318 SIR F. POLLOCK ON CERTAIN PROPERTIES OF SQUARE NUMBERS, ETC. 
therefore 2p -\- 1 = 2a 2 -\-2a-\-2b 2 ~\- 1 = (a + 1) 2 , a 2 , b 2 , b 2 , and the roots (a + 1),— a, b, 
pit ^ _j_ yji 
— b,= 1. If p he composed of 3 trigonal numbers, then p=a 2 -\-a-{-b 2 -\ — ^ — and 
2p-\-\=2a 2 -\-2a-\-2b 2 -\-m 2 -\-m-\-\, but wr+m-j-1 is of the form 4n 2 +2w+l? whose 
four roots (as already seen) are n+ 1, n, n , n, and if these roots be varied thus, 
+«+w+ 1 
a — n 
b-\-n 
b — n, 
the squares of these four roots will equal 2a 2 +2a + 2^ 2 + 4w 2 +2»+l, and the algebraic 
sum of these roots obviously may=l. It follows from this, that every possible odd 
number may be divided into integral square numbers (not exceeding 4), the algebraic 
sum of whose roots=l. 
I propose in a future communication to give a different proof of the Theorem C, 
and instead of proving the Theorem C. by Fermat’s proposition of the trigonal num- 
bers, I shall offer a proof of Fermat’s proposition of the trigonal numbers by the 
Theorem C; it is obvious that they are so connected that either may be proved from 
the other. 
I am not aware that the theorems A, B, or C, or the method above described of 
using a gradation-series, have ever been noticed before, and as they appear to add 
something (however little) to the theory of numbers, I have ventured to present them 
to the attention of the Royal Society. 
Note. — Numbers of the form 2w+2 {even numbers ) may be resolved into square 
numbers (not exceeding 4), the algebraic sum of whose roots may always equal 2, 
and so far they have an analogous property, but they do not possess the analogous 
property of being resolvable into roots whose algebraic sum will=2, 4, 6, 8, &c. 
