346 report — 1865. 



The cube of the Eulerian product is equal to the series |2( — l)'\2m + l)qh(»°-+»'i 

 (art. 124, equation 22) ; so that 



[2( — l)'" ? I(3m 2 +«)j 3 =i2( — l)"'(2»l + l)^fm2 + ,„), . . . . (F) 



a result which in an earlier memoir (Crelle, vol. xxi. p. 13, or translated in 

 Liouville, First Series, vol. vii. p. 85) Jacobi describes as " hitherto unparalleled 

 in analysis." Writing j 24 for q, and multiplying by q 3 , it becomes 



6-1 

 : 2l (-l) 2 bq*>, 



[*©*-]' 



the summations 2 X and 2 2 extending respectively to all positive uneven 

 numbers, and to all positive uneven numbers prime to 3. In this form it 

 expresses the theorem 



" The sum 2 ( ) extended to all compositions of any number N by 



\a x a 2 a z J 



the addition of three uneven squares a\, a\, a\, all of which are prime to 3, is 



m — \ 



( — 1) 2 m or according as N is or is not the triple of an uneven square." 

 Differentiating logarithmically, we find 



4-1 



S 1 2 2 (-l) 2 ft\b( a 2 -b 2 )q- 2 + 3b2 =0. 



This equation (in which all the exponents have the same quadratic form) 

 admits of immediate verification, elementary considerations sufficing to show 

 ft-i 



that the sum S( — 1) 2 ( - \b(a 2 — b 2 ) extended to every solution of the equa- 

 tion N=a 3 + 36 2 , is zero. Jacobi thus obtains a direct arithmetical proof of 

 the formula (F). (Crelle, vol. xxi. p. 15-18.) 



The square and the cube of the Eulerian product can also each of them be 

 represented in two different ways as the quotient of two elliptic products. 



Other formulae of Jacobi's are inferred from the fundamental equation (7) 

 in a somewhat more complicated way. Replacing v in that equation by certain 

 roots of unity, and multiplying two or more of the results together, Jacobi 

 obtains products winch can be expressed in more than one way by means of 

 elliptic products ; the formulas thus deduced are remarkable chiefly because 

 they lead to equations, not between two, but between three or more series, 

 the exponents of which have certain quadratic forms. 



Lastly, a few additional equalities are derived not from the fundamental 

 equation, but from the modular equations of the third and seventh orders. 

 The modular equation of the third order was brought by Legendre into the 

 form V7X + <i/k\=1 ; whence evidently </> 2 (w)</> 2 (3w) + ^ 2 (u))^ 2 (3a/) = l ; 

 writing for the functions (j> 2 and \p 2 their values given by equation (14), art. 124, 

 and changing q into q i , we find 



W 1 ( ]Y»+nN ~4(m 2 +3» 2 ) _ ^2«( 4 " ! +l) 2 +3(*i+l) 2 # 



The equation of the seventh order, in the form in which it has been put by 

 M. Gutzlaff*, 



admits of similar treatment, and furnishes as many as seven formulae on 

 account of the variety of expressions which the equations (27) and (28) allow 

 us to substitute for f and ^ in the equation 0(a>)<p(7w) + i£(w)^(7u>)=l. It 



* Crelle's Journal, vol. xii. p. 17JJ. 



