the Distribution of Electricity on Spherical Surfaces. 195 



series. However, nothing immediately tmus upon this, as the 

 expression is only used for obtaining an expression for fx in the 

 form of a definite integral, viz., equation (36), 



or, equation (39), 



f^ = 



bh r^ <^f(ri+i-i)^<'+w-») 



(H-6)(l-^)l \-t 



the latter of which gives (equation (115), in which I have written 



for a its value r — -) 

 \+b' 



J - {i+b){\-x)V\{\+b){\-x)} ^ii+j_^;/' 



where Z'(jo) is Legendre's function -T-logF^, which is develop- 

 able in the form 



Z'i^=log;>-|^-^ + ^-^, + &c., 



where B,, B3, &c. are Bernoulli's numbers. 



This is the starting-point of the present investigation ; and 

 attending to the equations 



log(l + A)Q,^_l 



A 2' 



0'^-' = 0(a^>l), 



log(l + A)_, 



log(l + A) 



we see that the development of Z'p becomes 



„, , , log(H-A)/0 0* 03 . \ 



which, observing that 



can be expressed under the more simple form 



02 



