SIE r. POLLOCK ON FEEMAT’S THEOEEM OF THE POLTOONAL NHMBEES. 413 
combined with an even square must occur somewhere in the series 
1, 5, 9, 13, &c., 
and the Table (if extended) must contain every possible case of odd and even numbers 
as exterior differences, combined with every possible and available middle difference ; 
for negative differences may be rejected, inasmuch as, if the roots be put according to 
then’ algebraic value, all the differences must be positive ; thus the roots and differences 
of 15 above were 
2 , - 3 , 5 
-1, 1, -2, 3; 
if the roots be placed according to their algebraic value, they would be — 2, — 1, 1, 3, 
and with the differences above 
1 , 2 , 2 
-2, -1, 1, 3; 
15 will therefore be found in the column above 5, and in the fourth place. The Table 
(extended indeffnitely) would therefore contain every possible odd number the sum of 
whose roots may equal 1. 
In connexion with the Table just mentioned, it may be well to state a theorem 
respecting the differences of the roots, by which, having obtained one division of an odd 
number into 4, or 3 squares (equal to, or greater than 1, and not more than 2 of them 
equal to each other), other modes of dhiding the odd number into 4 squares may 
generally be obtained. 
Theorem. 
If any number be composed of 3 squares, and the roots be aiTanged in the order of 
their algebraic value, if the tw'o differences between the adjoining roots differ by 3, or 
a multiple of 3, then by reversing the differences and obtaining roots whose algebraic 
sum shall equal the sum of the former roots, but whose differences shall be reversed, 
another form of di\ision into squares will be obtained ; that is, the sum of the squares 
of the roots thus obtained will be equal to the sum of the squares of the first roots. 
Example. 
[Note. — I use the symbol 2 to indicate that the numbers below it are to be 
considered as which are to be squared and added together; thus, 100 = 6^+8^; 
therefore 101 = 0, 1, 6, 8.] 
5 2 
The differences of 1, 6, 8 are 5, 2, which differ by 3. If, now, roots be obtained 
with differences 2, 5, and whose sum will equal 1 + 6-1-8 = 15, the sum of the squares 
2 5 
of these roots will equal 101. 2, 4, 9 are roots having the differences reversed, and 
their sum =15 ; therefore 2^+4^+ 9^=1* +6^+8^= 101. Again, leaving out 6 as a 
2 
1 7 
root, 65=0, 1, 8; the differences are 1, 7 ; the sum of the roots =9: — 2, 5, 6 are 
3 L 2 
