Let us call this, for short, " the element
QLOT." The attraction atP, due to the element QMOS which
A cuts out of the homoeoid, is less than the attraction at the
same point due to the element QLOT, and greater than that
due to the element whose trace is KMNS. But the attraction
at P, due to the first of these elements of spherical shells, is to
the attraction due to the other as the thickness of the first shell
is to that of the other, or as QTis to KS. (See Section 8.)
The limit of the ratio of QT to KS, as the solid angle of the
cone is made smaller and smaller, is unity ; therefore the limit
of the ratio of the attraction at P due to the element QMOS, to
the attraction due to the element of spherical shell whose trace
is QLNS, is unity. By a similar construction it is eas} 7 to show
that the limit of the ratio of the attraction at P, due to the
element which B cuts out of the homceoid, to the attraction due
to the portion of spherical shell whose trace is Q'L'N'S', is
unity.
But the attractions at P, due to the elements Q'L'N'S' and
QLNS, are equal in amount (since their thicknesses are the
same) and opposite in direction, so that if for the elements of
the homoeoid these elements were substituted, there would be no
resultant attraction at P. In order to get the attraction at P
in any direction due to the whole homteoid we may cut up the
inner surface of the homo?oid into elements, use the perimeter
of each one of these elements as the directrix of a conical sur-
18 THE ATTRACTION OF GRAVITATION.
face having its vertex at P, and find the limit of the sum of the
attractions due to the elements which these conical surfaces cut
from the homceoid. Wherever we have to find the finite limit of
the sum of a series of infinitesimal quantities, we may without
error substitute for any one of these another infinitesimal, the
limit of whose ratio to the first is unit}'. For the attractions at P
due to the elements of the homoeoid we may, therefore, substi-
tute attractions due to elements of spherical shells, which, as we
have seen, destroy each other in pairs. Hence our proposition.
A shell bounded by two concentric spherical surfaces gives a
special case under this theorem.
13. Sphere of Variable Density. The density of a homo-
geneous body is the amount of matter contained in the unit
volume of the material of which the body is composed, and this
may be obtained by dividing the mass of the body b} T its volume.
If the amount of matter contained in a given volume is not
the same thi-oughout a body, the body is called heterogeneous,
and its density is said to be variable.
The average density of a heterogeneous body is the ratio of
the mass of the body to its volume. The actual density p at
any point Q inside the body is defined to be the limit of the
ratio of the mass of a small portion of the body taken about Q
to the volume of this portion as the latter is made smaller and
smaller.
The attraction, at any point P, due to a spherical shell whose
density is the same at all points equidistant from the common
centre of the spherical surfaces which bound the shell but dif-
ferent at different distances from this centre, may be obtained
with the help of some of the equations in Article 9.
Since p is independent of 6 and <, it may be taken out from
under the signs of integration with regard to these variables,
although it must be left under the sign of integration with re-
gard to r.
Equations 19 to 24 inclusive hold for the case that we
are now considering as well as for the case when p is constant,
THE ATTRACTION OF GRAVITATION.
19
so that the attraction at all points within the cavity enclosed by
a spherical shell whose density varies* with the distance from the
centre is zero.
If P is without the shell, the attraction is
r r i , / r i47rp?- 2 dr
I AcZr = I ^ ,
Jr Jr. cr
or, if p=/(r),
I
The mass of the shell is evidently
[30]
[31]
and [30] declares that a spherical shell whose density is a
function of the distance from its centre attracts at all outside
points as if the whole mass of the shell were concentrated at the
centre.
If r = 0, we have the case of a solid sphere.
14. Attraction due to any Mass. In order to find the attrac-
tion at a point P (Fig. 12), due to any attracting masses J/'. we
may choose a system of rectangular coordinate axes and divide
P
X'
v/
^
I/ "
^\
/
FIG. 12.
J/"' up into volume elements. If p is the average density of one
of these elements (Ar'), the mass of the element will be pAi 1 '.
Let Q, whose coordinates are x', y\ z', be a point of the ele-
20 THE ATTRACTION OF GRAVITATION.
merit, and let the coordinates of P be x, y, z. The attraction
at P in the direction PQ due to this element is approximately
pAv'
^=5, and the components of this in the direction of the coordi-
PQ
nate axes are
pAv' , pAv' , , pAv' ,
cos a', t~ cos/3', and ^r cosy', [32]
PQ' P& PQ-
where a', /3', 7' are the angles which PQ makes with the positive
directions of the axes.
It is easy to see that
PL x'x
and, similarly, that
, and cos 7 ' =
Moreover,
and this we will call r 2 .
The true values of the components in the direction of the
coordinate axes of the attraction at P, due to all the elements
which go to make up Jf', are, then,
v _ limit
~A'=0
_ C C C p(x'-x}dx'dy'dz' . r3 ,
J J J [(*'- ^) 2 + (y'- y)*+ (z 1 - ^) 2 ]i '
T r_ limit X^pA?/(?/' y)
"~Av'=0
- r r r p^-y^ay^ r33 ,
J J J [(aj'_a!)+(y'-y)+('-2)]r
_ limit X^P A ^'( 2 ' 2: )
r r r P (^-g)
J J J [( B r_)i+(y'-
THE ATTRACTION OF GRAVITATION. 21
where p is the density at the point (#', y [ ', z'), and where the
integrations with regard to x', t/', and z' are to include the whole
of M'.
The resultant attraction at P, due to Jf ; , is
[34]
and its line of action makes with the coordinate axes angles
whose cosines are
The component of the attraction at the point (x, y, z) in a-
direction making an angle e with the line of action of R is
.Rcose. If the direction cosines of this direction are A', //.', v',
we have
cose = AA'+ M/X'+ w'.
15. The quantities X, Y, Z, and R, which occur in the last
section, are in general functions of the coordinates #, ?/, andz of
the point P. Let us consider X, whose value is given in [33 A ] .
x i x
If P lies without the attracting mass J/', the quantity '-
is finite for all the elements into which J/' is divided. Let L
be the largest value which it can have for an\- one of these
elements, then X is less than L I I j pdx'dy'dz', or iJ/', and
this is finite. If P is a point within the space which the attract-
ing mass occupies, it is easy to show that, whatever physical
meaning we may attach to X, it has a finite value. To prove
this, make P the origin of a system of polar coordinates, and
divide 3/' up into elements like those used in Section 10. It
will then be clear that
X= C C Cp8m*0coa<ttdrd0d<l>, [3G]
where the limits are to be chosen so as to include all the at-
tracting mass. Since sin 2 0cos< can never be greater than
22 THE ATTRACTION OF GRAVITATION.
unit} r , X is less than I I I pdrd6d(f>, which is evidently finite
when p is finite, as it always is in fact.
The corresponding expressions,
Y= C C Cps'm 2 0s'm<j>drd0d<l>, [37]
and Z = C C Cpsiu0cosOdrdOd<f>, [38]
can be proved finite in a similar manner ; and it follows that
X, Y, Z, and consequently R, are finite for all values of x, y,
and z.
As a special case, the attraction at a point P within the mass
of a homogeneous spherical shell, of radii ?- and r 1? and of den-
sity p, is
where r is the distance of P from the centre of the shell.
16. Attraction between Two Straight Wires. Let AK and
BK' (Fig. 13) be two sti-aight wires of lengths I and I' and of
line-densities p. and /*' ; and let KB = c. Divide AK into
K ............... B K'
M M'
FIG. 13.
elements of length A#, and consider one of these MM', such
that AM=x. The attraction of BK' on a unit mass concen-
trated at M would be (Sections 2 and 5), pA --- - I If,
therefore, the whole element MM' whose mass is /xA were con-
centrated at Jif, the attraction on it, due to BK', would be
[40]
THE ATTRACTION OF GRAVITATION. 23
The actual force, due to the attraction of BK', with which the
whole wire AK is urged toward the right, is
limit X^'..fA,v.r 1 1
c-x
= , pc i L. _v,
Jo \x (i + i'+c) x (l + c )/
- ""'[kg * ~* '-i -~ C ]r ""' ' g ( 'c(i + r+J ' [41]
17. Attraction between Two Spheres. Consider two homo-
geneous spheres of masses M and M' (Fig. 14), whose centres
C and C' are at a distance c from each other. Divide the sphere
M' into elements in the manner described in Section 9. The
attraction due to M at any point P' outside of this sphere is, as
we have seen, . and its line of action is in the direction
PC.
FIG. 14.
Let P'=(r, 0, <) be any point in the sphere Jf', and let
CP' = y. The attraction of M in the direction PC on an
element of mass prsintfAr A0A</> supposed concentrated at P is
component of this para ii e l to the
line C'C is P- The force
24 THE ATTRACTION OF GRAVITATION.
which the whole sphere M ' is urged toward the right by the
attraction of M is, then,
W C C Cp^^QdrdOdff)^ rcosfl) .- . 2 -,
J J J y 8
where the integration is to be extended to all the elements
which go to make up M' . It is proved in Section 9 that the
M'
value of this triple integral is , so that the force of attraction
c 2
MM'
between the two spheres is -
18. Attraction between any Two Rigid Bodies. In order to
find the force with which a rigid body M is pulled in any direc-
tion (as for instance in that of the axis of x) by the attraction
of another body M' , we must in general find the value of a
sextuple integral.
Let M be divided up into small portions, and let Am be the
mass of one of these elements which contains the point (x, y, z} .
The component in the direction of the axis of x of the attrac-
tion at (x, y, z) due to M' is
p(x' x)dx'dy'dz'
and this would be the actual attraction in this direction on a
unit mass supposed concentrated at (#, ?/, z). If the mass A?>i
were concentrated at this point, the attraction on it in the direc-
tion of the axis of x would be
Am C f f p(x'-x)dx'dy'dz' , .,
JJJ \_( X <- x y+(y>-yy+(z>-zy-]i
The actual attraction in the direction of the axis of x of M'
upon the whole of M is, then,
limit V A C C C P( X '~ ^dx'dy'dz'
Am i
THE ATTRACTION OF GRAVITATION. 25
If p' is the density at the point (#, ?/, z) , and if the elements
into which M is divided are rectangular parallelopipeds of di-
mensions A#, Ay, and Az, the expression just given may be
written
p'p (x 1 x) dx dy dz dx'dy'dz' p . . -,
where the integrations are first to be extended over M' and
then over M.
EXAMPLES.
1. Find the resultant attraction, at the origin of a system of
rectangular coordinates, due to masses of 12, 16, and 20 units
respectively, concentrated at the points (3, 4), ( 5, 12), and
(8, 6). What is its line of action ?
2. Find the value, at the origin of a system of rectangular
coordinates, of the attraction due to three equal spheres, each of
mass m, whose centres are at the points (a, 0, 0), (0,6,0),
(0, 0, c) . Find also the direction-cosines of the line of action
of this resultant attraction.
. 3. Show that the attraction, due to a uniform wire bent into
the form of the arc of a circumference, is the same at the centre
of the circumference as the attraction due to an}' uniform
straight wire of the same density which is tangent to the given
wire, and is terminated by the bounding radii (when produced)
of the given wire.
4. Show that in the case of an oblique cone whose base is
any plane figure the attraction at the vertex of the cone due to
any frustum varies, other things being equal, as the thickness
of the frustum.
5. Find the equation of a family of surfaces over each one of
which the resultant force of atti-action due to a uniform straight
wire is constant.
6. Using the foot-pound-second system of fundamental units,
and assuming that the average density of the earth is 5.G, com-
pare with the poundal the unit of force used in this chapter.
26 THE ATTIIACTION OF GRAVITATION.
7. If in Fig. 2 we suppose P moved up to A, the attraction
at P becomes infinite according to [7], and yet Section 15
asserts that the value, at any point inside a given mass, of the
attraction due to this mass is always finite. Explain this.
8. A spherical cavity whose radius is r is made in a uniform
sphere of radius 2 r and mass ra in such a way that the centre
of the sphere lies on the wall of the cavity, i^ind the attraction
due to the resulting solid at different points on the line joining
the centre of the sphere with the centre of the cavity.
9. A uniform sphere of mass m is divided into halves by the
plane AB passed thi'ough its centre C. Find the value of the
attraction due to each of these hemispheres at P, a point on the
perpendicular erected to AB at (7, if CP= a.
10. Considering the earth a sphere whose density varies only
with the distance from the centre, what may we infer about the
law of change of this density if a pendulum swing with the same
period on the surface of the earth and at the bottom of a deep
mine ? What if the force of attraction increases with the depth
at the rate of th of a dyne per centimetre of descent?
n
11. The attraction due to a cylindrical tube of length h and
of radii 72 and R^ at a point in the axis, at a distance c from
the plane of the nearer end, is
27rp[Vc 2 +^ 1 2 -V^+^ 2 +V(c +70 2 +^o 2 -V(c +/0 2 +^i 2 ]-
[Stone.]
12. A spherical cavity of radius b is hollowed out in a sphere
of radius a and density p, and then completely filled with
matter, of density p . If c is the distance between the centre
of the cavity and the centre of the sphere, the attraction due
to the composite solid at a point in the line joining these two
centres, at a distance d from the centre of the sphere, is
13. The centre of a sphere of aluminum of radius 10 and of
density 2.5, is at the distance 100 from a sphere of the same
THE ATTRACTION OF GRAVITATION. 27
size made of gold, of density 19. Show that the attraction
due to these spheres is nothing at a point between them, at a
distance of about 26.6 from the centre of the aluminum sphere.
[Stone.]
14. Show that the attraction at the centre of a sphere of radius
r, from which a piece has been cut by a cone of revolution
whose vertex is at the centre, is irpr sin 2 a, where a is the
half angle of the cone. [Stone.]
15. An iron sphere of radius 10 and density 7 has an eccentric
spherical cavity of radius 6, whose centre is at a distance 3
from the centre of the sphere. Find the attraction due to
this solid at a point 25 units from the centre of the sphere,
and so situated that the line joining it with this centre makes
an angle of 45 with the line joining the centre of the sphere
and the centre of the cavity. [Stone.]
16. If the piece of a spherical shell of radii r and i\, inter-
cepted by a cone of revolution whose solid angle is w and whose
vertex is the centre of the shell, be cut out and removed, find
the attraction of the remainder of the shell at a point P situated
in the axis of the cone at a given distance from the centre of
the sphere. If in the vertical shaft of a mine a pendulum be
swung, is there any appreciable error in assuming that the only
matter whose attraction influences the pendulum lies nearer the
centre of the earth, supposed spherical, than the pendulum
does ?
17. Show that the attraction of a spherical segment is, at its
vertex,
, / f 1 1 l*h }
-sVirJ'
where a is the radius of the sphere and h the height of the
segment. .
18. Show that the resultant attraction of a spherical segment
on a particle at the centre of its base is
2irhp , [3 a 2 - 3 ah + h-- (2 a - h) 3 M] .
3 (a A) 1
28 THE ATTRACTION OF GRAVITATION.
19. Show that the attraction at the focus of a segment of a
paraboloid of revolution bounded by a plane perpendicular to
the axis at a distance 6 from the vertex is of the form
' 20. Show that the attraction of the oblate spheroid formed
by the revolution of the ellipse of semiaxes a, &, and eccen-
tricity e, is, at the pole of the spheroid,
(
j 1
e- (. e
and that the attraction due to the corresponding prolate spheroid
is, at its pole,
47iy>a(l e'
2e ' 1-e
21. Show that the attraction at the point (c, 0, 0), due to
the homogeneous solid bounded l>y the planes x = a, x = 6, and
by the surface generated by the revolution about the axis of x
of the curve y =/(#) , is
2 7r pf 5 jl -- ^^ - I
y. i [(c-ao'+c/s) 8 ]*)
dx.
- 22. Prove that the attraction of a uniform lamina in the form
of a rectangle, at a point P in the straight line drawn through
the centre of the lamina at right angles to its plane, is
i ab
4 //, sin
where 2 a and 2 6 are the dimensions of the lamina and c the
distance of P from its plane. [See Todhunter's Analytical
Statics.']
THE NEWTONIAN POTENTIAL FUNCTION. 29
CHAPTER II.
THE NEWTONIAN POTENTIAL JUNCTION IN THE CASE
OP GRAVITATION.
19. Definition. If we imagine an attracting body M to be
cut up into small elements, and add together all the fractions
formed by dividing the mass of each element by the distance of
one of its points from a given point P in space, the limit of this
sum, as the elements are made smaller and smaller, is called the
value at P of " the potential function due to 3f."
If we call this quantity V, we have
where A??i is the mass of one of the elements and r its distance
from P, and where the summation is to include all the elements
which go to make up M.
If we denote by ~p the average density of the element whose
mass is A??t, and call the coordinates of the corner of this ele-
ment nearest the origin a;', y', z', and those of P, a, y, z, we may
write
Am =
and
= C C C
J J J
_
\_(x'-xy+(y'-y) 2 +(z'-zy]S
where p is the density at the point (V, y', z'} , and where the
triple integration is to include the whole of the attracting mass M.
As the position of the point P changes, the value of the quan-
tity under the integral signs in [47] changes, and in general V
is a function of the three space coordinates, i.e., V=f(x, y,z).
To avoid circumlocution, a point at which the value of the
30 THE NEWTONIAN POTENTIAL FUNCTION
potential function is F is sometimes said to be "at potential
F ." From the definition of Fit is evident that if the value at
a point P of the potential function due to a system of masses
M t existing alone is FI, and if the value at the same point of
the potential function due to another system of masses M~ 2 exist-
ing alone is F 2 , the value at P of the potential function due to
M { and M z existing together is V= Fi + F 2 .
20. The Derivatives of the Potential Function. If P is a
point outside the attracting mass, the quantity
which enters into the expression for V in [47], can never be
zero, and the quantity under the integral signs is finite every-
where within the limits of integration ; now, since these limits
depend only upon the shape and position of the attracting mass
and have nothing to do with the coordinates of P, we may dif-
ferentiate V with respect to either x, ?/, or z by differentiating
under the integral signs. Thus :
= f f f p(x'-x)a
JJJ [(x'xy 2 +(y'
x}dx'dy'dz'
where the limits of integration are unchanged by the differen-
tiation. The dexter integral in this equation is (Section 14)
the value of the component parallel to the axis of x of the
attraction at P due to the given masses, so that we may write,
using our old notation,
D,V=X, [49]
and, similarly, D y V=Y, [50]
D,V=Z. [51]
The resultant attraction at P is
2 , [52]
IN THE CASE OF GRAVITATION. 31
and the direction-cosines of its line of action are :
D V D V D V
cosa = *- , cos/3 = j^-, and cosy = * [53]
UK R
It is evident from the definition of the potential function that
the value of the latter at any point is independent of the par-
ticular system of rectangular axes chosen. If, then, we wish to
find the component, in the direction of any line, of the attraction
at any point P, we may choose one of our coordinate axes
parallel to this line, and, after computing the general value of
V, we may differentiate the latter partially with respect to the
coordinate measured on the axis in question, and substitute in
the result the coordinates of P.
21. Theorem. The results of the last section may be summed
up in the words of the following
THEOREM.
To find the component at a point P, in any direction PAT, of
the attraction due to any attracting mass M, ice may divide the
difference betiveen the values of the potential function due to M at
P 1 (a point beticeen P and K on the straight line PA") and at P
by the distance PP'. The limit approached by this fraction as
P' approaches P is the component required.
"We might have arrived at this theorem in the following way :
If X, I", Z are the components parallel to the coordinate axes
of the attraction at any point P, the component in any direction
PA" whose direction-cosines are A, /A, and v, is
XX + p. V+ vZ = XD Z V+ p.D y V+ i'D, V. [54]
Let x, y, z be the coordinates of P, and x -f A#, y -f Ay,
z -\- Az those of P', a neighboring point on the line PA".
If ^and V are the values of the potential function at P and
P' respectively, we have, by Taylor's Theorem,
V = V+ A* D Z V+ Ay .D, V+ \z - D, V + c,
where e is an infinitesimal of an order higher than the first.
32
THE NEWTONIAN POTENTIAL FUNCTION
but
therefore, pQ
= XD * V + t* D *
and this (see [54]) is the component in the direction PK of
the attraction at P : which was to be proved.
22. The Potential Function everywhere Finite. If P is a
point within the attracting mass, the sum whose limit expresses
the value of the potential function at P contains one apparently
infinite term. That V is not infinite in this case is easily
proved by making P the origin of a system of polar coordinates
as in Section 15, when it will appear that the value of the
potential function at P can be expressed in the form
V P = C C C P rsin6drd0d<l>',
[57]
and this is evidently finite.
Although V P is everywhere finite, yet when we express its
value by means of the equation [47], the quantity under the in-
tegral signs becomes infinite within the limits of integration,
when P is a point inside the attracting mass. Under these cir-
cumstances we cannot assume without further proof that the
result obtained by differentiating with respect to x under the
integral signs is really D X V. It is therefore desirable to com-
IN THE CASE OF GRAVITATION. 33
pute the limit of the ratio of the difference (A X F) between the
values of V at the points P'=(:c-|- Az, ?/, z) and P=(a?, y, z),
both within the attracting mass, to the distance (A#) between
these points. For convenience, draw through P (Fig. 15) three
lines parallel to the coordinate axes, and let Q = (#', y', z').
Let PQ = r, P*Q = r', and X'PQ = f.
Then
r' 2 = ^-^-(AiE) 2 2 r Aa cos i/^,
np' /j*
where cos \/ = ,
-, A...F C C rA 1\ pdx'dy
ana = i | ( -
Ax J J J \r' rj Ax
m rcf(.=-
J J J Vr'r 2 + rr'-
Ax
'2 ? Ax cos \l/ (Ax) *\ p dx' dy' dz'
rr' 2 J Ax
Therefore
D F= ^ m ^
; ' cos ^. [58]
This last integral is evidently the component parallel to the
axis of x of the attraction at P, so that the theorem of Article
21 may be extended to points within the attracting mass.
It is to be noticed that p is a function of a:', y', and z', but not
a function of x, y, and z, and that we have really proved that the
derivatives with regard to x, ?/, and z of
34
THE NEWTONIAN POTENTIAL FUNCTION
where F is any finite, continuous, and single-valued function of
a;', ?/', and z', can always be found by differentiating under the
integral signs, whether (x,y,z) is contained within the limits of
integration or not.
23. The Potential Function due to a Straight Wire. Let
fj. be the mass of the unit length of a uniform straight wire AS
(Fig. 16) of length 21. Take the middle point of the wire for
the origin of coordinates, and a line drawn perpendicular to the
wire at this point for the axis of x.
The value of the potential function at any point P (x, y) in
the coordinate plane is, then, according to [47],
=r
J-i
If r = AP =
and r' = BP =
whence y =
ig
4J
, we may eliminate x and y from [59] and
express V P in terms of r and r'.
Thus:
It is evident from [GO] that if P move so as to keep the sum
of its distances from the ends of the wire constant, V P will
IN THE CASE OF GRAVITATION. 35
remain constant. P's locus in this case is an ellipse whose
foci are at A and B.
From [59] we get
n y _
L-'* r P
= - 1
*L
cos 8 1 cos 8'
= f cos 8 + cos 8' 1,
asj_ J
and this (Section G). is the component in the direction of the
axis of x of the attraction at P.
24. The Potential Function due to a Spherical Shell. In
order to find the value at the point P of the potential function
due to a homogeneous spherical shell of densit}' p and of radii r
and *!, we may make use of the notation of Section 9.
