13 REPORT— 1873. 



of ways in which N can be expressed in the form a*+b^-\-c^-^2cf+3e'' ; [2^4] in 

 the form 2a^+2b^+4c^ &c. Then 



40320 [P] + 20160 [l'=2] + 6720 [I'S] + 10080 [1*2^] 

 + 1680 [1'4] + 3360 [1323] + 336 [1^5] + 6040 [r^2^] 

 + 840 [P24] + 1120 [P3'] + 56 [P6] + 1680 [12^3] 

 + 168 [125] +280 [134] + 8[17] + 2520 [2^] 

 + 420 [2^4] + 560 [23^] + 28 [26] + 56 [35] 

 + 70 [4^] +[8] = x(m, 



xM being the sum of the cubes of all the factors of « (= J N) which are such that when 

 « is divided by any of them the quotient is odd; viz. ;^(>j) = 2a^, a being any factor of 



n such that - is odd. 

 a 



Example.— Take N=96; therefore |N=12 = 1.12 = 2.6 = 3.4, so' that the only 



factors that have oddcofactors are 12 and 4, whence x(sN) = 123+43 = 1792. And 



06 = 49+25+9+9+1 + 1 + 1 + 1 = 81+9 + 1 + 1 + 1 + 1+1 + 1 



= 49+9+9+9+9+9 + 1 + 1 = 25+25+9+9+9+9+9+1 

 = 25+25+25+9+9+1 + 1 + 1. 

 Therefore 



[1=24] = 1, [1=6]= 1, [125]=2, [232] = 1, and 840+56+336+560=1792. 



(ix) Every number that is the sum of six odd squares is of the form 8«+6 ; 

 and if the lialf of such a number, being of the form 4«+3, be resolved in any man- 

 ner into two factors, one must be of the form 4rt+l and the other of the form 

 4«+3. Adopting a notation similar to that described in (viii), if 2s denotes any 

 number of the form 8w+6, 



720 [r] + 360 [1^2] + 120 [1^3] + 180 [P2=] + 30 [P4] + 60 [123] + 6 [15] 

 + 90 [2^] + 15 [24] + 20 [3=] + [6] = U(s), 



where |(s) = sum of the squares of all ths factors of s that ai-e of the form 4«+3, 

 — sum of the squares of all those that are of the form 4w+l. 



Examples. — Take 2s =30, then 



s=l . 15=3 .5; .-. ^(s) = 152+3=-5=-P = 208, and K208) = 26. 



And 30 = 25+1 + 1+1 + 1+1 = 9+9+9+1 + 1+1. 



Therefore [15] = 1, [3=] = 1, and 6+20 = 26. 

 Take 2s=270, then 



s = 135, and ^(s) = 135=+27^+15=+3=-45=-9=-5=-l = 17056, 



so that ^l(s) =2132. And it will be found that the decomposition into squares gives 



[P2] = l, [P3] = l, [P2=]=7, [P4]=2, [123]=5, [15]=2, [3»]=1, 

 and 



360+120+1260+60+300+12+20 = 21-32. 



(*) The above are the principal theorems proved, which were illustrated by several 

 other examples. The paper concluded with an algebraical proof of the identity 



(l-2x+2x*-2x' + ...y + (2xi+2x^-+2x'^-+...y = (l+2x+2x'+2x'+ ...y, 



which resulted ii'om the development of a process indicated by Gauss in his memoir 

 "Zur Theorie der neuen Transscendenten," Werke, t. iii. p. 447. 



[Since the paper, of which the above is an abstract, was read, Prof. H. J. S. Smith, 

 who kindly looked through it at the author's request, has pointed out to him that 

 most of the theorems contained in it had been previously published by Jacobi, 

 Eisenstein, and himself, though expressed in a somewhat different form. For 

 references see Prof. Smith's Report on the Theory of Numbers, Part VI. art. 127 

 (British Association Report, 1865, pp. 335-338).] 



