SIE F. POLLOCK OX FEEMAT’S THEOEEM OF THE POLYGONAL NUMBEE8. 411 
rior differences of the roots of the four squares into which 2n-\-l may be divided. Before 
explaining the Table, it is proper to state that if an odd number be divisible into 4 square 
numbers, three of them must be odd, and one of them even, or one of them must be odd, 
and 3 of them even, otherwise their sum cannot be an odd number ; it follows from this 
that if the difference between any two of them be an odd number, the difference between 
the other two must be an even number, and vice versa,', for let 
then if ci ^ — must equal 2^+1; if possible let c^—d^=2r, then 
— fZ'=2j9+2r ; add 2¥-\-2d^ (an even number) to each, and 
will be an even number, which by the hypothesis it is not; if, therefore, cd—lf be an 
even number, & — d^ cannot also be an even number, and therefore must be an odd 
one. If, therefore, the four roots of the squares into which any odd number may be 
di\ided are arranged in any order there will be three differences ; the two exterior 
differences will be one odd, the other even ; the middle difference may be either odd or 
even. 
The Table is arranged thus : — the lowest row of figures is the series 1, 6, 9, 13, 17, &c. 
(4« — 1); the next row above is the series of natural numbers, 0, 1, 2, 3, 4, »&c. (w), &c. ; 
the next row is 1, 3, 5, 7, 9, &c. (2^+1) the odd numbers; each of the odd numbers is 
the first term in a series increasing upwards by the numbers, 2, 4, 6, 8, 10, &c., forming 
an arithmetic series of the second order (the fii’st and second differences being respectively 
2 each) ; when the number in the lowest row cannot be divided into 2 squares, the 
arithmetic series is not formed, and the square spaces are marked with an asterisk, but 
when the number 4w4-l is dmsible into two square numbers, the roots of these squares 
constitute the two exteiior differences of the roots into which the odd numbers may be 
divided, and also of the roots into which each term of the series increasing upward may 
be divided ; the middle difference of the roots will be the smaller half of the sum of the 
2 roots of the square numbers into which 4^+1 may be divided, with a negative sign, 
and will increase by 1 in each successive term of the upward series. 
For example, in the Table take the number 29 in the lowest row, 7x44-1 = 2 9, 7 is 
the number above it, and 7x2-1-1 = 15 the odd number, which is the first term of the 
seiies 15, 17, 21, 27, 35, &c. Now 29 is composed of 2 square numbers, 4 and 25, whose 
roots are 2 and 5, 24-5 = 7 ; the smaller half is 3, and 2, — 3, 5 will be the differences 
of the roots of the squares into which 15 may be divided, and whose sum will equal 1 ; 
thus 
2 , - 3 , 5 
-1, 1, -2, 3; 
the roots when squared and added together equal 15, and the other terms of the series 
follow in like manner, obeying the law indicated ; thus 
5 , - 2 , 2 
— 3, 2, 0, 2 when squared and added . . =17 
2 , - 1 , 5 
— 2, 0, —1, 4 when squared and added . =21 
3 L 
MDCCCLXI. 
