66 Proceedings of the Royal Irish Academy. 



''Jp-a ; or, - P-,i niust be greater than any of the expressions 



'JP-U UP-2, llP-3, ' ' . 



It is useful to call these expressions the weight-functions of the original 

 coefficients. 



(4) JN'ext, we consider the effect of on any term containing only 

 two of the original coefficients. By § 17, the general expression for 

 such a term will be 



y fat-lV^^ 1 fp^' 

 at-l[ n I \t_\g^j' 



where t is any positive integer from zero to infinity. In order that 

 this shall be always less than unity, that part of it which is governed 

 by t must be less than unity. I^ow, by the exponential value of 

 r (^i^ + 1 ) when t is large, the expression 



— , that IS, {.<")}' 



takes a similar exponential form. Hence, ii^ = Jc, and t is very large, 

 the coefficient becomes 



~7t4^t.t{h- 1) t'{t{h- 

 so that the general term may be written 



««/2^"W^-iK^-ir"(-?-»)')' 



and if the term is to be ultimately always less than unity, we must 

 have 



the inequality being numerical. Hence, if - p-n is to be made the 

 subject of the invert, it must always be numerically greater than 



