Note on an Overlooked Theorem. 
273 
Had we omitted the elements h^^, on starting, the result would have 
been nugatory, namely — 
«22 '^'33 ^'44 I • 1 ^11 ^22 1 = 1^11 ^22 I • I «I3 «24 «3I ^42 | • 
If on the other hand we had in addition inserted b^^, h^^ we should have 
obtained for 
I ^22 «33 ^^44! • I ^11 ^22 I 
an expression consisting of six terms of a similar form beginning with 
(^^j^ <^i4 • 1^31 <^32| 
\ct21 (122 (''22, ^24 \(^41 ^42 I • 
^12 ^13 ^14 
K2 K K 
4. It is thus evident that the product of two determinants of the 
^th and ^th orders is expressible as a sum of products of two determinants : 
(1) of the {p - l)th and {q + l)th orders, (2) of the {p - 2)th and (g + 2)th 
orders, (3) of the (^ — 3)th and (g + 3)th orders, and so on, the number of 
results being p - q. 
In Sylvester's similar theorem, which has received every attention 
from writers of text-books, q is equal to p, and the orders of the determi- 
nant factors in the expansion are the same as those in the original 
product. 
Of course any one of the expansions obtained for the given product 
can be equated to any other, and there is thus originated a series of 
identities similar to Schweins' of 1825 (see Philos. Magazine, xviii (1884), 
pp. 416-427). 
