TRANSACTIONS Or THE SECTIONS. 11 



be expressed as the sum of two unequal odd squares, and, if of the' form 2(4m— l)'r*, 

 it can be so expressed in HV'C''^)"!} ways — the letters meaning as in (ii). 



(v) Consider any number N, and let a be the number of waj'S in which it can be 

 expressed as the sum of four squares, all different (a^+6^ + c^-f cf-) ; a, the number 

 of ways, two of the four squares being identical (2a^+6^+c^) ; 0,3, when two pairs 

 of squares are identical (2«^+26^J ; a^, when three squares are identical (Sa-+b^) ; 

 and a^ when all four are identical (4o^j. Let /3„ /S,, /Sj be similar quantities de- 

 noting respectively the number of ways in which N can be expressed as the 

 sum of three squares, with none, two, or three identical, 



Let y, y^, and S be similar quantities for two squares and one square, 



(a= + 6=, 2a^ a") ; 

 then 



48a+24a,+12a,,-l-8a3 + 2a,4-24/3-|-12/3,+4/33 + 6y + 3y2 + 5 



= the sum of the factors of N, if N be odd, 

 and 



= 3 X (the sum of the factors of n) if N is even, and = 2^n, n being odd. 



Generally, several of the quantities a, a^, &c. will vanish ; and some must always, 

 for two of the three /Sg, y^, 8 must be zero ; also a^ and b vanish unless N is a square ; 

 a^a, a^, and 72 vanish if N is odd ; and the letters a^, /Sj, y^, and S can only have the 

 values or 1. 



Examples. — Take N=81 ; the factors are 1, 3, 9, 27, 81, of which the sum 

 = 121. And 



81=36+25+16+4=64+9+4+4=64+16+1 = 49+16+16=36+36-1-9. 



Therefore 



a=l, a, = l, /3=1, i3,=2, S=l, and 48+24+24+12x2+l=12L 



Take N=68=2M7 ; and the sum of the factors of 17=18, which multiplied by 

 3=54. And 68=49+9+9+1 = 25+25+9+9=36+16+16=64+4. 

 Therefore 



«2=1> "22=1) i32=l, V = l, and 24+12+12+6=54. 



(vi) A considerable reduction takes place when N is of the form 8m+7, in 

 which case the formula merel}' becomes 



48a+24a^+8a3 = sum of the factors of N. 



Example.— Take N=63; sum of factoids = 104, and 



63=49+9+4+1=36+25+1 + 1=25+25+9+4=36+9+9+9. 



Therefore , 



a=l, 0^=2, 03=1, and 48+48+8 = 104. 



(vii) Let A denote the number of ways in which any number N of the form 

 8«+4 can be expressed as the sum of four odd squares, all different; A^ the num- 

 ber of ways when two are identical ; A^^ when two pairs are identical ; A3 and A^ 

 when three and four respectively are identical. Then 



24A+ I2A2+ 6A22+4A3 4- A^ = sum of factors of |-N. 



Example. — Take N = 84, sum of factors of :^N=.32. And 



84=49+25 + 9+1 = 25+25+25+9=81-1-1-1-1-1-1. 

 Therefore 



A=l, A3=2, and 24 + 8=32. 



(viii) Let [P] denote the number of ways in which any number N, divisible by 

 8, can be expressed as the sum of eight odd squares, all different ; [1*^2] the num- 

 ber of ways when a pair are identical &c., so that, e. g. [P23] denotes the number 



