Expressimj Solutions of Laplace's Equation. 



211 



But 



-2a: 



U-u)(x—/3) oc-(3 ( cT . _ ^^ ) 





\cc'" /3"v' 



yt 



Hence coefficient o£ u"^ in (52 

 --2^/^ + 1 V(^-^ + y^2_2^^ + 



(v/l-M^ 



1)^ 



(t-ix-^/t^'-'2t^i + l) 



-'2tiii + l)»'J 



- (-!)"+'_ r(t-^i^yf'-'2t^i + i)"'-~(t-iii-yt^-2tiii + l)-^-] 



/f'-'iti^i + i L ;: ':;jn J 



(1-/^^)2 



_(-i) 



m + \ 



^^|,n(f-,>)--i + "''"'-.!»'"-^' (,-,„)-^(^---2M + l)+...|.(56) 

 (1 — At^)2 '^ ^- J 



The coefficients of the various powers of t in this are of the 

 form 



: X rational integral function of /i, 



(i-/<^)5 



the coefficient of V^~^ being simply a constant multiple of 



1 



m' 

 (l-a2)2 



It follows that these functions become infinite when 

 /i=±l. 



Hence the coefficients of i'"~^, t'"-''^, ... are multiples of 



Q:;;-i, Q;:;-2,.-.Qr, Q?- 



If we write down the known expressions 



Q..(^)=iP„(y^)los^-W»_i(/.) . . (57) 



w^here W,j_i is an algebraic polynomial of degree n — 1, we 

 see, on differentiating ni times, where m>n, that the logarithm 

 must disappear and the points yLt= +1 are simple poles. 

 20. Let us now return to our general solution for V, 



^^ ^ cos . 



sm 



*) j.«(p,y /«(-')• 



P2 