C r f prsinB dr dQ d^ C C CP T d>/ d
= JJJ y~ ~ = J J J 'c
If P lies within the cavity enclosed by the shell, the limits of
y are (r c) and (r-j-c), whence
F=27rp(r 1 2 -r 2 ). [02]
If P lies without the shell, the limits of y are (c r) and
(c + y) 5 whence
F=-7rp (ri3 ~ r 3) - [63]
3 c
If P is a point within the mass of the shell itself, at a dis-
tance c from the centre, we may divide the shell into two parts
36
THE NEWTONIAN POTENTIAL FUNCTION
by means of a spherical surface drawn concentric with the given
shell so as to pass through P. The value of the potential func-
tion at P is the sum of the components due to these portions of
the shell ; therefore
3 c
[64]
If we put these results together, we shall have the following
table :
c>r Q
r <c< i\
r><c
F=
27r/D(r 1 2 r 2 )
'*P\ri -J 3c ^o
4 7Tp f r 3 r 3\
n vi r o )
OC
? \
D C V=
31 o j
\ c ~ /
~ 3c 2 ^ ~
D 2 V =
_47rp/2r ( 3 \
S *P( r s r a\
3 \ c 3 )
3J
If we make F, -D C F, 'and D c 2 Fthe ordinates of curves whose
abscissas are c, we get Fig. 17.*
Here LNQS represents F", and it is to be noticed that this
curve is everywhere finite, continuous, and continuous in direc-
tion. The curve OABC represents D C V. This curve is every-
where finite and continuous, but its direction changes abruptly
when the point P enters or leaves the attracting mass. The
three disconnected lines OA, DE, and FG represent D?V.
If the density of the shell instead of being uniform were a
function of the distance from the centre \_p =/(?')] 5 we should
have at the point P, at the distance c from the centre of the
8 P here '
[65]
?/
See Thomson and Tait's Treatise on Natural Philosophy.
IN THE CASE OF GRAVITATION.
37
From this it follows, as the reader can easily prove, that the
value of the potential function due to a spherical shell whose
density is a Junction of the distance from the centre only is
FIG. 17.
constant throughout the cavity enclosed b}- the shell, and at all
outside points is the same as if the mass of the shell were con-
centrated at its centre.
25. Equipotential Surfaces. As we have already seen, V is, in
general, a function of the three space coordinates [_V=f(x,y,z)\,
and in any given case all these points at which the potential
function has the particular value c lie on the surface whose
equation is T7 - ~,
V=f(x,y,z) = c.
Such a surface is called an " equipotential " or " level " sur-
face. By giving to c in succession different constant values,
the equation V c yields a whole family of surfaces, and it is
always possible to draw through any given point in a field of
force a surface at all points of which the potential function has
the same value. The potential function cannot have two differ-
ent values at the same point in space, therefore no two different
surfaces of the family F"=c, where V is .the potential function
due to an actual distribution of matter, can ever intersect.
105460
38 THE NEWTONIAN POTENTIAL FUNCTION
THEOREM.
If there be any resultant force at a point in space, due to any
attracting masses, this force acts along the normal to that equi-
potential surface on which the point lies.
For, let V=f(x, y,z) = c be the equation of the equipotential
surface drawn through the point in question, and let the coordi-
nates of this point be X Q , y , Z Q . The equation of the, plane
tangent to the surface at the point is
and the direction-cosines of any line perpendicular to this plane,
and hence of the normal to the given surface at the point
cos a = x " , [66 A ]
cos/3 =
and cos y = z<)
But if we denote the resultant force of attraction at the point
(x , 2/ , ZQ) by R, and its components parallel to the coordinate
axes by X, Y, and Z, these cosines are evidently equal to
X Y Z
, , and respectively, so that a, /?, and y are the direction-
R R H
angles not only of the normal to the equipotential surface at the
point (a: , j/ , z ) , but also [35] of the line of action of the re-
sultant force at the point. Hence our theorem.
Fig. 18 represents a meridian section of four of the system
of equipotential surfaces due to two equal spheres whose sec-
tions are here shaded. The value of the potential function due
to two spheres, each of mass M, at a point distant respectively
T! and r 2 from the centres of the spheres, is
IN THE CASE OF GRAVITATION.
39
and if we give to V in this equation different constant values,
we shall have the equations of different members of the system
of equipotential surfaces. Any one of these surfaces ma}* be
easily plotted from its equation by finding corresponding values
FIG. 18.
of r^ and r 2 which will satisfy the equation ; and then, with the
centres of the two spheres as centres and these values as radii,
describing two spherical surfaces. The intersection of these
surfaces, if they intersect at all, will be a line on the surface
required.
If 2 a is the distance between the centres of the spheres,
2 V
V = - gives an equipotential surface shaped like an hour-
a
glass. Larger values of V than this give equipotential sur-
faces, each one of which consists of two separate closed ovals,
one surrounding one of the spheres, and the other the other.
Values of V less than give single surfaces which look more
a
and more like ellipsoids the smaller V is.
Several diagrams showing the forms of the equipotential
surfaces due to different distributions of matter are given at
40 THE NEWTONIAN POTENTIAL FUNCTION
the end of the first volume of Maxwell's Treatise on Electricity
and Magnetism.
26. The Value of V at Infinity. The value, at the point P,
of the potential function due to any attracting mass M has
been defined to be
_ limit
Let r be the distance of the nearest point of the attracting
mass from P, then
or . [67]
r
M
The fraction has a constant numerator, and a denominator
n
which grows larger without limit the farther P is removed from
the attracting masses ; hence, we see that, other things being
equal, the value at P of the potential function is smaller the
farther P is from the attracting matter ; and that if P be moved
away indefinitely, the value of the potential function at P
approaches zero as a limit. In other words, the value of the
potential function at "infinity" is zero.
27. The Potential Function as a Measure of Work. The
amount of work required to move a unit mass, concentrated at
a point, from one position, P t , to another, P 2 , by any path, in
face of the attraction of a system of masses, Jf, is equal to
FIG. 19.
Vi V-i, where V\ and F 2 are the values at PI and P 2 of the
potential function due to M.
To prove this, let us divide the. given path into equal parts
of length As, and call the average force which opposes the
IN THE CASE OF GRAVITATION. 41
motion of the unit mass on its journey along one of these
elements AB (Fig. 19), F. The amount of work required to
move the unit mass from A to B is .FAs, and the whole work
done by moving this mass from P x to P 2 will be
limit "\.^ ^'T-T A
**2**
As As is made smaller and smaller, the average force opposing
the motion along AB approaches more and more nearly the
actual opposing force at A, which is D,V: therefore
limit
As =C
= - f P *A v- ds = r, -
J Pv
It is to be carefully noticed that the decrease in the potential
function in moving from Pj to P 2 measures the work required
to move the unit mass from P l to P 2 . If P 2 is removed farther
and farther from J/, F 2 approaches zero, and V\ V., approaches
Fj as its limit, so that the value at any point Pj, of the poten-
tial function due to any system of attracting masses, is equal
to the work which would be required to move a unit mass, sup-
posed concentrated at P n from Pj to " infinity" by any path.
The work (IF) that must be done in order to move an attract-
ing mass J/' against the attraction of any other mass J/, from
a given position by any path to " infinity," is the sum of the
quantities of work required to move th6 several elements (A?n f )
into which we may divide J/', and this may be written in the
form
limit ' _J>dxd>,dz_
w _
-m*
pp'dxdydzdtfdy'dz 1
CCCCCC pp
= J J J J J J {.(*'-*)*
W is called by some writers "the potential of the mass M'
with reference to the mass J/" ; by others, the negative of ir is
called "the mutual potential energy of Jfand M'."
lu many of the later books on this subject, the word
42 THE NEWTONIAN POTENTIAL FUNCTION
"potential" is never used for the A'alue of the potential func-
tion at a point, but is reserved to denote the work required to
move a mass from some present position to infinity. If V is
the value of the potential function at a point P, at which a
mass m is supposed to be concentrated, mV is the potential
of the mass m. If we could have a unit mass concentrated
at a point, the potential of this mass and the value of tlie poten-
tial function at the point would be numerically identical.
28. Laplace's Equation. We have seen that the value of the
potential function and the component in any direction of the
attraction at the point P are always finite functions of the space
coordinates, whether P is inside, outside, or at the surface of
the attracting masses. We have seen also that by differentiating
V at any point with respect to any direction we may find the
always finite component in that direction of the attraction at
the point. It follows that D X V, D y V, D Z V are everywhere
finite, and that, in consequence of this, the potential function
is everywhere continuous as well as finite.
If P is a point outside of the attracting masses, the quantity
under the integral signs in [48], by which dx'dy'dz' is multi-
plied, cannot be infinite within the limits of integration, and we
can find D^V by differentiating the expression for D X V under
the integral signs.
In this case
Dj* V = C C p^'-ft) 2 -?^ dx'dy'dz', [69]
and similarly,
D? V = 3 (V ' ~?^P dx'dy'dz', [70]
D* V = -~ P dx'dy'dz'. [71]
Whence, for all points exterior to the attracting masses,
=0. [72]
This is Laplace's Equation.
IX THE CASE OF GRAVITATION. 43
The operator (D x 2 + D* + Z>/) is sometimes denoted l)y the
symbol V 2 , so that [72] may be written
V 2 F=0. [73]
The potential function, due to every conceivable distribution
of matter, must be such that at all points in emptv space
Laplace's Equation shall be satisfied.
29. The Second Derivatives of the Potential Function are
Finite at Points within the Attracting Mass. If the point P
lies within the attracting mass, F"aiid D X V are finite, but the
quantity under the integral signs in the expression for D X V
becomes infinite within the limits of integration, and we cannot
assume that D/V may be found by differentiating D Z V under
the integral signs. In order to find D?V under these circum-
stances, it is convenient to transform the equation for D X V*
Let us choose our coordinate axes so as to have all the attract-
ing mass in the first octant, and divide the projection of the
contour of this mass on the plane yz into elements (dy'dz').
Upon each one of these elements let us erect a right prism,
cutting the mass twice or some other even number of times.
Consider one of the elements cly'dz' whose corner next the
origin has the coordinates 0, ?/', and z'. The prism erected on
this element cuts out elements ds^ f?.*>, ds s , c?s 4 , ds 2n from the
surface of the attracting mass and that edge of the prism which
is perpendicular to the plane yz at (0, //', z') cuts into the
surface at points whose distances from the plane of yz are
n <*3i 5? " 2-i? an d out f the surface at points whose dis-
tances from the same plane are a*, a, 6 , (t., n . At every one
of these points of intersection draw normals towards the interior
of the attracting mass, and call the angles which these normals
make with the positive direction of the axis of .r, ,, cu, a 3 . 2n .
It is to be noticed that a,, a n , a 5 , o 2n _! are all acute, and that
a 2 , cz 4 , a 6 , cu,, are all obtuse. The element dy'dz' may be re-
garded as the common projection of the surface elements
44
THE NEWTONIAN POTENTIAL FUNCTION
dsj, ds 2 , ds s , ' ds 2n , and, so far as absolute value is concerned,
the following equations hold approximately :
dy'dz'=
= ds 2 cosa 2 = ds s cosa 3 = = ds 2n cosa 2n .
But cfa/'cZz', ds l5 ds 2 , ds 3 , etc., are all positive areas, and cosa 2 ,
cosa 4 , cosa 6 , etc., are negative, so that, paying attention to
signs as well as to absolute values, we have
dy'dz'=
cosa 4 = etc.
FIG. 20.
Now
J.\.V=
'dy'dz
and in order to find the value of this expression by the
use of the prisms just described, we are to cut each one
of these prisms into elementary rectangular parallelepipeds by
planes parallel to the plane of yz ; we are to multiply the
values of every one of these elements which lies within the
attracting mass by the value of pDJl j at its corner next
V rJ
the origin [i.e., at (a %/ , ?/', z')] ; and we are to find the limit of
the sum of these as dx' is made smaller and smaller. We are
then to compute a like expression for each of the other prisms,
and to find the limit of the sum of the whole as the bases of the
IN THE CASE OF GRAVITATION. 45
prisms are made smaller and smaller and their number corres-
pondingly increased.
Wherever the function is a continuous and finite function
of a?', we have
r r
hence, if the elementary prisms cut the surface of the attracting
mass only twice,
x'= o a
*=,
and, in general,
= f fdy
JJ
+ C CC-DJpdx'dy'dz 1 [76]
Zl Pi P PS
COSajdtej-l -- -COSa.,C?S<H -- COS03<is 3 H ----
\ ? 'l 7 '2 r Z
y'dz', [77]
P* P
where is the value of the quantity - at the point where the
r k " T
line y = y', z = 2;' t cuts the surface of the attracting mass for the
frth time, counting from the plane yz.
In order to find the value of the limit of the sum which occurs
in this expression, it is evident that we may divide the entire sur-
face of the attracting mass into elements, multiply the area of each
element by the value of ^ at one of its points, and find the
r
limit of the sum formed by adding all these products together ;
but this is equivalent to the surface integral of ^ - taken all
over the outside of the attracting mass, so that
D X V= cosads +dx'dy'dz\ [78]
46 THE NEWTONIAN POTENTIAL FUNCTION
where the first integral is to be taken all over the surface of the
attracting mass and the second throughout its volume. This
expression for D x Vis in some. cases more convenient than that
of [48].
We have proved this transformation to be correct, however,
only when is finite throughout the attracting mass. If P is a
point within the mass, - is infinite at P. In this case surround
r
P by a spherical surface of radius e small enough to make the
whole sphere enclosed by this surface lie entirely within the
attracting mass. This is possible unless P lies exactly upon
the surface of the attracting mass. Shutting out the little
sphere, let V 2 be the potential function due to the rest (T 2 ) of
the attracting mass ; then, since P is an outside point with re-
gard to T 2 , we have, by [78],
D x V 2 =f~ cosa da' + f cos a ds + fff~r- dx'dy'dz', [79]
where the first integral is to be extended over the spherical
surface, which forms a part of the boundary of the attracting
FIG. 21.
mass to which V 2 is due ; the second integral is to be taken
over all the rest of the bounding surface of the attracting mass ;
and the triple integral embraces the volume of all the attracting
mass which gives rise to V 2 .
As e is made smaller and smaller, V 2 approaches more and
more nearly the potential function F, due to all the attracting
mass.
/p
- cos a ds', cos a can never be greater than 1
nor less than 1 , so that if ~p is the greatest value of p on the
IN THE CASE OF GRAVITATION. 47
surface of the sphere, the absolute value of the integral must be
P C
less than -I ds' or 47rpe, and the limit of this as e approaches
zero is zero. The second integral in [79] is unaltered by any
change in . If we make P the origin of a system of polar
coordinates, it is evident that the triple integral in [79] may be
written C C C
\ \ \ D x ' P -rs'mOdrdOd<l>, [80]
and the limit which this approaches as e is made smaller and
smaller is evidently finite, for, if r = 0, the quantity under the
integral sign is zero.
Therefore,
JSo D X V 2 = D X V= / cosads + fff^dx'dy'dz', [81]
and [79] is true even when P lies within the attracting mass.
Under the same conditions we have, similarly,
^dx'dy'dz', [82]
and
D z V= cos yds + dx'dy'dz 1 . [83]
Observing that in these surface integrals r can never be zero,
since we have excluded the case where P lies on the surface of
the attracting mass, and that the triple integrals belong to the
class mentioned in the latter part of Section 22, we will differ-
entiate [81], [82], and [83] with respect to x, y, and z respec-
tively, by differentiating under the integral signs. If the results
are finite, we may consider the process allowable.
Performin the work indicated, we have
D*V= Cpcosa-D x (-]ds+ C
J \rj J
C C CD,( } -\ D,' p . dx'dy'dz'. [86]
J J */ '
48 THE NEWTONIAN POTENTIAL FUNCTION
and by making P the centre of a system of polar coordinates
and transforming all the triple integrals, it is easy to show that
the values of D X V, D*V, D?V here found are finite whether
P is within or without the attracting mass. This result* is
important.
30. The Derivatives of the Potential Function at the Surface
of the Attracting Mass. Let the point P lie on the surface
of the attracting mass, or at some other point or surface where
p is discontinuous. Make P the centre of a sphere of radius e,
and call the piece which this sphere cuts out of the attracting
mass T! and the remainder of this mass T 2 . Let Fi and V 2 be
the potential functions due respectively to TI and T 2 ? then
V=V i + V 2 , D x V=D x V i +D x V 2 ,
and the increment [A(Z> Z F")] made in D X V by moving from P
to a neighboring point P', inside Tj, is equal to the sum of the
corresponding increments [A^^Fi) and A(Z> X F 2 )] made in
D x Vi and D,V 3 .
With reference to the space T 2 , P is an outside point, so that
the values at P of the first derivatives of F" 2 with respect to a;,
y, and z are continuous functions of the space coordinates and
Let d<a be the solid angle of an elementary cone whose vertex
is at any fixed point in T t used as a centre of coordinates.
FIG. 22.
The element of mass will be pi^dwdr. The component in the
direction of the axis of x of the attraction at due to 7\ is the
* Lejeune Dirichlet, Vbrlesungen iiber die im umgekehrten Verhdltniss des
Quadrats der Entfernuntj ivirkenden Krafte.
Riemann, Schwere, Electricitdt, und Magnetismus.
IX THE CASE OF GRAVITATION. 49
limit of the sum taken throughout J\ of - , where a is
r
the cosine of the angle which the line joining with the element
in question makes with the axis of x. The difference between
the limits of o> is not greater than 4?r, and the difference be-
tween the limits of r is not greater than 2e. If, then, K is the
greatest value which pa has in T^
It follows from this that if P' is a point within 7\ so that
PP'< e, the change made in D x Vi by going from P to P' is far
less than lG7r*e; but this last quantity can be made as small as
we like by making small enough, so that
limit \ / T\ -IT \ i\
p p / ^ A(Z> I F 1 ) = 0,
whence
limit A / T~> TT"V limit / T~ TT"\ i limit / T~ rr\ /~v
A (D x F) = pp/=o A ( D * F + PP'^o A (A V*) = 0,
and -D^F varies continuously in passing through P. In a similar
manner, it may be proved that D y V and D Z V are everywhere,
even at places where the density is discontinuous, continuous
functions of the space coordinates.
The results of the work of the last two sections are well illus-
trated by Fig. 17. We might prove, with 'the help of a trans-
formation due to Clausius,* that the second derivatives of the
potential function are finite at all points on the surface of the
attracting matter where the curvature is finite, but that these
derivatives generally change their values abruptly whenever the
point P crosses a surface at which p is discontinuous, as at the
surface of the attracting masses. The fact, however, that this
last is true in the special case of a homogeneous spherical shell
suffices to show that we cannot expect the second derivatives
of V to have definite values at the boundaries of attracting
bodies.
* Die Potentialfuncllon nnd das Potential, 19-24.
50
THE NEWTONIAN POTENTIAL FUNCTION
31. Gauss's Theorem. If any closed surface T drawn in a
field or force be divided up into a large number of surface
elements, and if each one of these elements be multiplied by the
component, in the direction of the interior normals of the force
of attraction at a point of the element, and if these products be
added together, the limit of the sum thus obtained is called the
'' surface integral of normal attraction over T"
If any closed surface T be described so as to shut in com-
pletely a mass m concentrated at a point, the surface integral
of normal attraction due to m, taken over T, is 4?rm; and, in
general, if any closed surface T bo described so as to shut in
completely any system of attracting masses J/, the surface in-
tegral over T of the normal attraction due to M is
In order to prove this, divide T up into surface elements,
and consider one of these ds at Q. The attraction at Q in the
direction QO, due to the mass m concentrated at 0, is -^- =
QO 2 r 2
The component of this in the direction of the interior normal is
*w?
cosa, and the contribution which ds yields to the sum whose
m cos a ds
Connect
limit is the surface integral required is
T~
every point of the perimeter of ds with by a straight line,
thus forming a cone of such size as to cut out of a spherical
surface of unit radius drawn about an element c?o>, say. If we
draw about a sphere of radius r = OQ, the cone will intercept
on its surface an element equal to r^-dta. This element is the
IN THE CASE OF GRAVITATION. 51
projection on the spherical surface of ds ; hence d$cosa = rvicu,
approximately, and the contribution of the element ds to our
surface integral is mcZu>. But an elementary cone may cut the
surface more than once ; indeed, any odd number of times. Con-
sider such a cone, one element of which cuts the surface thrice
in SL, S 2 , and S 3 . Let 0/5^ OS 2 , and OS S be called r 1? r 2 , and
r 3 respectively, and let the surface elements cut out of T by the
cone be dsi, eZs 2 , and cZs 3 , and the angles between the line S 3
and the interior normals to T at >Sj, S z , and S 3 be a 1? a 2 , a 3 . It is
to be noticed that when the cone cuts out of T 7 , the corresponding
angle is acute, and that when it cuts in, the corresponding angle
is obtuse, aj and a 3 are acute, and a, 2 obtuse. If we draw about
three spherical surfaces with radii r l5 r 2 , and r 3 respectively,
the cone will cut out of these the elements rfdw, r.?dw. and
r s -d<a. In absolute size, c/Sj = ?- 1 2 c?w secoj, ds. 2 = ?vdo> seca 2 , and
ds 3 = r^dta seca 3 , approximately, but ds. 2 and rdw are both posi-
tive, being areas, and seca 2 is negative. Taking account of
sign, then, cZs 2 = i" 2 dw sec a 2 , and the cone's three elements
3 - ield to the surface integral of normal attraction the quantity
/ ds, COS a, . d* COS ao . C^., COS a,\ , 7 , 7 , ,
(m - + - ) = m (aw w + aw) = m w.
v -f >v v y
However many times the cone cuts T, it will yield ??ic7o) to the
surface integral required : all such elementary cones will yield
then m ^ (7w = m-lw if T is closed, and, in general, ra0, when
is the solid angle which T subtends at 0.
If, instead of a mass concentrated at a point, we have any
distribution of masses, we may divide these into elements, and
apply to each element the theorem just proved ; hence our gen-
eral statement.
If from a point without a closed surface T an elementary
cone be drawn, the cone, if it cuts T 7 at all, will cut it an even
number of times. Using the notation just explained, the con-
tribution which any such cone will yield to the surface integral
taken over T of a mass m concentrated at is
52 THE NEWTONIAN POTENTIAL FUNCTION
/ dSi COS ci! . ds 2 COS a 2 , ds 3 COS a 3 , ds 4 COS a 4 \
( 2 2 ' T2 ' 2 I " ' J
\ 1 2 4 /
= m( da) -j- dco dw + d<o ) = m- = 0,
and the surface integral over any closed surface of the normal
attraction due to any system of outside masses is zero.
The results proved above may be put together and stated in
the form of a
THEOREM DUE TO GAUSS.
If there be any distribution of matter partly within and partly
without a dosed surface T, and if M be the sum of the masses
which T encloses, and M' the sum of the masses outside T, the
surface integral over T of the normal attraction N toward the
interior, due to both M and M', is equal to ^irM. If V be the
potential function due to both M and M', ive have
(*Nds = CD H
It is easy to see that if a mass M be supposed concentrated
on the surface of any closed surface T whose curvature is every-
where finite, the surface integral of normal attraction taken
over Twill be 2TrM ; for all the elementary cones which can be
drawn from a point P in the surface so as to cut Tonce or some
other odd number of times lie on one side of the tangent plane
at the point, and intercept just half the surface of the sphere of
unit radius whose centre is P.
