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



than 10 other series by which this is crossed and associated and connected, 

 and the number cannot in any case be less than 6. A series of the second 

 kind gives rise to other series crossing it in the same manner mutatis mu- 

 tandis^ which result is so obvious that it is not necessary further to dwell 

 upon it. A series of the third kind has all the differences of its roots the 



Same in each term. Let 



. 13 



-7, .2, 4, 14 



be a term in a diagonal series at 



the top row of the system. 



Thus 





11 



13 



16 







9 2 10 









8, 13, 2, 2 



— 7, 2, 4, 14 



6, 15, 4, 4 





9 





17 





9 2 10 





9 2 10 



2 



-8, 1, 3, 13 





— 6, 3, 5, 15 



When it reaches to the left and to the right, the second place, as 3 — 1 = 2 

 and 5 — 3 = 2, it furnishes two vertical series ; at the tenth row it furnishes 

 two more ; at the twelfth row two more. "When it gets into the column 

 whose index is 11 and then 9, before it reaches the margin or outer edge, 

 it furnishes two horizontal series ; and after it has passed the margin at 9 and 

 11 it furnishes two more, and at 21 it furnishes another. Here are six new 

 vertical and five new horizontal series ; besides which it furnishes at least 

 two other diagonal series which cross it. 



Having stated the properties which belong to the square, if the first odd 

 number in it be of the form (4^ + 2).?i+ 1, and that whether it be 2?i+ 1, 

 6?i+l> 107i + l, &c., or whatever be the value of certain squares maybe 

 found which I distinguish as A, B, C, H, P, Q, W, M, N, AM, AN, R, S, 

 T, V, &c., which are connected together by a community of roots where 

 the series cross each other in a manner that is invariable. 



The Diagram No. 3 is another example of the manner in which certain 

 of the terms in the different series communicate with each other, by the 

 roots being common to both, at the point where they cross. AM passes 

 diagonally to am, down to an, and up to AN. AN, in like manner, goes 

 down to an and up to am and AM. If the first term of the square be an 

 odd number of the form (4^ + 2) ?i+l, the roots from AM are in an 

 — (2i? + l), 1, (?^— 2^+1), n. The roots from AN are -(2^ + 1), -1, 



(w— 2j9 4-l)j The indices of the diagonal series are ( 2;z— (4^+ 1)/ 

 and the algebraic sum of the roots is the one or the other, according as 1 

 is + or — ; but xlM also passes to the series from R, and AN to the 

 series from S, and go to R and S, and thus go up to W. AM also reaches 

 the vertical line from M, and passes up to M, as AN does to N. Lastly, 

 AM goes to N thus, and AN to W in a similar manner. Whatever be the 



