of the hyperelliptic functions. 221 



p. 146), variation in which is equivalent to alteration by additive 

 constants of the functions g>$; this fix, z) is however not a co- 

 variant of the polynomial 



f{x) = X + \ x x + ... + \ 2p+2 x 2p+2 , 



whose square root appears in the hyperelliptic integrals. In 

 order to avoid this complication it is therefore desirable in the 

 original differential equations to define the functions <py with the 

 covariantive form f(x, z) = 2a x p+1 a z p+1 , where f(x) = a x - p+2 . To 

 be in unison with the usual notation we put therefore 



the changes in the functions |jp.y are easily found by calculating 

 the difference between the covariantive f (x, z) and the simplified 

 Abelian form used 



f(x, z) = Ix^-z 1 [2\2i + X^+i (x + z)]. 



In terms of the coefficients Ci the operator is 



where 



p = Sra - aa + x (7 t^ + ^* +l) a^ + tH *» d^ P 



the effect of this operator on the implicit forms of the differential 

 equations, which are identical in regard to e lt e 2 , e 3 , e 4 (Acta Math, 

 xxvii. p. 144), is the same as the operator 



d_ d_ j d_ d 



de 1 de 2 de 3 3e 4 ' 



As in virtue of the second elementary transformation the differ- 

 ential equations also allow the operator 



JB= 2 (2p + 2-r)C r+1 ^ r +t[(p-X + l)^- h , 

 r =o 0\J r 



+ (^-^+i)s-^-.] al ^+2irw a — ;> 



where K = (p — \ + l)^_ 1)M ,„ jP + ..., 



they must also allow the operator 



Q = l(P,B) = \(PR-RP), 



VOL. XII. PT. III. 15 



