206 [June 19, 



tions) in a square, in such manner that the division into 4 squares 

 of certain terms in each series, may produce a division into 4 squares 

 of every term of other series, and thus each term in the -whole square 

 will at last be divided into 4 squares, and the first term will be so 

 divided into 4 square numbers that two of the roots will be equal to 

 each other; two of them will diifer by 1, and the algebraic sum of all 

 the roots will be equal to 1 . 



It is not offered (at present) as a proof that it must be so, but as 

 a method by which that result may always (in fact) be obtained. 



If any odd number 2n-\- 1 be increased by 2, 4, 6, 8, 10, &c., the 

 (2n+l)th term will be (2w+l) 2 ; other terms will have a distinct 

 arithmetic relation to n, and w+ 1, and the whole series will be such 

 that, if the pih term can be divided into square numbers whose roots 

 shall equal 2p l, then every term of the whole series can be so 

 divided that the roots of the (p+ l)th term will be 2p+ 1, and so on 

 through the whole series. 



1 3 5 7 9 11 



Let 27, 29, 33, 39, 47, 57, &c. be such a series, with the odd 



2- 



322 

 numbers as indices of the sums of the roots, 39= 2+1+3 + 5, 



and the sum of the roots is 7, and the differences of the roots, placed 

 in arithmetic order, will be 3 . 2 . 2 ; then 29 will have roots with 



322 

 the same differences, the sum being 3, 3, 0, 2, 4 = 29, and 57 will 



322 



have roots 1, 2, 4, 6 = 57. The other numbers in the series will 

 have the differences reversed, but the sums of the roots will be re- 

 spectively as the odd numbers placed as the index of each. 



If any odd number be increased by 4, 8, 12, 16, &c., so as to form a 

 series, 2rc+l, 2rc + 5, 2n+ 13, 2ra + 25, pt\\ term (2n+ (p ) 2 , +p*)> 

 it will have in the first term 4 roots, 2 of which differ by ] ; in the 

 2nd term, 4 roots, 2 of which differ by 3 ; in the 3rd term 4 roots, 

 2 of which differ by 5 ; in the rath term 4 roots, 2 of which differ 

 by 2n 1 ; the other two roots will be common to all the terms. 

 If these odd numbers, 1, 3, 5, &c., be made indices of the 1st, 2nd, 

 3rd, &c. terms, and any one term can be found having 2 roots dif- 

 fering by the index of that term, then the roots of all the other 

 terms may be found. Let 27 increase by 4, 8, 12, &c., 

 13579 

 27 31 39 51 67, &c. ; 



