ON THE THEORY OF NUMBERS. 345 



2j/ iyi 9 (6'»+D 2 +(6»+i^ _ 2( _ 1 )'"+»V/ 2 < 6 '"+ 1,2 + 2 < 6 "' 2 , 



which, however, is only a particular case of the general formula (A). 



The Eulerian product is also of importance in the theory of the partition 

 of numbers. If it be developed in a series proceeding by powers of q, the 

 coefficient C(m) of the mth power of </ in the development, expresses the number 

 of ways in which m can be composed by the addition of unequal numbers, or 

 by the addition of equal or unequal uneven numbers. Euler observed that his 

 fractional expression of the product furnishes a recurring formula for the 

 calculation of C(m), and the same thing is true of each of Jacobi's fractions ; 

 the simplest of the seven recurring formulas being that arising from the 

 fraction (D), viz., 



the summation extending to all positive or negative values of s for which 

 m — 3s 2 is not negative, and e representing 1, or 0, according as m is or is not 

 of the form |(3n 2 + ?i). 



CO + CO 



The equation U(l—q'")= S (— l) m gi(»»*f»0 is memorable historically as 

 1 -oo 



the earliest example of the introduction of a Theta function into analysis*. 

 It expresses the theorem 



" The excess of the number of ways in which a given number can be com- 

 posed by the addition of an even number of unequal numbers above the 

 number of ways in which it can be composed by the addition of an uneven 

 number of unequal numbers is ( — 1)'" or 0, according as the given number is, 

 or is not, of the form |(3m 2 +m)." 



Of this theorem Jacobi has given an arithmetical demonstration, repro- 

 ducing Euler's proof of the analytical formula. 



co i co 



The logarithmic differential of 11(1 — 2'") is I,<b(m)q'", where #(m), as 



1 ?1 



in art. 125, is the sum of the divisors of m : Euler thus obtained the equation 

 co +co +00 



2*(m)2"'X 2 ( — 1)'"25(3»< 2 +»0=| S (— l)"'+\3m 2 + m)<p< 3 "' 2 -l'">, 



1 — 00 — 00 



which supplies a recurring formula for the calculation of 4>(m), viz., 



S(-lWm-^p)=E(m), 



the summation extending to all positive or negative values of s for which 



3 g 2 _L s ... 

 m— ■ is positive, and E(m) representing ( — l) s+, w, or 0, according as m 



is, or is not, of the form |(3s 2 + s)f. 



* In the year 1750 or 1751. Nov. Comm. Petropol. vol. iii. p. 155. 



f On the equations n(l+g'") = n 1 _\ m _ i, Tl{l-q"') = 1(-l)"'gi( 3 '"' ! +" 1 ), and their 



iv. of the Acta is omitted in the collection) ; Introductio in Analysin Infinitorum, part 4, 

 cap. 16 ; Waring, Philosophical Transactions for 1788, p. 388 ; Legendre, Theorie des 

 Nombres, ed. 3, vol. ii. p. 128 ; Jacobi, Fundamenta Nova, p. 185, Crelle, vol. xsxii. p. 164, 

 vol. xsxvii. p. 67, 73 (or Matheniatische Werke, vol. i. p. 345, vol. ii. p. 73, 79). 



