58 
ANALYTICAL GEOMETEY OF (n) DIMENSIONS. [ll 
Consider, in general, the system of equations 
dxJJ + a,d x JJ... + d Xn u = 0 (3). 
d Xi U + X 2 d X2 U... + \ n d Xn U = 0. 
Suppose 2 U = S (a 2 X 2 + 22 (a/9) x a xp, so that d X) U='£ (sa) x a (4), (5). 
The equations (3) may be written 
{«i (l 2 ) + a, (12) ... + a n (In)} + ... + x n \a x (nl) + a, (n2) ... +a n (n 2 )} =0 (6), 
&c. 
and forming the reciprocals of these, also replacing d x JJ, d x JJ... by their values, we 
have 
x x (l 2 ) + x 2 (12) + . 
«i (l 2 ) +«,(12)+. 
. x n (In), . 
. 7 n (In), . 
. x x (?il) + x 2 (n2) . 
. cq (nl) + a„ (n2) . 
• + SB n (n-) 
. + a n (n 2 ) 
= 0... 
...(7). 
\ (l 2 ) + X. 2 (12) + . 
■ K (1^), • 
. Xj (nl) + X 2 (n2) . 
• + (id) 
From these, assuming 
(l 2 b 
(21), 
(12),. 
(2 2 ), • 
.. (In) 
.. (2n) 
+ 0 
(8) 
ou 
(n2), . 
• fa 2 ) 
we obtain, for the reciprocal system of (3), 
x 1} 
x 2> . 
• X n 
= 0 
(9). 
«i, 
«2, •• 
. a,i 
Xi, 
x„ • 
• ^7» 
Now, suppose the equations (3) represent the system (2) ; their number in this case 
must be n — r. Also if 0 represent any one of the quantities a, /3 ... X, we have 
M+M-.+iA = o (10), 
K]6 X +a .,6. 2 ... -p K n e n =o. 
By means of these equations, the system (9) may be reduced to the form 
Aj a’j + A 2 x 2 . 
• + A n x n , 
.. K x x x + K 2 x 2 . 
• + K n X n , X r +1, 
X., , • 
..x n 
= 0 
0 
, 
0 
, 1, 
°V+2 , • 
. 7 n 
6 
, . 
0 
Xr+2, • 
which are satisfied by the equations (1). Hence the reciprocal system to (2) is (1), 
or (1), (2) are reciprocals to each other.