62 Proceedings of the Royal Irish Academy. 



always vanishes unless m is a. multiple of n, and then it = 9i Y, 

 where F is a constant which need not be considered at the moment. 

 By adding together the various series just given, 



Xi + Xo + . . . . = tiSi + toSo + + L2S_2 .... 



Kow, >S',. = 0, unless r be a multiple of 7i ; and it has just been proved 

 that when r is a multiple of 71, t.,. always vanishes, unless r = 0. 

 Hence ^_,.aS'_, vanishes, unless r = ; that is, the only term which 

 remains in the sum of the several series is the second one, namely, 

 t^S^, and this 



= 7% 



Thus the sum of the values of x obtained from [x]V2/ equal 

 to 



By multiplying Xi, X2, . . . together two at a time, and adding, 

 we have 



+ [t^U^a^l-"- t^t_^^a%-^] + {^iL3:S«^J-^+ 2!o^-22«°^""+ ^-i^-22(«J)-'} 



+ [t^t_^a}l-^ + t^t_{la%-'^ + t.^t_^'%a-^h-'\ + etc. 

 Since l.aH' = S„,Sr - S„ 



and 2'^{ahy" = - 



'm+r 



'2m 



and vanishes unless 7)i be a multiple of 71, all the terms within 



the large brackets, except the third and the (r + biy* terms, vanish. 

 Substituting the values of t in the latter, and remembering that 



if k be a positive integer, we find that also the (r + hiy^ terms vanish. 

 Hence the only term which remains is the third ; so that 



71 



^-2+ - nr- + 



2 2 



Thus the sum of the products of ^1, x^, oc^ . . . . taken two at 

 a time is equal to p_2. 



