136 Proceedings of the Royal Irish Academy. 



We next suppose that, as a function of z, ip varies as e*'^ ; then (82) is 

 -equivalent to 



fx [ cl'Iclr'' - r-Kijclr - I' \ '^p - 2C'lpirdxP/dr = 0. (84) 



It will now be convenient to substitute 



Ir = 2a, 2api/fiP = k, (85) 



when the equation becomes 



(dyda' - a-hl/da - 4)-i// - l^kad^jda = 0. (86) 



Solving this in a series of the form 



xL = S^l^a" = S — > 



the law connecting coefficients is 



(n + 4:){n + 2ynAn^i - 8{n + 2)nA„^2 + 16(1 - nJc)An = 0, (87) 



or Bn.i - 2Bn,o. + (1 - nk)B,, = 0. (88) 



There are evidently solutions whose initial terms are respectively 1, a*, 

 o^ log a, a^. As Tp/r and r~^d\p/dr must be finite when r vanishes, the solutions 

 with which we are concerned are those whose first terms are a^ a*. 



The latter is 



^' = d * 2 ^iri ^ (3 + 4,1.) ^ . (4 . 20^) ^ + (5 + 60^- + 32^0 ^ 

 + (6 + 140^ + 264:k') -^ + (7 + 280k + 1216k' + 384:k') 



\iii 'im 



+ (8 + 504/t + 4128A;^- + 4464Z;^ 



a 



+ (9 + 840^" + 11520A-^ + 28000/^^ + 6144A;*) 



|_9L10 

 + (10 + 1320^^ + 27984A;2 + 125840/;^ + 92640/1'^) 



+ (...+ 739136^'* + 122880A;5) nj^ +(...+ 2283840^:0 -^f.^g 



+ (... + 4 . 8 .12 . 16 . 20 . 24^^'=) , , .f ,\ . + . . . (89) 



13 I 14 ^ 



