[ 44 ] 



VI. On the Representation of an Uneven Number as a Sum of 

 four Squares, and as the Sum of a Square and two Triangular 

 Numbers, By J. W. L. Glaisher, M.A., F.R.S.* 



I, AT the end of a paper entitled " Verification of an Ellip- 

 -£*- tic Transcendent Identity," published in the Philoso- 

 phical Magazine for June 1874, I have reproduced in an ex- 

 Eanded form a proof (due to Gauss, but the steps of which he 

 ad only briefly indicated) of the identity 



(l + 2q + 2q i + 2q» + &c.y=(l-2q + 2q i -2q» + &c.y 



+ (2^+2^ + 2^ + &c.) 4 ; 



and it occurred to me subsequently that it would be interesting 

 to demonstrate the truth of the identity by showing, directly, 

 from arithmetical considerations, the equality of the coefficients 

 of like powers of x on the two sides of the equation. 



The coefficients of the even powers of x are obviously the 

 same ; and the theorem to be proved is that, N being any uneven 

 number, the number of representations f of 4N as the sum of 

 four uneven squares is equal to twice the number of represen- 

 tations of N as the sum of four, or a less number of, uneven 

 squares. It is convenient in what follows to have a name for 

 a decomposition of a number as a sum of four squares, irre- 

 spective of the order in which they are written; and I shall 

 call such a decomposition a resolution ; so that a? + b 2 + c 2 + cf 2 , 

 a 2 + c 2 + b 2 + d 2 , &c. are all the same resolution. 



Suppose N = a 2 + & 2 + c 2 + d 2 is a resolution of N, then we 

 can derive from it two resolutions of 4N into uneven squares, 

 viz. 



I. ( a + b + c + d) 2 , or II. ( a + b + c-d) 2 , 

 + ( a + b-c-d) 2 , + ( a + b-c + d) 2 , 



+ ( a — b — c + cl) 2 , +( a — b + c + cl) 2 , 



+ (-« + 6-c + d) 2 , + (-a + b + c + d) 2 ; 



and it is easily seen that no other set of combinations formed 

 by addition and subtraction of the elements a, b, c, d will give 

 a resolution not included in these two. 



If any one of the elements, say a, = 0, then I. and II. be- 



* Communicated by the Author. 



f "In counting the number of compositions by addition of squares, two 

 compositions 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." — Professor H. J. 

 S. Smith, " Report on the Theory of Numbers," art. 127 (British Associa- 

 tion Report, Birmingham Meeting, 1865, p. 337). 



