11] 
55 
11. 
CHAPTERS IN THE ANALYTICAL GEOMETRY OF (») 
DIMENSIONS. 
[From the Cambridge Mathematical Journal, vol. iv. (1843), pp. 119—127.] 
Chap. 1. On some preliminary formulae. 
1 take for granted all the ordinary formulae relating to determinants. It will be 
convenient, however, to write down a few, relating to a certain system of determinants, 
which are of considerable importance in that which follows: they are all of them 
either known, or immediately deducible from known formulae. 
Consider the series of terms 
(1). 
x. 2 
A: 
k, 
K, 
kn 
A 
.K 
being e< 
vertical rows selected, and the quantities contained in them formed into a determinant, 
this may be done in 
n (n — 1) (q + 2) 
1.2 n — q — 1 
so obtained will be represented by the notation 
different ways. The system of determinants 
by the notation 
x 1 , 
x 2 .. 
A., 
A, ... 
... A H 
k, 
K,... 
...k n 
ained 
by equating 
Xi , 
¿r 2 ... 
A, 
a 2 ... 
...A n 
k 
K, ... 
...k u 
(2), 
= 0 
(3).