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(2m+l) ; and mr, ms, mt, mv the roots of the odd number N-|-2m(2m 1) ; the sum of the roots in the first case being 4m + 1, and in the second 4m 1. So that giving to m successively the values 0, 1,2, 3, &c. in the general form N + 2m(2m+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 + 2.1.1, 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. ; or N, N + 1.2, N + 2.3, N + 3.4, N+4.5, N + 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 will be N+/?(|j+l), and the sum of its roots will be 2/j + 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 p(p--) is less than 2w+l, the number 2n + l can be resolved into squares the sum of whose roots is 2p + 1 ; otherwise it cannot.
2. The form of the roots of 2+l maybe found of which the algebraic sum is any possible odd number 2p+l except 1, provided all the odd numbers less than 2+l possess the property of having the algebraic sum of their roots =1. For if from 2n+l,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