From Gauss's Theorem it follows immediately that at some
parts of a closed surface situated in a field of force, but en-
closing none of the attracting mass, the normal component of
the resultant attraction must act towards the interior of the
surface and at some parts toward the exterior, for otherwise
the limit of the sum of the intrinsically positive elements of the
surface, each one multiplied by the component in the direction
of the interior normal of the attraction at one of its own points,
could not be zero. In other words, the potential function,
whose rate of change measures the attraction, must at some
IN THE CASE OF GRAVITATION. 53
parts of the surface increase and at others decrease in the direc-
tion of the interior normal.
From this it follows that the potential function cannot have a
maximum or a minimum value at a point in empty space ; for
if al such a point Q the potential function had a maximum value,
we could surround Q by a small closed surface, at every point
of which the potential function would increase in the direction
of the interior normal, and this would be inconsistent with the
fact that the surface integral of normal attraction taken over
the surface, which would contain no matter, must be zero.
Similarly it may be shown that the potential function cannot
have a minimum value at a point in empty space.
If the potential function be constant over a closed surface
which contains none of the attracting mass, it has the same
value throughout the interior ; for if this were not the case,
some point or region Q within T would have a value greater or
less than the surrounding region, and we could enclose Q b}* a
closed surface to which we could apply the course of reasoning
just used to show that V cannot attain a maximum value at a
point in empty space.
32. Tubes of Force. A line which cuts orthogonally the dif-
ferent members of the system of equipotential surfaces cor-
responding to any distribution of matter is called a ''line of
force," since its direction at each point of its course shows the
direction of the resultant force at the point. If through all
points of the contour of a portion of an equi potential surface
lines of force be drawn, these lines lie on u surface called a
FIG. 24.
"tube of force." "\Ve may easily apply Gauss's Theorem to a
space cut out and bounded by a portion of a tube of force and
two equipotential surfaces ; for the sides of the tube do not con-
54 THE NEWTONIAN POTENTIAL FUNCTION
tribute anything to the surface integral of normal attraction, and
the resultant force is all normal at points in the equipotential
surfaces. If w and w' are the areas of the sections of a tube of
force made by two equipotential surfaces, and if F and F' are
the average interior forces on o> and w', we have
F<a+F'u' = [87]
if the tube encloses empty space, and
F<a+F'w'=4;irm [88]
when the tube encloses a mass m of attracting matter.
33. Spherical Distributions. In the case of a distribution
about a point in spherical shells, so that the density is a
function of the distance from this point only, the lines of force
are straight lines whose directions all pass through the central
point. Every tube of force is conical, and the areas cut out of
different equipotential surfaces by a given tube of force are pro-
portional to the square of the distance from the centre.
Consider a tube of force which intercepts an area ty from a
spherical surface of unit radius drawn with O as a centre, and
apply Gauss's Theorem to a box cut out of this tube by two
equipotential surfaces of radii r and (?* + Ar) respectively.
Let AOB (Fig. 25) be a section of the tube in question.
The area of the portion of spherical surface w which is repre-
sented in section at ad is r 2 ^, and the area of that at be is
(r-f- Ar) 2 i/f. If the average force acting on w toward the inside
cf the box is F, the average force acting on w' toward the inside
of the box will be (F-\- A r jF), and the surface integral of
normal attraction taken all over the outside of the box is
r + Ar) V = -/" M^' r 2 ) , [89]
IN THE CASE OF GRAVITATION. 55
If the tube of force which we have been considering be ex-
tended far enough, it will cut all the concentric layers of matter,
traverse all the empty regions between the layers, if there are
such, and finally emerge into outside space.
If we choose r so that the box shall contain no matter, the
surface integral taken over the box must be zero.
In this case,
therefore, F= ^ [90]
and F= --+/*. [91]
From this it follows that in a region of empty space, either
included between the two members of a system of concentric
spherical shells of density depending only upon the distance
from the centre, or outside the whole system, the force of attrac-
tion at different points varies inversely as the squares of the
distances of these points from the centre.
Suppose that the box (abed) lies in a shell whose density is
constant ; then the surface integral of normal attraction taken
over the box is equal to 4^- times the matter within the box. In
this case the quantity of matter inside the box is
"- "- or
where e is an infinitesimal of an order higher than the first.
Therefore,
whence F=-' + ^, [92]
-
and V=--- ~ P r + /*. [93]
r o
56 THE NEWTONIAN POTENTIAL FUNCTION
If the box lies in a shell whose density is inversely propor-
tional to the distance from the centre, we shall have
limit A r (JV?) _ _ 4 AA rg -.
Ar = Ay L. J ' L^J
whence F=Zir\ + [95]
and V=- -2irXr + /t. [96]
In general, if the box lies in a shell whose density is /(r) , we
shall have
r, [97]
whence F = 4 - ij f/( r ) ^ . dr. [98]
r r* /
In order to learn how to use the results just obtained to de-
termine the force of attraction at any point due to a given
spherical distribution, let us consider the simple case of a single
shell, of radii 4 and 5, and of density [Ar] proportional to the
distance from the centre.
At points within the cavity enclosed by the shell we must
have, according to [90] and [91],
F= and V=-+p;
r 2 r
But the force is evidently zero at the centre of the shell, where
r is zero, so that c must be zero everywhere within the cavitv,
and there is no resultant force at any point in the region. The
value, at the centre, of the potential function due to the shell is
evidently
5 [99]
= C
and it has the same value at all other points in the cavity.
In the shell itself it is easy to see that we must have at all
points,
F^-icXi* and V=---^+p.'. [100]
~
IX THE CASE OF GRAVITATION. 57
In order to determine the constants in this equation, we may
make use of the fact that F and V are everywhere continuous
functions of the space coordinates, so that the values of F and
V obtained by putting r = 4, the inner radius of the shell, in
[100], must be the same as those obtained by making r = 4 in
the expressions which give the values of F and V for the cavity
enclosed by the shell. This gives us
c' = 2567rX and ^' = ^2^,
3
so that for points within the mass of the shell we have
F= -TT\r>, [101]
r 2
and
For points without the shell we have the same general expres-
sions for F and V as for points within the cavity enclosed by
the shell, namely,
F=\ and F=-- + m, [103]
r r
but the constants are different for the two regions.
Keeping in mind the fact that F and V are continuous, it is
easy to see that we must get the same result at the boundary of
the shell, where r= 5, whether we use [103], or [101] and [102].
This gives
k = 3G9 TT\ and m = ;
so that for all points outside the shell we have
, -IT SGOvrA. n (i-i
and V = 110-.)]
r
These last results agree with the statements made in Section
13, for the mass of the shell is 3GD-A.
The values, at every point in space, of the potential function
aud of the attraction due to any spherical distribution may be
58
THE NEWTONIAN POTENTIAL FUNCTION
found by determining, first, with the aid of Gauss's Theorem,
the general expressions for F and V in the several regions ;
then the constants for the 'innermost region, remembering that
there is no resultant attraction at the centre of the Sj'stem ; and
finally, in succession (moving from within outwards), the con-
stants for the other regions, from a consideration of the fact
that no abrupt change in the values of either F or V is made by
crossing the common boundary of two regions.
This method of treating problems is of great practical im-
portance.
34. Cylindrical Distributions. In the case of a cylindrical
distribution about an axis, where the density is a function of
the distance from the axis only, the equipotential surfaces are
concentric cylinders of revolution ; the lines of force are straight
lines perpendicular to the axis ; and every tube of force is a
wedge.
If we apply Gauss's Theorem to a box shut in between two
equipotential surfaces of radii r and ? + Ar, two planes perpen-
dicular to the axis, and two planes passing through the axis,
" FIG. 26.
we have, if ij/ is the area of the piece cut out of the cylindrical
surface of unit radius by our tube of force,
to = r'if/, CD' = (r + Ar) i/r,
and for the surface integral of normal attraction taken over the
box,
F<o + F'<' = ilf* r (r.F). [10G]
If our box is in empty space,
so that
F= and F=clogr + /.
[107]
IN THE CASE OF GRAVITATION.
If the box is within a shell of constant density p,
\jr \(r F) = 4 7n/< pr Ar,
so that F=Z 2irr and V=
59
[108]
35. Poisson's Equation. Let us now apply Gauss's Theorem
to the case where our closed surface is that of an element of
volume of an attracting mass in which p is either constant or a
continuous function of the space coordinates. We will consider
three cases, using first rectangular coordinates, then cylinder
coordinates, and finally spherical coordinates.
FIG. 27.
I. In the first case, our element is a rectangular parallelepiped
(Fig. 27). Perpendicular to the axis of a; are two equal sur-
faces of area Ay-Az, one at a distance x from the plane yz, and
one at a distance x + Ax from the same plane. The average
force perpendicular to a plane area of size A//Az, parallel to the
plane yz, and with edges parallel to the axes of ?/ and z, is evi-
dently some function of the coordinates of the corner of the
element nearest the origin.
That is, if P=(.r, ?/, z), the average force on PP^ parallel to
the axis of x is X=f(x, ?/, z), and the average force on PiP 7 in
the same direction is /(o; + A.T, ?/, z) = X + A x X, so that PP^
and P\P- yield towards the surface integral of interior-normal
attraction taken over the element, the quantitv A.rAyAz
A.r
Similarly, the other two pairs of elementary surfaces yield
60
THE NEWTONIAN POTENTIAL FUNCTION
A#AvAz 2 and
A?/
*
Az
, and, if p is the average
density of the matter enclosed by the box, we have
[A X" AY" A Z~}
1 z 1 = 47rp n AceA?/Az. [109]
Ace. Ay Az J
This equation is true whatever the size of the element Ace A?/ Az.
If this element is made smaller and smaller, the average nor-
mal force [X] on PP 4 approaches in value the force [-D X F] at
P in the direction of the axis of x ; Y and Z approach respec-
tively the limits Z^Fand D X V; and p approaches as its limit
the actual density [p] at P.
Taking the limits of both sides of [109], after dividing by
AccAyAz, we have
or V'F=-47rp, [110]
which is Poisson's Equation. The potential function due to any
conceivable distribution of attracting matter must be such that
at all points within the attracting mass this equation shall be
satisfied.
For points in empty space p = 0, and Poisson's Equation
degenerates to Laplace's Equation.
II. In the case of cylindrical coordinates, the element of vol-
ume (Fig. 28) is bounded by two cylindrical surfaces of revo-
FIG. 28.
lution having the axis of z as their common axis and radii r and
r + A? 1 , two planes perpendicular to this axis and distant Ax
IN THE CASE OF GRAVITATION.
61
from each other, and two planes passing through the axis and
forming with each other the diedral angle A0.
Call R, 0, and Z the average normal forces upon the elemen-
tary planes PP G , PP 2 , and PP S respectively, then the surface
interal of normal attraction over the volume element will be
[111]
[112]
= 47r/o (vol. of box) ;
whence, approximately,
r A0
vol. of box
The force at Pin direction PP 5 is D r V, in direction PP 4 is D t V,
and perpendicular to LP in the plane PLP^ is - D F, so that
if the box is made smaller and smaller, our equation approaches
the form ^_^ [113]
Fio. 29.
III. In the case of spherical coordinates, the volume element
is of the shape shown in Fig. 29. Let OP=r, ZOP=6, and
62 THE NEWTONIAN POTENTIAL FUNCTION
denote by < the diedral angle between the planes ZOP and
ZOX. Denote by R, , and 3? the average normal forces on the
faces PP 6 , PP 3 , and PP 2 respectively ; then the surface integral
of normal attraction over the elementary box is approximately
- sin0A0A</> \(rR) r A0 Ar A^ r A< Ar . A fl (sin0 )
= 4irp .(vol. of box) ; [114]
whence ^. M^) + _j_. A,^ + _J_. A,(sin^.@)
r 2 Ar r sin A< r sin A0
vol. of box [U5]
The force at P in the direction PP 5 is Z> r F, in the direction
PPi is -- D& V. and in the direction PP 4 is --D 6 V; there-
r sm r
fore, as the element of volume is made smaller and smaller, our
equation approaches the form
= -4^ r 2 sin 6. [116]
This equation, as well as that for cylinder coordinates, might
have been obtained by transformation from the equation in
rectangular coordinates.
36. Poisson's Equation in the Integral Form. In [109] X
may be regarded as a function of x, ?/, 2, Ay, and Az, which ap-
proaches Z^Fas a limit when Ay and Az are made to approach
zero, and it may not be evident that the limit, when Aic, Ay, and
Az are together made to approach zero, of the fraction - is
^jhtV
D X 2 V. For this reason it is worth while to establish Poisson's
Equation by another method.
It is shown in Section 29 that the volume integral of the
AA
quantity D x ( - ), taken throughout a certain region, is the sur-
IN THE CASE OF GRAVITATION. 63
face integral of ^coso. taken all over the surface which bounds
r
the region. In this proof we might substitute for - any other
function of the three space coordinates which throughout the
region is finite, continuous, and single-valued, and state the
results in the shape of the following theorem :
If T is any closed surface and U a function of x, y, and z
which for every point inside T has a finite, definite value which
changes continuously in moving to a neighboring point, then
ff CD I U-dxdydz= Cl T cosads, [117]
f f Cn y U- dx dydz=- Cucos fids, [11 8]
f f CDJJ- dx dijdz = - ClJcosyds, [119]
and
where a, /?, and y ai-e the angles made by the interior normals
at the various points of the surface with the positive direction
of the coordinate axes, and where the sinister integrals are to be
extended all through the space enclosed by T, and the dexter
integrals all over the bounding surface.
If we apply this theorem to an imaginary closed surface which
shuts in any attracting mass of density either uniform or vari-
able, and if for U in [ 1 1 7] , [118], and [H'J] we use respectively
D^V, D y V, and D,V, and add the resulting equations together,
we shall have
fff WF+ D;V+ D?V)<lxudz
= - f ( D z Fcos a 4- D y Vcos ft + D, Vcos 7 )ds. [ 1 20]
The integral in the second member of this equation is evi-
dently (see ["><>]) the surface integral of normal attraction taken
over our imaginary closed surface, and this by Gauss's Theorem
is equal to TT times the quantity of matter inside the surface,
so that
64 THE NEWTONIAN POTENTIAL FUNCTION
f C '('(D*V+D*V+ D;V) dxdydz
= 4:* C C Cpdxdydz. [121]
Since this equation is true whatever the form of the closed
surface, we must have at every point
XV V+ D y 2 V+D*V=-4; TT/).
9
For if throughout any region V Fwere greater than kirp, we
might take the boundary of this region as our imaginary surface.
In this case every term in the sum whose limit gives the sinister
of [121] would be greater than the corresponding term in the
dexter, so that the equation would not be true. Similar reason-
ing shuts out the possibility of VF's being less than 4-irp.
37. The Average Value of the Potential Function on a Spheri-
cal Surface. If, in a field of force due to a mass m concentrated
at a point P, we imagine a spherical surface to be drawn so as
to exclude P, the surface integral taken over this surface of the
value of the potential function due to m is equal to the area of
the surface multiplied by the value of the potential function at
the centre of the sphere.
To prove this, let the radius of the sphere be a and the dis-
tance [OP] of P from its centre c. Take the centre of the
sphere for origin and the line OP for the axis of x. Divide the
surface of the sphere into zones by means of a series of planes
cutting the axis of x perpendicularly at intervals of Ax % . The
area of each one of these zones is 2-n-adx, so that the surface
integral of is
dx _ f"
-2caj
and the value of this, since the radical represents a positive
... . 47ra 2 m , . , ...
quantity, is , which proves the proposition.
IN THE CASE OF GRAVITATION. 65
The surface integral of the potential function taken over the
sphere divided by the area of the sphere is often called "the
average value of the potential function on the spherical sur-
face."
If we have any distribution of attracting matter, we may
divide it into elements, apply the theorem just proved to each
of these elements, and, since the potential function due to the
whole distribution is the sum of those due to its parts, assert
that:
The average value on a spherical surface of the potential
function due to any distribution of matter entirely outside the
sphere is equal to the value of the potential function at the
centre of the sphere.
It follows, from this theorem, that if the potential function is
constant within any closed surface S drawn in a region T, which
contains no matter, so as to shut in a part of that region, it will
have the same value in those parts of T which lie outside S.
For, if the values of the potential function at points in empty
space just outside S were different from the value inside, it would
always be possible to draw a sphere enclosing no matter whose
centre should be inside /5, and which outside S should include
only such points as were all at either higher or lower potential
than the space inside S ; but in this case the value of the poten-
tial function at the centre of the sphere would not be the average
of its values over its surface.
The value of the potential function cannot be constant in un-
limited empty space surrounding an attracting mass 37, for, if
it were, we could surround the mass by a surface over which the
surface interal of normal attraction would be zero instead of
The average value on a spherical surface of the potential
function [F], due to any distribution [J/"] of attracting matter
wholly within the surface, is the same as if M were concen-
trated at the centre of the space which the surface encloses.
For the average values [T", and T^,-|-A r T',] of T" on con-
centric spherical surfaces of radii r and r -f Ar may be written
66 THE NEWTONIAN POTENTIAL FUNCTION
- ( Vds (or I FcZw, if d<a is the solid angle of an ele-
4irr 2 J 4-irJ
mentary cone with vertex at O, which intercepts the element ds
from the surface of a sphere of radius r), and I (F+A r F)do> ;
47T*/
whence A r F = I A r V- do>,
4:TrJ
and D r V =
Now I D r V' a 2 do) is the integral of normal attraction taken
over the spherical surface, whence, by Gauss's Theorem,
^ rr . n
and F = +0,
r
since V = 0, for r = oo .
38. The Equilibrium of Fluids at Rest under the Action of
Given Forces. Elementary principles of Hydrostatics teach us
that when an incompressible fluid is at rest under the action of
any system of applied forces, the hydrostatic pressure p at the
point (cc, y, z) must satisfy the differential equation
dp = P (Xdx + Ydy + Zdz}, [122]
where X, Y, and Z are the values at that point of the force
applied per unit of mass to urge the liquid in directions parallel
to the coordinate axes.
For, if we consider an element of the liquid [AaAyAz]
(Fig. 27) whose average density is p and whose corner next
the origin has the coordinates (&, y, z) , and if we denote by p x
the average pressure per unit surface on the face PP 2 P 4 P 3 , by
p x + A x p x the average pressure on the face P 1 P 5 P 7 P M and by
X n the average applied force per unit of mass which tends to
move the element in a direction parallel to the axis of x, we
have, since the element is at rest,
= (p x
A ,
or
Aa;
IN THE CASE OF GRAVITATION. 67
If the element be made smaller and smaller, the first side of
the equation approaches the limit pX, and the second side the
limit D;P, where p is the hydrostatic pressure, equal in all direc-
tions, at the point P.
This gives us D x p = P X. [1 23]
In a similar manner, we may prove that
D y p = P Y,
and D 2 p = pZ;
whence dp = D x p dx -f D^p cly + D z pdz
= p(Xdx + Ydy + Zdz} .
If in any case of a liquid at rest the only external force
applied to each particle is the attraction due to some outside
mass, or to the other particles of the liquid, or to both together,
X, Y, and Z are the partial derivatives with regard to x, y, and
2 of a single function F, and we may write our general equation
in the form
dp = p(D z V- dx + D y V- dy + D Z V- dz) = p. dV,
whence, if p is constant,
p = p V+ const., [124]
and the surfaces of equal hydrostatic pressure are also equi-
potential surfaces.
According to this, the free bounding surfaces of a liquid at
rest under the action of gravitation only are equipotential.
EXAMPLES.
1. Prove that a particle cannot be in stable equilibrium under
the attraction of any system of masses. [Earnshaw.]
2. Prove that if all the attracting mass lies without an equi-
potential surface S on which V= c, then V= c in all space
inside S.
3. Prove that if all the attracting mass lies within an equi-
potential surface S on which V=C, then in all space outside AS
the value of the potential function lies between C and 0.
68 THE NEWTONIAN POTENTIAL FUNCTION
4. The source of the Mississippi River is nearer the centre of
the earth than the mouth is. What can be inferred from this
about the slope of level surfaces on the earth?
5. If in [59] x be made equal to zero, V becomes infinite.
How can you reconcile this with what is said in the first part of
Section 22?
6. Are all solutions of Laplace's Equation possible values of
the potential function in empty space due to distributions of
matter ? Assume some particular solution of this equation
which will serve as the potential function due to a possible dis-
tribution and show what this distribution is.
7. If the lines of force which traverse a certain region are
parallel, what may be inferred about the intensity of the force
within the region ?
8. The path of a material particle starting from rest at a
point P and moving under the action of the attraction of a given
mass Mis not in general the line of force due to M which passes
through P. Discuss this statement, and consider separately
cases where the lines of force are straight and where they are
curved.
9. Draw a figure corresponding to Figure 17 for the case of
a uniform sphere of unit radius surrounded by a concentric
spherical shell of radii 2 and 3 respectively.
10. Draw with the aid of compasses traces of four of the
equipotential surfaces due to. two homogeneous infinite cylinders
of equal density whose axes are parallel and at a distance of
5 inches apart, assuming the radius of one of the cylinders to
be 1 inch and that of the other to be 2 inches.
11. Draw with the aid of compasses meridian sections of
four of the equipotential surfaces due to two small homogeneous
spheres of mass m and 2m respectively, whose centres are 4
inches apart. Can equipotential surfaces be drawn so as to lie
wholly or partly within one of the spheres ? What value of the
potential function gives an equipotential surface shaped like
the fio-ure 8 ? Show that the value of the resultant force at the
O
point where this curve crosses itself is zero.
EN THE CASE OF GRAVITATION. 69
12. A sphere of radius 3 inches and of constant density p. is
surrounded by a spherical shell concentric with it of radii 4
inches and 5 inches and of density fir, where r is the distance
from the centre. Compute the values of the attraction and of
the potential function at all points in space and draw curves to
illustrate the fact that V and D r V are everywhere continuous
and that Z> r 2 Fis discontinuous at certain points.
13. A very long cylinder of radius 4 inches and of constant
density /A is surrounded by a cylindrical shell coaxial with it
and of radii 6 inches and 8 inches. The density of this shell is
inversely proportional to the square of the distance from the
axis, and at a point 8 inches from this axis is p. Use the Theo-
rem of Gauss to find the values of F, D r V, and D r 2 V at differ-
ent points on a line perpendicular to the axis of the cylinder at
its middle point. If the value of the attraction at a distance
of 20 inches from the axis is 10, show how to find p.
14. Use Dirichlet's value of D X V, given by equation [78],
to find the attraction in the direction of the axis of x at points
within a spherical shell of radii r and TI and of constant den-
sity p.
15. Are there any other cases except those in which the
density of the attracting matter depends only upon the distance
from a plane, from an axis, or from a central point, where sur-
faces of equal force are also equipotential surfaces? Prove
your assertion.
16. Prove that if a mass MI be divided up into elements, and
if each one of these elements be multiplied by the value at one
of its own points of the potential function F 2 due to another
mass 3/ 2 , the limit of the sum of these infinitesimal products will
be equal to the limit of the sum extended over 3/ 2 of the product
of the masses of its elements by the corresponding values of the
potential function due to 3/j. That is, show that
F 2 .fLV, =
f
.'
where the sinister integral is to embrace all J/i and the dexter
all Jf 2 . [Gauss.]
