of the hyperelliptic functions. 225 



II. 



Passing now to the modular relations referred to in the 

 introduction of this paper, we use the formula, easily proved by 

 direct multiplication of the series (as in the writer's Abelian 

 Functions, p. 472) 



a- (u + t)a(u-t) = l l % (u) X (t), 



where as usual 



1# 



2 \q 



= S gau 2 +2i'iv(n+iq')+iTrT(ii+^q') i + niq(n+iq') 



^ (u) differs from a (u) only in having 2au 2 for an?, ^iriv for 2iriv, 

 and 27riT for nrir, and ^ e (u) denotes ^ u ; - ( ) , the summation 



L w; no,_ 



in regard to e extending to the 2^ terms obtained by giving to 

 the p elements of e the values and 1. We have, if 



d i d 2 



D^vp — J)ij = 



dhdtpdudtp' dtidtj' 



&ij (M) = ~ 2^{u) [ m<T ( u + ') & ( u ~ *)]*=»> 

 Qw («) = - K-VT-, [D^Pa (u + t)a (u - t)] t=0 . 



Consider then any one of our differential equations in which 

 the right side is linear in the functions %>y, say an equation 



Q^vp (u) = A Kpvp + tAij^ij (u) ; 



this is equivalent to 



J)^Pa (u + t)<r (u -t) + 2A kpvp a (u + t)<r (u - 1) 



- tAijDVa (u + t)<r (u -0 = 0, 



where after differentiation t is to be put zero ; substituting the 

 above expression for o(n + t)<r(u — t) and equating to zero the 

 coefficients of each of the 2 p functions ^ e (u), we obtain the 

 equations 



Vw (0) + 2A^ vp % (0) - XAqW (0) = 0, 



which are the modular equations in question. 

 These relations can be reduced ; we have 



S- e (u) = 2 e 2au2+i7T ' v (n+W+ftrfr(»+j€y _ gov? ^ (^ g^ 



