572 



bers 2, 4, 6, 8, 10, &c., forming an arithmetic series of the second 

 order (the first and second differences being respectively 2 each) ; 

 when the number in the lowest row cannot be divided into 2 squares, 

 the arithmetic series is not formed, and the squares are marked with 

 an asterisk, but when the number 4n+ 1 is divisible into 2 square 

 numbers, the roots of these squares constitute the two exterior dif- 

 ferences of the roots into which the odd number may be divided, and 

 also of the roots into which each term of the series increasing upward 

 may be divided ; the middle difference of the roots will be the smaller 

 half of the sum of the 2 roots of the square numbers into which 

 4n+ 1 may be divided, with a negative sign, and will increase by 1 in 

 each successive term of the upward series. 



For example, in the Table take the number 29 in the lowest row, 

 7x4 + 1 = 29, 7 is the number above it, and 7x2 + 1 = 15 the odd 

 number, which is the first term of the series 15, 17, 21, 27, 35, &c. 

 Now 29 is composed of 2 square numbers, 4 and 25, whose roots 

 are 2 and 5, 2 + 5 = 7; the smaller half is 3, and 2, 3, 5 will be 

 the differences of the roots of the squares into which 15 may be 

 divided, and whose sum will equal 1 ; thus 



2, -3, 5 

 -1, 1, -2,3; 



the roots when squared and added together equal 15, and the other 

 terms of the series follow in like manner, obeying the law indicated ; 



thus 



5, -2, 2 

 3, 2, 0, 2 when squared and added . =17 



2, -1,5 

 2, 0, 1, 4 when squared and added . =21 



5,0,2 

 4, 1, 1, 3 when squared and added . =27 



2, 1,5 

 3, 1, 0, 5 whose squares .... =35 



The proof of all this depends on a property of numbers mentioned 

 in the Philosophical Transactions for 1854, vol. cxliv. p. 317. 



If any number be composed of two triangular numbers, it will also 

 equal a square and a double triangular number. If 



it will be of the form 2 + a + 6 2 , and may be assumed equal to 



