336 report — 1865. 



(2) M=42,(-ip ll _i-=42 J*L 



=42,2,(-l) 2 9 I 



^ f =1+8 ^ IT(S)^ =1+8S " (i + (-i)YT- 



=l+24S n S«S2»-16S,2^f=l + 8[2-(— l)"]S„Sj^\ 

 4« 2 K 2 _ 16 vg" _ 16v ?"(1 + ? 5 ") 



=163,2^2". 



(4) 



V— 1 



( 5) w_i + -un H ^--i*(-i)T^ ; 



= l + 42„2*(-l) 2 (4a' 2 -a 2 ) 2 ». 



1/-1 



(0) w-«,'!t_a ( (-i)" r ^c 



7 7T 3 1 + 2" 1—2" 



S'-1 g — 1 



=42„2 5 [(-l) 3 -(-1) 2 ]^ 



m 16K' 16 n'g" 



(7) __ l+162„ 1 _ ( _ ]L)BgB 



= l + 162 w 2 d (-l)" +, ^V- 



=2562,, 2S'Y"- 



Of these formuke, the first two are the analytical expression of the prin- 

 cipal theorems relating to the composition of numbers by the addition of two 

 squares (see art. 95 of this Report) ; the others may be paraphrased as 



1'ulloWS*. 



(3) "The number of representations of any number N as a sum of four 

 squares is eight times the sum of its divisors if N is uneven, twenty-four 

 times the sum of its uneven divisors if N is even." 



(4) "The number of compositions of the quadruple of any uneven number 

 N by the addition of four uneven squares is equal to the sum of the divisors 

 of N." 



(5) " The number of representations of any number N as a sum of six 



s— 1 

 squares is 42(— 1) 2 (4g' 2 — S 2 ), S denoting any uneven divisor of ~N, §' its 

 conjugate divisor. In particular if N = 1, mod 4, the number of represen- 



,5-1 Sj-l 



tations is 122(-1) 2 i 2 ; if N = -l, mod 4, it is -202(-l) 2 a 2 ." 



(6) " The number of compositions of the double of any uneven number N 



* The expansions of (1)X(2), (1-)X(4), (3)x(2), (3)x(4), are also given in sections 40 

 and 41 of the ' Fundamenta' ; and may be similarly interpreted. 



