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XIV. On certain Properties of square numbers and other quadratic forms, with a Table 
of odd numbers from 1 to 191, divided into 4, 3 or 2 square numbers, the algebraic 
sum of whose roots ( positive or negative) may equal I, by means of which Table all 
the odd numbers up to 9503 may be resolved into not exceeding 4 square numbers. 
By Sir Frederick Pollock, F.R.S., Lord Chief Baron. 
Received, Dec. 20, 1853, — Read, Dec. 22, 1853, — Revised by the Author, Nov. 1854. 
Some years ago, in examining the properties of the triangular or trigonal numbers 
ri [ji — 1 \ 
0, 1, 3, 6, 10, 15, &c. 
I observed that every trigonal number was composed of 4 trigonal numbers, viz. 
3 times some prior trigonal number plus the next in the series, either immediately 
before or after that prior number ; 
thus 45=10+10+10+15 
55 = 15 + 15 + 15 + 10; 
or generally, as all numbers are of the form 2n—l or of 2n, all trigonal numbers 
are of one of the 2 forms, 2 n I 2 — n, 2 n 2 -\-n, 
rd + n 
2 n — n— 
X 3- 
and 
2 w 2 +w= 
n 2 + n 
2 ~ 
X 3 
I found also that all the natural numbers in the interval between any two con- 
secutive trigonal numbers, might be composed of 4 trigonal numbers, having the sum 
of their bases or roots constant , viz. the sum of the roots or bases of the 4 trigonal 
numbers which compose the first of the 2 trigonal numbers. 
This will be best explained by an example : the roots or bases are placed over the 
numbers, and it will be observed their sum is constant in the same interval. 
