360 report — 1865. 



members, the last two by multiplying together the series in their left-hand 

 members. Combining these identities with the formulas IX. and X., and 

 attending to the equation £(0)=^,, we obtain 



i k ;/ M°)i (i-if- 1 ) 2 < A > 



qo i n co ,» -i 



122 E 00t»= 1 + 8 2 2 j . . (B) 



of which the second (art. 127, equation 3) may be written in the form 



co 



122.E(H) 5 »=(l + 2 5 + 2r/ + 2r/ + ...) 3 , .... (B') 

 



in which it expresses the arithmetical theorem of Gauss, to which we have 

 already referred (art. 130). 



It appears from the equations (A) and (B) that the generating functions of 

 F(m) and E(«) are elliptic series; and M. Hennite, in two important memoirs 

 (Comptes Bendus, Aug. 5, 18(31, or Liouville, New Series, vol. vii. p. 25, and 

 Comptes Bendus, July 7, 1862) suggested, as it would seem, by these equations, 

 has succeeded in deducing the second of them, and others of the same character, 

 from the general expansions of elliptic functions, without having occasion to 

 consider the special modules which admit of complex multiplication. He has 

 thus discovered a new and comparatively elementary method of arriving at the 

 formulae of M. Kroneckcr ; to whom indeed this method was already known, as 

 his expressions of the generating functions of E(h) and F(«) indicate, and as 

 he has himself expressly stated in a note published after the appearance of 

 M. Hermite's first memoir (Monatsberichte, May 2G, 1802, pp. 307, 308). M. 

 Hermite's method is an extension of that employed by Jacobi (see art. 127), 

 and depends on the developments of doubly periodic functions in series pro- 

 ceeding by sines or cosines of the multiples of the argument. To this set of 

 developments, however, M. Hermite adds a second obtained by dividing the 

 product of two Theta functions by a third. A series of the first set, and one 

 of the second (both alike containing only sines, or only cosines, and only even, 

 or only uneven multiples of the argument), are then multiplied together. The 

 non-periodic part of the product (or its integral, taken from the limit to tt) 

 is a function of q only, and if the product can be formed in more than one 

 way, we obtain different expressions of this function, a comparison of which 

 supplies in each ease an arithmetical formula. We take the following example 

 from M. Hermite's first memoir ; and, with him, we write for brevity, 0, 9,, 



H, Hl for e(^), e/^f), H(?^), H^), and 6, B v Vl for 0(0), 



0,(0), H,(0)*. ff 



Multiplying together the three pairs of series 



t = 2 2""cos2«.r, 



— co 



(i) 



* The developments of elliptic functions, in series proceeding by sines or cosines of 

 the multiples of the argument, which are employed in this article, will be found in the 

 Fundanienta Nova (sections 40-42), or in M. Hermite's second memoir (Comptes Eendus, 

 July 7, 1862). 



