516 Mr Baker, On a certain system of [May 2, 



(2) Let 



a(lt)= % e au i +2niv(n+r)+iirT(n+ry i +2niq(n+r) 

 n l ...n P 



be, as usual, the general hyperelliptic theta function of p variables, 

 with any characteristic, the quantities a^ occurring in the quad- 

 ratic form aiC~ satisfying the equation 



ns* <c,a z,c TT»,a [ x [ z dxdz %VS + F (x, z) 



22,a ij u i ' Uj =1J ZC - ^— ^-— ', 



where * F (z, z) = 2/0) = 2s 2 , 



£*<*.)" 





further let p, (u) = - ^ log a (u), p ijM (u) = -^ <p, (u), 



and let e 1} e 2 , e 3 , e 4 be any quantities whatever. 



Then we have the relation 

 8 (e 2 - e 3 ) (e 3 - ej (e t - e 2 ) (e 4 - e x ) (e t - e 2 ) (e 4 - e 3 ) 2222 %> (u) . e^V" V" V" 3 



K fiv p " H 



A(A " 



[^( e4)ei )-4( e4 -^22p (tt)ej- 1 ^- 1 ] 



Afi 



+ («s - «i) K - « a ) [^ («». «i) -4(« 3 - ^) 2 22 p («) <£- V" 1 ] 



[F (« 4 , e 2 ) - 4 (« 4 - e 2 ) 2 22 p («) e*" 1 e^" 1 ] 



+ K - e 2 ) (« 4 - e 3 ) [F (e, , e 2 ) - 4 ( Cl - e 2 ) 2 22 p^ («) ej " * *~ 1] 



tF( e4 , e3 )-4( e4 - e3 ) 2 22^ (m)^" 1 ^- 1 ], 



A ^ A,x 4 3 



wherein 



f^x i ~ 1 dx p 



u? a = '- , y- =f(x) = (x, l)^ +2 , m» = S Ui Xr ' 0r , 



Jal/ r=l 



and F (x, z) may have its most general form. The summation for 

 each of the symbols X, //,, v, p is from 1 to p. 



The deduction of the explicit forms of the differential equa- 

 tions, by equation of like powers of e l5 e 2 , e 3 , e 4 on the two sides, 

 is a question in the calculus of symmetric functions. 



(3) For the case of 2 variables we put down the differential 

 equations in two forms, (a) the explicit form, in accordance with 

 § 1, (/?) a symbolical form, itself of some interest. 



* Compare, e.g., Math. Annalen, l. (1898), p. 471. 



