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I.
PREFACE. THE
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&-i LIBRARY 1VERSITY OF CALIFORNIA. Received Accessions No.
^
*-/1/-44
N>
I.
PREFACE. THE
exhaustive
character
the
of
late
Professor
Maxwell's work on
Electricity and Magnetism has necessarily reduced all subsequent treatises on these
subjects to
the rank of commentaries.
Hardly any
advances have been made in the theory of these branches of physics during the last thirteen years of which the first
But
suggestions may not be found in Maxwell's book. the very excellence of the work, regarded from
the highest physical point of view, is in some respects a hindrance to its efficiency as a student's text-book.
Written as
under the conviction of the paramount importance of the physical as contrasted with it
is
the purely mathematical aspects of the subject, and therefore with the determination not to be diverted
immediate contemplation of experimental to the development of any theory however fas-
from the facts
cinating, the style is suggestive rather than didactic,
and the mathematical treatment
what unfinished and
obscure.
is
occasionally some-
It is possible, therefore,
that the present work, of which the first volume is now offered to students of the mathematical theory
of electricity,
may be
of service
as an introduction
Its aim or commentary upon, Maxwell's book. to, is to state the provisionally accepted two-fluid theory,
and to develop
it
into its mathematical consequences,
PREFACE.
VI
regarding that theory simply as an hypothesis, valuable so far as it gives formal expression and unity to experimental facts, but not as embodying an accepted physical truth. The greater part of this
volume
is
accordingly
occupied with the treatment of this two-fluid theory as developed by Poisson, Green, and others, and as The success of Maxwell himself has dealt with it. this theory in formally
explaining and co-ordinating only equalled by the artificial
experimental results is and unreal character of the postulates upon which is
based.
bilities,
The
electrical fluids are physical
it
impossi-
only as the basis of mathematical
tolerable
calculations, and as supplying a language in which the facts of experience have been expressed and
results
calculated
and
anticipated.
These
results
being afterwards stated in more general terms may to suggest a sounder hypothesis, such for
serve
instance
as
we have
offered
to
us in the displace-
ment theory of Maxwell. In the arrangement of the treatise the first three chapters are devoted to propositions of a purely mathematical character, but of special and constantly recurring application to
an arrangement
it
is
electrical
hoped that the
able to proceed with the in
due course with as
By such reader may be
theory.
development of the theory
little
interruption as possible
from the intervention of purely mathematical processes. Few, if any, of the results arrived at in these three chapters contain anything new or original in them, and the methods of proof have been selected with a
PEEFACE.
Vll
view to brevity and clearness, and with no attempt at
any unnecessary modifications of demonstrations
already generally accepted. All investigations appear to point irresistibly to a state of polarisation of some kind or other, as the
accompaniment of
electrical action,
and accordingly the
properties of a field of polarised molecules have been considered at considerable length, especially in Chapter XI, in connection with the subject of
physical
specific induction
posite dielectric,
and Faraday's hypothesis of a comand in Chapter XIV, with reference
to Maxwell's displacement theory.
last-mentioned hypothesis
is
The value of the
now
than any
universally recogas of more promise generally regarded other which has hitherto been suggested in
the
of placing electrical theory
nised,
and
way
it is
physical basis.
upon a sound
CONTENTS. CHAPTER
I.
GREEN S THEOREM. ABT. 1-2.
PAGE Green's Theorem
3.
Generalisation of Green's
5.
Correction for Cyclosis Deductions from Green's
6-17.
Theorem
...
1-2 .
.
.
3-5
5-6
Theorem
.
CHAPTER
.
6-19
.
II.
SPHEEICAL HAEMONICS. Harmonics
18-19.
Definition of Spherical
20-24.
General Propositions relating to Spherical Harmonics Zonal Spherical Harmonics
25-34. 35-37.
38-39.
.
.
Expansion of Zonal Spherical Harmonics Expansion of Spherical Harmonics in general
CHAPTER
.
.
.
.
.
.
.... ....
20-21 21-26 27
cfo
an
-
or
V*K u
are given within the limits of the triple integrals,
1
DEDUCTIONS FROM GREENES THEOREM.
4.]
K
where
is
and constant
positive
lower degree than
1
we have only
For,
for each surface,
15
and Ku* of
.
to replace
V2 u
by the more general
ex-
pression
d
.
