130 Sir F. Pollock on the Mysteries of Numbers [Apr. 19, 



their roots being equal to the index ; the roots of W may also be obtained 

 from A by taking its roots down Tertieally, making n, n successively 

 n—l, n-\- 1, Qi—2, n-\-2y &c., until the square is reached, in which the 

 index is 2 n-{-2 p + l, and 2 n-\-2p-\-l being the arithmetical sum of all 

 the roots of A; Akere becomes a term in a series which moves diagonally, 

 and on being carried up towards the left will give the roots of W. When 

 the indices of the squares through which this vertical series passes equal 

 2 n + l,by making p+1, p, one positive and the other negative, the term 

 becomes a term in another series which moves diagonally upwards towards 

 the left. This must occur both when the index equals 2 n— I and when 

 it equals 2 n-\-l ; and as the indices increase downwards uniformly by 2, 

 it follows that 2 n—1 and 2 n-\-l will be the indices of contiguous terms 

 of the vertical series, and therefore two contiguous term.s will become 

 terms of series moving diagonally upwards to the left ; and as these two 

 series are contiguous to each other, their terms found in the first series 

 (that is, the series in the top line) will also be contiguous. 



These two terms I have designated as AM and AN. AM comes 

 from the term where the judex equals 2n+l, and AN from, the terni 

 where the index equals 2 n — 1 ; the roots of AM are 



0, 2p + l, n—2 _p + 2, n, 



those of AN are 



0, 2 + n—2jD, n, 



and being arranged thus. 







— 2p + l, 0, n— 2^ + 2, n, 



-2p-\-l, 0, n-22:), n, 



will move diagonally downwards to the left ; and as each of these have two 

 roots that differ in one case by 2 p, iu the other by 2p + 2, the terms 

 in these series that are parallel to those terms in the first vertical series 

 which have their external indices respectively 2p and 2p + 2, the terms 

 of AM and AN will (I say) in these places become terms in series mov- 

 ing vertically, and on being followed up to the series in the top row will 

 be found to give the roots of M ^nd N. The roots of M and N may be 

 obtained by another method as follows : — As the algebraic sum of the 

 roots of A equals its index, therefore A is a term in a diagonal series 

 moving downwards towards the left ; and as two of its roots, p, p + 1, differ 

 by 1, it follows that whenever the value of n has been so altered that 

 2 n±l equals the index by making p + one positive and the other 

 negative, the term becomes a term in another series, which series will move 

 at right angles to the series last mentioned. Now taking the series which 

 moves to the left of A, it is clear that this result will obtain in two places ; 

 first, when the altered value of n makes 2 n—1 equal the index, and 

 secondly, when the altered value of n makes 2 equal the index. 



