226 



Dr Baker, On the differential equations 



where v = co 1 u; the differentiations in regard to u are linearly 

 obtainable from those in regard to v, and we clearly have 



dv x dv,t dr^ ' dv K 2 dr KK ' 



that is the modular equations involve only differentiations of the 

 first and second orders in regard to the quantities t Am . 



Conversely the relations can be used to express the algebraic 

 moduli, that is the coefficients in the binary form occurring in the 

 hyperelliptic equation, in terms of transcendental constants, in 

 various forms ; there are for instance in case p = 2 twenty such 

 modular equations. For larger values of p, the results can, 

 presumably, be used to express transcendental conditions that 

 the theta functions should be hyperelliptic. It is clear that the 

 original differential equations are a consequence of the modular 

 equations. 



One complication in working out the modular equations in 

 detail arises from the presence of the factor e au% in the theta 

 function. In case p = 1 we may start from the equation of 

 Weierstrass's theory 



and use the known facts 



V_ 

 2©' 





n 3 q m 



12r)(o 



24 2 



nq- 



za-t 



q^ q (v^) = ~d^ 2 -i\9^ 4 l 



Then we eventually find, proceeding just as in the general case, 

 that the linear differential equation 



„ d?y dy V , -, - 5 n s x 



of Fuchsian type, existing only within a circle of radius unity, has 

 the two integrals 



= 0, 



Vi 



1 + 2 2 a? 



n=l 



2 2 X 2{n+i) 

 ra=0 



x 1 ' 2 n(i -x m ) 

 i 



x T ^U(l-x m ) 

 i 



III. 



Taking for one variable the differential equation 

 p" - 6g> 2 = - \\ \, + ±\\ 3 + X 2 p, 

 giving g>' 2 - 4^ 3 = — (\„A, 4 — 5XA3) p + X,^ 2 -1- constant ; 



