5G 
ANALYTICAL GEOMETRY OE (71) DIMENSIONS. 
[XI 
The U 'p equations represented by this formula reduce themselves to 
1.2 (n — q + 1) ^ 
(n — q) independent equations. Imagine these expressed by 
(1) = 0, (2) = 0 (n-q) = 0 (4), 
any one of the determinants of (2) is reducible to the form 
©! (1) + ® 2 (2) ... + ©n_2 (n-q) (5), 
where @ 1} @ 2 ...© n _ 5 are coefficients independent of x ly x 2 ...x n . The equations (3) may 
be replaced by 
X x «! + Xo« 2 + • 
. X n « M , 
/¿1«! + . . . , 
■ • • TjXj. + ... 
= 0 
(6), 
Xj-d.! + X 2 dL 2 + . 
. \ n x n , 
y 1 A 1 + ..., 
... «1^!+ ... 
XiA, + X 2 A 2 4-. 
• A« Kn> 
g l K 1 + ..., 
r 1 K l + ... 
and conversely from (6) we may deduce (3), unless 
Ai, A 2 , ... A n 
AL > A*2 > • • • /Li 
Li > t 2 , ... r n 
= 0 
(7). 
(The number of the quantities A, fi...r is of course equal to ft.) The equations (3) 
may also be expressed in the form 
(8), 
Xi 
, cc 2y 
• x n 
Xi-d-! + .. 
• y X/j j\. 2 2) • 
• AiA n . 
. 4- to x K n 
A^A^ 4-. 
• COqKiy “f - ... (OqK2, . 
• A 9 A n . 
• o) q R n 
the number of the quantities X, g ... <a being q. 
And conversely (3) is deducible from (8), unless 
Xj, ... a) 1 
Xg, ... co q 
= 0 
(9). 
Chap’. 2. On the determination of linear equations in x ly x 2y ... x n which are 
satisfied hy the values of these quantities derived from given systems of linear equations. 
It is required to find linear equations in x ly ... x n which are satisfied by the values 
of these quantities derived—1. from the equations %&! = 0, 23' = 0 ... (S' = 0 ; 2. from the 
equations gT = 0, 23" = 0 ... Jt" = 0; 3. from = 0, 23'" = 0 ... 2Bt'" = 0, &c. &c., where 
= A 1 x 1 + A 2 ac 2 ...+A n x n , (1), 
23' = I?i#i + B.x 2 ... + B n x n , 
and similarly 23",..., 23"',..., &c. are linear functions of the coordinates«!, x 2y ... x,