Gamma Function to an Electrostatic Problem. 257 



-The following relations for the electric density at any point 

 on the surface of either of a pair of spheres in contact can be 

 established by a somewhat laborious calculation depending 

 upon the principle of inversion ; they are not given in any 

 book with which I am acquainted : — 



Taking the general case when the spheres are of unequal 

 radii a and b, let P be any point oh the surface of the one of 

 radius a, and the point of contact of the spheres ; and let <£ 

 be the angle between OP and that tangent-line at which 

 lies in the plane containing OP and the line of centres. 



Also let Lb-sz -. and let p=— — — . Then if p be the 



n a + u A fx r - 



density at P, and V the potential of the system 



irpa 1 1 1 1 1 - r 



-^r- = 1 — - — — — -^==^ — . . .ad inf. 



V >/l+(l-^)/> 8 N /l + 2i2~^)/> 2 VI + 3(3 -/*)/>* 



1 1 1 



H == -\ . + . . . ad inf. 



;2 



\/l + (1 + a0/> 2 n/1 + 2(2 +/*)/? n/1 + '&$+?)$> 



For the other sphere the same expression holds except that 



a must now be ;, and b must take the place of a on the 



r a + b' l 



left-hand side. 



This series cannot, I believe, be summed in terms of any 



known functions. The total number of terms is shown by the 



calculation to be even, so that for the point of contact, where 



<f) and therefore p = 0, it gives p = 0. At the other extremity 



of the diameter <f> = ~ and p = -s so that for this point We 



obtain (after reduction) 



4cirpa 1 2 r 1 1 1 



.v 



V {2^? + 4 2 ~^ + &=? + - • '} 



The series in brackets can be summed by Poisson's method, 

 and we ultimately obtain for the end density 



,«£«*?. ...... (7) 



If the spheres be equal, /* = i ; and if we now put 

 m=2 V2 sin <£, 

 the expression for the density at any point becomes 

 Airpa 1 _^ _1 _1 ! 1 -_ 



V Vl+m 2 Vl + 3m' 2 ^1 + bm 2 Vl + lU/re 2 