Tr
du
.
dx^(K dx') and
d
+
d
du.
.
(K
dy^
)
dy'
.
du^
Tr
.
+ dz^(K dz'),
or
V*KU,
Q u by
and every step in the process applies as 14.] Again,
if
S
before.
be a closed surface, or series of closed surfaces o- be a function having arbitrarily
external to each other, and if
assigned values at each point on u satisfying the condition (1)
=
(2)
Vu= 2
u
(3)
is
a-
S,
at each point on
S
t
at each point in external space,
of lower degree than
For there must be an satisfy the conditions
J.
infinite variety of functions
(4)
/ /
UcrdS
trary quantity differing from zero,
than
there always exists a function
=
and
E
where
JE,
U
(5)
is
is
U
which
any
arbi-
of lower degree
J.
For any such function the integral Qu must be greater than There must therefore be some one or more of such functions for which this integral is not greater than for any other. Let u be any such function. Let u + u' be any other function satisfying zero.
(4)
and
(5),
Then -,-
oc
o-,
it
and
for
and
V2 u =
and that
preceding,
0.
may
at all points external to S, and that
we may make
QU + U
'
=
Qu + Q u
f >
=
-=
cr
V2 ^ =
and
by as
CuV
This theorem
be extended to the case in which
being zero, has assigned values at space.
=
can be shewn by the same process as in Art. 10 that
properly choosing E before,
which therefore Tlu'trdS
all
points
also, as in
y
2
u,
the
instead of
in the external
DEDUCTIONS FROM GREEN'S THEOREM.
16
[15.
Again, as there always exists a function u satisfying the conditions, so it can be shewn that it has single and determinate value at For,
all
external points. there be
if possible, let
two functions u and
u' of degree f 7
less
than
i,
at each point
both satisfying the conditions, so that
on
external points.
and
8,
V 2 u = V 2 */,
or
=
u',
-7-
dv
u) =
V 2 (u
sJ
=
-7
dv
at all
Then
= 0, and therefore an
-=-
=
da
-=, dx
&c.,
and u
since both vanish at
infinite distance.
We
can shew also by the same process that there exists a 15.] function u satisfying the condition that y 2 ^ at all points
=
in the internal space,
=
and
o-
(JvV
adS ffFor
that
if
condition
ua-dS=E might
=
were
at all points on 8, provided 0.
not
the
satisfied,
condition
be satisfied by making u a constant, in
which case Q u would not have a minimum value greater than In fact, if V 2 u zero, and the proof would fail. everywhere
dS and therefore -=- cannot be equal to ; dv JJ dv rr points on 8 unless / / a dS = 0.
within at all
If 8,
8, / / -r-
V 2 w,
instead of being zero,
the problem
maybe
is
z
satisfying
th
degree
the condition
coefficient
differential
every partial
ft
of
u,
as
will also satisfy the condition
^^ +
y..
dx*dydz
v
~ -Q
For since the order of partial differentiation follows that
is indifferent, it
dM^
***
= 0. be taken as origin of rectangular coordinates, and let the coordinates of
19.] Let any point
H
P
be
Let $
be any Let OH be any axis drawn from and designated by Ji, and let Q be any point in this axis, and let OQ = p. Ije ^ f> *7j C b e the coordinates of P x, y, z.
function of #, y,
referred parallel to the axes
the ratio
through
0.
to
Q
(#, y, z)
z.
as
origin,
with axes
Then the limiting value
of
1
21
DEFINITION.
9.]
as p is indefinitely diminished, is denoted
by
IJ#fe*;4 It is clear that
d
is itself
w d(j>
or
^<MW),
generally a function of x,y z\ and therefore 9
axis OH', denoted
by
h',
be drawn from 0, we
may
if
another
find
by a
similar process
d
and so on
for
any number of axes.
If u be any function
V2 u
=
0,
and
if
h^ ^2
,
of #, y,
^
...
z
satisfying
the condition
denote any number of axes drawn
from the origin, and the expression
A.A, '" JL. dl\ dh 2
dh{
be found according to the preceding definition, then
WU ^\'''~dhi For
by
let 119
m lt
>
% be the direction cosines of the axis h v
definition
du
But by hypothesis
,
du~
du dx
V
2
n 1\
i l
dy u
=
du ~J~
dz
0.
