61] 
ON GEOMETRICAL RECIPROCITY. 
379 
Let i be a point in the pole conic, and when i is considered as belonging to 
the first figure, let il 1 be considered as the corresponding line in the second figure 
(ij being the point of contact on the polar conic). 
Then if j be another point in the pole conic, in order to determine which of 
the tangents is the line in the second figure which corresponds to j considered as a 
point of the first figure, let il 2 be the other tangent through I: the points of contact 
of the tangents through j may be marked with the letters J lf J 2 , in such order that 
IiJ 2 , I. 2 Ji meet in the line of contact of the two conics, and then jJ x is the required 
corresponding line. Again, I and i, as before, if A be a tangent to the polar conic, 
then, marking the point of contact as J lf let J 2 be so determined that I X J 2 , I 2 J X 
meet in the line of contact of the conics: the tangent to the polar conic at J 2 will 
meet the pole conic in one of the points where it is met by the line B, and calling 
this point j, B considered as belonging to the second figure will have j for its 
corresponding point in the first figure. Similarly, if the point of contact had been 
marked J 2 , J x would be determined by an analogous construction, and the tangent at 
J 1 would meet the pole conic in one of the points where it is met by the line B 
(viz. the other point of intersection); and representing this by j', B considered as 
belonging to the first figure would have j' for its corresponding point in the second 
figure, that is, considered as belonging to the second figure, it would have j for its 
corresponding point in the first figure (the same as before). 
Similar considerations apply in the case where a tangent A of the polar conic, 
considered as belonging to one of the figures, has for its corresponding point in the 
other figure one of its points of intersection with the polar conic; in fact, if A 
represents the line il 1 , then A, considered as belonging to the second figure has i for 
its corresponding point in the first figure, which shows that this question is identical 
with the former one. 
To appreciate these constructions it is necessary to bear in mind the following 
system of theorems, the third and fourth of which are the polar reciprocals of the 
first and second. 
If there be two conics having a double contact, such that K is the line joining 
the points of contact, and k the point of intersection of the tangents at the points of 
contact: 
1. If two tangents to one of the conics meet the other in i, and j, j 1 respectively, 
then, properly selecting the points j, j lf the lines ij ly meet in K. And 
2. The line joining the points of intersection of the tangents at i, j lf and of 
the tangents at i 1} j passes through k. Also 
3. If from two points of one of the conics, tangents be drawn touching the other 
in the points I, I 1 and J, J x , then, properly selecting the points J, J 1} the lines 
IJ lt I X J meet in K. And 
4. The line joining the points of intersection of the tangents at I X J X and of the 
tangents at I ly J passes through k. 
48—2