314 SIR F. POLLOCK ON CERTAIN PROPERTIES OF SQUARE NUMBERS, ETC. 
And a similar table will show the corresponding decrease of the sum of the squares. 
Sum of roots. Increase and decrease. Decrease of sum of squares. 
3 1 2 
5 2 4 
7 3 6 
9 4 8 
&c. &c. &c. 
The use that may be made of these theorems will best appear by an example or two. 
51 is composed of 4 square numbers, 25, 16, 9, 1, whose roots are 5, 4, 3, 1. 
4 + 3 — 5 — 1 = 1 
54-3-4-1 = 3 
5 + 4 — 3 — 1 = 5 
5 + 4 + 1 — 3 = 7 
Then by Theorem A. square numbers which compose 53, 55, 57, and 59 may be 
obtained, and by Theorem B. those which compose 49, 47, and 45 by adding to or 
subtracting from the roots ; thus 
5 
+ 4 - 3 - 
1 = 5 
= 2x3 — l,w = 3 
3 
3 3 
3 
being deducted or added, 
2 
1 6 
4 
become the new roots, the sum of whose 
squares = 57 = 5 1+2x3. 
5 
+ 4 + 1 - 
3 = 7 
= 2 X 3 + 1,22 = 3 
3 
3 3 
3 
2 
1 — 2 
6 
are the new roots, the sum of whose 
squares = 45 = 51 - 
-2x3. 
Again, 5 5 1 
0 are roots of 
' squares 
which compose 51, 
5 
+ 5 + 1 
0=11 
= 2 X 6 — 1,72 = 6 
6 
6 6 
6 
1 
1 — 5 
6 
the squares of these new roots = 63 = 51 
+ 12 (51+2X6). 
Also, 
5 
+ 5-1 
0 = 9 
= 2x5 — 1,72 = 5 
5 
5 5 
5 
0 
0 6 
5 
the squares of these new roots = 61 by 
Theorem A. ; by Theorem B. the 
squares 
which compose 41 and 43 may be found, 
and thus the square numbers (not exceeding 4) which compose 51 being given, square 
numbers not exceeding 4 maybe discovered, which compose 41, 43, and all the inter- 
mediate odd numbers up to 63. 
This method of obtaining the square numbers that compose a succession of odd 
numbers, suggested that if a method similar to what was observed in the trigonal 
numbers were adopted as to the square numbers, the series of odd numbers might be 
