148 Royal Society. 



them increased by 1, and the others or other be each diminished by 

 1, the sum of the squares of the roots thus increased or diminished 

 will be a2+J- + c'--|-rf- + 2. This he found to be only a particular 

 case of more general theorems. 



Theorem A. — If the sum of the roots a, b, c, d=2n—l, and n be 

 added to each of the less set, and subtracted from each of the greater, 

 the increase in the sum of the squares of the new roots will be 2w. 



Theorem B. — If the sum of the roots =2/1+1, and 7i be added to 

 each of the less set and subtracted from each of the greater, the 

 dimimition in the sum of the squares of the new roots will be 2n. 



By means of these he shows — ■ 



Theorem C. — If any four squares be assumed which compose an 

 odd number, these may be diminished till four squares are attained 

 the sum of whose roots will equal 1 . 



By applying the first of these theorems to four roots, the sum of 

 whose squares is an odd number, the author deduces, in a tabular 

 form, the squares (four or less) which compose the odd numbers 

 from 21 to 87 ; and remarks that there does not appear to be any 

 limit to this mode of continuing to increase the sum of four squares 

 by 2 each time. As, however, although this may render it probable 

 that every odd number is composed of four, three, or two squares, it 

 falls very short of a mathematical proof, unless it can be shown that 

 the series can be continued by some inherent property belonging to 

 it, he proceeds to examine the series, in order to ascertain what 

 approach can be made to such a proof. 



Adopting a method similar to that observed in the triangular num- 

 bers, the author forms what he terms the series of Gradation, by 

 means of which the series of squares which compose the odd num- 

 bers may be advanced by steps or stages which increase regularly 

 and obey a certain law, and at which this series is, as it were, com- 

 menced anew from roots of the form n,n,n,n-\-\, ov n—\,n,n,n; 

 the form of the sum of the squares of these roots being 4«-+2w-f 1, 

 and the series of gradation 1, 3, 7, 13, 21. 31, 43, 57, 73, &c. On 

 this principle a more extended table of the odd numbers resolved 

 into squares (not exceeding four in number) is constructed. On 

 this the author remarks that it is complete to the 96th odd number 

 (191), that is, there are in this table square numbers which will 

 form the odd numbers in succession, whose roots (some -(- , some — ) 

 = 1 ; and therefore the expression 4«-+2w-l-l up to 4if- + 2n-\-\dl 

 may be divided into 4 or 3 squares, whatever be the value of n. 

 The numbers in the table exactly fill up the interval between 



479^ 472, 472, 48'-=S931, 

 and 472, 48«, 48'^ 482 = 9121, 



whose diflference =190, the difference between the first term and 

 the last term in the table : it will therefore resolve into square num- 

 bers any odd number up to 9121 + 190=9211. 



With reference to the mode in which the intervals in the table 

 may be filled up, the author states the following general theorems re- 

 lating to the sums of three square numbers, by means of which the 