Therefore
vi*!, dx are severally equal to zero.
v -, 2
v
dy
2
dz
Therefore
and therefore by successive steps
v *^_*
J*
o.
Then
SPHERICAL HAEMONICS.
22 20.]
-
is
[20.
If
a spherical harmonic function of degree
d
1
.
,l
For
(
*1
Similarly
(I)
!* -_.! + 1
(-)
=-
d*
5
-^
+ -^-
*-"*^ 3z 1
1
and
2
whence
W + ^^ + *
J
d!
cte
2'
3
1_"
"
r3
r
Whatever be the
21.]
_ " __ 3
3(s
"*
r5
""
"*"
r
directions of the i-axes k lt ^ 2
function
JL
d
\
__
"
r3 ,
... /^,
the
M
~dh^""dh^ \ r
where
M
any constant,
(i+
degree
For
is
it is
is
a spherical harmonic function of
1).
evidently a homogeneous function of that degree,
and since
it
follows that
If
we
write this function in the form
1 1
of of
M, the direction cosines of the axes To fix the ideas we may conceive a
fi
-^ Y is a function
l}
>
&2
,
i
t
.*h it and those
sphere from the centre of which are drawn in arbitrarily given directions the ^-axes ... H^ Then OHi, Off2 ...OHi cutting the sphere l9 2, r.
mff
,
if 1^.
be any radius, at every point on OQ or OQ produced has a definite numerical value, being a function of the di-
Offly
rection cosines of
or
H
P
OQ
OP.
If h lt / 2
,
...
hi
OH^ and of OQ, and independent of r be the fixed axes of any harmonic, P any
...
SPHERICAL HARMONICS.
22.] variable point,
axes
h-fr
^2
Y
i
P
at
23
spoken of as the harmonic at
is
P
with
%i-
5
Since each axis requires for the determination of
its
direction
two independent quantities, Yi will be a function of the two variable magnitudes determining the direction of r and the 2i arbitrary constant magnitudes determining the directions of the 2- axes. also be expressed in terms of the ^-cosines t may
Y
fjiiy
H2
,
...
fa of the angles
- cosines of
the angles
2
and an expression
much
made by
for
Y
i
r with the 2-axes
and the
made by the axes with each form
in this
other,
be found without
may
difficulty.
22.] If
T
t
be a spherical harmonic function of degree
and if r = \/ 2 +^ 2 + ^ 2 , then monic function of degree i. For by differentiation
r 2i+l
V
i
(i+
I),
will be a spherical har-
= (2 * + 1) r '- x V + r"+1 2
1
i
Similar expressions hold for
Adding
these expressions, and remembering that
we obtain
V
1
2
(r**
V) t
=
(2i+
1) (2
1
+ 2) r
2 *'- 1
V
i
j
4 -t-
\
24
SPHERICAL HARMONICS.
and
=-+>". 5+^-5 = V V 2
[23.
0.
i
Therefore
= and
and
r* is
We
i+ ^
0,
a homogeneous function of as, y, z of degree therefore a spherical harmonic function of degree i. Fi is
i:
Y.
have seen that -~j , as above defined,
monic function of degree
(i
+
is
a spherical har-
1).
It follows then that or ri Y. is
a spherical harmonic function of degree
i.
23.] Every possible spherical harmonic function of integral positive degree, i, can be expressed in the form r Yi if suitable i
fi ^ 2 ... & t determining Yt l9 a of the i ih degree be function homogeneous Hi
directions be given to the axes
For
if
contains
-
-
2
of the degree
arbitrary constants.
2 contains
i
In order that
V 2 ^ may
-
-
leaving
^
'-
V 2 /^
being
arbitrary constants.
V 2^
all values of #, y, and z, must be separately zero.
- relations between the constants in
-^- -2
Therefore
L-
*-
it
*
be zero for
the coefficient of each term in
This involves
.
,
2i+l
or
2
H^
them independent.
of
Therefore every possible harmonic function of degree i is to be found by attributing proper values to these 2i-f 1 constants. But the directions of the ^-axes ^2 ... ^ involve 2 i arbitrary
^
constants,
,
,
making with the constant M, 2 i -f
therefore always possible to choose the e-axes the constant M, so as to make ....
r
M
d d d TnT'Twr -:/* dfi dhi r dh^ 2
.
