SIR F. POLLOCK OK FEEMAT’S THEOEEM OF THE POLYGONAL NUMBEES. 417 
2 _ — 
(2w— CP + 1), (2 w— i>)+2Q? + l); so that, whenever 2(^+1) is composed of not exceed- 
ing 2 squares, the term is composed of not exceeding 4 squares; but the (2w + l)th 
2 y"—-. _ 2 
term will also equal the 2wth term will equal (2w— 2), (2w— 1) 
-f- |^8w— 2 
increasing. 
, and so on, — the series of arithmetic numbers decreasing by 2, instead of 
An example in actual figures will better illustrate this. 
Series. 
Eoots. 
Numbers. 
Eoots. 
Numbers. 
19= 
0, 
1 
® 
also = 
1,0 
21 = 
1, 
2 
= 
0,1 
® 
27 = 
2, 
3 
= 
1,2 
@ 
37 = 
3, 
4 
= 
2,3 
@ 
51 = 
4, 
5 
= 
3,4 
® 
69 = 
5, 
6 
= 
4,5 
@ 
91 = 
6, 
7 
® 
= 
5,6 
® 
117 = 
7, 
8 
= 
6,7 
® 
147 = 
8, 
9 
® 
= 
7,8 
@ 
181 = 
9, 
10 
® 
8,9 
® 
The first of these cannot be continued usefully, because the number becomes negative 
after the 10th term, the other series continues. 
219 = 
10, 11 
&c. 
-2 
= 
9,10 
&c. 
® 
See. 
If the odd number be increased by 4, 8, 12, 16, &c., the series obtained will have 
2 
similar properties ; its 2wth term will be 0, 1, n, n, the roots n, n will diminish by 1 in 
each preceding term, and 1 will be an arithmetic number increasing by 2, as appears 
below in the case of the odd number 19. 
* If the form of the odd mimher he In + 3, the arithmetic number is 8w-t-4. 
