378 
ON GEOMETRICAL RECIPROCITY. 
[61 
In the particular case where these lines and points are respectively identical (the 
identity of the lines implies that of the points and vice versa) we have the theory of 
“ reciprocal polars.” Here, where it is unnecessary to define whether the points or 
lines belong to the first or second figures, the line corresponding to a point and the 
point corresponding to a line are spoken of as the polar of the point and the pole of 
the line, or as reciprocal polars. 
“The points which lie in their respective polars are situated in a conic, to which 
the polars are tangents.” Or, stating the same theorem conversely, 
“ The lines which pass through their respective poles are tangents to a conic, the 
points of contact being the poles.” 
To determine the polar of a point, let two tangents be drawn through this point to 
the conic, the points of contact are the poles of the tangents; hence the line joining 
them is the polar of the point of intersection of the tangents, that is, “ The polar of 
a point is the line joining the points of contact of the tangents which pass through 
the point.” 
Conversely, and by the same reasoning, 
“The pole of a line is the intersection of the tangents at the points where the 
line meets the conic.” 
The actual geometrical constructions in the several cases where the point is within 
or without the conic, or the line does or does not intersect the conic, do not enter 
into the plan of the present memoir. 
Passing to the general case where the lines and points in question are not identical, 
which I should propose to term the theory of “Skew Polars” (Polaires Gauches), we 
have the theorem, 
“ Considering the points in the first figure which are situated in their respective 
corresponding lines in the second figure, or the points in the second figure which are 
situated in their respective corresponding lines in the first figure, in either case the 
points are situated in the same conic (which will be spoken of as the * pole conic ’), 
and the lines are tangents to the same conic (which will be spoken of as the * polar 
conic’), and these two conics have a double contact.” This theorem is evidently identical 
with the converse theorem. 
The corresponding lines to a point in the pole conic are the tangents through 
this point to the polar conic; viz. one of these tangents is the corresponding line 
when the point is considered as belonging to the first figure, and the other tangent 
is the corresponding line when the point is considered as belonging to the second figure. 
The corresponding points to a tangent of the polar conic are the points where 
this line intersects the pole conic; viz. one of these points is the corresponding point 
when the line is considered as belonging to the first figure, and the other is the 
corresponding point when the line is considered as belonging to the second figure.