70 THE NEWTONIAN POTENTIAL FUNCTION
17. Two uniform straight wires of length I and of masses m^
and m 2 are parallel to each other and perpendicular to the line
joining their middle points, which is of length y. Show that
the amount of work required to increase the distance between
the wires to y 2 by moving one of them parallel to itself is
" 72 1 ?/ VZ 2 + 2/ 2 Hog - - I ' ' [Minchin.j
y Jy=vi
18. Show that if the earth be supposed spherical and covered
with an ocean of small depth, and if the attraction of the par-
ticles of water on each other be neglected, the ellipticity of the
ocean spheroid will be given by the equation,
_ The centrifugal force at the equator
9
19. A spherical shell whose inner radius is r contains a mass
m of gas which obeys the Law of Boyle and Mariotte. Find
the law of density of the gas, the total normal pressure on the
inside of the containing vessel, and the pressure at the centre.
20. If the earth were melted into a sphere of homogeneous
liquid, what would be the pressure at the centre in tons per
square foot ? If this molten sphere instead of being homo-
geneous had a surface density of 2.4 and an average density of
5.6, what would be the pressure at the centre on the supposition
that the density increased proportionately to the depth?
21. A solid sphere of attracting matter of mass m and of
radius r is surrounded by a given mass M of gas which obeys
the Law of Boyle and Mariotte. If the whole is removed from
the attraction of all other matter, find the law of density of the
gas and the pressure on the outside of the sphere.
22. The potential function within a closed surface S due to
matter wholly outside the surface has for extreme values the
extreme values upon S.
23. If the potential functions V and V due to two systems
of matter without a closed surface have the same values at all
points on the surface, they will be equal throughout the space
enclosed by the surface.
IN THE CASE OF GRAVITATION. 71
24. The potential function outside of a closed surface due to
matter wholly within the surface has for its extreme values two
of the following three quantities : zero and the extreme values
upon the surface.
25. Prove that if R is the distance from the origin of coordi-
nates to the point P, and if V P is the value at P of the potential
function of any system of attracting masses within a finite dis-
tance of the origin, the limit as R is made infinite of V P -P is
equal to M, the whole quantity of attracting matter.
72 THE POTENTIAL FUNCTION
CHAPTER III.
THE POTENTIAL FUNCTION IN THE CASE OP
KEPULSION.
39. Repulsion, according to the Law of Nature. Certain
physical phenomena teach us that bodies ma}' acquire, by
electrification or otherwise, the property of repelling each other,
and that the resulting force of repulsion between two bodies is
often much greater than the force of attraction which, ac-
cording to the Law of Gravitation, every body has for every
other body.
Experiment shows that almost every such case of repulsion,
however it may be explained physically, can be quantitatively
accounted for by assuming the existence of some distribution of
a kind of" matter," every particle of which is supposed to repel
every other particle of the same sort according to the " Law of
Nature," that is, roughly stated, with a force directly propor-
tional to the product of the quantities of matter in the particles,
and inversely proportional to the square of the distance between
their centres.
In this chapter we shall assume, for the sake of argument,
that such matter exists, and proceed to discuss the effects of
different distributions of it. Since the law of repulsion which
we have assumed is, with the exception of the opposite direc-
tions of the forces, mathematically identical with the law which
governs the attraction of gravitation between particles of pon-
derable matter, we shall find that, b}~ the occasional intro-
duction of a change of sign, all the formulas which we have
proved to be true for cases of attraction due to gravitation
can be made useful in treating corresponding problems in
repulsion.
IN THE CASE OF REPULSION.
73
40. Force at Any Point due to a Given Distribution of
Repelling Matter. Two equal quantities of repelling matter
concentrated at points at the unit distance apart are called
" unit quantities" when they are such as to make the force of
repulsion between them the unit force.
If the ratio of the quantity of repelling matter within a small
closed surface supposed drawn about a point P, to the volume
of the space enclosed by the surface, approaches the limit p when
the surface (always enclosing P) is supposed to be made smaller
and smaller, p is called the "density" of the repelling matter
at P.
In order to find the magnitude at any point P of the force due
to any given distribution of repelling matter, we may suppose
the space occupied by this matter to be divided up into small
elements, and compute an approximate value of this force on the
assumption that each element repels a unit quantity of matter
concentrated at P with a force equal to the quantity of matter
in the element divided by the square of the distance between P
and one of the points of the element. The limit approached by
this approximate value as the size of the elements is diminished
indefinitely is the value required.
FIG. 30.
Let Q (Fig. 30), whose coordinates are x', ?/, z', be the
corner next the origin of an element of the distribution. Let'p
be the density at Q and Ax-'A^/'Az' the volume of the element;
then the force at P due to the matter in the element is approxi-
74 THE POTENTIAL FUNCTION
f) ^VT* &.1I A 2i
mately equivalent to a force of magnitude " acting in
ftf
the direction QP, or a force of magnitude ^ _ acting
Ptf
in the direction PQ. If the coordinates of P are cc, y, 2, the
component of this force in the direction of the positive axis of x
p Ace' Aw' Az' (#' x)
18 FT-i - V* , / , va . / f xv 1a and the f orce at P parallel
[('-z) 2 + (y'- y) 2 + (' ) a ]i
to the axis of a; due to the whole distribution of repelling
matter is
'''' -,
where the triple integration is to be extended over the whole
space filled with the repelling matter. For the components of
the force at P parallel to the other axes we have, similarly,
Y= - C C C p(y'-y)dx'dy'dz' r
J J J i (x >- x y+ (y <-yy +(z <- z yy
and
7= CCC p(z'-z)dx'dy'dz' r .-,
<><
If we denote by V the function
pdx'dy'dz'
which, together with its first derivatives, is everywhere finite
and continuous, as we have shown in the last chapter, it is easy
to see that
X=-D.V, Y=-D y V, Z = -D M V, [127]
Y, [128]
and that the direction-cosines of the line of action of the re-
sultant force at P are
-'-' and -
IN THE CASE OF REPULSION. 75
It follows from this (see Section 21) that the component in
any direction of the force at a point P due to any distribution
M of repelling matter is minus the value at P of the partial
derivative of the function V taken in that direction.
The function Fgoes by the name of the Newtonian potential
function whether we are dealing with attracting or repelling
matter.
In the case of repelling matter, it is evident that the resultant
force on a particle of the matter at any point tends to drive that
particle in a direction which leads to points at which the poten-
tial function has a lower value, whereas in the case of gravita-
tion a particle of ponderable matter at any point tends to move
in a direction along which the potential function increases.
41. The Potential Function as a Measure of Work. It is
easy to show by a method like that of Article 27 that the
amount of work required to move a unit quantity of repelling
matter, concentrated at a point, from P l to P 2 , m face of the
force due to any distribution J/ of the same kind of matter, is
V 2 FI, where V l and F 2 are the values at P l and P., respec-
tively of the potential function due to M. The farther P l is
from the given distribution, the smaller is Fi, and the less does
F 2 F! differ from F 2 . In fact, the value of the potential
function at the point P 2 , wherever it may be, measures the work
which would be required to move the unit quantity of matter by
any path from " infinity" to P 2 .
42. Gauss's Theorem in the Case of Repelling Matter. If a
quantity m of repelling matter is concentrated at a point within
a closed oval surface, the resultant force due to m at any point
on the surface acts toward the outside of the surface instead of
towards the inside, as in the case of attracting matter.
Keeping this in mind, we may repeat the reasoning of Article
31, using repelling matter instead of attracting matter, and sub-
stituting all through the work the exterior normal for the in-
terior normal, and in this way prove that :
76 THE POTENTIAL FUNCTION
If there be any distribution of repelling matter partly within
and partly without a closed surface T, and if M be the whole
quantity of this matter enclosed by T, and M' the quantity out-
side T, the surface integral over T of the component in the di-
rection of the exterior normal of the force due to both M and M'
is equal to 4 irM. If V be the potential function due to M and
Jf', we have
43. Poisson's Equation in the Case of Repelling Matter. If
we apply the theorem of the last article to the surface of a
volume element cut out of space containing repelling matter,
and use the notation of Article 35, we shall find that in the case
of rectangular coordinates the surface integral, taken over the
element, of the component in the direction of the exterior
normal is
[130]
A?/ Az
where X is the average component in the positive direction of
the axis of x of the force on the elementary surface AyAz, and
where Y and Z have similar meanings. It is evident that if
the element be made smaller and smaller, X, Y", and Z will
approach as limits the components parallel to the coordinate
axes of the force at P. These components are D X V, D y V,
and D,V; so that if we divide [130] by AxAyAz and then
decrease indefinitely the dimensions of the element, we shall
arrive at the equation
V 2 F=-47r/>. [131]
By using successively cylinder coordinates and spherical co-
ordinates we may prove the equations
[132]
and
sin
sin 0, [133]
IN THE CASE OF REPULSION. 77
so that Poisson's Equation holds whether we are dealing with
attracting or repelling matter.
44. Coexistence of Two Kinds of Active Matter. Certain
physical phenomena may be most conveniently treated mathe-
matically by assuming the coexistence of two kinds of "matter"
such that any quantity of either kind repels all other matter of
the same kind according to the Law of Nature, and attracts all
matter of the other kind according to the same law.
Two quantities of such matter may be considered equal if,
when placed in the same position in a field of force, they are
subjected to resultant forces which are equal in intensity and
which have the same line of action. The two quantities of
matter are of the same kind if the direction of the resultant
forces is the same in the two cases, but of different kinds if the
directions are opposed. The unit quantity of matter is that
quantity which concentrated at a point would repel with the
unit force an equal quantity of the same kind concentrated at
a point at the unit distance from the first point.
It is evident from Articles 2, 14, and 40 that m units of one
of these kinds of matter, if concentrated at a point (a, ?/, z) and
exposed to the action of m^ ? 2 , m 3 , ... m k units of the same
kind of matter concentrated respectively at the points (x^ 1/1, zj,
(a? z , ?/ 2 , z 2 ), (* 3 , y 3 , z a ), ... (* t , y t , z*), and of m i+1 , w t+2 , ... m n
units of the other kind of matter concentrated respectively at
the points (x k+l , y t+1 , z* + i), (**-j-2 2/*+2' * + *)> (#n V^ O
will be urged in the direction parallel to the positive axis of x
with the force
* + mV^i^, [134]
where r t is the distance between the points (*, y, z) and
(a:,., ?/,., z,.) .
If we agree to distinguish the two kinds of matter from each
o ^
other by calling one kind " positive " and the other kind " neg-
ative," it is easy to see that if every m which belongs to positive
78 THE POTENTIAL FUNCTION
matter be given the plus sign and every m which belongs to
negative matter the minus sign, we may write the last equation
in the form
X=-',
The result obtained by making m in [135] equal to unity is
called the force at the point (#, y, z).
In general, m units of either kind of matter concentrated at
the point (#, y, z) , and exposed to the action of any continuous
distribution of matter, will be urged in the positive direction of
the axis of x by the force
n*ri
; [136]
in this expression, p. the density at (V, ?/', z') , is to be taken
positive or negative according as the matter at the point is
positive or negative : m is to have the sign belonging to the
matter at the point (x, y, z) : and the limits of integration are to
be chosen so as to include all the matter which acts on m.
With the same understanding about the signs of m and of p,
it is clear that the force which urges in any direction s, m units
of matter concentrated at the point (#,?/, z) is equal to in -D 8 V,
where Fis the everywhere finite, continuous, and single-valued
function
r f r p (x 1 - *) dx' dy' dz'
J J J [(x'-x
and that mV measures the amount of work required to bring up
from " infinity" by any path to its present position the m units
of matter now at the point (#, y, z) .
If we call the resultant force which would act on a unit of
positive matter concentrated at the point P "the force at P,"
it is clear that if any closed surface T be drawn in a field of
force due to any distribution of positive and negative matter so
as to include a quantity of this matter algebraically equal to Q,
IN THE CASE OF REPULSION. 79
the surface integral taken over T of the component in the direc-
tion of the exterior normal of the force at the different points of
the surface is equal to 4?rQ.
It will be found, indeed, that all the equations and theorems
given earlier in this chapter for the case of one kind of repelling
matter may be used unchanged for the case where positive and
negative matter coexist, if we only give to p and m their proper
signs.
It is to be noticed that Poisson's Equation is applicable
whether we are dealing with attracting matter or repelling mat-
ter, or positive and negative matter existing together.
EXAMPLES.
1. Show that the extreme values of the potential function
outside a closed surface S, due to a quantity of matter algebrai-
cally equal to zero within the surface, are its extreme values
on S.
2. Show that if the potential function due to a quantity of
matter algebraically equal to zero and shut in by a closed sur-
face S has a constant value all over the surface, then this con-
stant value must be zero.
80
SUKFACE DISTRIBUTIONS.
CHAPTER IV.
SUEPACE DISTRIBUTIONS. -GEEEFS THEOEEM.
45. Force due to a Closed Shell of Repelling Matter. If a
quantity of very finely-divided repelling matter be enclosed in a
box of any shape made of indifferent material, it is evident
from [127] and from the principles of Section 38 that if the vol-
ume of the box is greater than the space occupied by the repel-
ling matter, the latter will arrange itself so that its free surface
will be equipotential with regard to all the active matter in
existence, taking into account any there may be outside the box
as well as that inside. It is easy to see, moreover, that we
shall have a shell of matter lining the box and enclosing an
empty space in the middle.
That any such distribution as that indicated in the subjoined
diagram is impossible follows immediately from the reasoning
of Section 37. For ABC and DEF are parts of the same
equipotential free surface of the matter. If we complete this
surface by the parts indicated by the dotted lines, we shall
enclose a space void of matter and having therefore throughout
a value of the potential function equal to that on the bounding
GREEN'S THEOREM. 81
surface. But in this case all points which can be reached from
by paths which do not cut the repelling matter must be at the
same potential as 0, and this evidently includes all space not
actually occupied by the repelling matter ; which is absurd.
Let us consider, then (see Fig. 32), a closed shell of repelling
matter whose inner surface is equipotential, so that at every
point of the cavity which the shell shuts in, the resultant force,
due to the matter of which the shell is composed and to am-
outside matter there may be, is zero.
Let us take a small portion w of the bounding surface of the
cavity as the base of a tube of- force which shall intercept an
FIG. 32.
area Von an equipotential surface which cuts it just outside the
outer surface of the shell, and let us apply Gauss's Theorem to
the box enclosed by w, o>', and the tube of force. If F' is the
average value of the resultant force on <o', the only part of the
surface of the box which yields anything to the surface integral
of normal force, we have
F'a>' = 4 TT??I,
where m is the quantity of matter within the box. If we multi-
ply and divide by o>, this equation may be written
jp" = i^.^. [137]
If <o be made smaller and smaller, so as always to include a
given point A, w' as it approaches zero will always include a
point B on the line of force drawn through A, and F' will ap-
proach the value F of the resultant force at B.
The shell may be regarded as a thick layer spread upon the
82 SURFACE DISTRIBUTIONS.
inner surface, and in this case the limit of may be consid-
w
ered the value at A of the rate at which the matter is spread
upon the surface. If we denote this limit by <r, we shall have
If B be taken just outside the shell, and if the latter be very
thin, ^Q ("-7) evidently differs little from unity; and we see
that the resultant force at a point just outside the outer sur-
face of a shell of matter, whose inner surface is equipotential,
becomes more and more nearly equal to 4?r times the quantity
of matter per unit of surface in the distribution at that point as
the shell becomes thinner and thinner.
The reader may find out for himself, if he pleases, whether or
not the line of action of the resultant force at a point just out-
side such a shell as we have been considering is normal to the
shell.
It is to be carefully noticed that the inner surface of a closed
shell need not be equipotential unless the matter composing the
shell is finely divided and free to arrange itself at will.
When the shell is thin, and we regard it as formed of matter
spread upon its inner surface, a- is called the "surface density"
of the distribution, and its value at any point of the inner sur-
face of the shell may be regarded as a measure of the amount of
matter which must be spread upon a unit of surface if it is to
be uniformly covered with a layer of thickness equal to that of
the shell at the point in question.
46. Surface Distributions. It often becomes necessary in the
mathematical treatment of physical problems, on the assump-
tion of the existence of a kind of repelling matter or agent, to
imagine a finite quantity of this agent condensed on a surface
in a layer so thin that for practical purposes we may leave the
thickness out of account. If a shell like that considered in the
last section could be made thinner and thinner by compression
GREEN S THEOREM.
83
while the quantity of matter in it remained unchanged, the
volume density (/o) of the shell would grow larger and larger
without limit, and tr would remain finite. A distribution like
this, which is considered to have no thickness, is called a sur-
face distribution.
The value at a point P of the potential function due to
a superficial distribution of surface densit}* a- is the surface
integral, taken over the distribution, of -, where r is the dis-
r
tance from P.
It is evident that as long as P does not lie exactly in the
distribution, the potential function and its derivatives are always
finite and continuous, and the force at any point in any direc-
tion may be found by differentiating the potential function
partially with regard to that direction.
If p were infinite, the reasoning of Article 22 would no
longer apply to points actually in the active matter, and it is
worth our while to prove that in the case of a surface distri-
bution where a- is everywhere finite, the value at a point P of
the potential function due to the distribution remains finite, as
P is made to move normally through the surface at a point of
finite curvature.
To show this, take the point (Fig. 33), where P is to cut
the surface, as origin, and the normal to the surface at as
84 SUKFACE DISTRIBUTIONS.
the axis of x, so that the other coordinate axes shall lie in
the tangent plane.
If the curvature in the neighborhood of is finite, it will be
possible to draw on the surface about a closed line such that
for every point of the surface within this line the normal will
make an acute angle with the axis of x.
For convenience we will draw the closed line of such a shape
that its projection on the tangent plane shall be a circle whose
centre is at and whose radius is V, and we will cut the area
shut in by this line into elements of such shape that their pro-
jections upon the tangent plane shall divide the circle just
mentioned into elements bounded by concentric circumferences
drawn at radial intervals of Aw, and by radii drawn at angular
distances of A^>.
.If a;, 0, are the coordinates of the point P, #', y', z' those
of a point of one of the elements of the area shut in by the
closed line, and a the angle which the normal to the surface
at this point makes with the axis of x, the size of the surface
element is approximately - , where w 2 = 2 /2 -f ?/' 2 , and the
cosa
value at P of the potential function due to that part of the sur-
face distribution shut in by the closed line is
The quantity
(ru a- sec a
cos a -v(x x') 2 + u?
u J
is always finite, for, whatever the value of the quantit\- under
the radical sign in the last expression may be when a;, a;', and u
are all zero, it cannot be less than unity, and therefore Vi must
be finite even when P moves down the axis of x to the surface
itself.
If V and V 2 are the values at P of the potential functions
due respectively to all the existing acting matter and to that
GREEN'S THEOREM. 8-3
part of this matter not lying on the portion of the surface shut
in by our closed line, we have F= Vi + V 2 , and, since P is a
point outside the matter which gives rise to F 2 , the latter is
finite ; so that V is finite.
The reader who wishes to study the properties of the deriva-
tives of the potential function, and their relations to the force
components at points actually in a surface distribution, will find
the whole subject treated in the first part of Riemanu's Schwere,
Electricitiit and Magnetismus.
Using the notation of this section, it is easy to write down
definite integrals which represent the values of the potential
function at two points ou the same normal, one on one side of
a superficial distribution, and at a distance a from it, and the
other on the other side at a like distance, and to show that the
difference between these integrals may be made as small as we
like by choosing a small enough. This shows that the value of
the potential function at a point P changes continuously, as P
moves normally through a surface distribution of finite super-
ficial density. If matter could be concentrated upon a geo-
metric line, so that there should be a finite quantity of matter
on the unit of length of the line, or if a finite quantity of matter
could be really concentrated at a point, the resulting potential
function would be infinite on the line itself, and at the point.
47. The Normal Force at Any Point of a Surface Distribu-
tion. In the case of a strictly superficial distribution on a
closed surface where the repelling matter is free to arrange
itself at will, the inner surface of the matter (and hence the
outer surface, which is coincident with it) is equipotential, and
the resultant force at a point B just outside the distribution is
normal to the surface and numerically equal to 4 ^ times the
surface density at B. This shows that the derivative of tho
potential function in the direction of the normal to the surface
has values on opposite sides of the surface differing by 4 -a-,
and at the surface itself cannot be said to have any definite
value.
86 SURFACE DISTRIBUTIONS.
It is easy, however, to find the force with which the repelling
matter composing a superficial distribution is urged outwards.
For, take a small element to of the surface as the base of a tube
of force, and apply Gauss's Theorem to a box shut in by the
surface of distribution, the tube of force, and a portion to' of
an equipotential surface drawn just outside the distribution.
Let F and F' be the average forces at the points of <a and to'
respectively, then the surface integral of normal forces taken
over the box is F'u'Fu, and this, since the only active
matter is concentrated on the surface of the box (see Section
31), is equal to 27ro- w, where <r is the average surface density
at the points of the element to. This gives us
Now let the equipotential surface of which to' is a part be
drawn nearer and nearer the distribution ; then
lirn =1, lim -F" = 47rtr , and F=
F is the average force which would tend to move a unit quan-
tity of repelling matter concentrated successively at the differ-
ent points of w in the direction of the exterior normal, but the
actual distribution on to is wtr , so that this matter presses on
the medium which prevents it from escaping with the force
27r<r 2 to; and, in general, the pressure exerted on the resisting
medium which surrounds a surface distribution of repelling
matter is at any point 2^0^ per unit of surface, where a- is the
surface density of the distribution at the point in question.
We may imagine a superficial distribution of matter which is
fixed, instead of being free to arrange itself at will. In this
case the surface of the matter will not be in general equipoten-
tial, but, if we apply Gauss's Theorem to a box shut in by a
slender tube of force traversing the distribution, and by two
surfaces drawn parallel to the distribution and close to it, one
on one side and one on the other, we may prove that the
GREEN'S THEOREM.
87
normal component of the force at a point just outside the dis-
tribution differs by 4 TTOT from the normal component, in the same
sense, of the force at a point just inside the distribution on the
line of force which passes through the first point.
48. Green's Theorem. Before proving a very general theorem
due to Green,* of which what we have called Gauss's Theorem
is a special case, we will show that if T is any closed surface
and U a function of x, ?/, and z, which for every point inside T
is finite, continuous, and single-valued,
f f fa, U- dx dy dz = Cu- D n x . cZs,
[140]
where the first integral is to include all the space shut in by T 7 ,
and the second is to be taken over the whole surface, and where
D n x represents the partial derivative of a; taken in the direction
of the exterior normal.
To prove this, choose the coordinate axes so that T shall lie
in the first octant, and divide the space inside the contour of the
Fio. 34.
projection of T on the plane yz into elements of size fb/dz. On
each of these elements erect a right prism cutting T twice or
some other even number of times. Let us call the values of U
at the successive points where the edge nearest the axis of .T of
* George Green, An E*sa>/ on the Application of Mnthf mntiral Analysis to
lite Theories of Elect ricitj and Maynetism. Nottingham, 1828.
