206 ^- FlJJiSAWA. 



The complementary function is 



aiicl, a>s H^-2 can not contain irrational powers of c, we must hîive 

 C'=0, C"=0. Hence, —ç— being even, 



(125) H^-s^ 2^\;i- l)7j(H--?i-6)e*»'"-^^-^ + 2^^^7r-2)0i2-9)ç^""'"^^ 



+ 2^~hiin + 1 )(;r + ?i - G)e^"("-^'+^- 2'^-^vK« + 8)ç*"'"+^>-2 



-2^^M?i-8)c^"'"+^'+2 



and thence, 



(126) E,j_2 = - 2^-^;i(7i - 8)ç="("-^'-=*- 2^^?i(m + 8)e^''("-3)+i 



+ 2''-7yi(« + l)(n2 + n- 6)ç^"'"+"-^ + 2'"~%n^- 2){n''- 9)ç^«("+i'-i 

 + 2*-7(;i - 1 )n{ii^ - ;i - Q)$Mn+i)+^^ 



When is odd, we find likewise (or, more sim])ly, by writing — // 



in place of n in (125) and (12()),) 



(127) H,_ = -2''-')i{n- ;il)ç-"("-3'--2_2î^5„^^^+ 3^^5«(«-3)+2 



+ 2^^7?.(7J + 1 )(7r + », - 6)çi«("+^>-^ + ^2^-'^^uP-2){n-- 9)ç-^"f"+i' 

 + 2^^??(« - 1 ){n^—n — 6)ç^"("+i^+^ 



(128) E^_2 = - 2P-^7i(M - 1 ){n^- n - 6)ç*"<«-i^5 _ 2^- 6^,^2 _ 2)^,^2 _ (j^ -^«(»-d-i 



- 2^-^w(n - l){n^+n - 6)e^"<"-^'+H 2*-^n(n + 8)c*"<"+^^^ 



+ 2''-^7î.(??.-3)l^*"^"^^'+\ 



Next, consider i^y s which satisfies the differential equation 

 where 



