364 report— 1865. 



because the coefficient of q" in the expansion of ^q~H is a sum containing 

 two units for every solution of the equation 4u=ac — b 2 , in which, a, b, c 

 being positive and uneven, &<«<c, and one unit for every such solution in 

 which either & = «<c, or b«x=c; and because (bj' reasoning similar to that 

 already employed in this article) it is ascertained that this sum is equal to 

 the number of seduced forms of determinant — 4», i. e. to F(4h). Eliminating 

 I, we obtain, finally, 



CO GO 



1 2F(K)2 B =2[A(»i)-r(ra)]2" J 02F(u) 2 "=Sr' {n)q'\ 

 1 1 



or 



XL . . F(«)+2F(n-l 2 ) + 2F(n-2 2 ) + ... = A0O-r(n), 



XII. . . F(»)-2F(n-l 2 ) + 2F(n-2 2 )-... = r'(«)- 

 These equations are equivalent to the formulae I., II., III., V., VI. of M. 

 Kronecker ; these five, therefore (and with them, according to M. Kronecker, 

 the remaining three, IV., VII., VIII.), are deducible by analytical transfor- 

 mations from the single equation 



GO 



^cosa?e?a?=2F(»V*. 



'o w 1 



135. M. Kronecker asserts that the formula) I. -VIII. are independent, i. e., 

 that none of them can be deduced from the others by means of the elementary 

 equations satisfied by the functions F and G ; and that all the similar relations, 

 which are supplied by the theory of complex multiplication, may be obtained, 

 with the help of those elementary equations, by combining the eight formula?. 

 And it is certain that the system of the eight formulas does, in this sense, ex- 

 plicitly contain all the relations of similar form, which have been subsequently 

 given by M3I. Hermite and Joubert. Thus, many of these relations are par- 

 ticular cases of the formulas XL and XII., or of the combinations (XL) + (XII.) 

 (in M. Joubert' s memoir, the formulas 1, 2, 3, those of page 28, and the first 

 of page 29 ; also the first two formulas in M. Hermite's memoir (Liouville, 

 new series, vol. vii. p. 25) are of this land) ; others, again, are immediately 

 deducible from the two formulae 



4F(«) + 8F(h-4 2 ) + 8F(m-8 2 ) + ...=<J>(«), « = 3, mod 8, 

 8F(«-2 2 ) + 8F(n-6 2 )+8F(«-10 2 ) + . . .=*(«), n=7, mod 8, 

 combined by addition or subtraction with V., VI., and VII. But each of 

 these two formulas results from the combination f(V.) + |(VI.)—2(IV.), sim- 

 plified by means of the elementary equations satisfied by F and G. In this 

 way the formulas 4—9 of M. Joubert's memoir, and the third formula in M. 

 Hermite's memoir (Liouville, ibid, p. 36), may be obtained. Lastly, the 

 equation 



6G (^) + 6G (^-) + • • . = |(3*(»)-*(»)) 



n = 1, mod 4 

 (M. Joubert, p. 30) arises from the combination (IV.)— |(V.). 



M. Joubert's formulas, however, as they are given in his memoir, are not 

 immediately comparable to those of M. Kronecker. He rejects from the 

 modular equation of the uneven order n, the factors due to the square divisors 

 of n (see art. 125 of this report), and, in consequence, those derived classes 

 whose coefficients have any common divisor with n are excluded from his 

 enumerations. At the same time, the numerical functions, depending on the 



