84 THEORY OF COLLINEATIONS. 
through S and S, are transformed into the pencils through S, 
and S., respectively, K must be transformed into K,. 
The line SS, is a common ray of the two pencils through S 
and S,; considered as a ray of the pencil through S it is 
transformed into the tangent to K at S,. Since S is trans- 
formed into S, and S, into S., the line SS, is transformed into 
the line S,S,. Hence S, is the point where the tangent to K 
at S, cuts K,. 
Consider a point P on K and its corresponding point P, on 
K,. The lines SP and S,P, since they meet on K, are corre- 
sponding rays of the two pencils through S and S,._ But S is 
transformed into S, and P into P,; hence SP and S,P, are 
corresponding lines of the same two pencils. Therefore, S,P 
and S,P, are the same straight line since they both correspond 
to SP. Hence P and P,, corresponding points on the two 
conics K and K,, are collinear with S,. Hence we have the 
important result: * 
*Reye-Holgate, Geometry of Position, p. 187. 
