11] ANALYTICAL GEOMETRY OF (n) DIMENSIONS. 
59 
Theorem. Consider the equations 
(&' = 
0, 
33' = 
= 0 
= 0) .... 
.. (12), 
(&" = 
0, 
33" = 
= 0 
-B" 
= 0), 
(»'" = 
0, 
33'" = 
= 0 
..m" 
' = 0), 
&c. 
l. The 
equations 
d Xi U , 
d x JJ 
<4.tr 
= 
0, 
da 
^ , 
dxjd > • 
.. 4„*7 
= 0, . 
.. (13), 
A', 
A',.. 
4* 
A", 
• • A n " 
GI, 
GI,.. 
GI 
0/', 
0 2 ",. 
• O n " 
& c. 
which are the reciprocals of these systems, represent taken conjointly the reciprocal of 
the system of equations (3) of the same chapter. 
Let this system, which contains ti — {(n — r) + (n — r') + ...} equations, be represented by 
The reciprocal system is 
cc 1 x 1 + a 2 x 2 ... 
Ai x 1 + ¡3 t x 2 ... 
-f“ Xji — 
+ An — 
0 
0. 
(11), 
£1 "b £2 ■ 
+ 
Vfx 
3* 
11 
0. 
i/", C?£C a ^ 
«1 , «2 
... d Xn L 
... CL n 
= 0 
(15), 
t., & 
... ?„ 
containing (n — r) + (n — r f ) + &c.... equations. 
Also, by the formulae in Chap. 2, 
a 1 x 1 + ... + a n x n = A/gt' + /¿/33' + ... o-/<?5' (A, /x ... a, r' in number). 
Ai x x + ... + x n = A/gt' + /j, 2 'W + ... <r 2 '(§i' 
£i#i ... + x n = Xe'^L + fie 33 / + ... (16), 
writing 6 = 71— {(71 — r) + (71 — r') + ...}. 
Also, assuming any arbitrary quantities 772 ••• Vn • <f>i, <&> •■•</>«. (the number of 
sets being (r' — 8),) such that 
77^1 ... +7] n x n = A+/¿0 +1 'i3 r + ... O0 +1 <5' -• 
</>! ¿Tj ... + = A/ £J' + /x/ 33'+... cr/ C5'. 
(1?),