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,