520 



Mr Baker, On a certain system of 



[May 2, 



terms; as a practical example I have obtained the expansion of 

 the function a 12 (u 1} u 2 ) up to terms of the ninth order, assuming 

 only the terms of the first order to be ^t 1 . It is probably better 

 however that the linear partial differential equations, which for 

 any specified function are more convenient than these, should first 

 be deduced from them*. 



From either point of view it is clear that the differential 

 equations, regarded as given, are sufficient entirely to define the 

 theta functions. 



(4) The case of three variables may be dealt with more 

 cursorily. Putting 



A = gfegb - g>3lg>22 + g£ ~ fpS&ll, 



f(x) = \ + \x + . . . + \x 6 + 4<x 7 , 

 F {a;, z) = sW [2X L + \ 2i+1 (x + z)\ 



i=0 



the fifteen differential equations are 

 1) ^ 3333 - 6$, = \X, + X 6 x + 4>y, (2) g^ - 6fap„ = \ 6 y - 2|+ 6z, 



3) g>333i - 6g?33^ 31 = \z - 277, 



4) £> 3322 - 4g& - 2p 33 #> 22 = £\ B y + X 6 £ - 2?;, 



5) ^332i - 4^3 2 g) 31 - 2^33^21 = lX 5 z, (6) ^3ii - 4g& ~ 2p to p 11 = 2A, 



7) ^3222 - 6gbfta = ~ A, - ^X 3 # + X^ + X" 5 2 ~ 6£ 



8) g>3221 - 4^2^21 - 2fMfc = ~ $\ + M - 2A, 



9) g>3211 - 4^1^21 - ty*$U = X + i\Z, 



0) Pan - 6Mn = X o« - iM + X 2 Z, 



1) P2222 ~ 6f& = -^X+^W-SX^+X.y+X^+X^ - 3X 6 £+ 1 2A, 



2) g> 222 i - 6g> 21 g> 22 = - 2A - I \X 6 - fXi* + X 3 £ + X v - 1 \ 5 £ 



3) #> 22U - 4g4 - 2£> 22 g> 11 = - -|A. \ 6 - 2X x- lX$ + X 2 z + JX^, 



4) Pan - 6pinpu = ~ i V^ 5 - 2\^ + f V - \ \£ + Xtf, 



5) 0>im - 6^ = JAA - I X X 4 + 4 V - 3\ | + M + X 2 £ 



From the first twelve of them we find the three independent 

 equations 



S = R i /P > T=QR/P, U = Q*/P, 



For the case of p = l, see Weierstrass, Werke, i. p. 7. 