88 SURFACE DISTRIBUTIONS.
any one of these prisms cuts T, t/i, U^ U 3 , ... U 2n respectivel} 1 ;
the angles which this edge makes with exterior normals drawn
to T at these points, a l5 03, a 3 , ... cu n Vand the elements which
the prism cuts from the surface T, ds^ ds. 2 , ds s , ...ds 2n . It is
evident that wherever a line perpendicular to the plane yz cuts
into T, the corresponding value of a is obtuse and its cosine
negative, but wherever such a line cuts out of T, the correspond-
ing value of a is acute and its cosine positive.
Keeping this in mind, we shall see that although the base of
a prism is the common projection of all the elements which it
cuts from T, and in absolute value is approximately equal to
an} T one of these multiplied by the corresponding value of cos a,
yet, since dxdy, ds^ ds 2 , etc., are all positive areas and some of
the cosines are negative, we must write, if we take account of signs,
If the indicated integration with regard to x in the left-hand
member of [140] be performed and the proper limits introduced,
we shall have
/ /+
U. A +L\ -],[141]
where the double sign of integration directs us to form a quan-
tity corresponding to that in brackets for every prism which
cuts T, to multiply this bj- the area of the base of the prism,
and to find the limit of the sum of all the results as the bases of
the prisms are made smaller and smaller.
Since we may substitute for dydz any one of its approxi-
mate values given above, we may write the quantity within
the brackets
U\ cos ai dSi + U 2 cos a. 2 ds 2 + U 3 cos a 3 ds s + ,
and this shows that the double integral is equivalent to the sur-
face integral, taken over the whole of T, of 7 cos a, whence we
may write
CCCD x U'dxdydz=Cucosads, [Hi]
GREEN'S THEOREM. 89
where the first integral is to be taken all through the space shut
in by T, and the second over the whole surface.
Let P(x, y, z) be anj point of T, a, /?, and y the angles
which the exterior normal drawn to ? at T> makes with the
coordinate axes, and P' a point on this normal at a distance
An from P. The coordinates of P' are
x 4- An cos a, y -\- An cos/?, z 4- An cos y,
and if W=f(x, y, z) be any continuous function of the space
coordinates,
IF" f(f 11 y\
" p J \- K i Ui z ) ?
W P ' =f(x-\- An cos a, ?/ + An cos/?, z 4- An cosy)
=/(#, y, z) + An cos a D z f+ An cos/? D y f
and 4-AncosyZ) z /+(An) 2 Q,
~\V W *
, p = cos a A/4- cos /? A/+ cos 7 ' -D./+ AH Q,
whence
1 i m = D n Wp = cos a D x f+ cos ft D y f-\- cos y D 2 f. [143]
If, as a special case, W= x, we have Z> n x=cosa; so that
[142] may be written
fffPxU- dxdydz = CuD n 'x da, [144]
which we were to prove.*
Green's Theorem, which follows very easily from this result.
may be stated iu the following form :
If U and V are any two functions of the space coordinates
which together with their first derivatives with respect to these
coordinates are finite, continuous, and single-valued throughout
the space shut iu by any closed surface T, then, if n refers to
an exterior normal,
* This theorem has been virtually proved already in Sections 29 and .'JO.
90 SURFACE DISTRIBUTIONS.
= Cu- D n V- ds - C C Cu- V 2 F- dxdydz [145]
= Cv-D n U-ds- C C Cv-V 2 U-dxdydz, [146]
where the triple integrals include all the space within T and the
single integrals include the whole surface.
Since D X U- D X V=D X (U- D X V)-U-D X 2 V,
we have J I I D x U'D X V- dx dydz
= C CCD X (U' D x V)dxdydz- C C Cu-D x 2 V- dxdydz;
but, from [144],
CC CD x (U-D x V)dxdydz= Cu-D x V> D n x-ds,
whence C C C(D X U-D X V) dxdydz
= Cu- D x V-D n x-ds- CCCU'D*V-dxdydz. [147]
If we form the two corresponding equations for the deriva-
tives with regard to ?/ and z, and add the three together, we shall
obtain an expression which, by the use of [143], reduces im-
mediately to [145], Considerations of symmetry give [146].
If we subtract [146] from [145], we get
f ff ( U- V 2 V- V- V 2 U) dxdydz
[148]
In applying Green's Theorem to such spaces as those marked
T in the adjoining diagrams, it is to be noticed that the walls
of the cavities, marked S' and S", as well as the surfaces,
GREEN S THEOREM.
91
marked S, form parts of the boundaries of the spaces, and that
the surface integrals, which the theorem declares must be taken
FIG. 35.
over the whole boundaries of the spaces, are to be extended
over S' and S" as well as over S. We must remember, how-
ever, that an exterior normal to T at S' points into the cavity C'.
49. Special Cases under Green's Theorem. I. If in [148]
V be the potential function due to any distribution either of
repelling matter or of positive and negative matter existing
together, whether this matter is within or without T 7 , and if
U= 1, we have
and 47T CCC P dxdydz= f[-D M F]<7s. [149]
The triple integral on the left-hand side of the equation is the
whole amount of matter (algebraically considered, where we have
both positive and negative matter) within 7\ and the dexter is
the surface integral taken over T of the force in the direc-
tion of the exterior normal; so that [14D] expresses Gauss's
Theorem.
II. If in [145] we make U equal to F, and let this represent
as before the potential function due to any distribution of actual
matter within or without 7\ we shall have
|" C Cffdxdydz = fv- D n Yds + 4* f ffp Vdxdydz, [150]
where 11 is the resultant force at the point (x, y, z) .
92 SURFACE DISTRIBUTIONS.
III. If in [145] we make U= V= u, any function which
within the closed surface T satisfies the equation V 2 w = 0, we
shall have
C C (\(D x u}-+(D y uY+(D,uy-\dxdydz = fu-D n u.ds.[\5\-\
IV. If in [148] Fis the potential function due to two distri-
butions of active matter, M { inside the closed surface T and M 3
outside it, and if U= - where r is the distance of the point
(#, ?/, z) from a fixed point 0, we must consider separately the
two cases where is respectively without T and within T.
A. If is without T 7 , V 2 f - ] =0 for points within the sur-
w
face. Also, V V= 47iy>, so that
ds - JV- D n () ds = -
FIG. 36.
The triple integral is evidently equal to the value at the point
of the potential function due to M 1 alone. If we call this Fi,
siud notice (see [143]) that D n r at any point of T is the cosine
of the angle 8 between r and the exterior normal to J 7 , we have
D n V , FcosS ,
If 7 1 is a surface eqnipotential with respect to the joint action
of 3/j and J/2, and if we denote by V, the constant value of V
on T 7 , we have
""cos 8 ,
ds =
GREEN'S THEOREM. 93
and it is easy to show, by the reasoning used in Section 31,
that j - ds = 0, whence
J r
V l = --LC*>sZd8. [153]
B. If is a point inside 7 1 , whether or not it is within M^
and -if T is equipotential with respect to the action of M l and
J/ 2 , we will surround by a small spherical surface s' of
radius r', and apply [148] to the space lying inside T and with-
out the spherical surface. In doing so, it is to be noticed that
s' forms part of the boundary of the region we are dealing with,
and that an exterior normal to the region at s' will be an interior
normal of the sphere.
FIG. 37.
we have
2/l\
Since for all points of the region we are considering V [ - 1 = 0.
W
ave
f^T*- C* M _ r./WlW f r D ,,(Tw
J r J r 1 J \rj J \r' J
= _47r C C C^dxdydz; [154]
or, since ds' = r' 2 d<a' , where dot' is the area which the elementary
cone whose base is ds' and vertex intercepts on the sphere
of unit radius drawn about 0,
Uds + V.f c -^ds - r' fa V- do,'-
[155]
94 SURFACE DISTRIBUTIONS.
. It is easily proved, by the reasoning of Section 31, that
'cosS ,
J'
and it is clear that if r' be made smaller and smaller, the third
integral of [155] approaches the limit zero. If V' is the average
value of Fon the surface s',
J V'd w =
and as r' is made smaller and smaller, this approaches the value
47rF , where V is the value of Fat 0. The value, when r 1 is
zero, of the triple integral in [155] is evidently FI, and we
have
f*^ds + 47rF,-47rF = -47rF 1 . [156]
If F 2 is the value at of the potential function due to M, 2
alone, V = Fi + F 2 , so that [156] may be written in the form
V s - F 2 = - -- -ds. [157]
4W r
If T is not equipotential with respect to the action of M\ and
3f 2 , we have
4 TT F, = C-^ds - C VD n (-} ds. [158]
J r c/ \ry
V. If in [148] we make U=-, where r is the distance of
the point (x,y,z) from a fixed point 0, and if F=v, a function
which within the closed surface T satisfies the equation Vv = 0,
we shall have
4 TTV = f vZ> B (T\ ds - f^ ds, [159]
J \rj J r
if is within T, and
f^ds= CvD H (}d8, [160]
J r J \rj
if is outside T.
GREEN'S THEOREM. 95
50. Th? Surface Distributions Equivalent to Certain Volume
Distributions. Keeping the notation of IV. in the last article,
let T be a closed surface equipotential with respect either to
the joint action of two distributions of matter, M l inside T and
J/2 outside it, or (when M 2 equals zero) to the action of a
single distribution within the surface ; and let Fj, F" 2 , and V
be the values of the potential functions due respectively to M l
alone, to M 2 alone, and to 3/i and M. 2 existing together. If a
quantity of matter were condensed on T so as to give at every
D V
point a surface density equal to - , the whole quantity of
4?r
matter on the surface would be
and this, by [149], is equal in amount to J/i. Let us study the
effect of removing MI from the inside of T and spreading it in
a superficial distribution J/,' over 2\ so that the surface density
D V
ut every point shall be In what follows, it is assumed
4 TT
that we have two distributions of matter, one inside the closed
surface and the other outside. It is to be carefully noted, how-
ever, that by putting 3L equal to zero in our equations, all our
results are applicable to the case where we have an equipotential
surface surrounding all the matter, which may be all of one kind
or not.
The value, at any point 0, of the potential function due to
the joint effect of J/ 2 and the surface distribution J//, would be
4 ir
If is an outside point, we have, by [153],
F = F 2 + F 1 ,
so that the effect at any point outside an equipotentinl surface
of a quantity J/, of matter anyhow distributed inside the Mir-
face is the same as that of an equal quantity of matter dis-
tributed over the surface in such a way that the superficial
96 SURFACE DISTRIBUTIONS.
density at ever}- point is "- , where V is the value of the
4 -
potential function due to the joint action of M l and any matter
(M 2 ) that may be outside the surface.
If is an inside point, we have, by [157],
F =F 2 + F,-F 2 = F., [161]
which shows that the joint effect of M 2 and M is to give to all
points within and upon the surface the same constant value of
the potential function which points upon the surface had before
M t was displaced by MJ. If, therefore, Jf/ and M 2 exist without
.Mi, there is no force at any point of the cavity shut in by T;
or, in other words, the force due to M alone is at all points
inside T equal and opposite to that due to M 2 .
If M! and M 2 exist without J/i', the cavity enclosed by T is, in
general, a field of force. MI acts as a screen to shield the space
within T from the action of M 2 .
The surface of MI is equipotential with respect to all the
active matter, so that there is no tendency of the matter com-
posing the surface distribution to arrange itself in any different
manner upon T.
51. The potential function F, due to any distribution of
matter whose volume density p is everywhere finite, satisfies the
following conditions :
(1 ) F and its first space derivatives are everywhere finite and
continuous, and are equal to zero at an infinite distance from
the attracting mass.
(2) If R is the distance from the origin of coordinates to the
point P,
limit / T7 - -n\ -it-
R = ( F P'- R ) = - af '
where M is a definite constant.
(3) Except at the surface of the attracting mass, or at some
other surface where p is discontinuous, .
V 2 F=-47rp,
where p is to be put equal to zero outside of the attracting mass.
GREEN'S THEOREM. 97
It is easy to show from Green's Theorem that for a given
value of p as a function of x, y, and z, only one function which
will satisfy these three conditions exists.
Suppose, for the sake of argument, that there are two such
functions, Faiid F', and put u = V V. It is evident that u
satisfies conditions (1) and (2), and that V~(M) = except where
p is discontinuous. Parallel to each surface of discontinuity,
and very near to it, draw two surfaces, one on each side, so as
to shut in the places where Vu is not zero, and draw a spherical
surface about the origin, using a radius R large enough to
O ' ~ O ~
enclose all the surfaces of discontinuity.
If now we apply [151] to that part of the space inside the
spherical surface and not shut in by the barriers which we have
drawn, and if we notice that each pair of parallel barriers to-
gether yields nothing to the surface integral, we shall have
f f f[(A) 2 + (D, uY + (>^) 2 ] dxdydz = Cu -D n u- ds,
where the dexter integral is to be extended over the spherical
surface only.
If dw is the solid angle of the infinitesimal cone which inter-
cepts the element ds from the spherical surface, we have
I uD n uds = R 2 f uD R ud<a.
Now since u satisfies condition (2) above, it is easy to show
that if we make R grow larger and larger, this surface integral
approaches the value zero as a limit, for u approaches the value
- and D R u the value ^-, so that the whole integral ap-
-/I /t
proaches the value' , which, when R i.s made infinite,
Jv
approaches the value zero.
If we embrace all space in our sphere, we shall have
. ) 2 ] dxdydz = 0,
whence D t u = 0, D f ? = 0, D,u = 0.
98 SURFACE DISTRIBUTIONS.
Therefore u is constant in all space, and since it is zero at
infinity, must be everywhere zero, so that V= V.
52. Thomson's Theorem or Dirichlet's Principle. We will now
prove a theorem* which is usually called Dirichlet's Principle
by Continental writers, but which in English books is generally
attributed to Sir W. Thomson. This theorem, in its simplest
form, asserts that there always exists one, but no other than
this one, function, v, of x, y, z, which (1) is finite, continuous, and
single-valued, together with its first space derivatives, through-
out a given closed region L ; (2) at every point of the region
satisfies the equation V 2 v = 0; and (3) at every point on the
boundary of the region has any arbitrarily assigned value, pro-
vided that this can be regarded as the value at that point of a
single- valued function which has derivatives finite, continuous,
and single-valued all over this boundary.
There is evidently an infinite number of functions which
satisfy the first and third conditions. If, for instance, the equa-
tion of the bounding surface S of the region is F(x, ?/, z) = 0,
and if the value of v at the point (#, y, z) upon this surface is to
be /(#, y, z} , any function of the form
*(, y, z) -F(x, y, z} +/(, y, z)
would satisfy the third condition, whatever finite function $
might be.
If we assign to the function to be found a constant value C
all over S, v ^C will satisfy all three of the conditions given
above.
* Green, An Essay on the Application of Mathematical Analysis to the
Theories of Electricity and 'Magnetism. Gauss, Allgemeine Lehrsdtze in Bezie-
httny an f die im verkehrten Verhaltnissc des Quadrats der Entfernung wirkenden
Anziehunrjs- und Abstossunyskrafte. Thomson, Reprint of Papers on Electro-
statics and Magnetism. Pirichlet, Vbrlesungen iiber die im umgekehrten Ver-
hdltniss des Quadrats der Entfernnncj wirkenden Krafte. Also, Thomson and
Tail's Natural l^n'/oso/ihi/, and several papers by Dirichlet in Crelle's Jour-
nal and in the Comptes Rendus.
GREEN'S THEOREM. 99
If the sought function is to have different values at different
points of /S, and if for u in the integral
Q = (D*uy + (VyuY + (D,w) 2 ] dxdydz,
which is to be extended over the whole of the region, we substi-
tute any one of all the functions which satisfy conditions (1)
and (3), the resulting value of Q will be positive. Some one at
least of these functions (i>) must, however, yield a value of Q
which though positive, is so small that no other one can make Q
smaller. Let h be an arbitrary constant to which some value
has been assigned, and let iv be any function which satisfies
condition (1) and is equal to zero at all parts of $, then
U=v-+-hw will satisfy conditions (1) and (3), and conversely,
there is no function which satisfies these two conditions which
cannot be written in the form U= v -\- hw, where h is an arbi-
trary constant, and iv a function which is zero at S and which
satisfies condition (1).
Call the minimum value of Q due to v, Q c , and the value of Q
due to U, Qv, then
( C C(
(D x v D z w +D y v-D y w + D,v - D f ic) dxdydz
+ h? [( D * w Y + ( D v w Y + ( A') 2 ] dxdydz,
which, since iv is zero at the boundary of the region, may be
written, by the help of Green's Theorem,
Q v - Q t = -2/t C C Cw
Now since Q v is the minimum value of Q, no one of the infi-
nite number of values of Q U Q, formed by changing h and w
under the conditions just named can be negative ; but if V s u
were not everywhere equal to zero within Z/. it would be easy
to choose v: so that the coefficient of '2h in the expression
for Qu Q v should not be zero, and then to choose h so that
Qv Q should be negative. Hence V 2 v is equal to zero through-
100 SURFACE DISTRIBUTIONS.
out L, and there always exists at least one function which satis-
fies the three conditions stated above.
There is only one such function ; for if beside v there were
another u = v + hw, we should have, since the coefficient of h is
zero when V 2 (w) = 0,
and, that Q u may be as small as Q v , JiQ must be zero, whence
either h = or O = 0, and if O = 0, w is zero. Therefore,
u = v, and there is only one function which in any given case
satisfies all the three conditions given above.
By applying the same reasoning to the space outside a closed
surface S and inside a spherical surface of large radius R which
is finally made infinite, it is easy to prove that there always exists
in the space outside a closed surfaced one and only one function
v which (1) has a given value at every point of $, (2) satisfies
the equation V 2/ y = 0, (3) together with its first derivatives, is
finite and continuous outside $, and (4) is such that the limit,
as R becomes infinite, of Rv is a definite, finite constant.
These theorems help us to prove other theorems, of which two
are of considerable interest for us.
I. If a function v=f(x,y,z), together with its first space
derivatives, is finite and continuous in all space outside a sur-
face $, and outside this surface satisfies the equation V 2 t> = 0,
and if v\ r x? + y 2 + z 2 approaches a definite, finite, constant
limit as .the point (x, y, z) moves away from the origin to in-
finity, then this function may be considered to be the potential
function of a surface distribution of matter upon S.
In order to prove this, we will first apply [160] to v', the
function which has on S the same value as v, which inside S is,
with its first derivatives, finite and continuous, and which satisfies
the equation V 2 v'= ; and use the space inside S as our region.
This gives
f
J
where n refers to the exterior normal of S.
GREEN'S THEOREM. 101
If we now apply [159] to the function v, using as a field the
space outside S and within a spherical surface S 1 of large radius
M, drawn about the point as centre, we shall have
s- (T^ f?s + _L fvds' + JL CD a v
J r R-J RJ
where n is made to refer to the same normal as before bv a
change of sign in the first two integrals. If now we combine
the two equations just obtained, and make R infinite, so that tbe
last two integrals of the second equation shall vanish, we shall
have
VO = + -T f(AX-A^) ,
<7rJ r
which is the value at an outside point of the potential function due
to a superficial distribution of surface density (D n v' D n v)
4ir
spread upon S.
It is to be noticed that the letter r refers to a point without
S in each of the last three equations. Instead of one closed
surface we might have several, as it is eas}- to prove by intro-
ducing as many Dirichlet's functions as there are surfaces.
We will state the second theorem, leaving the proof, wh/ch is
almost identical in form with the one just given, for the reader.
II. If a function v' = F(x,y,z} satisfies the equation V 2 f'=
throughout the space enclosed by a closed surface , and within
this space, together with its first derivatives, is everywhere
finite and continuous, it may be considered to be the potential
function within this space of a surface distribution on S.
The superficial density of this distribution will be found to be
J-(D n t,'-A,v),
where v is the function which has the same value on S that ?
has, and outside S satisfies the equation v~('-') = an< -l tnc other
conditions given above.
It follows, from these theorems, that we may assign any con-
tinuously arranged arbitrary values to the potential function at
102 SURFACE DISTRIBUTIONS.
the different points of a closed surface , make these values the
common values on the surface of the functions v and v', and
assert that a distribution of matter on S of surface density
o- = (D n v' D n v) would give rise to a potential function
4?r
having the chosen values on S. In this case v and v' would be
the values in regions respectively without and within S of the
potential function due to this surface distribution. It is, then,
always possible to distribute matter in one and only one way
upon a given closed surface so that the value of the potential
function due to the matter shall have given values all over the
surface.
EXAMPLES.
1. Prove that there always exists one, but no other than this
one, function, v, which, together with its first space derivatives,
is finite, continuous, and single-valued everywhere within a given
region L, has values at the boundary of the region equal to
those of an arbitrarily chosen, finite, continuous, and single-
valued function, f(x, y, 2), and satisfies at every point in L the
equation
D X (K- D x v) + D y (K- D tj v} + D,(K> D z v^ = 0,
where K is a function positive within L.
2. If the potential function due to a certain distribution of
matter is given equal to zero for all space external to a given
closed surface S and equal to <(#, ?/, z), where < is a continu-
ous single-valued function zero at all points of S, for all space
within S ; there is no matter without $, there is a superficial dis-
tribution of surface densit
upon S, and the volume density of the matter within S is
A?*].
[Thomson and Tait.]
ELECTROSTATICS. 103
CHAPTER V.
ELEOTEOSTATIOS.
53. Introductory. Having considered abstractly a few of
the characteristic properties of what has been called "the New-
tonian potential function," we will devote this chapter to a very
brief discussion of some general principles of Electrostatics.
By so doing we shall incidentally learn how to apply to the treat-
ment of practical problems many of the theorems that we have
proved in the preceding chapters.
In what follows, the reader is supposed to be familiar with
such electrostatic phenomena as are described in the first few
chapters of treatises on Statical Electricity, and with the hypoth-
eses that arc given to explain these phenomena.
Without expressing any opinion with regard to the physical
nature of what is cnlled electrification, we shall here take for
granted that whether it is due to the presence of some sub-
stance, or is only the consequence of a mode of motion or of a
state of polarization, we may, without error in our results, use
some of the language of the old "Two Fluid Theory of Elec-
tricity " as the basis of our mathematical work.
The reader is reminded that, among other things, this theory
teaches that :
(1) Every particle of a body which is in its natural state con-
tains, combined together so as to cancel each other's effects at
all outside points, equal large quantities of two kinds of elec-
tricity with properties like those of the positive and negative
" matter" described in Section 44.
(2) Electrification consists in destroying in some way the
equality between the amounts of the two kinds of electricity
which a body, or some part of a body, naturally contains, so
that there shall be an excess or charye of one kind. If the
104 ELECTROSTATICS.
charge is of positive electricity, the body is said to be posi-
tively electrified ; if the charge is negative, negatively electrified.
Either kind of electricity existing uncombined with an equal
quantity of the other kind, is called free electricity.
(3) When a charged body A is brought into the neighborhood
of another body B in its natural state, the two kinds of elec-
tricity in every particle of B tend to separate from each other,
one being attracted and the other repelled by A's charge, and
to move in opposite directions.
In general, a tendency to separation occurs in all parts of the
body, whether it is charged or not, where the resultant electric
force (the force due to all the free electricity in existence) is
not zero. This effect is said to be due to induction.
