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



by 2, 4, 6, 8, &c. . . . {2n), and with the exception of one term they are all 

 in the diagonal lines (see diagram No. 1), in which the single red line 

 passes through the first terms, and the double red line shows where the 

 terms of the series (the second, third, &c.) belonging to that (as a first 

 term) are to be found ; the red ink numbers indicate their order. 

 1 2 3 4 5 6 7 8 



The Nos. 203, 205, 209, 215, 223, 233, 245, 259, &c., &c., compose 

 the series ; the terms increase by 2, 4, 6, 8, 10, &c. . . .2n. A.ny number 

 in the diagonal from 161 may be the first term of a similar series. 



The diagonal from 163 will give all the other first terms of the third set 

 of series. 



In order to explain the indices which appear in the diagram No. 1, and 

 to show in what manner the series are connected with, and pass into each 

 other, it is necessary to point out the properties of the two series which 

 compose the square, and of the third series, which is a necessary result. 



All of them have this property in common, — -that if you can discover the 

 roots of the squares which compose any term of the series with reference 

 to the nature of the series, and the order in the series of that term, then 

 you know the roots of every term in the series, both before and after that 

 term. The first series increases by 4, 8, 12, &c. The indices of the terms 

 of a series so increasing I have made 1, 3, 5, 7, &c., and they are put at 

 the top as being common to all the series that are horizontal. 



For this reason, if two numbers differ by 1, as ^ + 1, and the larger 

 be increased by 1, and the smaller be diminished by 1, and the process be 

 continued, the result will be 



n, ?^+l, 



n — \, n + 2j 



n—2, n + S, 



n — 3, w + 4, 



&c. &c. 



n — (p — l) n-{-p. 



If these be treated as roots, the sums of the squares will be 



2?2' + 2w+l, . 

 2w' + 2w + 5, 

 2n'' + 2n-\-lS, 

 2n^ + 2n + 25, 

 &c. &c. 

 2<+2?z + 2/-2^+l. 



The sums of the squares increase by 4, 8, 12, &c. 



