40 Mr. J, W. L. Glaislior on the Representation oj 



so that a=«, @=b, y=d, 8=—c; and, generally, we must 

 have a 2 , /3 2 , <y 2 , 8 2 = cr, b 2 , c 2 , d 2 . The second case in effect 

 includes the other (in which the resolutions are of the same 

 class) by supposing the sign of 8 changed. 



The presence of zero values of a, b, (fee. in no way affects the 

 argument (except by reducing to equivalence the two classes 

 I. and II.) ; so that generally we see that every resolution of 

 N gives rise to two resolutions of 4N, unless a zero value occurs, 

 when there is but one. It is also evident that the transformed 

 resolutions are of the same form as regards equality of squares 

 as the original resolutions : viz. if a 2 , b 2 , c 2 , d 2 are all different, 

 the squares in the transformations are all different; if a 2 = b 2 , 

 two squares in each of the transformations are equal, and so 

 on. This is true also if one of the squares be zero. If two be 

 zero, the transformation converts c 2 + $ into P 2 + P 2 + Q 2 + Q 2 ; 

 and if three be equal, into d 2 + d 2 + d 2 + (l 2 as before mentioned. 



We have now to show that every resolution of 4N as the 

 sum of four uneven squares A 2 , B 2 , C 2 , D 2 , may be derived 

 from a resolution of N of the form a 2 + /3 2 + 7 2 + 8 2 by a trans- 

 formation of class I. or class II. ; that is, that we can always 

 find integer values of a, B, y, 8 that satisfy the system of 

 equations 



a + /S + Y-S=±A, 



* + £-7 + S=±B, 



«— /3 + y + S=±C, 



— a + /3 + 7 + S=±D. 



Taking the positive signs throughout for the first series of 

 values, and changing the sign of J) for the second, 



u=i( A + B + C-D) or si( A + B + C + D), 



£=i( A + B-C + D) „ i( A + B-C-D), 



7 = i( A-B + C + D) „ i( A-B + C-D), 



S=i(-A + B + C + D) „ j(-A + B + C-D); 



and, since D is uneven, one (and only one) of these systems 

 gives integer values to a, /3, 7, 8. Thus there is no resolution 

 of 4K into uneven squares that is not derivable by transforma- 

 tion from a resolution of N ; and it only remains to connect the 

 representations of N with those of 4N. 



1°. Consider the resolutions in which no one of the squares 

 a 2 , b 2 , c 2 , d 2 is equal to zero ; then, since each resolution of K 

 gives rise by transformation to resolutions of the same form as 

 regards equality of squares, each resolution of both N and 4N 

 produces the same number of compositions, and therefore also 

 of representations ; so that the number of representations of 



