the Distribution of Electricity upon Spherical Surfaces. 125 

 and consequently 



(x(^.f)y-(ro.|))'=A(Y(.4)) 



Whence also 



i(l+6)M,.,= (^*)'log(l + A)(0{0(l + i)-M)''. 



"NVe have by the general theorem, 



^ = loge' = log(l + A)e'«; 



and consequently whenever n <\c 2, 



log(l + A)0"=0. 



But i <|c: 2, and the function {0{0{l + b) — b}y contains only 

 0' and the superior 'powers; it is therefore reduced to zero by 

 the operation log (1 + A), and wc have 



and the theorem in question is thus proved. The foregoing ex- 

 pressions for ©2) ®3» &c. show that these functions all divide by 

 b, and moreover that when i is even and greater than 2, then 

 that ®i divides by b-. The equation 



©<=!^^^{(l+0(l + 4))'-(-0(l+i))'} 



gives generally for the term in 0^ involving //, the expression 



H 



And it is to be shown, first, that the coefficient vanishes for a = ; 

 and next, that when i is even and > 2, the coefficient also va- 

 nishes for « = 1. Putting « = 0, the coefficient is 



!=^^'((i+or-(-o)'), 



which ia equal to 



!5^^t£)(i^^_(_yi,0- 



