ON THE THEORY OP NUMBERS. 337 



i'-l S-} 



by the addition of six uneven squares is 1 i g -2[(—l) 2 —( — 1) 2 ]3 2 ;ifN=l, 



£-1 



mod 4, this number is zero ; if N" = — 1, mod 4, it is — ^ 2(— 1) 2 <5V 



(7) The number of representations of any uneven number as a sum of eight 

 squares is sixteen times the sum of the cubes of its divisors ; for an even num- 

 ber it is sixteen times the excess of the cubes of the even divisors above the 

 cubes of the uneven divisors." 



(8). " If N is any number whatever, the number of compositions of 8N by 

 the addition of eight uneven squares is equal to the sum of the cubes of those 

 divisors of N whose conjugates are uneven." 



In counting the number of compositions by addition of squares, two com- 

 positions are to be considered as different if, and only if, the same places in 

 each are not occupied by the same squares ; but in counting the number of 

 representations we have to attend also to the signs of the roots of the squares. 

 Thus each composition by the addition of four squares, none of which is zero, 

 is equivalent to sixteen representations. Only one or two of the preceding 

 theorems are enunciated in the published writings of Jacobi : see Crelle's 

 Journal, vol. iii. p. 191 ; vol. xii. p. 167. Some of the others have been 

 given by Eisenstein (Crelle, vol. xxxv. p. 135), who had also obtained purely 

 arithmetical demonstrations of them from the theory of quadratic forms con- 

 taining several indeterminates. "In my investigations," he says, "these 

 theorems are proved by purely arithmetical considerations, and appear as 

 special cases of more general theorems ; at the same time we see why these 

 developments close with the eighth power ; since, in fact, eight is the greatest 

 number of indeterminates for which only one class of forms, represented by 

 a sum of squares, appertains to the determinant — 1." 



In the second of the notes to which we have just referred (Crelle, vol. xii. 

 p. 167), Jacobi has given an arithmetical demonstration of the theorem (4). 

 It consists in a kind of translation of the analytical proof into an arithme- 

 tical one ; and is of great interest and importance, as the first example of a 

 new method, and as having suggested important researches to MM. Liouville 

 and Kronecker (see Liouville's Journal, New Series, vol. vii. p. 48 ; M. Kro- 

 necker, * Monatsberichte,' May 26, 1862, p. 307). 



2K^ 

 The doubly periodic functions of argument — - obtained by dividing any 



Theta function by any other, or the product of any two of them, by the pro- 

 duct of the other two, admit of development in series proceeding by sines or 

 cosines of multiples of the argument x. These developments, which, unlike 

 the developments of the Theta functions themselves, are not convergent for all 

 values of x, real or imaginary, will be found for the most part in section 39 

 of the ' Fundamenta Nova ' ; and the complete system has been given by M. 

 Hermite (Comptes Eendus, July 7, 1862). One, which we require in this 

 place, will serve as an example of the rest, 



kK . 2Kx ^, oi" sin vx \ 

 --sin am =2„ 2 B1U,1 L / 



2k *• 1-2* [■ (A) 



= 2* 2j sin Sx . gi". ) 



It is from these developments that the expansions (1) ... (8) of the 



powers of a/ _ and a/ __ are deduced Thus, writing - for x in (A), 



we find, since sin am K=l, 



1865. 2 a 



