932 Proceedings of Royal Society of Edinburgh. [sess. 
so that the theorem is seen to he the Extensional of a manifest 
identity.* 
Sylvester, J. J. (1851, Aug.). 
[On a certain fundamental theorem of determinants. Philos. 
Magazine , ii. pp. 142-145 : Collected Math. Papers, i. 
pp. 252-255.] 
After a characteristic introductory paragraph about the import- 
ance of the new theorem and his reasons for publishing it, 
Sylvester proceeds — 
“The theorem is as follows: — Suppose that there are two 
determinants of the ordinary kind, each expressed by a square 
array of terms made up of n lines and n columns, so that in 
each square there are n 2 terms. Now let n he broken up in 
any given manner into two parts p and q, so that p + q = n. 
Let, firstly, one of the two given squares he divided in a given 
definite manner into two parts, one containing p of the n given 
lines, and the other part q of the same; and secondly, let the 
other of the two given squares be divided in every possible 
way into two parts, consisting of q and p lines respectively, 
so that on tacking on the part containing q lines of the 
second square to the part containing p lines of the first 
square, and the part containing p lines of the second 
square to the part containing q of the first, we get hack 
a new couple of squares, each denoting a determinant 
different from the two given determinants : the number of 
such new couples will evidently he 
n(n - 1) ... (n -p + 1) . 
1-2 ... p ; 
and my theorem is that the product of the given couple of 
determinants is equal to the sum of the products ( affected with 
the proper algebraical sign ) of each of the new couples formed 
as above described 
The same is then stated in symbols, viz., 
See Trans. P.S.U., xxx. p. 4. 
