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

 the number 2n + 1 composed of squares the algebraic sum of whose 

 roots is 2/7+1. 



It thus appears that any odd number 2w+l can be divided into 

 squares the sum of whose roots will equal 3, 5, 7, &c. (any possible 

 odd number except 1) if the odd numbers below it can be divided 

 into squares the sum of whose roots =1 ; and if it can be shown 

 that its roots in some form will equal 1, then the theorem M will be 

 true for that number and for every number below it. 



This is illustrated by an example, and then another theorem, 

 called " Theorem Q," is stated. In this a series of roots and odd 

 numbers is formed by making the 1st and 3rd differences of the 

 roots constant, but reversed every alternate term, and increasing or 

 diminishing the middle difference by 1 each term ; or the middle 

 difference is made constant and the 1st and 3rd vary. The sums of 

 the roots thus oecome constant in every term of the series, but the 

 sums of the squares of the roots increase, as in theorem P, by the 

 even numbers 2, 4, 6, 8, &c.', so that the increase at any number of 

 terms p is p(p+ 1), or the double of a triangular number. 



By the application of these theorems to a variety of examples, it 

 is shown how any odd number may be composed of four squares, 

 such that the algebraic sum of their roots may equal 1 . 



The theorems P and Q, it is considered, connect every odd num- 

 ber with every other odd number, so as to make it impossible if one 

 odd number be composed of four squares, but that every other odd 

 number should likewise be so. It is pointed out in what manner 

 every possible combination of numbers which can furnish the differ- 

 ences of the roots of any squares, not exceeding four, which can 

 make an odd number, and the sum of which roots = 1, can be derived 

 from the gradation series, that is from 4/> a + 2/>+ 1. The combined 

 effect of the theorems P and Q is therefore to prove that every 

 odd number must be composed of not exceeding four square num- 

 bers. 



The author goes on to show that every number is composed of 

 not exceeding three triangular numbers, by proving that if every 

 odd number 2n+ 1 can be divided into four square numbers the sum 

 of whose roots = 1, then n will be composed of not exceeding three 

 triangular numbers. This, is done by taking the differences of the 



B2 



