on two Spherical Surfaces. 55 



<£>(x) — the question depends on the solution of the functional 

 equations 



att>(x)+— <!>(-^)=h, 



rv c — x \c—x) 



a 2 / a 2 \ 



where of course the x of either equation may be replaced by a 



different variable. 



It is proper to consider the meaning of these equations : for 



a point on the axis, at the distance x from the centre of the 



first sphere, or say from the point A, the potential of the 



a 2 (a 2 \ 

 electricity on this spherical surface is acfyx or — <£( — ), accord- 



x \ x / 



ing as the point is interior or exterior ; and, similarly, if x 



now denote the distance from the centre of the second sphere 



(or, say, from the point B), then the potential of the electricity 



on this spherical surface is bQx or —<!>(— j, according as the 



point is interior or exterior ; <f>(x) is thus the same function 

 of (x, a, b) that &(x) is of (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 



= a$x+ <£( ); 



c — x \c — 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 



c — x^\c—x) v y 



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 re- 

 spectively. 



It is to be added that the potential of an exterior point, dis- 

 tances from the points A and B =x and c—x respectively, is 



= ^) + ^*(-> 



x \x/ c — x \c — xj 



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. 



