an Uneven Number as a Sum of four Sqtiares. 47 



4N as a sum of uneven squares is double the number of repre- 

 sentations of N. 2°. If one of the squares is zero, no alteration 

 is made in the number of compositions ; but each composition 

 of N" gives rise to 8 representations only instead of 16, and the 

 two classes become identical; so that in this case also the num- 

 ber of representations of 4N is double the number of repre- 

 sentations of 3JJ, 3°. If two squares are zero (so that the other 

 two must be^ different), the number of representations pro- 

 duced by each resolution of K is 12 .4, while the corresponding 

 resolution of 4N (being of the form P 2 + P 2 + Q 2 + Q 2 ) gives 

 rise to 6 . 16 ; and, 4°, if E" is an uneven square, this resolu- 

 tion gives 8 representations, while the corresponding resolu- 

 tion d 2 + d? + d 2 + d 2 of 4N gives 16 representations. Thus, 

 universally the number of representations of 4K" as the sum of 

 four uneven squares is equal to double the number of repre- 

 sentations of 2? as a sum of four, or a less number of, squares. 

 I here add an example of the transformation. Take N = 1 1 7 * ; 

 then 



82 + 6 2 -f4 2 + l 2 gives4N=19 2 + 9 2 + 5 2 + l 2 . . (I.) 



= 17 2 + ll 2 + 7 2 + 3 2 , . (II.) 



9 2 + 4 2 + 4 2 4 . 2 2 ?J 4N=19 2 + 7 2 + 7 2 + 3 2 . . (I.) 



= 15 2 + 11 2 + 11 2 + 1 2 . (II.) 



l0 2 + 3 2 + 2 2 + 2 2 „ 4N=17 2 + 9 2 + 7 2 + 7 2 . . (I.) 



= 13 2 + 13 2 + ll 2 + 3 2 . (II.) 



72 + 6 2 + 4 2 + 4 2 4N=21 2 + 5 2 + l 2 + l 2 . . (I.) 



= 13 2 + 13 2 + 9 2 + 7 2 , . (II.) 



& + & + & + & „ 4N=21 2 + 3 2 + 3 2 + 3 2 . . (I.) 



= 15 2 + 9 2 + 9 2 + 9 2 ,. . (II.) 

 10 2 + 4 2 + l 2 - „ 4N=15 2 + 13 2 + 7 2 + 5 2 , 

 8 2 + 7 2 + 2 2 „ 4N=17 2 + 13 2 + 3 2 + l 2 , 

 9 2 + 6 2 „ 4N=15 2 + 15 2 + 3 2 + 3 2 . 



Apart from the verification of the elliptic transcendent iden- 

 tity, the connexion between the separate resolutions of N and 

 the corresponding resolutions of 4N is interesting. '. 

 . II. . The following investigation relates to the representation 

 of an uneven number as the sum of one square and two trian- 

 gular numbers. 



* The number of representations of an uneven number as the sum of 

 - four squares is eight times the sum of its divisors, and 117 is chosen on 

 account of the number of its divisors. 



