ON THE THEORY OF NUMBERS. 319 



with [V.] — j[IV.]; and with II. if n is unevenly even. If n is the qua- 

 druple of an uneven number, its left-hand member may be written in the 

 form 



[e(0 + ,e(-_ 1 .) +2E («_ 2 =) + ....] 



+ HF(»-l ! ) + iF(»-3=) + |F(»-5')+...]; 



the sum of the first of these series is §X( t)=|X(h) ; the second series, 



coinciding with f[I.]~ |[V.], has for its sum fX(») ; the two series together 

 are therefore equal to 2X(«). Lastly, the formula, if true for any even 

 number, is also true for its quadruple ; for if n=0, mod 8, 



E(») + 2E(>j-1 2 )+2E(h-2 2 ) + . . . 



= [E@ + 2E(«-r) +2 E(L'- 2 =) + ...] 



+ [2E(»-1 2 ) + 2E(h-3 2 ) + 2E(h-5 2 ) + . . .], 



and every term of the second series is zero, because the numbers n— l 2 , 

 n— 3 2 , . . are all =7, mod 8. 



Of the preceding formulae those which contain the functions ¥ and W, are 

 to be regarded as of a more abstruse character than those which only contain 

 X, 4>, and $'. The latter, in fact, are deducible from known theorems of 

 arithmetic. Thus if we multiply the formula X. by 12, the right-hand 

 member becomes 8[2 + ( — l)"]X(?i), or the number of representations of n 

 as a sum of four squares (see art. 127). Consequently 12E(m) is the number 

 of representations of n as a sum of three squares ; for, assuming that this is 

 so for 1, 2, 3 ... «—l, we may infer from the formula X. that it is so for n. 

 Thus the celebrated theorem of Gauss*, which connects the number of re- 

 presentations of a number n as a sum of three squares, with the number of 

 classes of quadratic forms of det. — n, is contained in the formula X.; and 

 conversely, that formula is itself deducible from the theorem of Gauss com- 

 bined with the other and more elementary theorem, which connects the 

 number of representations of n as a siun of four squares with the sum of the 

 uneven divisors of n. 



131. Demonstration of the Formula; of M. Kronecher. — We shall first de- 

 monstrate the formula V. For this purpose, we consider the equation 

 / 8 0'> 1— .r) = 0, obtained by writing x for K 2 , and 1—x for \ 2 in the modular 

 equation / 3 (^ 2 ,\ 2 )=0 of an uneven order n (art. 125). We shall determine 

 the order of this equation by two different methods ; first, by ascertaining the 

 dimensions of f(x, 1— x), when a? is increased without limit ; secondly, by 

 assigning its roots, and the multiplicity of each of them ; a comparison of 

 these two determinations will give the formula V. 



(i.) Letx=f(Q); then 



f s (x^-x)=n[m-f(?^~y\, 



because (art. 125) 



t * Disq. Aritk. art. 291. Legendre bad discovered particular cases of the theorem by 

 induction. Hist, de l'Ac. de Paris, 1785, p. 530 sqq. Theorie des Noinbres, ed. 3, vol. i. 

 troisieme partie. 



t Since <p\u>+2)=$%u>) (equation i. Table A), the systems of values represented by 



