112 



PEOF. A. E. H. LOVE ON THE TRANSMISSION OF 



the summation refers to integral values of n, and E B and !/< denote the functions 

 determined by the equations 



E B ( 2 ) = (1 ^-Y e = 2V-V''<- +w "z- ( ' +lw K ll+1/t (w), 



\z az/ z 



sn 



. . . (19) 



= 2 



Here 



2 cos 



' Mn+1/2) "J-n-V 1 (2)-^ ( -' w ^ + , /J ^)}. (20) 



The function II can be expanded in a series of similar form. In the region r < ? 

 the series is known (cf. MACDONALD ( 7 )) to be 



II = 2 



3 * c <, .... (21) 



and in the same region we have the known result that 



V / O v 

 ^ \l 



n = 1 





The proof of this result involves the known equations 



'*(*), .... (23) 



which hold also when \[/- is replaced by E, and the equations 



)^5 a=I J (2n+l)P. 



of 



/* 



-=i. (24) 



In like manner we may obtain the equations 



811' " 



/j -^ = 2 (2n+ ] 



OP )! = 1 



. (25) 



where C n and C' B are new constants, determined in terms of the constants of types 

 B n and B',, by the equations 



. . . . (26) 