In our work we shall assume all this to be true, and proceed
to apply the principles stated in Section 44 to the treatment of
problems involving distributions of electricity. We shall find it
convenient to distinguish between conductors, which offer prac-
tically no resistance to the passage of electricity through their
substance, and nonconductors, which we shall regard as prevent-
ing altogether such transfer of electricity from part to part.
54. The Charges on Conductors are Superficial. When elec-
tricity is communicated to a conductor, a state of equilibrium is
soon established. After this has taken place, there can be no
resultant force tending to move any portion of the charge
through the substance of the conductor, for, by supposition, the
conductor does not prevent the passage of electricity through
itself.
Moreover, the resultant electric force must be zero at all
points in the substance of a conductor in electric equilibrium ;
for if the force were not zero at an_y point, electricity would
be produced by induction at, that point, and carried away
through the body of the conductor under the action of the
inducing force.
From this it follows that the potential function V, due to all
the free electricity in existence, must be constant throughout
ELECTROSTATICS. 105
the substance of any single conductor in electric equilibrium,
whether or not the conductor be charged, and whether or not
there be other charged or uncharged conductors in the neigh-
borhood. Different conductors existing together will in general
be at different potentials, but all the points of any one of these
conductors will be at the same potential.
Wherever V is constant, y~I"=0, and hence, by Poisson's
Equation, p = 0, so that there can be no free electricity within
the substance of a conductor in equilibrium, and the whole
charge must be distributed upon the surface. Experiment
shows that we must regard the thickness of charges spread upon
conductors as inappreciable, and that it is best to consider that
in such cases we have to do with really superficial distributions
of electricity, in which the conductor bears a rough analogy to
the cavity enclosed by the thin shells of repelling matter de-
scribed in the preceding chapter.
The surface density at any point of a superficial distribution
of electricity shall be taken positive or negative, according as
the electricity at that point is positive or negative, and the force
which would act upon a unit of positive electricity if it were
concentrated at a point P without disturbing existing distribu-
tions shall be called "the electric force" or "the strength of
the electric field at P."
It is evident, from Sections 45 and 4G, that the electric force
at a point just outside a charged conductor, at a place where
the surface density of the charge is <r, is 47ro-, and that this is
directed outwards or inwards, according as o- is positive or nega-
tive.
In other words, D n V, the derivative of the potential function
in the direction of the exterior normal, is equal to 4 Tnr. and
the value of Fat a point P just outside the conductor is greater
or less than its value within the conductor, according as the
surface density of the conductor's charge in the neighborhood of
P is negative or positive.
It is to be carefully noted that, although the surface of a con-
ductor must always be equipoteutial, the superficial density of
106 ELECTROSTATICS.
the conductor's charge need not be the same at all parts of the
surface. We shall soon meet with cases where the electricity
on a conductor's surface is at some points positive and at others
negative, and with other cases where the sign of the potential
function inside and on a conductor is of opposite sign to the
charge.
It is evident, from the work of Section 47, that the resistance
per unit of area which the nonconducting medium about a con-
ductor has to exert upon the conductor's charge to prevent it
from flying off, is, at a part where the density is cr, 27rcr.
55. General Principles which follow directly from the Theory
of the Newtonian Potential Function. If two different distribu-
tions of electricit}-, which have the same system of equipoten-
tial surfaces throughout a certain region, be superposed so as to
exist together, the new distribution will have the same equipo-
tential surfaces in that region as each of the components. For,
if YI and V 2 , the potential functions due to the two components
respectively, be both constant over any surface, their sum will
be constant over the same surface.
Two distributions of electricity, which have densities every-
where equal in magnitude but opposite in sign, have the same
system of equipoteutial surfaces, and, if superposed, have no
effect at any point in space.
Two distributions of electricity, arranged successively on the
same conductor so that at every point the density of the one
is m times that of the other, have the same system of equipo-
tential surfaces, and the potential function due to the first is
everywhere m times as great as that due to the second.
If the whole charge of a conductor which is not exposed to
the action of any electricity except its own is zero, the super-
ficial density must be zero at all points of the surface, and the
conductor is in its natural state. For if cr is not everywhere
zero, it must be in some places positive and in others negative ;
and, according to the work of the last section, the potential
function F, due to this charge, must have, somewhere outside
ELECTROSTATICS. 107
the conductor, values higher and lower than F , its value in the
conductor itself. But this would necessitate somewhere in empty
space a value of the potential function not lying between T'J) and
0, the value at infinity ; that is, a maximum in empty space if
T^ is positive, and a minimum if T^, is negative ; which is
absurd.
A system of conductors, on each of which the charge is null,
must be in the natural state if exposed to the action of no out-
side electricity. For, by applying the reasoning just used to
that conductor in which the potential function is supposed to
have the value most widely different from zero, we may show
that the surface density all over the conductor is zero, so that
no influence is exercised on outside bodies ; and then, suppos-
ing this conductor removed, we may proceed in the same way
with the system made up of the remaining conductors.
If a charge J/ of electricit}-, when given to a conductor, ar-
ranges itself in equilibrium so as to give the surface density
<r =/(*', y. 2) and to make the potential function V I ' r
' r
constant within the conductor, ;) charge J/", if arranged on the
conductor so as to give at every point the density rr= f(x, >/,z)
would be in equilibrium, for it would give everywhere the poten-
tial function ( I * = T , and this is constant wherever T',
J r
is constant.
Only one distribution of the same quantity of electricity J/on
the same conductor, removed from the influence of all other
electricity, is possible ; for, suppose two different values of sur-
face density possible, cr 1 =j\(x, y, z) and <r 2 =f->(x*y,z), then
o- 2 = f->(x*y*z) is a possible distribution of the charge ^F.
Superpose the distribution o- 2 upon the distribution o-, so that
the total charge shall be equal to zero ; then the surface density
at every point is rr l o- 2 , and this must be zero by what we have
just proved, so that o- : = <r.,.
Since we may superpose on the same conductor a number of
distributions, each one of which is by itself in equilibrium, it is
108 ELECTROSTATICS.
easy to see that if the whole quantity of electricity on any con-
ductor be changed in a given ratio, the density at each point
will be changed in the same ratio.
56. Tubes of Force and their Properties. We have seen that
a unit of positive electricity concentrated at a point P just out-
side a conductor would be urged away from the conductor or
drawn towards it, according as that point on the conductor which
is nearest P is positively or negatively electrified. If we regard
lines of force drawn in an electric field as generated by points
moving from places of higher potential to places of lower poten-
tial, we may say that a line of force proceeds from every point
of a conductor where the surface density is positive, and that a
line of force ends at every point of a conductor where the sur-
face density is negative. No line of force either leaves or
enters a conductor at a point where the surface densit}' is zero,
and no line of force can start at one point of a conductor where
the electrification is positive and return to the same conductor
at a point where the electrification is negative. No line of force
can proceed from one conductor at a point electrified in any way
and enter another conductor at a point where the electrification
Las the same name as at the starting-point. A line of force
never cuts through a conductor so as to come out at the other
side, for the force at every point inside a conductor is zero.
Lines and tubes of force are sometimes called in electrostatics
lines and tubes of " induction."
When a tube of force joins two conductors, the charges Q 15
Q 2 of the portions S^ S 2 which it cuts from the two surfaces are
FIG. 38.
made up of equal quantities of opposite kinds of electricity.
For if we suppose the tube of force to be arbitrarily prolonged
ELECTROSTATICS. 109
and closed at the ends inside the two conductors, the surface
integral of normal force taken over the box thus formed is zero,
for the part outside the conductors yields nothing, since the re-
sultant force is tangential to it, and there is no resultant force
at any point inside a conductor. It follows, from Gauss's
Theorem, that the whole quantity of electricity (Qi-f-Q L .) inside
the box must be zero, or Q, = Q 2 , which proves the theorem.
If <TI and o- 2 are the average values of the surface densities of
the charges on Si and S-, respectively, we have ' 1 >S' 1 = Q 1 and
0-282 = Q%, whence
<r s = -<rif- [162]
A 2
The integral taken over any surface, closed or not, of the
force normal to that surface is called by some writers the flow
of force across the surface in question, and by others the induc-
tion through this surface.
If we apply Gauss's Theorem to a box shut in by a tube
of force and the portions ,, S' 2 which it cuts from any two
equipotential surfaces, we shall have, if the box contains no
electricity,
F 2 S z -Fi>Si=Q, [163]
where FI and F 2 are the average values, over Si and S respec-
tively, of the normal force taken in the same direction (that in
which V decreases) in both cases. In other words, the flow of
force across all equipotential sections of a tube of force con-
taining no electricity is the same, or the average force over an
equipotential section of an empty tube of force is inversely pro-
portional to the area of the section.
When a tube of force encounters a quantity m of electricity
(Fig. 39), the flow of force through the tube on passing this
110 ELECTROSTATICS.
electricity is increased by 4-jrm. If, however, the tube encoun-
ters a conductor large enough to close its end completely, a
charge m will be found on the conductor just sufficient to reduce
to zero the flow of force (/) through the tube. That is,
m = -
47T
It is sometimes convenient to consider an electric field to be
divided up by a system of tubes of force, so chosen that the flow
of force across any equipotential surface of each tube shall be
equal to 4?r. Such tubes are called unit tubes; for wherever
one of them abuts on a conductor, there is always the unit quan-
tity of electricity on that portion of the conductor's surface which
the tube intercepts. In some treatises on electricity the term
" line of force" is used to represent a unit tube of force, as
when a conductor is said to cut a certain number of " lines of
force."
It is evident that m unit tubes abut on a surface just outside
a conductor charged with m units of either kind of electricity,
if the superficial density of the charge has everywhere the same
sign. These tubes must be regarded as beginning at the con-
ductor if m is positive, and as ending there if m is negative.
If a conductor is charged at some places with positive elec-
tricity and at others with negative electricity, tubes of force
will begin where the electrification is positive, and others will
end where the electrification is negative.
It is evident that no tube of force can return into itself.
FIG. 40.
57. Hollow Conductors. When the nonconducting cavity,
shut in by a hollow conductor K (Fig. 40), contains quantities
ELECTROSTATICS. Ill
of electricity (ra^ w 2 , m 3 , etc., or ^ ?n) distributed in any way,
but insulated from A", there is induced on the walls of the cavity
a charge of electricity algebraically equal in quantity, but oppo-
site in sign, to the algebraic sum of the electricity within the
cavity.
Call the outside surface of the conductor S and its charge
3/ , the boundary of the cavity , and its charge 37,., and sur-
round the cavity by a closed surface <S, every point of which lies
within the substance of the conductor, where the resultant force
is zero. Now the surface integral of normal force taken over
S is zero, so that, according to Gauss's Theorem, the algebraic
sum of the quantities of electricity within the cavity and upon
$ { is zero. That is,
j/ ; + Wl + ms + Ml3 -f .. . = 3fi + V (m) = 0, [1 G4]
and this is our theorem, which is true whatever the charge on
S is, and whatever distribution of free electricity there may
be outside K. If the distribution of the electricity within the
cavity be changed by moving m 1? m.>, etc., to different positions,
the distribution of J/j-on S ( will in general be changed, although
its value" remains unchanged.
If K has received no electricity from without, its total charge
must be zero ; that is,
If a charge algebraically equal to M be given to A",
3f o = J/_J/..
The combined effect of ^ (m ), the electricity within the cavity,
and MI, the electricity on the walls of the cavity, is at all points
without Si absolutely null. For, if we apply [!">'>] to S, any sur-
face drawn in the conductor so as to enclose .\. we shall have />,.!'
everywhere zero, since the potential function is constant within
the conductor; this shows that 1'j, the potential function due to
112 ELECTllOSTATICS.
all the electricity within , must be zero at all points without S ;
but S may be drawn as nearly coincident with S { as we please.
Hence our theorem, which shows that, so far as the value of the
potential function in the substance of the conductor or outside
it, and so far as the arrangement of M and of M ', any free
electricity there may be outside K, are concerned, M ( and N (m)
might be removed together without changing anything. The
potential function at all points outside S t is to be found by con-
sidering only M and M' .
If Si happens to be one of the equipotential surfaces of ^ (m)
considered by itself, M i will be arranged in the same way as a
charge of the same magnitude would arrange itself on a con-
ductor whose outside surface was of the shape $ if removed
from the action of all other free electricity.
The potential function ( F 2 ) due to M and M' is constant
everywhere within S ; for if we apply [157] to a surface ,
drawn within the substance of the conductor as near S as we
like, we shall have
F S -F = 0,
whicli proves the theorem.
The potential function within the cavity is equal to F" 2 + Fi,
where Vi is the potential function due to M t and ^ (w) Of these,
F" 2 is, as we have seen, constant throughout K and the cavity
(Section 31) which it encloses, while V\ has different values in
different parts of the cavity, and is zero within the substance of
the conductor.
Suppose now that, by means of an electrical machine, some
of the two kinds of electricity existing combined together in a
conductor within the cavity be separated, and equal quantities
(g) of each kind be set free and distributed in any manner
within the cavity.
The value of Fi within the cavity will probably be different
from what it was before, but F" 2 will be unchanged ; for the
ELECTROSTATICS. 113
quantity of matter in the cavity is unchanged, being now, alge-
braically considered,
so that M t is unchanged, although it may have been differently
arranged on <, in order to keep the value of Fi zero within
the substance of the conductor. If now a part of the free
electricity in the cavity be conveyed to S t in some way, the sub-
stance of the conductor will still remain at the same potential as
before. For, if I units of positive electricity and n units of
negative electricity be thus transferred to S { , the whole quantity
of free electricity within the cavity will be N (m) l + n, and
that on S { will be M { -\-l-n: but these are numerically equal,
but opposite in sign, and the charge on S t , if properly arranged,
suffices, without drawing on M to reduce to zero the value of
Fi in K. Since M and M' remain as before, Y 2 is unchanged,
and the conductor is at the same potential as before. So long
as no electricity is introduced into the cavity from without K,
no electrical charges within the cavity can have any effect out-
side S^
Most experiments in electricity are carried on in rooms, which
we can regard as hollows in a large conductor, the earth. Fg,
the value of the potential function in the earth and the walls of
the room, is not changed by anything that goes on inside the
room, where the potential function is V V l + V 2 . Since we
are generally concerned, not with the absolute value of the poten-
tial function, but only with its variations within the room, and
since Fo remains always constant, it is often convenient to dis-
regard F, altogether, and to call 1\ the value of the potential
function inside the room. When we do this we must remember
that we are taking the value of the potential function in the
earth as an arbitrary zero, and that the value of I", at a point in
the room really measures only the difference between the values
of the potential function in the earth and at the point in ques-
tion. When a conductor A in the room is connected with the
114 ELECTROSTATICS.
walls of the room by a wire, the value of V\ in A is, of course,
zero, and A is said to have been put to earth.
58. Induced Charge on a Conductor which is put to Earth.
Suppose that there are in a room a number of conductors, viz. :
AI charged with M 1 units of electricity, and A 2 , A 3 , A, etc.,
connected with the walls of the room, and therefore at the po-
tential of the earth, which we will take for our zero. If the
potential function has the value p l inside A^ every point in the
room outside the conductors must have a value of the potential
function lying between pj and 0, else the potential function must
have a maximum or a minimum in empty space. If 2h is posi-
tive, there can be no positive electricity on the other conductors ;
for if there were, lines of force must start from these conductors
and go to places of lower potential ; but there are no such places,
since these conductors are at potential zero, and all other points
of the room at positive potentials. In a similar way we may
prove that if 2h is negative, the electricity induced on the other
conductors is wholly positive.
Now let us apply [158] to a spherical surface, drawn so as
to include A l and at least one of the other conductors, but with
radius a so small that some parts of the surface shall lie within
the room. If we take the point at the centre of this surface,
we shall have
47rF 2 = - fD r V-ds + ~ Cvds. [165]
aJ o?J
If M is the whole quantity of electricity within the spherical
surface, there must be a quantity M outside the surface, either
on the walls of the room or on conductors within the room.
The value at of the potential function, V 2 , due to the elec-
M
tricity without the sphere, is less in absolute value than ,
for it could only be as great as this if all the electricity outside
the sphere were brought up to its surface.
By Gauss's Theorem,
ELECTROSTATICS. 115
therefore, f Fds = 47ra[Jtf +aF 2 ]. [166]
Now, if MI is positive, the integral is positive, for all parts of
the spherical surface within the room yield positive differentials,
and all other parts zero, so that the second side of the equation
is positive. But a V. 2 is of opposite sign to M, and is less in
absolute value ; hence, M is positive, and the total amount of
negative electricity induced on the other conductors within the
spherical surface by the charge on A^ is numerically less than
this charge, unless some one of these conductors surrounds A s ;
in which case the induced charge comes wholly on this conduc-
tor, while the other conductors, and the walls of the room, are
free. Some of the tubes of force which begin at A l end on the
walls of the room, provided these latter can be reached from
AI without passing through the substance of any conductor.
59. Coefficients of Induction and Capacity. If a number of
insulated conductors, A^ A M A^ etc., ai - e in a room in the pres-
ence of a conductor A l charged with J/j units of electricity, the
whole charge on each is zero ; but equal amounts of positive and
negative electricity are so arranged by induction on each, that
the potential function is constant- throughout the substance of
every one of the conductors.
Let the values of the potential functions in the system of con-
ductors be PJ, 2hi Pzi P\i etc- Since each conductor except A\ is
electrified, if at all, in some places with positive electricity, and
in others with negative electricity, some lines of force nni-4
start from, and others end at, every such electrified conductor.
so that there must be points in the air about each conductor at
lower and at higher potentials than the conductor itself. But
the value of the potential function in the walls of the room is
zero, and there can be no points of maximum or minimum poten-
tial in empty space ; so that PI must be that value of the poten-
tial function in the room most widely different from zero, and
Pzi PAI P^ e tc-, must have the same sign as p t .
The reader may show, if he likes, that both the negative part
116 ELECTHOSTATICS.
and the positive part of the zero charge of any conductor, ex-
cept An is less than J/i.
Letp u be the value of the potential function in a conductor
A L charged with a single unit of electricity, and standing in
the presence of a number of other conductors all uncharged
and insulated. Then if Pui Pizt Pu-> etc., are, under these cir-
cumstances, the values of the potential functions in the other
conductors, A^ A^ A, etc., the potential functions in these
conductors will be M l p 12 , M 1 p l3 , M^pu, etc., if A l be charged
with Mi units of electricity instead of with one unit. This is
evident, for we may superpose a number of distributions which
are singly in equilibrium upon a set of conductors, and get a
new distribution in equilibrium where the density is the sum of
the densities of the component distributions, and the value of
the resulting potential function the sum of the values of their
potential functions.
If A! be discharged and insulated, and a charge 3/ 2 be given
to A 2 , the values of the potential functions in the different con-
ductors may be written
etc.
If now we give to A l and A 2 at the same time the charges J/i
and M., respectively, and keep the other conductors insulated,
the result will be equivalent to superposing the second distribu-
tion, which we have just considered, upon the first, and the con-
ductors will be respectively at potentials,
MiPii + MaPsLj MiPu + M 2 p<a, M 1 p 13 + M 2 p 2R , etc. [167]
If all the conductors are simultaneously charged with quanti-
ties MI, M 2 , Jfy, J/4, etc., of electricity respectively, the value
of the potential function on A k will be
V k = M lPlk + M 2 p, k + 3/3^3, + + M kPut + M n p HM [108]
"Writing this in the form V k =, a k + M k p kk , we see that if the
charges on all the conductors except A k be unchanged, a* will be
constant, and that every addition of units of electricity to
ELECTROSTATICS. 117
the charge of A k raises the value of the potential function in
it by unity. If we solve the n equations like [168] for the
charges, we shall get n equations of the form
M* = T 7 ! q lt + V, q, k + F 3 q, k + - + V k q kk + - + V n q nk , [169]
where the ^'s are functions of the p's.
If all the conductors except^ are connected with the earth,
3/ A = V k q kk , and q kk is evidently -the charge which, under these
circumstances, must be given to A k in order to raise the value
of the potential function in it by unity. It is to be noticed that
# u and are in general different.
p kk
The charge which must be given to a conductor when all the
conductors which surround it are in communication with the
earth, in order to raise the value of the potential function with-
in that conductor from zero to unity, shall be called the
capacity of the conductor. It is evident that the capacity of a
conductor thus defined depends upon its shape and upon the
shape and position of the conductors in its neighborhood.
60. Distribution of Electricity on a Spherical Conductor.
Considerations of symmetry show that if a charge J/ be given
to a conducting sphere of radius r, removed from the influence
of all electricity except its own, the charge will arrange itself
uniformly over the surface, so that the superficial density shall
be everywhere o- = - o -
The value, at the centre of the sphere, of the potential function
due to the charge 3/on the surface is , and, since the potential
r
function is constant inside a charged conductor, this must be
the value of the potential function throughout the sphere. If M
is equal to r, = 1 ; hence the capacity of a spherical conductor
r
removed from the influence of all electricity except its own. is
numerically equal to the radius of its surface.
118 ELECTROSTATICS.
61. Distribution of a Given Charge on an Ellipsoid. It is
evident from the discussion of bomoaoids in Chapter I. that a
charge of electricity arranged (on a conductor) iu the form of
a shell, bounded by ellipsoidal surfaces similar to each other
(and to the surface of the conductor) , and similarly placed,
would be in equilibrium if the conductor were removed from the
action of all electricity except its own. We may use this prin-
ciple to help us to find the distribution of a given charge on a
conducting ellipsoid.
Let us consider a shell of homogeneous matter bounded by
two similar, similarly placed, and concentric ellipsoidal surfaces,
whose semi-axes shall be respectively a, &, c, and (l-|- a )<^
(1 +a)6, (1 +a)c. If any line be drawn from the centre of
the shell so as to cut both surfaces, the tangent planes to these
two surfaces at the points of intersection will be parallel, and
the distance between the planes is jpa, where p is the length
of the perpendicular let fall from the centre upon the nearer of
the planes.
If p is the volume deusit}- of the matter of which the shell is
composed, the mass of the shell is M=^Trabc [(1 + a) 3 1] p,
and the rate at which the matter is spread upon the unit of sur-
face is, at any point, cr = pS, where 8 is the thickness of the
shell measured on the line of force which passes through the
point in question. Eliminating p from these equations, we have
M8
(1701
If, now, in accordance with the hypothesis that the thickness of
the electric charge on a conductor is inappreciable, we make a
smaller and smaller, noticing that 8 differs from pa by an infini-
tesimal of an order higher than the first, we shall have for a
strictly surface distribution,
0- = -*-. [171]
4irabc
If the equation of the surface of the ellipsoidal conductor is
*L + yL+?L-i
a? ^ W ^ <? "' '
ELECTROSTATICS. 119
we have
! = (ETZT?.
p \a 4 b* c 4 '
and
This last expression shows that, as c is made smaller and
smaller, o- approaches more and more nearly the value
M
[172]
47ra& x 1--
\ a 2 b-
and this gives some idea of the distribution on a thin elliptical
plate whose semi-axes are a and 6.
For a circular plate, we may put a = 6 in the last expression,
which gives
[173]
for the surface density at a point r units distant from the centre
of the plate.
