Royal Society. 859 



The general statement of the method of proof may be made thus : 

 two theorems are introduced which connect every odd number with 

 the gradation series, 1, 3, 7, 13, &c., of which the general term is 

 « + w* + l or 4p'- + 1p-\-\ (that is, the double of a triangular number 

 + 1), each term of which series can be resolved into four squares, the 

 algebraic sum of the roots of vihich, p, p,p, p-{-\, or p—\, p,p, p 

 may manifestly be = 1 . By these theorems it is shown that every 

 odd number is divisible into four squares, having roots capable of 

 forming as the sum of the roots 1, 3, 5, 7, &c. up to the greatest 

 possible sum of the roots. 



As the four square numbers which compose an odd number must 

 obviously be three of them even and one odd, or three odd and one 

 even, the differences of the roots among themselves must be the first 

 odd and the third even, or vice versd ; and therefore these roots must 

 have the sum of the first and third differences an odd number ; the 

 middle difference may be either odd or even. 



The first of the theorems referred to, called by the author " Theo- 

 rem P," is in substance this : — 



Let r, s, t, V be the roots the squares of which compose any odd 

 number N, such that r + s + t-{-v=:l, and let each of these roots be 

 increased by m; then r+m, s+m, t-\-m, v-\-m will be the roots of 

 the odd number N + 2m(2OT+l) ; and m — r, m— s, m — t, m — v the 

 roots of the odd number N-i-2m('2»i — 1) ; the sum of the roots in 

 the first case being 4/n+l, and in the second 4m — 1. So that 

 giving to m successively the values 0, 1,2, 3, &c. in the general form 

 N + 2>n(2?n+l), a series will be formed in which the sums of the 

 roots will be 1,3, 5, 7, 9, &c., and the sums of their squares N, 

 N + 2.I.1, N + 2.1.3, N + 2.2.3, N + 2.2.5, N + 2.3.5, 

 N + 2.3 .7, N + 2,4.7,&c.; orN,N + 1.2, N + 2 . 3, N + 3.4, 

 N + 4 . 5, N-l-5 .6, N + 6 . 7, N + 7 . 8, &c. So that if p be the 

 distance of any odd number in this series from N, the number vdll be 

 N +/?(/?+ 1), and the sum of its roots will be '2p-\-l. 



The conclusions to be drawn from this theorem are then stated: — 



1. The greatest sum of the roots of the squares into which any 

 odd number can be divided may be obtained : for let 2« + 1 be any 

 odd number, and 2p-{- 1 the odd number to which the algebraic sum 

 of its roots is required to be equal; then if p is such that ^(^+1) 

 is less than 2n+l, the number 2n+l can be resolved into squares 

 the sum of whose roots is '2p + \ ; otherwise it cannot. 



2. The form of the roots of 2«-fl maybe found of which the 

 algebraic sum is any possible odd number 2/3+1 except 1, provided 

 all the odd numbers less than 2w+l possess the property of having 

 the algebraic sum of their roots =1. For if from 2n+ i, p{p-\-l) 

 be taken, there will remain an odd number (N in Theorem P) such 

 that, according to the condition stated, the algebraic sum of its 

 roots = 1 ; and in the series of roots and odd numbers formed from 

 these roots according to theorem P, p terms from N will be found 

 the number 2«+l composed of squares the algebraic sum of whose 

 roots is 2p-\-\. 



It thus appears that any odd number 2n + 1 can be divided into 



