416 SIE F. POLLOCK ON FEEMAT’S THEOEEM OF THE POLYGONAL NUMBEES. 
It may also be composed of one square and one arithmetic number in two ditferent 
ways, thus: — 
Arith. 
numbers. 
Boots. 
Numbers. 
Boots. 
19= 
0 
or = 
g 
1 
21 = 
@) 
1 
= 
2 
25 = 
@ 
2 
= 
g 
3 
81= 
@ 
3 
= 
4 
39= 
(g) 
4 
= 
g 
5 
49= 
@ 
5 
= 
g 
6 
61 = 
(g 
6 
= 
g 
7 
75= 
@ 
7 
8 
91 = 
@ 
8 
= 
g 
9 
109= 
@ 
9 
= 
® 
10 
129 = 
g 
10 
= 
11 
151 = 
g 
11 
= 
® 
12 
175 = 
g 
12 
= 
13 
201 = 
g 
13 
= 
® 
14 
229= 
(§) 
14 
= 
® 
15 
259= 
g 
15 
z= 
® 
16 
291 = 
g 
16 
z= 
@ 
17 
325= 
g 
17 
= 
® 
18 
361 = 
g 
18 
= 
® 
19 
399= 
g 
19 
= 
20 
Again, if a number of the form 4w+l be increased by 2, 6, 10, 14, &c., the series formed 
2 2 — 
will have its (2n+l)th term =0, 0, 2n, 2w+l; its (2w)th term will be =2n—l, 2n, 
-f- ; the (2?i— l)th term will be equal to (2n—2), 2w— 1+ the (2n--_p)th term 