'
or
r
Y
1
^
in ,
h2
It is
all. ,
. . .
li
i
and
SPHERICAL HARMONICS.
24-]
25
equal to any given spherical harmonic function of degree i. Therefore ri i is a perfectly general form of the spherical har-
Y
monic function of positive integral degree i. Again, every possible spherical harmonic function of negative
Y
integral degree
For
if
V
i
it
(i+l),
(i+l) can be expressed in the form -j~
be any spherical harmonic function of degree follows from Art. 22 that r 2i+l 7i is a spherical
harmonic function of degree follows
by the former part of
i. Hence, i being integral, it this proposition that r 2i+l 7i can
i always be expressed in the form r Yi by suitably choosing the axes of Yit and therefore that Vi may be expressed in the form
,
Therefore ri Yi and -~^ are the most general forms of the
harmonic functions of the integral degrees
spherical (i +
Y
i
1
is
where
i
)
i
and
respectively.
defined as the surface spherical harmonic of the order is
i
always positive and integral; r
the solid harmonics of the order 24.] If J^ and
Y
i
i,
'Y-
and
-^
are called
i.
Yj be any two surface spherical harmonics 0, and referred to the same or different
with the same origin axes,
and of orders
i
and j
respectively,
and
if
/ /
Y Yj dS i
be
found over the surface of any sphere with centre 0, then
YjdS=
0,
unless
i
= j.
Let H. and Hj be the solid spherical harmonics of degrees and j respectively corresponding to the surface harmonics Yi and J,, so that = r*Yit ffj = rJYj. t
i
H
Make U and
U' equal to fft and Hj respectively in the equation of Green's theorem taken for the space bounded by the aforesaid spherical surface, then
ZONAL SPHEKICAL HARMONICS.
26
rrH JJJ
(
because sphere
and
r
dx
^ d_Ei
^'
d
dx
dy
V 2^
and
V
2
dy
ffj are
[25.
dffjd dz d
each zero at every point within the
;
similarly,
being the radius of the sphere
that
;
is
or
j)
(i
therefore either or
i=j,
The points in which the axes h^ h^...h i 25.] Definition. drawn from any origin meet the spherical surface of radius unity round
When
as centre are called the poles of the axes h^ ^2 ... h it these poles coincide, the corresponding spherical har,
all
monics are called zonal spherical harmonics solid and superficial respectively, referred to the common axis, and the surface spherical
harmonic of order
i is
in this case written
Q
t
>
be the cosine of the angle between r and the common axis in the case of the surface zonal harmonic Q t of order i, then If
Qi
is
ju
the coefficient of
e
in ascending powers of
i
in the expansion of
e.
ZONAL SPHERICAL HARMONICS.
25.]
OA
Let
POA
be
and
axis,
OP
let
be r and the angle
0.
OA
In
common
be the
27
M
take a point
at the
V
distance p from 0. Then if be i the solid zonal harmonic of degree (i
+
1)
corresponding to the sur-
harmonic from definition that face zonal
Q i}
follows
it
o
F-/*vJL i
when
is
p
Let p
Fi e-3.
~dp PM
made equal
to zero after differentiation.
= er and let cos =
Then
F
a,
But 2f
*
_ "
>
or
or
naF(P) by the
last Article.
Suppose that/ is originally greater than
& i
and
i
#,
da;
then
ultimately.
ZONAL SPHERICAL HARMONICS.
2Q.]
31
And, by Art. 25,
i=f{&+
31.] Considered as a function of
-
.
2
coefficient
the zonal harmonic Q*
of order
i,
and
We can prove the (a)
As proved
derived by the expansion of
//,
is
called the Legendre's
P
frequently written { following properties of the coefficients P. is
above, if
p
=
.
1,
l-e Hence,
if p =
1,
P = t
1
for all values of
1
Hence,
if
^
If u
,
In the
OP =
limit,
r,
if
OQ = r + dr. Q
be taken near
enough to P, the force of repulsion may be considered constant, as m' moves Fig.
