312 SIR F POLLOCK ON CERTAIN PROPERTIES OF SQUARE NUMBERS, ETC. 
8x9 
9 x 10 
2 
3 
4 
4 
4 
= 15 
= 36 
6 
IO 
IO 
IO 
3 
3 
4 
5 
15 
37 
6 
6 
10 
15 
/ 
2 
4 
4 
5 
15 
38 
3 
10 
10 
15 
2 
3 
5 
5 
15 
39 
3 
6 
15 
15 
2 
3 
4 
6 
15 
40 
3 
6 
10 
21 
l 
4 
5 
5 
15 
41 
1 
10 
15 
15 
1 
4 
4 
6 
15 
42 
1 
10 
10 
21 
1 
3 
5 
6 
15 
43 
1 
6 
15 
21 
2 
2 
4 
7 
15 
44 
3 
3 
10 
28 
4 
4 
4 
5 
17 
= 45 
10 
IO 
IO 
15 
3 
4 
5 
5 
17 
46 
6 
10 
15 
15 
3 
4 
4 
6 
17 
47 
6 
10 
10 
21 
3 
3 
5 
6 
17 
48 
6 
6 
15 
21 
2 
4 
5 
6 
17 
49 
3 
10 
15 
21 
3 
3 
4 
7 
17 
50 
6 
6 
10 
28 
2 
4 
4 
7 
17 
51 
3 
10 
10 
28 
2 
3 
5 
7 
17 
52 
3 
6 
15 
28 
1 
4 
6 
6 
17 
53 
1 
10 
21 
21 
1 
4 
5 
7 
17 
54 
1 
10 
15 
28 
4 
5 
5 
5 
= 19 
— 55 
io 
15 
15 
15 
lOx 11 
2 
5 5 5 6 
From 55 to 66 (=15, 15, 15, 21) the constant sum of the bases will be 19, and 
this may be continued without limit. 
If the law by which this can be continued were discovered and proved, it would 
furnish the means of proving Fermat’s theorems of the polygonal numbers ; but not 
being aware of any law by which the series that fills up the intervals could be con- 
tinued, I turned my attention to the square numbers as containing (apparently) a 
greater variety of theorems, and as being (certainly) of all quadratic forms that which 
