132 Sir F. Pollock on the " Mysteries of Numbers'' [Apr. 19, 



If the term AM descends till n—2p-\-2 =2p+l, it rises to the left in 

 another diagonal series and goes on till it crosses M, where it is found that 

 the roots are always the same as those which arise from M, descending by 

 means of its two equal roots. The term AN does the same with respect 

 to N. 



If AM descend to the margin at am, and one step further into an, and 

 AN descends to an, they will be found to have the same roots, and they 

 will be 



from AM 





~2p+l, 1, n- 



2p-{- 1, n 



from AN 









The only difference being that in the one case 1 is positive, in the other it 

 is negative ; but whatever be the value of p or n, in this portion or term 

 of the square 1 is always one of the roots. 



In crossing the two horizontal series from R and S, it will always be 

 found that at the points of intersection the roots of AM correspond with 

 the roots of the series from R, and the roots from AN correspond with 

 those from S. 



Diagram No. 4 exhibits the way in which B, C, P, and Q are connected 

 together. The roots of A at the {n—p)i\i square will always be —(^+1), 

 — j?> n, n, andp+ l,p must be one of them odd, the other even ; therefore, 

 whether n be odd or even, an odd number will be formed by — (p+ 1), n, 

 or —p, n. The series from A descending diagonally has the roots n, n 

 decreasing, but the negative roots increasing. The differences will con- 

 tinue the same ; and when the series arrives under that index which cor- 

 responds to the odd number — (i> + 1 ) n, or —p, n, it becomes a term in 

 a horizontal series which goes to P, two terms of which are always 

 jp, p + 1 . W, in descending diagonally, has its index on reaching PB 1 . 

 When A has descended so as to reach the even number of the two, 

 ( — (p+1), —p, n), it rises in a vertical series to Q; and the three 

 series, WR, BP, CQ, always intersect in the same point or small 

 square, H. 



The paper then exhibits in a Diagram (No. 5) all the roots which 

 arise from applying the method to the odd No. 161, and shows that the 

 roots 



of A would be 3, 2, 16, 16 



ofW „ 3, 2, 6, 16 



ofM „ 8, 3, 10, 10 



ofN „ 7, 2, 12, 12 



of AM „ 5, 0, 10, 16 



of AN „ 5, 0, 12, 16 



It shows the roots of the squares marked B, C, P, Q, R, S, and many 

 others j and, finally, it shows that 161 would be composed of the squares 



