1898.] 



with two-dimensional fluid motion. 



389 



Consider now the function 



MS/*)- 



Kr ~ f* 



where f r = {ol,X + A-)/(7>-£ + &/•)- and the summation extends to 

 every substitution of the group. When one of the p fundamental 

 substitutions of the group, as given by the positions of the circles 

 on the f plane, is put into the form £' = (*£+ P)/(yZ+$), with 

 a.8 — fiy = 1, there is an ambiguity as to the signs of the quantities 

 a, /3, 7, 8, which must be settled beforehand in order that the 



Ng2. 



function A, (£, fi) may be definite. There are then 2 P functions 

 \ (£, fi), differing according to the conventions adopted in the case 

 of the p fundamental substitutions. It is known (Oamb. Phil. 

 Proceedings, viii (1895), p. 332, or loo. cit. (k), p. 368) that anyone 

 of these functions has, in the region II, beside £ = oo , p zeros 

 fi 1} ..., fi p which are given by the equations 



<" ■* +...+<" "* = k ig, + K), (r = 1, ... , p\ 



where m 1} ..., m p are the zeros of the theta function ® {v^ m ), and 



(gi + h), >'->(9p + K) 



is an aggregate of p integers, each of which may be taken to be 

 either even or odd according to the conventions, explained above, 

 adopted for the signs of a, /3, <y, 8 in the p fundamental sub- 

 stitutions. 



Hence we deduce, if the conventions be taken so that each of 

 g r + h r is an even integer, the functions 



\{Z,a), V(£c) 



vanish respectively in (a 1} ..., a p ) and (c ly .... c p ). More generally, 

 from the results developed above, such as v r ar > Cr = — \, we have, the 