The charge M distributed according to this law on both sides
of a circular plate of radius a raises the plate to potential
dr irM
V = ^ C
a /o
so that'the capacity of the plate is
2. [174]
7T
62. Spherical Condensers. If a conducting sphere A of radius
r (Fig. 41) be surrounded by a concentric spherical conducting
shell B of radii r, and r and charged with m units of electricity
while B is uncharged and insulated, we shall have
(1) the charge m uniformly distributed upon S, the surface
of the sphere ;
(2) an induced charge m (Section 57) uniformly distributed
upon $<, the inner surface of B ;
120 ELECTHOSTATICS.
(3) a charge +m (since the total charge of B is zero) uni-
formly distributed on S , the outer surface of B.
FIG. 41.
The value at the centre of the sphere of the potential function
7J7 *??? 77?
due to all these distributions is V A = ---- 1 -- , and this is
r r t r
the value of V throughout the conducting sphere. The value of
7/i
the potential function in B is V B =
' a
If now a charge M be communicated to jB, this will add itself
to the charge m already existing on S , and the charge on /S, : will
be undisturbed. The values of the potential functions in the
conductors are now
- r m m m + M . rr m + M
If now B be connected with the earth so as to make V B = 0,
the charges on S and S { will be undisturbed, but the charge on
Wb 7/1
S will disappear. V A is now equal to ----
r r t
If A were uncharged, and B had the charge 3f, this charge
would be uniformly distributed upon S , for, since the whole
charge on S is zero, the whole charge on $ ( must be zero also.
It is easy to see that S and /S^must both be in a state of nature,
for if not, lines of force must start from S and end at ,-, and
others start at S t and end at S, which is absurd.
ELECTROSTATICS.
121
If A were put to earth by means of a fine insulated wire
passing through a tiny hole in .B, and if B were insulated aud
charged with M units of electricity, we should have a charge x
on S, a charge x on S^ and a charge M-\-x on S . To find
ic, we need only remember that F A = - -+ -| - = 0, whence
x may be obtained. Ti r '"
If B be put to earth, and A be connected by means of the fine
wire just mentioned, with an electrical machine which keeps its
prime conductor constantly at potential Fi, A will receive a charge
y and will be put at potential Fi. To find y, it is to be noticed
that there is a charge y on 6',, aud no charge on , which is
?/ ?/
put to earth. V A = '- = Fi, whence y may be obtained.
If r = 99 millimeters and r,-= 100 millimeters, y= 9900 FJ.
If a sphere, equal in size to A but having no shell about it,
were connected with the same prime conductor, it too would
receive a charge z sufficient to raise it to potential F t , and z
would be determined by the equation Fi= - If r = 99. we have
r
z = 99 Fi ; hence we see that A, when surrounded by B at
potential zero, is able to take one hundred times as great a
charge from a given machine as it could take if B were removed.
In other words, B increases A's capacity one hundred fold.
A and B together constitute what is called a condenser.
FIG. 42.
If A of the condenser AB, both parts of which are supposed
uncharged, be connected by a fine wire (Fig. 42) with a sphere
122 . ELECTROSTATICS.
A' which has the same radius as A, and is charged to potential
Vit A and A' will now be at the same potential [F 2 ], and A will
have the charge cc, and A the charge y. The total quantity of
electricity on A at first was rFi, so that x + y = rV^ and
v =y=-- +_,
2 ~r~~r i\ r '
whence x and y may be found.
The reader may study for himself the electrical condition of
the different parts of two equal spherical condensers (Fig. 43) ,
FIG. 43.
of which the outer surface S of one is connected with an elec-
tric machine at potential Fi, and the inside of the other, $', is
connected with the earth. The two condensers, which are sup-
posed to be so far apart as to be removed from each other's
influence, illustrate the case of two Leyden jars arranged in
cascade.
63. Condensers made of Two Parallel Conducting Plates.
Suppose two infinite conducting planes A and B to be parallel
to each other at a distance a apart ; choose a point of the
plane A for origin, and take the axis of x perpendicular to the
planes, so that their equations shall be x = and x = a. Let the
planes be charged and kept at potentials V A and V B respectively.
It is evident from considerations of symmetry that the potential
function at the point P between the two planes depends only
upon P's x coordinate, so that
D,V=0, AF=0, n s F=0, D.*V=Q.
ELECTROSTATICS. 123
Laplace's Equation gives, then,
D,*V=0,
whence D Z V= 07, and V=Cx + D.
If x = 0, V= V A ; and if x = a, F= F* ; so that
F=F*-F* + F,, and D Z V=^-
The lines of force are parallel between the planes, and the
surface densities of the charges on A and B are
- and - -- - respectively.
If we take a portion of area S out of the middle of each plate,
Of / "I / TV" \
there will be a quantity of electricity on S A equal to ' - ,
and an equal quantity of the other kind of electricity on S B .
The force of attraction between S A and S B will be 2iro*'S, or
s (V H -V A Y
STT ft 2
If S B be put to earth, the charge that must be given to S A in
order to raise it to potential unity is
S '
In other words, the capacity of S A is inversely proportional to
the distance between the plates.
In the case of two thin conducting plates placed parallel to and
opposite each other, at a distance small compared with their
areas, the lines of force are practically parallel except in the
immediate vicinity of the edges of the plates ;* and we may infer
V B
FIG. 44.
* See Maxwell's Treatise on Electricity and Magnetism, Vol. I. Fig. XII.
124 ELECTROSTATICS.
from the results of this section that the capacity of a condenser
consisting of two parallel conducting plates of area $, separated
by a layer of air of thickness a, when one of its plates is put to
O
earth is very approximately - for large values of -
4?ra a
64. Capacity of a Long Cylinder, surrounded by a Concentric
Cylindrical Shell. In the case of an infinite, conducting cylinder
of radius r f , kept at potential V t and surrounded by a concentric
conducting cylindrical shell of radii r and r', kept at potential
V , we have symmetry about the axis of the cylinder, so that
D$ V= 0, and Laplace's Equation reduces to the form
whence, for all points of empty space between the cylinder and
its shell, V=C + Dlogr.
But F= Vi when r = r^ and V= V when r= r ,
hence V=
and
FIG. 45.
The surface densities of the electricity on the outer surface
of the cylinder and the inner surface of the shell are respectively
ELECTROSTATICS. 125
r '- r - v '~ v >.
4irrlo!f
so that the charge on the unit of length of the cylinder is
F F
! - -, and the charge on the corresponding portion of the
2 log!!
r t
inner surface of the shell is the negative of this. We may find
the capacity of the unit length of the cylinder by putting F" =
and Fj = 1, whence capacity = --
2 log 1
r t
If r in tins expression is made very large, the capacity of the
cylinder will be very small.
In the case of a fine wire connecting two conductors, r, will
be very small, and there will be no conducting shell nearer than
the walls of the room, so that the capacity of such a wire is
plainly negligible.
65. Specific Inductive Capacity. In all our work up to this
time we have supposed conductors to be separated from each
other by electrically indifferent media, which simply prevent
the passage of electricity from one conductor to another. We
have no reason to believe, however, that such media exist. in
nature. Experiment shows, for instance, that the capacity of
a given spherical condenser depends essentially upon the kind
of insulating material used to separate the sphere from its
shell, so that this material, without conducting electricity.
modifies the action of the charges on the conductors. Insu-
lators, when considered as transmitting electric action, are
sometimes called dielectrics.
Whatever may really be the physical natures of the sub-
stances, such as glass, parafline, ebonite, etc., which we com-
monly use as insulators, it has been shown that their behavior
would be fairly well accounted for on the supposition that they
126 ELECTROSTATICS.
are made up of truly insulating matter in which are imbedded,
at little distances from one another, small, conducting par-
ticles. It is evident that every such particle, if lying in a
field of force, would be polarized ; that is, one part would be
charged positively by induction, and the part most remote
from this would be charged negatively, and that these induced
charges would have some influence in determining the values
of the potential function at points in the dielectric and in the
conductors adjacent to it.
Using the notation of Section 62, let the part A of a spheri-
cal condenser be charged with m units of positive electricity
and separated from the part .B, which is put to earth, by a
spherical shell of radii r and r t made up of a given dielectric.
Let us first ask ourselves what the effect of the dielectric would
be if it consisted of extremely thin concentric conducting spheri-
cal shells separated by extremely thin insulating spaces. It is
evident that in this case we should have a quantity m of elec-
tricity induced on the inside of the innermost shell, a quantity
-\-m on the outside of this shell, a quantity m on the inner
surface of the next shell, a quantity +m on the outside of this
shell, and so on. If there were n such shells in the dielectric
layer, and n + 1 spaces, and if 8 were the distance from the
inner surface of one shell to the inner surface of the next,
and AS the thickness of each shell, the value, at the centre of
A, of the potential function due to the charges on these shells,
would be
1 1 1 1
y'=m\
A
r A3 + 8 r + 28 r-
1 1 r
-AS + 28
1 1
r + ?iS
i_
r AS + rtSj
(r AS-f 8) (r+28)(r AS+28)
This quantity lies between
t-n
ELECTROSTATICS. 127
2 g
but these differ from each other by less than e = TO AS ^-^ , so
that mA I , which is easily seen to lie between
Jr y?
G and //, differs from F/ by less than e. If, then, 8 is very
small in comparison with r and r i5 F a ' differs from m\(- -}
Vi r J
by an exceedingly small fraction of its own value.
This shows that the effect, at the centre of A, of such a
system of conducting shells as we have imagined would be
practically the same as if a charge m A were given to the
inner surface of the dielectric, and a charge +?A to its outer
surface, while the charges on the surfaces of the thin shells
within the mass of the dielectric were taken away. That is,
the value of the potential function in A would be
m(l A)( ] instead of
V r 'J
Such a system of shells introduced into what we have hitherto
supposed to be the electrically inert insulating matter between
the two parts of a spherical condenser would increase the capa-
city of the condenser in the ratio of 1 to 1 A. It is to be
noticed that A is a proper fraction : A = and A = 1 would
correspond respectively to a perfect insulator and to a perfect
conductor.
As Dr. E. II. Hall has suggested to me, the result given
above might be easily obtained by computing* the amount of
work done in moving a unit particle of electricity (supposed
to be concentrated at a point, and not to disturb existing dis-
tributions) from A to B. It is easy to see that the force at
any point in the mass of one of the thin conducting shells
would be zero, and that the force at any point in the space
between two shells would be exactly the same as if then' were
no shells in the dielectric. We have no reason to think that
there are any such differences between the values of the force
at contiguous points in the dielectric as this would indicate,
and the conception of the thin shells has been introduced only
* Mascart et Joubert, Lemons sur I'Elcctricite, 124.
128 ELECTROSTATICS.
because the effect of these shells can be more easily computed
thau that of a number of discrete particles.
"When, however, the dielectric between the parts of a spheri-
cal condenser is supposed to contain not a system of continuous
shells, but a number of separate conducting particles, these are
often regarded as forming a series of concentric layers, and it is
assumed that the sum of the charges induced on the inner
sides of the particles in the innermost layer is A'm, where
A' is a proper fraction, larger or smaller in different dielectrics
according as the particles are nearer together or farther apart,
and that the inner surfaces of all the other layers have each
the same charge, and the outer surface of every layer the cor-
responding positive charge -f- X'm. The effect of this kind of
dielectric, if made to replace a perfect insulator in our calcu-
lations, would be to increase the capacity of the condenser in
the ratio 1 to 1 p., where p. = A/A, and it is evident that the
same effect might be produced by a charge p.m on that
surface of the dielectric which touches A, and a charge + p,m
on that surface which is in contact with B.
Experiment shows that dielectrics used to separate and to
surround charged conductors behave, in many respects, as if
every surface in contact with a conductor had a charge opposite
in sign to that of the conductor, and in absolute value p. times
as great, p. being less thau unit}*, and constant for any one
dielectric. That is, if the dielectric separating from each other
a number of conductors be displaced by another, the capacities
of all the conductors will be changed in the same ratio, depend-
ing only upon the natures of the two dielectrics.
The ratio of the fraction , in the case of any dielectric to
1 A*
the same fraction in the case of 'air, for which p. is very
nearly the same as for what we call a vacuum, is called the
specific inductive capacity of the dielectric in question. This
ratio is greater than unity for all solid and liquid dielectrics
with which we are acquainted. The specific inductive capacity
of a perfect conductor would be infinite.
ELECTROSTATICS. 129
The following very clear statement of the effect produced by
changing the dielectric which envelops the parts of a condenser
made of two plates, is due to Dr. Hall, and is copied with his
permission :
"The fundamental fact concerning static electrical induction
as observed by Faraday is this,* that if the two plates of a
condenser, separated by air, receive respectively e l and e.,
units of electricity when charged to a certain difference of
potential, e.g., by connection with the poles of a battery of
many cells in series, the same two plates would, if anv other
medium were substituted for the air, other conditions remaining
unchanged, receive respectively Ke l and Ke units of electric-
ity, K being some quantity greater than unity. This quantity
.fiT is called the specific inductive capacity of the second medium.
" Now, since the difference of potential between A and B is
the same in these two cases, the 'electromotive intensity, 'f ?.e.,
the force exerted upon unit quantity of electricity, is the same
in the two cases at any given point lying in the region through
which the change of dielectric extends. If we were to attempt
to determine the surface densities of the charges of the conduc-
tors by means of the equation \
the values obtained would be the same for both cases. These
would be the actual values of the surface densities if air were
used, but would evidently not be the actual surface densities
for the other case. For this latter case, the values thus found
are called the 'apparent' surface densities, and bear to the
true densities the ratio 1 to A'.
" We must not conclude from this that A and 7? with charges
Ke { and I\e.> respectively in the second medium would act,
* Max well's Treatise on Flectn'riti/ and ^f^/}neti.nn, Art. f>2.
t Maxwell's Treatise nn Electricity and Magnetism, Art. 44.
t Maxwell's Treatise on E/ertriciti/ and Magnetism, First Edition, Art. 83.
See, also, Section 47 of this book.
130 ELECTROSTATICS.
in all electrical respects, like the same bodies with charges e^
and e 2 in air. Two spheres, A and -B, in air, with centres at
distance r from each other, and having charges e x and e 2
respectively, would attract each other with a force 1 2 , whereas
<r
the same two spheres with actual charges Ke^ and Ke 2 in a
medium of specific inductive capacity K would attract each
other with a force* ^ ?. This seems at first inconsistent
r
with the fact that the electromotive intensity at any point, as
stated above, is the same in both cases. The electromotive
intensity at any point, however, meaus the force that would be
exerted upon unit actual quantity of electricity at that point,
not the force that would be exerted upon unit apparent quan-
tity. S6 the average force exerted by A's charge upon B'a
charge in either of our two cases is for each actual unit of
B's charge. Hence, the total force exerted by A upon .B is
-L.2 for the first case, and '~ for the second case, as stated
r r~
before."
66. Charge induced on a Sphere by a Charge at an Outside
Point. The value at any point P of the potential function due
to m 1 units of positive electricity concentrated at a point A^ and
m 2 units of negative electricity concentrated at a point A 2> is
m l m 2
V = --- where ^ = A 1 P and r 2 = A 2 P.
1\ T 2
It is easy to see that if m 1 is greater than w 2 , so that ?% = Xm 2
where A > 1 , V will be equal to zero all over a certain sphere
which surrounds A z .
If (Fig. 46) we let A^=a, A 1 = 8 1 , A 2 = 8 2 , OD = r,
it is eas to see that
X 2 a, a <i a 2 A 2
2 _
n ,
A 2 1 K- 1 (A/ I)
* Maxwell's Treatise on Electricity and Magnetism, Art. 94.
ELECTROSTATICS. 131
and a = 8 -^^^l. [176]
1 2
If PR represents the force / t due to the electricity at A^ and
PQ the force / 2 due to the electricity at A 2 , the line of action of
the resultant force F (represented by PL} must pass through
the centre of the sphere, since the surface of the sphere is equi-
potential.
FIG. 46.
The triangles A 1 PO and A 2 PO are mutually equiangular, for
they have a common angle AfiP, and the sides including that
angle are proportional (r = 8 l &. 2 ). Hence, from the triangles
QPL and A t PA s ^ by the Theorem of Sines,
sn a
"sinC^-a,)'
7 SL ' - C 178 ]
Sin (03 ajy
, -r, af, am, aXm, ri-m
whence F= l = [I'^J
r 2 r 2 rf' 7'j 3
Now, according to Section oO, we ma}- distribute upon the
spherical surface just considered a quantity m 2 of negative elec-
tricity in such a way that the effect of this distribution at all
points outside the sphere shall be equal to the effect of the
charge m 2 concentrated at J.,, and the effect at points within
the sphere shall be equal and opposite to the effect of the charge
mi concentrated at A^. Since F is the force at P in the direc-
132 ELECTROSTATICS.
tion of the interior normal to the sphere, we shall accomplish
this if we make the surface density at every point equal to o-,
where
-(V-r) 2 m 1;
a
and if we now take away the charge at A^ the value of the po-
tential function throughout the space enclosed by our spherical
surface, and upon the surface itself, will be zero. If the spheri-
cal surface were made conducting, and were connected with the
earth by a fine wire, there would be no change in the charge of
the sphere, and we have discovered the amount and the distri-
bution of the electricity induced upon a sphere of radius r, con-
nected with the earth by a fine wire and exposed to the action
of a charge of ?% units of positive electricity concentrated at a
point at a distance 81 from the centre of the sphere.
If now we break the connection with the earth, and distribute
a charge m uniformly over the sphere in addition to the present
distribution, the potential function will be constant (although
no longer zero) within the sphere, and we have a case of equi-
librium, for we have superposed one case of equilibrium (where
there is a uniform charge on the sphere and none at A^) upon
another. The whole charge on the sphere is now
IT in, m ^ ,
and the value of the potential function within it and upon the
surface,
V M .mi_m
r 8 1 r
If the conducting sphere were at the beginning insulated and
uncharged, we should have M = 0, and therefore
- ^ ~ A and F= l - [181]
i / i
If we have given that the conducting sphere, under the influ-
ence of the electricity concentrated at A is at potential Fi, we
ELECTROSTATICS. 133
know that its total charge must be V l r , an( j jtg surface
density
\ rf
It is easy to see that the sphere and its charge will be at-
tracted toward A l with the force
g i /*s ^\a s r
and the student should notice that, under certain circumstances,
this expression will be negative and the force repulsive.
If m l = ?)i 2 , the surface of zero potential is an infinite plane,
and our equations give us the charge induced on a conducting
plane by a charge at a point outside the plane.
The method of this section enables us to find also the capacity
of a condenser composed of two conducting cylindrical surf aces,
parallel to each other, but eccentric ; for a whole set of the
equipotential surfaces due to two parallel, infinite straight lines,
charged uniformly with equal quantities per unit of length of
opposite kinds of electricity, are eccentric cylindrical surfaces-
surrounding one of the lines A.,, and leaving the other line A t
outside. "We may therefore choose two of these surfaces, dis-
tribute the charge of A l on the outer of these, and the charge
of A 2 on the inner, by the aid of the principles laid down in
Section 50, so as to leave the values of the potential function
on these surfaces the same as before. These distributions thus
found will remain unchanged if the equipotential surfaces are
made conducting.
The reader who wishes to study this method more at length
should consult, under the head of Electric Images, the works of
Gumming, Maxwell, Mascart. and Watson and Burbury, as well
as original papers on the subject by Murphy in the Philosophical
Magazine, 1833, p. 350, and by Sir "W. Thomson in the Cam-
bridge and Dublin Mathematical Journal for 1848.
134 ELECTROSTATICS.
67. The Energy of Charged Conductors. If a conductor of
capacity (7, removed from the action of all electricity except its
own, be charged with M l units of electricity, so that it is at
M
potential V\ 1 , the amount of work required to bring up to
C
the conductor, little by little, from the walls of the room, the
additional charge Am, is A W, which is greater than V\ &.M or
AJf, and less than ( FI + A^F) ^ M or
C
If the charge be increased from M l to M 2 by a constant flow,
the amount of work required is evidently
The work required to bring up the charge M to the conductor
at first uncharged is then
M 2 CV 2 MV
2(7 2 2
[185]
This is evidently equal to the potential energy of the charged
conductor, and this is independent of the method by which the
conductor has been charged.
If, now, we have a series of conductors A^ A 2 , vl 3 , etc., in the
presence of each other at potentials FI, F 2 , F 3 , etc., and having
respectively the charges M^ M 2 , M 3 , etc., and if we change all
the charges in the ratio of # to 1, we shall have a new state of
equilibrium in which the charges are xM^ xM 2 , xM 3 , etc. ; and
the values of the potential functions within the conductors are
icFi, o?F 2 , xV s , etc. The work (ATF) required to increase the
charges in the ratio x + Aaj instead of in the ratio x is greater
than
' (M l Aa?) ( FI) -(- (M s Ax) (a; F 2 ) + ( M 2 Aa?) (a; F:,) + etc. ,
or x &x[Mi FI + M. 2 F 2 + M z F 3 + etc.] ,
and less than
(x + Aaj) Aa- [JtfiFi +M 2 V 2 +M 3 V a + etc.] ;
ELECTROSTATICS. 135
hence the whole amount of work required to change the ratio
from to -^ is
W, - W, = x *~ x * [3/ t F + 3/, F 2 + 3/3 F 3 + etc.] . [186]
MI
If in this equation we put x l = and x 2 1 , we get for the
work required to charge the conductor from the neutral state to
potentials F 15 F 2 , F 3 ,
[187]
68. If a series of conductors A^ A. 2 , A 3 , etc., are far enough
apart not to be exposed to inductive action from one another.
and have capacities C^ C 2 , C s , etc., and charges 37, , 3/ 2 , 3/$, etc.,
so as to be at potentials F 1? F 2 , T 7 "., etc., where 3/i=dFi,
3/ 2 = C 2 F 2 , 3/ 3 = (7 3 F 3 , etc., we may connect them together by
means of fine wires whose capacities we may neglect, and thus
obtain a single conductor of capacity
The charge on this composite conductor is evidently
3/ x + 3/ 2 + 3/ 3 + - = ( 3/ ) ;
and if we call the value of the potential function within it F, we
shall have ^-^ ^-^
V'2^(C) =2/30;
whence F= ^^dl^i^l, [188]
^i ~r ^2 ~r t-'s ~r
a formula obtained, it is to be noticed, on the assumption that
the conductors do not influence each other.
The energy of the separate charged conductors before being
connected toether was
V
Jf\
136 ELECTROSTATICS.
and the energy of the composite conductor is
w , = i(3
~r ^2 ~r ^3 ~r
^
[190]
which is always less than E unless the separate conductors were
all at the same potential in the beginning.
EXAMPLES.
1. Show that in general the surface density of a charge dis-
tributed on a conductor is greatest at points where the convex
curvature of the surface of the conductor is greatest.
2. A hollow in a conductor is at the uniform potential Fi
when a charge is communicated to a conductor within the cavity
sufficient to raise this conductor to potential V 2 if it were in
empty space. Give some idea of the changes brought about by
this charge.
3. Show that a field of electric force consists wholly of
non-conductors bounded, if at all, by conducting surfaces.
4. Prove that if a distribution of electricity over a closed
surface produce a force at every point of the surface perpendic-
ular to it, this distribution will produce a potential function con-
stant within the surface.
5. Two conducting spheres of radii 6 and 8 respectively
are connected by a long fine wire, and are supposed not to be
exposed to each other's influences. If a charge of 70 units of
electricity be given to the composite conductors, show that 30
units will go to charge the smaller sphere and 40 units to the
larger sphere, if we neglect the capacity of the wire. Show
t? ^)
also that the tension in the case of the smaller sphere is -
Z007T
per square unit of surface.
