11] ANALYTICAL GEOMETRY OF (n) DIMENSIONS. 61 
Suppose X x , Xo,... X n determined by the equations 
[l 2 ] X, + [12] X 2 ... + [In] X n = 0 (24), 
[21] X x + [2 2 ]X 2 ... + [2n] X n = 0, 
[nl]X 1 + [n2]X 2 ...+[n 2 ]X n = 0; 
equations that involve the condition that 1c satisfies an equation of the order n — r, as 
will be presently proved. 
Then shall x x =X x ... x n — X n satisfy the system of equations, which is the reciprocal of 
X x , 
X 2 , . 
• • x n 
= 0 
(25) 
«1, 
«2, • 
• • 
: Pi, 
p2,- 
• • Pn 
To prove these properties, in the first place we must find the form of V. 
Consider the quantities % B , . ..f £ , (n — r) in number, of the form 
Ç A = A 1 x 1 + A 2 x 2 ...+A n x n , (26), 
= B x T B 2 x 2 ... T B n x n , 
— L x x x T L 2 x 2 ... T L n x n , 
where, if © represent any of the quantities A, B ... L, 
a x ® x + a 2 © 2 ... + a n % n =0, (27), 
/SA + /3 2 © 2 ... + © n = 0, 
Pi®i + pfflz • • • + Pn®n = 0, 
2 V = (A 2 )^ 2 + (B 2 ) & + ... + 2 (AB) &&+...« 2 (A 2 ) + 22 (AB) çj, 
Hence, if 2 V= 2 {a 2 } x 2 + 22 {a/3} x a xp (28), 
we have for the coefficients of this form 
{l 2 } = 2 (A 2 ) A 2 + 22 (AB) A X B X , {12} = 2 (A 2 ) A X A 2 + 2 (AB) (A X B 2 + A 2 B X ), 
and consequently the coefficients of 2U — 2kV are 
[1 2 ] = (1 2 )-&{1 2 }, [12] = (12) — k {12}. 
Hence, 6 representing any of the quantities a, /6 ... p, 
0 1 {l*}+0 2 {12}...+0 n {lw} = 0 . 
0 X {nl} + 0 2 {w2} ... + 6 n {n 2 } = 0; 
(29),