5-
from P to Q, and equal to mm'f(r)._ Therefore the work done by the force in moving the repelled particle
from
P
to
Q
is
mm'f(r) eoaQds, or mm'f(r)dr, and
independent of c/> if dr be given. Therefore the whole work done
from distance
r
to distance r
mm'
from
by the
force in the
is
motion
is
f(r\dr,
and depends upon rx and r2 and these quantities only. We have for simplicity considered m fixed at 0, but the proof evidently holds if both m and m' be moveable, and move from a ,
distance r^ to a distance r2 apart under the influence of the mutual repulsive force mm'f(r). If the mutual force had been attractive instead of repulsive, in other respects following the same law, the expression for the work done would be the same
as that for the repulsive force, but with reversed If in sign. case on the the for the any effecting integrations expression
work done prove
to be negative, this result must be interpreted as expressing the fact that positive work is done against, and not by, the force in the motion considered.
THE POTENTIAL.
42.]
In either
case,
the work done
47
whether the force be repulsive or
attractive,
proved to be a function of r- and r2 only, and independent of the course taken between the initial and final positions of
is
m.
We have thus shown that
if
f(r) be any continuous function of
the distance between the two particles m and m', a potential exists. At present, as above stated, we are concerned only with the case in
In that case the work done by the
which f(r) =-g
mutual force between
m
and
m', as their distance varies
from r
1
(*'2
^dr, that is
'-#
mm and
if
the force be attractive
mm,(1
1 }
//'' dV or
dv
+
dV
- + 4770dv
= 0*.
* The cases of finite and infinite p have been considered separately, with the view to their physical interpretations. There is no exception in any case to the
equation
v F + 47rp = 2
discontinuous,
i.e.
0,
when
because,
v2 F
p is infinite.
becomes
infinite
whenever
-, &c. are
dx
THEOREMS CONCERNING THE POTENTIAL.
51.]
55
The mean value over the surface of any sphere of the potential due to any matter entirely without the sphere is equal to 51.]
the potential at the centre.
For
let a be the radius of the sphere, r the distance of any 2 in point space from the centre, a do) an element of the surface. Then denoting by V the mean value of over the sphere, we
V
have
=
/y
47T
n n JTT
-=-a2 d
sum
where a
is
of the distribution, the potential
the radius.
a
We
have then
=
F,
or
-
= a.
^V
* t
THEOREMS CONCERNING THE POTENTIAL.
63.] If
$ be an
within of 8
is
it,
M -=
equipotential surface to a system of matter wholly be the potential of the system on S, the capacity
and j
V
where Jf
is
sum
the algebraic
of the matter of the
M
is also the enclosed system. For, by Art. 60, algebraic sum of a distribution over S which has potential at every point on it.
V
62.] If V be the potential of any distribution of matter over a closed surface S, and if
where
and
V
t
is
u'
=
-
the constant value of
= 0, and
V2 -
>
is
since
on
S,
T--&8-**9 Jby Art. 9
45,
zero at all points within S.
The equation
therefore becomes
=-8,
But
which proves the Secondly,
V
if
first
and
part of the proposition. for origin a point
we take
inside
apply Green's theorem to the external space, with
u and
u',
we
obtain
snce
of S, and
V and -
for
SPHERICAL HARMONICS APPLIED TO POTENTIAL. 67
64.]
mal
V2 - =
and as before
in this case,
measured inwards from
is
S,
dV
-r=R, dv
Hence the
potential at
case, the nor-
and therefore
V F=-47rp'. 2
also
any
Also in this
0.
internal point of the distribution
T>
^
over
4
differs
by
a constant quantity from that of the
external portion M', and therefore the force due to the distribu-
--JP
tion
over
8
is
4lT
Hence
it
equal to that due to the external portion
follows that the force at
any external point due
to the
-n
internal portion is equal to that due to the distribution
4?r
over 8, and the force at any internal point due to the external
-73
is
portion
equal to that of the distribution
P
of any distribution 64.] To express the potential at any point of matter in a series of spherical solid harmonics.
Take as origin any point 0. Let OP = f. Let be any point in the distribution. referred to OP as axis, be r, 0, I
J-iJo
^
QiP-^du.d.
in which the
first
>
if r
Finally, the potential at
of
the
\j+Q^
integral will be omitted
second will be omitted
all
P
a
when
r