Il] 
ANALYTICAL GEOMETRY OF (11) DIMENSIONS. 
57 
Also r', r"... representing the number of equations in the systems (1), (2)... and k the 
number of these given systems, 
Assume 
(n — r) + (n - r") + ...}► n - 1 or (k - 1) n + 1 ;)> / + r" +•... 
Q = \'®L' + pW+..., 
+fi'w+...= +fi'w+...= \"w+ii!"w +...=& c (2), 
the latter equations denoting the equations obtained by equating to zero the terms 
involving x x , those involving x 2 , &c.... separately. Suppose, in addition to these, a 
set of linear equations in A, p , /.i’’... so that, with the preceding ones, there 
is a sufficient number of equations for the elimination of these quantities. Then, 
performing the elimination, we thus obtain equations T' = 0, where T is a function of 
x x , x. 2 ... which vanishes for the values of these quantities derived from the equations 
(1) or (2) ...&c. The series of equations’T'= 0 may be expressed in the form 
a', 
33',. 
1 
A x ', 
A',. 
..O', 
A/',. 
0" 
.. W J , 
in', 
Bn, . 
:G n \ 
in", . 
.. 0 n ", 
4i", • 
..O'', 
A'",... B/", 
in", . 
. On", 
A w ¿> /// 
JTL n , . . . J. l n , 
0 ... 
(3). 
•(I), 
Chap. 3. On reciprocal equations. 
Consider a system of equations 
A x X x “4" AyX^ • • • T A,i^n = 0, 
K x x x + K 2 x 2 ... + K n x n = 0, 
(r in number). 
The reciprocal system with respect to a given function (U) of the second order 
in x x , x 2 ... x n , is said to be 
d Xl U, 
A x , 
d Xl U,. 
A-2, • 
. d Xn U 
• A n 
= 0 
(2), 
k x , 
k 2 ,.. 
. kn 
(n — r in number). 
It must first be shown that the reciprocal system to (2) is the system (1), or 
that the systems (1), (2) are reciprocals of each other. 
C. 
8