70 
Ml!. J. II. JEAXS OX THE EQUILIBRIUM 
It follows that X ^ constant value at the surface S, and this, l)y a suitable 
choice of the constant C, may he taken to Ije zero. Also it follows that cy/ 0 n = 0 
at all points on the surface S, where c/dn denotes differentiation with respect to the 
normal. We therefore have at every point of the surface S 
. dx/cn =0 .( 15 ). 
§ 4 . Let us denote the potential at a point inside the surface S by Y,-, and that at 
a. point outside S by Yq. Let us examine, as a trial solution for Y, 
Y, = 7Tp \C + [ (^ (I) + [ (f) {t]) dr] — £77!.( 16 ), 
I J 0 - 0 J 
1(f) 'A (^) j.(!')• 
Since 77 = the greatest value of [77 1 at any point on S is equal to the greatest 
radius, say which can be drawn from the origin to S. Since the series ^ (77) is, 
by hypothesis, convergent for all points on the boundary, it follows that the radius 01 
convergence of the power series (f) (77) must be greater than and hence that (77) is 
■V 
convergent at all points inside the surfaceS. Hence also (f) (77) c/77 i^^ust be convergent 
- 0 
at all j^oints inside S. The same is obviously true if i is written instead of 77, Hence 
it follows that if Y^ is defined by ecjuation ( 16 ), then Y; and its first differential 
coefficients will be finite and continuous at all ^joints inside S. 
In a precisely similar manner it can 1 )e shown that if Yq is defined by ecpiation ( 17 ), 
then Yq will be finite and continuous at all points outside S except at infinity, and 
that the first differential coefficients of Yq will be finite at all j^oints outside S. 
From what has been said it follows that A"q and Y^ are finite at the boundarv. 
Tlie value of Y- — Yq at the l)Oundary is 7rpy, and tliis vanishes by equation ( 14 ). 
Hence Y is finite and continuous at all points of space (except infinity). 
Since the series (f> {rj) and xjj [rj) have been supposed to be convergent on the 
boundary, it follows that the first differential coefficients of Y must be convergent on 
the boundary, and hence that these differential coefficients are finite at all })oints ot 
space. At the fioundary, 
0^ i/on — 0Vq/0;; = irp cx/o)2 = 0, 
by equation ( 15 ). Hence it follows that the first differential coefficients of Y are 
finite and continuous at all points of space. 
At a point inside S, 
vA" = vAb = 47 rp 010 ^ { c ; + ^<^(0 (^) — ^} = - 
and similarly at a point outside S, 
VA^ = YA^q = 47rp 
0^ 07; 
^ (^) d ^-\- I Q (77) (/77 I = 0 
