SIR F. POLLOCK ON CERTAIN PROPERTIES OF SQUARE NUMBERS, ETC. 315 
resolved into square numbers. In the trigonal numbers the successive bases of the 
4 trigonal numbers into which the terms of the trigonal series are each divisible, are 
0 0 0 1 
0 111 
1112 
12 2 2 
2 2 2 3 
2 3 3 3 &c. 
If instead of using these numbers as bases or roots of trigonal numbers they be 
squared and added together, they furnish a series (1,3,7, 13, 21, 31, &c.) whose 
general term is m 2 +m+ 1, or according as mis even or odd, 4w 2 +2w+l : this expres- 
sion is manifestly divisible into square numbers whose roots will be either n, n, n, n- f- 1 
or n— 1, n, n, n, and the sum of the roots will be 4w+l , and with reference to integral 
quantities will be a maximum. The terms of the series 1, 3,7, 13, &c., furnish steps, 
places, or positions at which the process of increasing the sum of the squares might 
commence again, and as far as any law of increase is applicable to one term it is 
applicable to all. I have therefore called this series 1, 3, 7, 13, &c. the gradation- 
series of this system of resolving the odd numbers into square numbers not exceed- 
ing 4. 
I have prepared a table in which the odd numbers from 1 to 191, respectively, 
are divided into square numbers not. exceeding 4, the algebraic sum of whose roots 
may be made equal to 1. 
This table of the odd numbers up to 191 is at the end of the paper; the terms of 
the gradation-series (as they occur) are distinctively denoted, and all the sets of roots 
of the odd numbers up to 191 are capable of forming 1 as their algebraic sum ; and 
by means of this series any odd number from up 1° 4w 2 +2w-f 191 inclu- 
sive, may be divided into not exceeding 4 square numbers, whatever be the value of n. 
The examination of this series led me to observe a remarkable property of odd 
numbers with reference to the square numbers (not exceeding 4) into which they 
may be divided, and which may be stated in the following theorem. 
Theorem C. 
Every odd number may be divided into square numbers (not exceeding 4), the 
algebraic sum of whose roots (positive or negative) will (in some form of the roots) 
be equal to every odd number from 1 to the greatest possible sum of the roots, or the 
theorem may be stated in a purely algebraical form thus : 
If there be 2 equations, 
a 2 -H 2 +c 2 +d 2 =2/z+l 
and a-\-b-\-c-\-d=2r J r \, 
a , b, c, d being each integral or nil, n and r being positive, and r a maximum, then if 
2 s 2 
