Full text: The collected mathematical papers of Arthur Cayley, Sc.D., F.R.S., late sadlerian professor of pure mathematics in the University of Cambridge (Vol. 11)

2 
ON THE DISTRIBUTION OF ELECTRICITY 
[706 
(x, b, a). Hence, first, for a point interior to the sphere A, if x denote the distance 
from A, and therefore c — x the distance of the same point from B, the potential of 
the point in question is 
= acf)X + 
b 2 
C — X 
$ 
— x) ’ 
and, secondly, for a point interior to the sphere B, if x denote the distance from B 
and therefore c—x the distance of the same point from A, the potential of the 
point is 
cx 
c — x 
</> 
(T 
c — x 
+ bA> (x). 
The two equations thus express that the potentials of a point interior to A and of 
a point interior to B are =h and g respectively. 
It is to be added that the potential of an exterior point, distances from the points 
A and B — x and c — x respectively, is 
cX 
= — 9 
x T 
Ò 2 ru 
+ ~ $ 
c—x 
b 2 
c — x. 
and that, by the known properties of Legendre’s coefficients, when the potential upon 
an axial point is given, it is possible to pass at once to the expression for the potential 
of a point not on the axis, and also to the expression for the electrical density at a 
point on the two spherical surfaces respectively. The determination of the functions 
cf>(x) and <L(«) gives thus the complete solution of the question. 
I obtain Poisson’s solution by a different process as follows:—Consider the two 
functions 
8XC + b 
and 
and let the nth functions be 
a? (c — x) 
c 2 — b- — ex’ 
b 2 (c — x) 
c 2 — a? — ex' 
a n x H - b, 
cx + d ’ 
OLX + ft 
yx + 8 5 
a n x + /3, 
suppose, 
suppose ; 
c n x + d, 
and 
y n x + B n 
respectively. 
Observing that the values of the coefficients are 
(a, b ) = ( — a 2 , cXc ), and (ot, /3 ) = ( — b 2 , 
b 2 c 
c, 
— c , 
b 2 
— c, & — cX 
so that we have 
a + d = a + B, = c 2 — a 2 — b 2 , ad — be = ctB — /3y, = cXb 2 , 
and consequently that the two equations 
(A, + !)■’ (a + d) 2 (A. + 1) J (a. + B) 2 
ad — be ’ 
aS — (3y ’ 
A, 
A.
	        
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