the Distribution of Electricity on Spherical Surfaces. 197 

 where for shortness, 



K = b + t, A'=(l+i)(l + /0), M'=b + {l+b)tO, 

 B= t, B'=l + (l+*)/0, B"= {\+b)tO. 

 And it may be remarked that 



A'=l+^' + B", B'=l+B", A"=6 + B". 

 We thence obtain 



^ hb log(l + A) r> (A^-B^A-^)<Z^ 



"~ 1 + * ^ Jo (A2 - 2 AB/A^ + B2«2)f 



A61og(l + A) r r 1 {M^-W-x'^)dt r\ (A"^-B"^^^)<^/ \ _ 



^ "L Jo (A'2 - 2A'B V + B''^A'2^5 ~ J^ (A"2 - 2A"B'V-^ + B''^^^)^ J 



and writing x=\, 



^~l + 6 A Jo(A2-2AB/;l + B2)5 



A lJo(A'2-2A'BV + B'2)^ Jo (A"^2-2A"BV + B"2)^J" 



The integrals in the foregoing expression are of the form 

 {Q + m)dt 



r 



'o(L + 2M^ + N<2)2 

 The value of the indefinite integral is 



1 (NG-MH)^ + MG-LH 

 LN-JVP (L + 2M< + NO* ' 



from which the value of the definite integral can be at once 

 found. It is easy, by means of the values to be presently given, 

 to verify that, in each of the three definite integrals, N G — M H = 0; 

 and the expression for the definite integral is therefore 



M G-LH r 1 \\ 



LN - W \ (L + 2M + N)* L' J ' 



In the first integral wc have 



G = 62, L = 6^ 



11 = 2^ M = i(l-/i), 



N = 2(l-/t*), 

 whence 



h'N-W=b\\-fj){\+ij,), MG-LH=~i^(l-<-^), 



L + 2M + N=/»2 + 2(l-yLt)(l-f-Z,), L=^.2; 



