222 Br Baker, On the differential equations 



which is easily found to be 



dC r ^ ' r/rNL dp* 



— % (2p + 2 — \ — fi — v - p) fp\p»l>' 



a 



while we easily find 



(P,Q) = -P, (Q,R) = -B, 



and (iP, — Q, iR) have the usual structure of the projective group 

 in one variable. 



In virtue of these transformations the differential equations 

 are all deducible from several of them. Thus for p = 2 the sigma 

 function formed with the covariantive polynomial f{x, z) is equal 

 to that formed with the reduced Abelian polynomial multiplied 

 by the factor 



exp [— -Jq (X^2 2 + I^WjjM! + \aWi a )]> 



and making the substitution to the coefficients C$ with binomial 

 coefficients as above, and putting for brevity 



the five differential equations are, for i = 0, 1, 2, 3, 4 in turn, 



- £Q« = Pi + G i+2 p 22 - 2G i+1 @ 21 + Ci%> u , 

 where Q {i) denotes in turn Q 2222 , Q 2221 , Qwn, Q2111, Qmi, and 

 p = G G 4 - 4CA + 3C 2 2 , P, = \ (G G 5 - 3C& + 2G 2 G 3 ), 



P 2 = i(G G 6 -9G 2 G 4 + 8G 3 ^; 



P 4 =C 6 C 2 -40 5 a 3 + 3C 4 2 , P 3 = \{G,G 1 -ZG,G 2 + 2G i G 3 ). 



These satisfy the condition of isobarism, are alike whether we 

 count from the beginning or the end, and they are all deducible 

 from the last of them by successive applications of the operator 

 denoted above by P. 



For j) = 3 we find that the sigma function formed with the 

 covariantive form f{x, z) is equal to that formed with the simpli- 

 fied Abelian form multiplied by the factor 



exp [- \ (3C 2 v 3 2 + 4tC 3 u s v 2 + Gja z u x + 9G 4 u 2 2 + 4C f 5 «t 2 t« 1 + 3CX 2 )] 



and the fifteen differential equations become, if in the second 

 parts of the right sides we abbreviate by writing down in order 

 only the coefficients of 



#>33, fifa, #>31, #>22, #>21, Pll 



and use A = %> 32 %> 21 - $> m p 22 + £J 31 2 - p a g? u , 



