540 Royal Society : — 



(not arithmetical of the 1st order) have a similar relation with regard 

 to the roots of the 4 squares of which they may he composed, that 

 is, those which are equidistant from the middle, or the middle term 

 (according as the number of" terms is even or odd), have the middle 

 roots the same, and the exterior roots have an arithmetical relation 

 to each other (varying with 'the distance from the centre), viz. the 

 one being less and the other greater by the same quantity. 



Thus, if any number of terms (exceeding 3) of either of the 2 

 series above-mentioned (viz. 1, 3, 9, 19, &c., or 1, 5, 13, 25, &c.), 

 and, beginning with the first tbrm, the differences be added " inverso 

 ordine," a new series will be ' obtained possessing the property in 



2 4 6' 8 

 question ; thus the first 7 J;e*ms of the 1st series are, 1, 3, 9, 19, 33, 

 10 12 :, ■ '■ 



51, 73; the differences' are^ 2,-G, 10, 14, 18, 22; if the differences 

 be added "inverso ordine", the series becomes 1, 23, 41, 55, G5, 

 71, 73, each term of whiehr^nay be divided into 4 squares, whose 

 roots will be as follows :— rf • . 



;i;l,J,2,7 



Here there is a middle terrn, a^ll 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, tlie 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 G 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 

 2 4 6 



1, 37, 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 



2 4 



1 27 49 

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



+2,0,3,6 



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 : — 



