in which 



ON THE THEORY OF NUMBERS. 329 



jirrbafl 



m=cin + bv + ab, 



(aciS ! +2bcnv+bdv-+2abc l L+2abdi> + al>-c) 



1 ^ ■* «T I UNO 



or 



V6 



/6\ . . :„ . 

 = 1-1 i-? a , if a is uneven, 



= [t) i~* a X i-K«-DC*-i), if b is uneven* ; 



the radical *J —i(a-\- 6a) represents that square root of —?'(« + 6a), of which 

 the real part is positive ; lastly, A is determined hy the equation 



A/^=0 o , o (O,a)= J!^l- T =0ai,c d (0,w), . . (33) 



V 7T V— 2(ff + 0to) 



which is a particular case of the formula (32) ; and M hy the equation 



E- ,J **° o\ {0,u,) (34) 



The formula supposes that 6 is different from zero and positive ; if 6 = 0, we 

 may suppose cr=cZ=l, so that w=c+a, and the formula of transformation is 



e„, F fe oj -«-?** ftfl4+H ., (*,•), (35) 



where ^=^#4 



The equations of the annexed Tahle, which, for any transformation of the 

 first order, express the relation subsisting between the given and the trans- 

 formed modulus, are also due to M. Hermite, and are of great importance in 

 the theory of the functions <f>(w) and ^(w) f . They may be obtained by 

 applying the formula of transformation (32) to the expressions of ^(w) given 

 by Jacobi (27). There are six cases, answering to the six solutions, of which 

 the congruence ad — 6c=l, mod 2 is susceptible. We add, in each case, the 

 value of the multiplier. 



* These determinations of the value of J coincide with those given hy M. Hermite in 

 Liouville's Journal, vol. iii. p. 20 ; where, however, it would seem that the formulae re- 

 lating to the two cases of " a pair" and "« impair" ought to be transposed. 



t " Sur la resolution de 1' equation du cinquieme degreV' Comptes Eendus, vol. xlvi. 

 p. 508 ; or in a separate reprint (including other memoirs from vols. xlvi. and xlviii.) with 

 the title " Sur la theorie des equations modulaires, et la resolution de 1'equation du cinqui- 

 eme degre," p. 4. 



