On an Uneven Number as a Sum of four Squares. 45 



come identical ; if also another, b, = 0, the resolution is of the 

 form P 2 + P 2 + Q 2 + Q 2 ; if further c = 0, the form is 



d 2 + d 2 + d 2 + (P. 



Confining our attention first to the case in which no one of 

 the quantities a, b, c, d is zero, we see that every resolution of 

 N gives rise to two resolutions of 4N : — one of class I., in which 

 the largest square is (a + b + c + d) 2 ; and the other formed from 

 it by changing the sign of any one letter throughout. Consider 

 any two resolutions of N, viz. a 2 + b 2 + e 2 + d 2 and a 2 + /3 2 + y 2 + S 2 ; 

 they give rise to four resolutions of 4JST, viz. two of class I. and 

 two of class II. ; and it is easily seen that the two resolutions 

 of class I. cannot be identical (and also that those of class II. 

 cannot be identical), unless a 2 , b 2 , c 2 , <P = u 2 , /3 2 , y 2 , S 2 *, which 

 is supposed not to be the case. But it requires further exami- 

 nation to see that one of the resolutions of class I. cannot be 

 equal to one of those of class II. ; viz. we have to show that, if 

 a 2 + /3 2 + 7 2 + S 2 be an uneven number, and if 



+ (-« + /3 + Y + 8) 2 

 = (a + b + c + dy + (a + b-c-d) 2 + (a-b-c + d) 2 



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



term for term, then it follows that a 2 , /3 2 ; y 2 , S 2 = a 2 , b 2 , c 2 , d 2 . 

 The conditions give 



* + & + y—8=±( a + b + c + d), 



a + /3— y + S=±( a + b — c—d), 



a— /3 + 7 + S=±( a— b — c + d), 



-ct + /3 + Y + 8=±(-a + b-c + d); 



and a few moments' consideration shows that the ambiguities 

 on the right-hand side must be all replaced by + or by — , 

 or that two must be + and two — . (For, ex. gr., take 

 the first three + and the last — , and we have by addition 

 a + /3 + y + S=2a; that is, an uneven number =2a.) Repla- 

 cing all the ambiguities by the same (say the + ) sign, the 

 equations are 



a + /3 + 7 — 8 = a + b + c + d, 



a + /3— y + S= a + b — c — d) 



a— /3-fy + S= a—b—c + d, 



—u + /3 + y + S=:—a + b — c + d, 



* By (&, W, <?, d 2 = ct 2 ,fi 2 , y*, b 2 is meant that the four former squares 

 are to be equal to the latter independently of order ; for example, « 2 =/3 2 , 

 6 2 =# 2 , c 2 = S 2 , d 2 =y 2 satisfies the equation. 



