SIR F. POLLOCK ON CERTAIN PROPERTIES OF SQUARE NUMBERS, ETC. 313 
is most familiar, and in which calculations or comparisons may be made with the 
greatest facility. 
Very lately I observed the following property of square numbers: — 
If any four square numbers a 2 , b 2 , c 2 , d 2 have their roots such that by making one 
or more positive and the rest negative, the algebraic sum of the roots may equal 1 ; 
then if the roots whose sum is one less than the others be each increased by 1, and 
the others be each decreased by 1, the sum of the squares of the roots thus increased 
and decreased will be equal to a 2 +6 2 +c 2 +cf 2 + 2. Let a-\-b—c—d=\, and let c and 
d become c+1, d-\- 1, and let a and b become a— 1, b— 1, then 
{a-\f+(b-\y+{c+\) 2 +{d+\y=a 2 +b 2 +c 2 +d 2 +2x{-a-b+c+d)+A, 
but 
2 X ( — a—b J r c J r d) = — 2, 
therefore the sum of the squares of the new roots = a 2 + 6 2 -l-c 2 4-^ 2 +2. If a — b 
— c—d— 1, the result is the same, decreasing a and increasing each of b, c and d by l . 
The theorem is more general (as might have been expected). 
Theorem A. 
For if instead of 1 the algebraic sum of the roots be equal to 2n— l,and the nega- 
tive roots be numerically increased by n and the positive roots be decreased by n, 
the increase in the sum of the squares of the new roots thus formed will be 2 n. 
Let a-\-b— c— d=2n — 1 , then {a — n) 2 -\-{b— n) 2 -\-{c-\-n) 2 -\-(d-\-n) 2 -=a 2 -\-b 2 -\-c 2 -\-d' 1 
— 2 an — 2bn-\-2cn-\-2dn-\-A:rd, but — 2 an — 2bn-\-2cn-\-2dn= — (2n — l)x2w= — An 2 
-\-2 n, .'.the sum of the squares of the new YOOte=a 2J r b 2 -\-c 2 -\-d 2 -\-2n. 
The following table shows the result of different algebraic sums of the roots, with 
the corresponding increase or decrease of roots and increase of the sum of the squares. 
Corresponding increase Increase of 
Sum of roots. or decrease of roots. sum of squares. 
1 1 2 
3 2 4 
5 3 6 
7 4 8 
9 5 10 
11 6 12 
&c. &c. &c. 
There is a similar theorem with respect to the decrease of the sum of the squares. 
Theorem B. 
If a-\-b— c— d=2n-\-\ (instead of 2n— 1), then if a and b be each diminished and 
c and d be increased by n, the sum of the squares of the new roots will be less by 2 n, 
and (a— n) 2 -\-{b— n) 2 -\-{c-\-n) 2 -\-(d-\-n) 2 will equal a^+^+^+e? 2 — 2 n. 
2 s 
MDCCCLIV. 
