SIE F. POLLOCK OK FEEILAT’S THEOEEM OF THE POLYGONAL NUMBEES. 419 
the same series, but decreasing instead of increasing ; and it is worthy of remark that 
the first term of the series is the sum of the root and the arithmetic number, viz. 15. 
If both the series decrease, as 
9, 8 , 7, 6 , &c., 
6 , 5, 4, 3, &c.. 
and the lower be considered as roots, the series is 
45 . 12 33 , 10 23, g 15, 
whose first term is 3, the difierence betw^een the arithmetic number and the root ; if the 
upper be considered as roots, the first term is 3, but negative, and the series would be 
— 3,2 — 1, 4 3, 6 9, g 17 , 10 2 / , 12 &c. 87, ,8 69, ig 53 , 14 39, &c. 
If the series be composed of 2 equal roots, increasing or decreasing each by 1, or of 2 
roots differing by 1 , and increasing or decreasing in like manner, then if the series of 
numbers differ by 2 , so that all the terms shall be odd, a series will be formed of the 2 nd 
or 3rd kind, whose second difference will be 4; thus if the numbers be 9, 11, 13, 15, &c., 
and the roots 3, 3, 4, 4, 5, 5, 6 , 6 , the series will be 27, ig 43 , 20 63 , 24 87, a series of the 
3rd kind having a second difference of 4, and the fii’st term will be the difference between 
the number and the sum of the roots, viz. 9 — (3 + 3); for 3 , 4 7,8 15 , 12 27 produces the 
series ; but if the numbers decrease by 2 , 
9, 7 5, 3, &c. 
(3,3), (4,4), (5,5), ( 6 , 6 ), 
the series will be 
27.12 39, ,g 55, 20 "^^5 
and the first term of that series is the sum of the roots and the odd number, viz. 
9+3+3=15, for 15, 19, 27, 39, &c. is the series. 
So if the roots, instead of being equal, differ by 1, thus, 
6 , 8 , 10 , 12 , &c. 
3, 4, 4, 5, 5, 6 , 6 , 7, &c., 
the series will be 
31,18 49,22 71, 26 97, &c., 
a series of the 2nd kind, whose first term is the difference between 6 and 3 + 4, viz. 1, 
and negative, and the series is 
-1, 2 1, 6 7,10 17, 14 31, IS 49,22 71, &c.; 
but if the numbers decrease, as 
12 , 10 , 8 , 6 , &c. 
3, 4, 4, 5, 5, 6 , 6 , 7, &c., 
the series is 
37,14 51,18 69, 22 91, &c, 
and the first term is 
19=12 + 3+4, 
19,2 21,6 27,10 37,14 51, &c. 
Some remarkable properties arise from connecting these series together, which I must 
reserve for a future communication. 
3 M 
MDCCCLXI. 
