944 
Proceedings of Royal Society of Edinburgh. [sess. 
Then observing the ‘ indices superieures 5 attached to the V s in this, 
we are asked to consider two possible cases. In the first place, we 
have to note that if no one of the i/^’s be identical with any one 
of the <£’ s, the term is a term of the expansion of the product on 
the left-hand side, and that the number of such terms in the 
expansion of each product on the right-hand side being 
(1.2.3 . . . n) • (1.2.3 . . . p) • (1.2.3 . . . hf^p) 
and the number of products 
n(ii -\)(n — 2) .... (n - p + 1) 
1.2.3 . . . p 
the total number of such terms is 
(1.2.3 . . . nf , 
which is exactly the total number on the left-hand side. In the 
second place, if one of the i/^s be identical with one of the <£’s, say 
^ i = ^P+h 5 
it is pointed out that there must exist another term in which, in 
place of having 
jfi lfp+h 
°i 6 p+h 
we shall have 
jfi jfp+h 
®p-\-h 
and that these two terms having necessarily different signs, must 
cancel each other. 
Sylvester, J. J. (1852, Oct.). 
[On Staudt’s theorems concerning the contents of polygons and 
polyhedrons, with a note on a new and resembling class of 
theorems. Philos. Magazine (4), iv. pp. 335-345 : Collected 
Math. Papers , i. pp. 382-391.] 
After a page of introduction, written in a light semi-historical, 
semi-critical style, Sylvester prepares the way for considering his 
main subject by giving as a basis two algebraical lemmas. The 
first he formulates as follows : — 
“ If the determinants represented by two square matrices 
are to be multiplied together, any number of columns may be 
