380 ON GEOMETRICAL RECIPROCITY. [61 
These theorems are in fact particular cases of two theorems relating to two conics 
having a double contact with a given conic. 
It may be remarked also that the corresponding points to a tangent of the pole 
conic are the points of contact of the tangents to the polar conic which pass through 
the point of contact of the given tangent, and the corresponding lines to a point of 
the polar conic are the tangents to the pole conic at the points where it is intersected 
by the tangent at the point in question. 
We have now to determine the corresponding lines to a given point and the 
corresponding points to a given line, which is immediately effected by means of the 
preceding results. 
Thus, if the point be given, 
“ Through the point draw tangents to the polar conic, meeting the pole conic in 
A 1 , A 2 and B lf B. 2 (so that A X B 2 and A 2 B 1 , intersect on the line joining the points 
of contact of the conics), then A. 2 B. 2 and A 2 B 2 are the required lines.” 
In fact A 1} B x and A 2 , B. 2 are pairs of points corresponding to the two tangents, 
so that A 1 B 1 and A 2 B. 2 are the lines which correspond to their point of intersection, 
that is, to the given point, and similarly for the remaining constructions. Again, 
“ Through the point draw tangents to the pole conic, and from the points of 
contact draw tangents to the polar conic, touching it in a lt a. 2 and j3 1} (3 2 (so that 
and a. 2 /3j intersect on the line joining the points of contact of the conic), then a 1 ^ 1 
and a. 2 B 2 are the required lines.” 
So that Aj, B 1} a 1 , /3 1 are situated in the same line, and also A 2) B 2 , a 2 , /3 2 . 
Again, if the line be given, 
“ Through the points where the line meets the pole conic draw tangents to the 
polar conic Ci, G 2 and D 1} D. 2 (so that the points C^Zt, and G 2 D 1 lie on a line passing 
through the intersection of the tangents at the points of contact of the tangents), 
then G 1 D l and C. 2 D 2 are the required points.” 
Again, 
“ At the points where the line meets the polar conic draw tangents meeting the 
pole conic, and let y 1} y 2 and B 1} B 2 be the tangents to the pole conic at these points 
(so that the points Yi<b and y. 2 $i lie on a line through the intersection of the tangents 
at the points of contact of the conics), then y lt Sj and Y2. are the required points ” ; 
so that C 2 , Dj, Yi, Sj pass through the same point and also C. 2 , D 2 , y 2 , B 2 . 
“ The preceding constructions have been almost entirely taken from Pliicker’s 
“ System der Analytischen Geometrie,” § 3, Allgemeine Betrachtungen fiber Coordinaten- 
bestimmung. I subjoin analytical demonstrations of some of the theorems in question. 
Using x, y, z to determine the position of a variable point, and putting for shortness 
f = ax + a'y + a"z, 
y = bx + b'y + b"z, 
£ = cx + c'y + c"z.