242 



71, 73, each term of which may be divided into 4 squares, whose 

 roots will be as follows : 



Here there is a middle term, all the terms equidistant from it have 

 the same middle roots, the terms next to the middle term have the 

 exterior roots, the one 2 less, the other 2 more, those next but one 

 4 less and 4 more, and the extreme terms 1 and 73 have their exte- 

 rior roots one 6 less and the other 6 more than the corresponding 

 roots of the other. 



If 8 terms of the series be taken as 1, 3, 9, 19, 33, 51, 73, 99, 

 and the differences be added " inverso ordine" the series becomes 

 0246 



1, 27, 49, 67, &c., the terms of which divided into 4 squares, so that 

 the differences of the exterior roots may correspond with the index, 

 will be 



02 4 6 8 10 12 14 



1 27 49 67 81 91 97 99 



0,0,1,0 -1,3,4,1 -2,4,5,2 -3,0,7,3 -4,0,7,4 -5,4,5,5 -6,3,4,6 -7,0,1,7 



+2,0,3,6 -1,4,5,5 -2,4,5,6 -1,0,3,9 



+ 1,1,4,7 0,1,4,8 



Here there is no middle term ; the terms equidistant from the centre 

 have the same middle roots, while the differences between the exte- 

 rior roots increase as the numbers 1, 3, 5, 7. 



The other series, 1, 5, 13, 25, &c., gives a similar result. If the 

 differences of the first 7 terms be added " inverso ordine," the new 

 series, with its indices and the roots of the 4 squares which compose 

 each term, will be as follows : 



Here there is a middle term ; the equidistant terms have the same 

 middle roots, the exterior roots are (next to the middle term) the 



