1862.] 



207 



but 39 = 2, 3, 1, 5, and 2, 3 differ by 5. Then the terms of the 

 whole series may be divided into 4 squares, 2 of which will be com- 

 mon to all the terms, and the other 2 will have the difference pointed 

 out by the index. The roots are placed below each term, and the 

 middle roots are common to all the terms : 



1 3 5 7 9 11 



27 31 39 51 67 87, &c. 



0,1,5,1 -1,1,5,2 -2,1,5,3 -3,1,5,4 -4,1,5,5 -5,1,5,6 



If an odd number (2n+ 1) be increased by 2, 6, 10, 14, &c., and 

 2n+l,2n + 3,2w + 9,2+19, 2n + (p I) 2 , (>-l) 2 , 1 (pih term) be 

 the resulting series, then, if the even numbers (beginning with 0) be 

 made indices, and any term in the series can be divided into 4 

 squares, 2 of them having their roots with the algebraic difference 

 pointed out by the index, then the other 2 roots will be common to 

 all the terms, and in a similar manner all the terms will have roots 

 corresponding with the index of each term. 



The series 2w+l, 2rc + 5, 2+13, 2ra + 25, &c. will have for its 



th term ri 2 , + w a + (l), for its nth. term is obviously 2w + (w I) 2 

 ); the (n-l)th term will be (>-l) 2 , + (w-l) 2 + 

 ; and going backwards to the first term, the roots (n 1), (n 1) 



decrease by 1, and the arithmetic number increases by 2 ; but this 

 obtains beyond the first term into a continuation of the series back- 

 wards ; thus, 



15 11 II 15 23 35 51 



5)l o(n)o 1(9)1 2(7)2 3(J)3 4(3)4 5(7)5. 



Instead of this mode of continuing the roots and arithmetic num- 

 bers, they may be applied thus : 



(T) 



23 



2@2 



11 



13 



17 







0^0 



11 



15 



2'T 2 



23 



303 



3^3 

 35 



51 



5^5 

 71 



95 



123 



5 



And whenever the arithmetic number is of the form 2a 2 -f-2-f25 2 +l, 

 (that is) is the sum of 2 triangular numbers multiplied by 2 and in- 

 creased by 1, then, by altering the even squares, the term may be 

 made to consist of 4 squares, as to which the roots of 2 of them 

 will differ by 1 . 



