326 PROFESSOR C. J. JOEY OY QUATERXIOXS AXD PROJECTIVE GEOMETRY. 
In other words, is the reciprocal in the wi-flat which contains the origin and 
the points a of the (m — l)-flat which contains the points a. 
For examjde, in three dimensions, 
^2 
1 + 
YfCiSai — Vo^So.-, 
(589) 
is the point in the j^lane o^^ao which is reciprocal to the line a^a-^. 
162. A comparison of the ecjuations (581) and (585) shows that the m points 
+ Vh^ h - ■ ■ im .(590) 
(of which ?g . . . i,n are at infinity) may be taken as defining the [m — l)-flat 
containing the points a. 
Hence, conversely, if [ci]^ is any function satisfying the equations of condition 
Mm — ; 
(591), 
it is the symbol of an (m — l)-flat. In fact, we can reduce this function to the form 
(585) and the j^roposition is evident by (590). 
163. The symbol of the fat reciprocal to [o]„, with respect to the auxiliary 
quadric (587), S. q® = 0, in an n-space is 
[a]m G 
(592), 
where Xl is the qiuoduct of'' n” mutually rectangular vector units in the n-space, or 
In fact, from (585) we obtain 
(593). 
[Ctfa = ( —)™ ^{iji, — x)ip,^+fra+2 ■ . . ffzhh • • • 
= + Xh)in-,^%^+2 . • . in = M»+l-™ 
(594); 
and n I — m defining points of this new {n — ?7?)-flat are (590) 
y a■^I, ... In .(595). 
But all these points are conjugates, with resjDect to the auxiliary quadric, of the 
m points (590) ; and therefore the flat is the reciprocal of the flat Mm- 
More symbolically, we have the relations 
Y„ [a]„. n = Y„_,„. [a],, n ; Y,„_, [«],„. O = - AJ . [a],„ G . (596), 
and in particular for three dimensions we deduce the relations 
[rA] = - {a'l/) ; {ab) = [o7/] . 
connecting a line and its reciprocal (compare p. 224). 
( 597 ), 
