Convergence of Infinite Series of Analytic Functions. 315 



The spores, so far as observed, are all of one kind ; they are ellip- 

 soidal in form, with longitudinal crests or ridges ; their dimensions are 

 9096 //, in length by 6570 /x in width. 



The most characteristic point in the structure of the new cone the 

 fertility of both dorsal and ventral lobes of the sporophyll is regarded 

 as more probably due to special modification than to the retention of a 

 primitive condition. 





" On the Convergence of Infinite Series of Analytic Functions." 

 By H. A. WEBB, B.A., Fellow of Trinity College, Cambridge. 

 Communicated by Professor A. R FORSYTE, Sc.D., LL.D., 

 F.E.S. Eeceived November 10, Bead November 24, 1904. 



(Abstract.) 

 Consider the differential equation 



. . ., 



\/Qo> Qi> Q2j . > are one-valued analytic functions of z, independent of 

 k, k is a constant, and the series defining Q is convergent for all values 

 of k, such that 



*|>R, 



except at the singularities of the functions 



Qo, Qi, Q 2 . - . 



Exclude points in the 0-plane indefinitely near these singularities. 

 The series 



t . .ad inf.), 



J 



where w < ^ are functions of z independent of k, can be formally and 

 uniquely constructed to represent any given particular integral of the 

 differential equation. 



For all values of z not excluded the series 



are convergent, and when k is very large, 



e ik <f>o + e- ik 

 is an approximate value of the integral. 



VOL. LXXIV, 



2 B 



