736 
DE. PLiiCKEE ON A NEW GEOMETET OF SPACE. 
containing its own pole, is determined by means of the poles of any two planes passing 
through that pole ; likewise a point, falling within its polar plane, is determined by 
means of the polar planes of any two points of its polar plane. A right line joining 
any two points of space is conjugate to the right line, along which the polar planes of 
both points intersect each other. If one of two conjugate right lines envelopes within 
a given plane a curve, the other describes a conical surface ; the vertex of the cone falls 
within the plane containing the enveloped curve. Generally if one of the two conju- 
gate right lines describes a configuration, the other one likewise describes such a sur- 
face. If one of the two surfaces degenerates into a cone, the other degenerates into a 
plane curve*. 
22. Anoint of space being given , to construct the plane which contains all rays of the 
complex passing through the point. 
Each ray intersecting two conjugate lines is a ray of the complex. Accordingly 
the only right line , starting from a given point and meeting any two conjugate is a 
ray of the complex. We obtain a new ray, starting from the same point, by means 
of each new pair of conjugate lines. All such lines constituting the plane corre- 
sponding to the given point, two pairs of conjugate lines are sufficient to determine 
that plane. 
A plane of space being given , to construct the point where meet all rays of the complex 
confined within the plane. 
Each right line joining the two points in which two conjugate right lines are inter- 
sected by a given plane being a ray of the complex, there will be obtained, within the 
given plane, as many rays as there are known pairs of conjugate lines. Any two such 
pairs are sufficient in order to determine the point within the plane corresponding to it 
where all rays meet. 
A plane is intersected by the two lines of each conjugate pair in two points ; the right 
lines joining two such points are rays of the complex converging all towards the point 
which corresponds to the plane. Again, the two planes passing through a point of space 
and meeting the two lines of a conjugate pair, intersect each other along a ray of the 
complex confined within the plane which corresponds to the point. 
23. After this geometrical digression, immediately indicated by analysis, we resume 
the analytical way. 
By putting in the general equation (9) of the plane corresponding to a given point 
tf'=0, y'= 0, z'= 0, 
we obtain 
Ax+By+Cz=0, (17) 
in order to represent the plane corresponding to the origin. 
* The peculiar kind of polar reciprocity we meet here was first noticed by M. Mo Bins in the 10th volume of 
‘ Crelle’s Journal,’ and was afterwards expounded by L. F. Magnus in his valuable work ‘ Sammlung von 
Aufgaben und Lehrsatzen aus der analytischen Geometrie des Eaumes,’ pp. 139-145. 
