whence also 
62 
ANALYTICAL GEOMETRY OF (n) DIMENSIONS. 
Oi [l 2 ] + ... 0 n [In] = 0 1 (l 2 ) + ... 9 n (111), 
[n 
9 1 [?il] + ... 6 n [n 2 ] = 9j (nl) + ... 0 n (?i 2 ). 
Hence, the equations for determining X 1 , ... X n may be reduced to 
Xi [«i (l 2 ) + • • • «n (1»)] + X* [«i (21)... + a n (2n)] ... + Z n [«! (ill) ... + a n (ii 2 )] = 0 ... (30), 
[ft (l 2 ) + ... ft (lw)]+ X 2 [ft (21) ... + ft (2ii)]... + Xn [ft (wl) .. • + ft t (ii 2 )] = 0, 
-^l [pi (l 2 ) + • • • Pn (lw)] + X 2 [p x (21)... + p n (2n)]... + X n [p n (ill) ... + p n (w 2 )] = 0. 
X x \r + 1, 1] + X% [r +1, 2]... + X n [i' +1, ii] = 0, 
= 0, 
X, [n, 1] + 
Xo [w, 2] ... + 
X n [n 2 ] 
= 0. 
Eliminating X 1 ...X n , since the first r equations do not contain k, the equation in 
this quantity is of the order n — r. 
Next form the reciprocals of the equations (25). These are 
d x JJ, d x JJ,...d x JJ =0 
■Ai , .ft ,... A. n 
(31). 
0 
0 
0 
which are evidently satisfied by x 1 = X 1 , x 2 = X 2 ... x n = X n . 
In the case of four variables, the above investigation demonstrates the following 
properties of surfaces of the second order. 
I. If a cone intersect a surface of the second order, three different cones may 
be drawn through the curve of intersection, and the vertices of these lie in the plane 
which is the polar reciprocal of the vertex of the intersecting cone. 
II. If two planes intersect a surface of the second order through the curve of 
intersection, two cones may be drawn, and the vertices of these lie in the line which 
is the polar reciprocal of the line of intersection of the two planes. 
Both these theorems are undoubtedly known, though 
them to any given place. 
I am not able to refer for