ELECTROSTATICS. 137
6. An uncharged sphere A, of radius r, occupies the centre
of the otherwise empty, equipotential cavity, enclosed by a
spherical shell B of radii r, and r , so large that the effect inside
the cavity of the charge induced on B by a charge m, communi-
cated to A from without, may be neglected. If the value of the
potential function within the cavity before A was charged was
O, at what potential is A now ? Ans. C + .
r
7. The first of three conducting spheres, ^.1, B, and (7, of
radii 3, 2, and 1 respectively, remote from one another, is
charged to potential 9. If A be connected with B for an
instant, by means of a fine wire, and if then B be connected
with C in the same way, C"s charge will be 3-G. [Stone.] If,
in the last example, all three conductors be connected at the
same time, C"s charge will be 4-.~>.
8. A charge of M units of electricity is communicated to a
composite conductor made up of two widely-separated ellipsoidal
conductors, of semiaxes 2, 3, 4 and 4, 0, 8 respectively, con-
nected by a fine wire. Show that the charges on the two ellip-
soids will be -M and - 37" respectively. [Stone.]
a ;">
9. Can two electrified bodies repel each other when no lines
of force can be drawn from one body to the other?
10. Two conductors, A and B, connected with the earth are
exposed to the inductive action of a third charged body. Do
A and B act upon each other? If so, how?
11. Show that two equal conductors similarly placed with re-
spect to each other always repel each other if raised to the same
potentials and insulated ; but that if the volume of the potential
function within the conductors differ never so little from each
other, they will repel each other at great distances, hut at very
near distances (supposing no spark to pass) they will attract
each other. [Cummings.]
12. The superficial density has the same sign at all points of
a conducting surface outside which there is no free electricity.
13. Show that r-i-B of the unit tubes of force proceeding
138 ELECTROSTATICS.
from an electrified particle, at a distance 8 from the centre of a
conducting sphere of radius r, which is put to earth, meet the
sphere if there are no other conductors in the neighborhood,
and that the rest go off to "infinity."
14. A charged insulated conductor A is so surrounded by a
number of separate conductors _B, (7, Z>, , which are put to
earth, that no straight line can be drawn from any point of A
to the walls of the room without encountering one of these other
conductors : will there be any induced charge on the walls of
the room? See Section 37.
15. Two uniform straight wires of equal density, each two
inches long, lie separated by an interval of one inch in the
same straight line. Find the equation of the equipotential sur-
faces due to these wires, and find what must be the density of a
superficial distribution of matter on one of these surfaces which
at all outside points would exert the same attraction as the
wires do.
1C. An insulated conducting sphere of radius r charged with
m units of positive electricity is influenced by ra units of posi-
tive electricity concentrated at a point 2r distant from the cen-
tre of the sphere. Give approximately the general shape of the
equipotential surfaces in the neighborhood of the sphere.
Give an instance of a positively electrified body whose poten-
tial is negative.
17. A conductor, the equation of whose surface is
^. + id + *1 = i
25 16 9
is charged with 80 units of electricity ; what is the density at a
point for which x = 3, y 3?
If the density at this point be a, what is the whole charge on
the ellipsoid ?
18. Prove that the capacity of n equal spherical condensers
when arranged in cascade is onlv about -th of the capacitv of
n
one of the condensers ; but that if the inner spheres of all the
ELECTROSTATICS. 139
condensers be connected together by fine wires, and the outer
conductors be also connected together, the capacity of the com-
plex condenser thus found is about n times that of a single
* O
one of the condensers.
19. Prove that if the charges of a system of conductors be
increased, the increment of the energy of the system is equal to
half the sum of the products of the increase in the charge first
conducted into the sum of the values of the potential function
within it at the beginning and the end of the process, or to half
the sum of the products of the increment of the value of the
potential function in each conductor into the sum of the original
and final charges on that conductor. [Maxwell.]
20. Prove that if the charges of a fixed system of conductors
be increased, the sum of the products of the original charge and
the final potential of each conductor is equal to the sum of the
products of the final charge and the original potential. [Max-
well.]
21. Discuss the following passage from Maxwell's Elementary
Treatise on Electricity :
' Let it be required to determine the equipotential surfaces
due to the electrification of the conductor C placed on an insu-
lating stand. Connect the conductor with one electrode of the
electroscope, the other being connected with the earth. Elec-
trify the exploring sphere,* and, carrying it by the insulating
handle, bring its centre to a given point. Connect the elec-
trodes for an instant, and then move the sphere in such a path
that the indication of the electroscope remains zero. This path
will lie on an equipotential surface."
'2'2. Prove that the coefficients of potential (;>) and induction
((j) treated in Article f>l> have the following properties :
(1) The order of the suffixes of any p or any / can be inverted
without altering the value of the coefficient, or. in other words,
Pu = Pu< and v = 7tt-
* A very small conducting sphere fitted with an insulating handle.
140 ELECTROSTATICS.
(2) All thep's are positive, but p lk is less than either p a or J9 M .
(3) Those q's whose two suffixes are the same are positive ;
the others are negative. That is, q kk and q tt are positive ; but
q u is negative and is, moreover, numerically less than either of
the others.
23. Prove that the following theorems (Maxwell's Elemen-
tary Treatise on Electricity) are contained in the statements of
the preceding problem :
(1) In a system of fixed insulated conductors, the potential
function in A k due to a charge communicated to A t is equal to
the potential function in A l due to an equal charge in A k .
(2) In a system of fixed conductors connected, all but one,
with the walls of the room, the charge induced on A k when A l
is raised to a given potential is equal to the charge induced on
AI when A k is raised to an equal potential.
(3) If in a system of fixed conductors, insulated and origi-
nally without charge, a charge be communicated to A k which
raises it to potential unity and A l to potential n, then if in the
same system of conductors a charge unity be communicated
to An and A k be connected with the earth, the charge induced
on A k will be n.
24. A condenser consists of a sphere A of radius 100 sur-
rounded by a concentric shell whose inner radius is 101 and
outer radius 150. The shell is put to earth, and the sphere has
a charge of 200 units of positive electricity. A sphere B of
radius 100 outside the condenser can be connected with the
condenser's sphere by means of a fine insulated wire passing
through a small hole in the shell. B is connected with A ; the
connection is then broken, and B is discharged ; the connection
is then made and broken as before, and B is again discharged.
After this process has been gone through with five times, what is
A's potential? What would it become if the shell were to be
removed without touching A?
25. Suppose the condenser mentioned in the last problem in-
sulated and a charge of 100 units of positive electricity given to
ELECTROSTATICS. 141
the shell. "What will be the potential of the sphere? of the
shell? If we then connect the sphere with the earth by a flue
insulated wire passing through the shell, what will the charge on
the shell be ? What will be the potential of the shell ? If next
A be insulated, and the shell be put to earth, what will be A's
potential ? What will be its potential if the shell be now wholly
removed ?
26. A spherical conductor of radius r is surrounded by a con-
centric conducting spherical shell of radii T^ and 7? , and the
outer surface of this shell is put to earth. If the inner conduc-
tor be charged, show the effect at all points in space of moving
the conductor so that it shall be eccentric with the shell. How
is the capacity of the system changed by this ?
27. Prove that if the spherical surfaces of radii a and 6,
which form a spherical condenser, are made slightly eccentric,
c being the distance between their centres, the change of elec-
. .~ ,. . , ,. ... -
trification at any point of either surface is
where is the Angular distance of the point from the line of
centres, and where the difference between the values of the
potential function on the two surfaces is unity.
28. Show that if an insulated conducting sphere of radius a
be placed in a region of uniform force (X), acting parallel to
the axis of , the function X x\ 1 + C satisfies all the
X x\ 1
L r J
_
conditions which the potential function outside the sphere must
satisfy, and is therefore itself the potential function. Show
that the surface density of the charge on the sphere is - .
[Watson and Burbury.]
142 ELECTROSTATICS.
MISCELLANEOUS PROBLEMS.
1. Prove that the attraction due to a homogeneous hemi-
sphere of radius r is zero, at a point in the axis of the hemisphere
g
distant r approximately from the centre of the base.
2. Show that the attraction at the origin clue to the homo-
geneous solid bounded by the surface obtained by revolving
one loop of the curve r 2 = a? cos 20, is -| -n-a.
3. If the earth be considered as a homogeneous sphere of
radius r, and if the force of gravity at its surface be $, show
that from a point without the earth, at which the attraction is
- g, the area 27rr 2 l - on the surface of the earth
n
will be visible.
4. A spherical conductor A, of radius a, charged with M
units of electricity, is surrounded by n conducting spherical
shells concentric with it. Each shell is of thickness a, and is
separated from its neighbors by empty spaces of thickness a.
Show that the limit approached by V A as n is made larger and
M
larger is (nat. log 2), and that for a finite number of shells
a
y M_ C ll+x ' n+l dx. [Stone.]
a Jo l+x
5. If two systems of matter (M and M'), both shut in by a
closed surface S, give rise to potential functions ( V and F') ,
which have equal values at every point of S, whether or not S
is an equipotential surface of either system, then V cannot
differ from V at any point outside $, and the algebraic sum of
the matter of either system is equal to that of the other. [See
Section 52, and Watson and Burbury's Mathematical Theory of
Electricity and Magnetism, 60.]
6. Show that if two distributions of matter have in common
an equipotential surface which surrounds them both, all their
equipotential surfaces outside this will be common.
ELECTROSTATICS. 143
7. Prove that if V be the potential function clue to any dis-
tribution of matter over a closed surface /S', and if a-' be the
density of a superficial distribution on /S', which gives nse to
the same value of the potential function at each point of S as
that of a unit of matter concentrated at any given point 0,
then the value at O of the potential function due to the first
distribution is I V-u-'-dS.
||KSS of
gtrbich
Boston.
'S3
Peirc,'s Three and Four Place Tables of Loga-
rithinic anil Trigonometric Functions. By JAMKS Mn.i.s PKIRCE,
University Professor of Mathematics in Harvard University. Quarto.
Cloth. Mailing Trice, 45 cts. ; Introduction, 40 cts.
Four-place tables require, in the long run, only half as much time
+ five-place tables, one-third as much time as six-place tables, and
one-fourth as much as those of seven places. They are sufficient
for the ordinary calculations of Surveying, Civil, Mechanical, and
Mining Engineering, and Navigation; for the work of the Physical
or Chemical Laboratory, and even for many computations of Astron-
omy. They are also especially suited to be used in teaching, as they
illustrate principles as well as the larger tables, and with far less
expenditure of time. The present compilation has been prepared
with care, and is handsomely and clearly printed.
Elements of the Differential Calculus.
With Numerous Examples and Applications. Designed for Use as a
College Text-Book. By W. E. BYERLY, Professor of Mathematics,
Harvard University. 8vo. 273 pages. Mailing Price, $2.15 ; Intro-
duction, 2.00.
This book embodies the results of the author's experience in
teaching the Calculus at Cornell and Harvard Universities, and is
intended for a text-book, and not for an exhaustive treatise. Its
peculiarities are the rigorous use of the Doctrine of Limits, as a
foundation of the subject, and as preliminary to the adoption of the
more direct and practically convenient infinitesimal notation and
nomenclature; the early introduction of a few simple formulas and
methods for integrating: a rather elaborate treatment of the. use of
infinitesimals in pure geometry : and the attempt to excite and keep
up the interest of the student by bringing in throughout the whoie
book, and not merely at the end, numerous applications to practical
problems in geometry and mechanics.
James Mills Peirce, Prof, of
Math., H'irvard I'uiv. (From the Har-
is general without being superficial ;
limited to leading topics, and yet with-
vard Register) : In mathematics, as in in its limits; thorough, accurate, and
other branches of study, the need is , practical ; adapted to the communica-
now very much felt of teaching which tion of some degree of power, as well
'59
as knowledge, but free from details
which are important only to the spe-
cialist. Professor Byerly's Calculus
appears to be designed to meet this
want. . . . Such a plan leaves much
room for the exercise of individual
judgment ; and differences of opinion
will undoubtedly exist in regard to one
and another point of this book. But
all teachers will agree that in selection,
arrangement, and treatment, it is, on
the whole, in a very high degree, wise,
able, marked by a true scientific spirit,
and calculated to develop the same
spirit in the learner. . . . The book
contains, perhaps, all of the integral
calculus, as well as of the differential,
that is necessary to the ordinary stu-
dent. And with so much of this great
scientific method, every thorough stu-
dent of physics, and every general
scholar who feels any interest in the
relations of abstract thought, and is
capable of grasping a mathematical
idea, ought to be familiar. One who
aspires to technical learning must sup-
plement his mastery of the elements
by the study of the comprehensive
theoretical treatises. . . . But he who is
thoroughly acquainted with the book
before us has made a long stride into
a sound and practical knowledge of
the subject of the calculus. He has
begun to be a real analyst.
H. A. Newton, Prof, of Math, in
Yale Coll., Xcw Haven : I have looked
it through with care, and find the sub-
ject very clearly and logically devel-
oped. 1 am strongly inclined to use it
in my class next year.
S. Hart, recent Prof, of Math, in
Trinity Coll., Conn.: The student- can
hardly fail, I think, to get from the book
an exact, and, ;it the same time, a satis-
factory explanation of the principles on
which the Calculus is based; and the
introduction of the simpler methods of
integration, as they are needed, enables
applications of those principles to be
introduced in such a way as to be both
interesting and instructive.
Charles Kraus, Techniker, Pard-
ubitz, Bohemia, Austria ; Indem ich
den Empfang Ihres Buches dankend
bestaetige muss ich Ihnen, hoch geehr-
ter Herr gestehen, dass mich dasselbe
sehr erfrcut hat, da es sich durch
grosse Reichhahigkeit, besonders klare
Schreibweiseund vorzuegliche Behand-
lung des Stoffes auszeichnet, und er-
weist sich dieses Werk als eine bedetit-
eride Bereicherung dermathematischen
Wissenschaft.
De Volson 'Wood, Prof, of
Math., Stevens' hist., Hoboken, ~N.J.:
To say, as I do, that it is a first-class
work, is probably repeating what many
have already said for it. I admire the
rigid logical character of the work,
and am gratified to see that so able a
writer has shown explicitly the relation
between Derivatives, Infinitesimals, and
Differentials. The method of Limits
is the true one on which to found the
science of the calculus. The work is
not only comprehensive, but no vague-
ness is allowed in regard to definitions
or fundamental principles.
Del Kemper, Prof, of Math.,
Hampden Sidney Coll., I 'a. : My high
estimate of it has been amply vindi-
cated by its use in the class-room.
R. H. Graves, Prof, of Math.,
Univ. of Jvorth Carolina: I have al-
ready decided to use it with my next
class ; it suits my purpose better than
any other book on the same subject
with which I am acquainted.
Edw. Brooks, Author of a Series
of Math. : Its statements are clear and
scholarly, and its methods thoroughly
analytic and in the spirit of the latest
mathematical thought.
i6o
Syllabus of a Course in Plane Trigonometry.
By W. K. BYEKI.Y. 8vo. 8 pages. Mailing Price, 10 cts.
Syllabus of a Course in Plane Analytical Geom-
etry. By \V. E. BYERLY. 8vo. 12 pages. Mailing Price, 10 cts.
Syllabus of a Course in Plane Analytic Geom-
etry {Advanced Course.) By W. E. BYERLY, Professor of Mathe-
matics, Harvard University. 8vo. 12 pages. Mailing Price, 10 cts.
Syllabus of a Course in Analytical Geometry of
Three Dimensions. By W. E. BYERLY. 8vo. IO pages. Mailing
Price, 10 cts.
Syllabus of a Course on Modern Methods in
Analytic Geometry. By W. E. BYERLY. 8vo. 8 pages. Mailing
Price, 10 cts.
Syllabus of a Course in the Theory of Equations.
By W. E. BYERLY. 8vo. 8 pages. Mailing Price, 10 cts.
Elements of the Integral Calculus.
By \V. E. BYERLY, Professor of Mathematics in Harvard University.
8vo. 204 pages. Mailing Price, $2.15; Introduction, $2.00.
This volume is a sequel to the author's treatise on the DitYerential
Calculus (see page 134), and, like that, is written as a text-book.
The last chapter, however, a Key to the Solution of Differential
Equations, may prove of service to working mathematicians.
H. A. Newton, Prof, of Math.,
Yale Coll. : \Ve shall use it in my
optional class next term.
Mathematical Visitor : The
subject is presented very clearly. It is
the first American treatise on the Cal-
culus that we have seen which devotes
any space to average and probability.
Schoolmaster, Lcndcn : The
merits of this work are as marked as
those of the Differential Calculus by
the same author.
Zion's Herald : A text-book every
way worthy of the venerable University
in which the author is an honored
teacher. Cambridge in Massachusetts,
like Cambridge in England, preserves
its reputation for the breadth and strict-
ness of its mathematical requisitions,
and these form the spinal column of a
liberal education.
A Short Table of Integrals.
To accompany BYE JULY'S INTEGRAL CALCULUS. By B. O.
PEIRCE, JR., Instructor in Mathematics, Harvard University. 16 pages.
Mailing Price, 10 cts. To be bound with future editions of the Calculus.
Elements of Quaternions.
By A. S. HARDY, Ph.D., Professor of Mathematics, Dartmouth College.
Crown, 8vo. Cloth. 240 pages. Mailing Price, $2.15; Introduction,
The chief aim has been to meet the wants of beginners in the
class-room. The Elements and Lectures of Sir W. R. Hamilton
are mines of wealth, and may be said to contain the suggestion
of all that will be done in the way of Quaternion research and
application : for this reason, as also on account of their diffuseness
of style, they are not suitable for the purposes of elementary instruc-
tion. The same may be said of Tail's Quaternions, a work of
great originality and comprehensiveness, in style very elegant but
very concise, and so beyond the time and needs of the beginner.
The Introduction to Quaternions by Kelland contains many exer-
cises and examples, of which free use has been made, admirably
illustrating the Quaternion spirit and method, but has been found,
in the class-room, practically deficient in the explanation of the
theory and conceptions which underlie these applications. The
object in view has thus been to cover the introductory ground more
thoroughly, especially in symbolic transformations, and at the same
time to obtain an arrangement better adapted to the methods of
instruction common in this country.
PRESS NOTICES.
Westminster Review : It is a
remarkably clear exposition of the sub-
ject.
The Daily Review, Edinburgh,
Scotland : This is an admirable text-
book. Prof. Hardy has ably supplied
a felt want. The definitions are models
of conciseness and perspicuity.
The Nation : For those who have
never studied the subject, this treatise
seems to us superior both to the work
of Prof. Tail and to the joint treatise by
Profs. Tail and Kelland.
New York Tribune : The Qua-
ternion Calculus Is an instrument of
mathematical research at once so pow-
162
erful, flexible, and elegant, so sweeping
in its range, and so minutely accurate,
that its discovery and development has
been rightly estimated as one of the
crowning achievements of the century.
The time is approaching when all col-
leges will insist upon its study as an
essential part of the equipment of young
men who aspire to be classified among
the liberally educated. This book fur-
nishes just the elementary instruction
on the subject which is needed.
New York Times : It is especially
designed to meet the needs of begin-
ners in the science. ... It has a way
of putting things which is eminently its
own, and which, for clearness and force,
is as yet unsurpassed. ... If we may
not seek for Quaternions made easy, we
certainly need search no longer for
Quaternions made plain.
Van Nostrand Engineering
Magazine : To any one who has
labored with the very few works ex-
tant upon this branch of mathematics,
a glance at the opening chapter of
Prof. Hardy's work will enforce the
conviction that the author is an in-
structor of the first order. The book
is quite opportune. The subject must
soon become a necessary one in all
the higher institutions, for already are
writers of mathematical essays making
free use of Quaternions without any
preliminary apology.
Canada School Journal, To-
ronto : The author of this treatise has
shown a thorough mastery of the Qua-
ternion Calculus.
London Schoolmaster : It is in
every way suited to a student who
wishes to commence the subject ab
initio. One will require but a few
hours with this book to learn that this
Calculus, with its concise notation, is a
most powerful instrument for mathe-
matical operations.
Boston Transcript : A text-book
of unquestioned excellence, and one
peculiarly fitted for use in American
schools and colleges.
The Western, St. Louis: This
work exhibits the scope and power of
the new analysis in a very clear and
concise form . . . illustrates very finely
the important fact that a few simple
principles underlie the whole body of
mathematical truth.
FROM COLLEGE PROFESSORS.
James Mills Peirce, Prof, of
Math., Harvard Coll. : I am much
pleased with it. It seems to me to
supply in a very satisfactory manner
the need which has long existed of a
clear, concise, well-arranged, and logi-
cally-developed introduction to this
branch of Mathematics. I think Prof.
Hardy has shown excellent judgment
in his methods of treatment, and also
in limiting himself to the exposition
and illustration of the fundamental
principles of his subject. It is, as it
ought to be, simply a preparation for
the study of the writings of Hamilton
and Tail. I hope the publication o/
this attractive treatise will increase the
attention paid in our colleges to the
profound, powerful, and fascinating cal-
culus of which it treats.
Charles A. Young. Prof, of
Astronomy, Princeton Coll. : I find it
by far the most clear and intelligible
statement of the matter I have yet
seen.
i6 3
Elements of the Differential and Integral Calculus.
With Examples and Applications. By J. M. TAYLOR, Professor of
Mathematics in Madison University. 8vo. Cloth. 249 pp. Mailing
price, $1.95; Introduction price, $i.So.
The aim of .this treatise is to present simply and concisely the
fundamental problems of the Calculus, their solution, and more
common applications. Its axiomatic datum is that the change of a
variable, when not uniform, may be conceived as becoming uniform
at any value of the variable.
It employs the conception of rates, which affords finite differen-
tials, and also the simplest and most natural view of the problem of
the Differential Calculus. This problem of finding the relative
rates of change of related variables is afterwards reduced to that of
finding the fimit of the ratio of their simultaneous increments ; and,
in a final chapter, the latter problem is solved by the principles of
infinitesimals.
Many theorems are proved both by the method of rates and that
of limits, and thus each is made to throw light upon the other.
The chapter on differentiation is followed by one on direct integra-
tion and its more important applications. Throughout the work
there are numerous practical problems in Geometry and Mechanics,
which serve to exhibit the power and use of the science, and to
excite and keep alive the interest of the student.
Judging from the author's experience in teaching the subject, it
is believed that this elementary treatise so sets forth and illustrates
the highly practical nature of the Calculus, as to awaken a lively
interest in many readers to whom a more abstract method of treat-
ment would be distasteful.
Oren Root, Jr., Prof, of Math.,
Hamilton Coll., N.Y.: In reading the
manuscript I was impressed by the
clearness of definition and demonstra-
tion, the pertinence of illustration, and
the happy union of exclusion and con-
densation. It seems to me most admir-
ably suited for use in college classes.
I prove my regard by adopting this as
our text-book on the calculus.
C. M. Charrappin, S.J., St.
Louis Univ. : I have given the book a
thorough examination, and I am satis-
fied that it is the best work on the sub-
ject I have seen. I mean the best
work for what it was intended, a text-
book. I would like very much to in-
troduce it in the University.
(Jan. 12, 1885.)
University of California
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