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DE. PLUCIvEE ON A NEW GEOMETEY OF SPACE. 
The plane BOA passing through O and an axis AB is represented by the equation 
The equation (12) being with regard to p and q of the first degree, indicates that all 
such planes, containing the different axes of the configuration, intersect each other along 
a given right line DD' passing through O. Hence all axes meet a fourth right line, 
itself confined within the hyperboloid. 
The complete determination of the hyperboloid presents no difficulties. We may for 
instance find its centre and its axes by determining the shortest distance of any two of 
the axes generating it. 
10. Let a congruency either of rays or axes be represented by two linear equations. 
In adding to these equations two new ones, likewise of the first degree, there exists only 
one ray or axis the coordinates of which satisfy simultaneously the four linear equations. 
Two new equations of this description are obtained if, among the rays or axes of the 
congruency, we select those either passing through a given point, or confined within a 
given plane. In the case of rays, let (fi, ?/, z') be a given point, then we get 
%!=rz ,J r%, 
y'=sz'~ J-<7 
in order to express that all rays meet in that point. Let 
t'x+u'y+v'z+ 1=0 
be the equation of a given plane, then we get 
ir-\-u!s-\-v— 0, 
t'g-\-u'(r- {-1=0 
in order to express that the rays lie within that plane, Again, in the case of axes, let 
(if, u', v ') be a given plane, then we get the new linear equations 
t'x + v'z t = 1 , —pv' -f- sr, 
or 
u'x-\-v'z u = 1, u'=qv'-\-z, 
in order to express that the axis is confined within that plane. Let in regarding x/, y\ z r 
as constant, t, u, v as variable, 
1 = 0 
represent a given point, then we get 
a?p+tfq+z!= 0, 
odvs-\-y'x ,-\- 1 = 0 
in order to express that the axes pass through that point. Hence 
In a congruency represented by the system of two linear equations , there is one single 
ray or axis passing through any given point of space, as there is one single ray or axis 
confined within a given plane. 
