( 390 ) 



9. A closer aequainta,nce \^ ith the form of ;S'/ is obtained b}^ 

 tracing' the piiichpoints on the double curve ; there can be two of 

 them on p.^ and two on AV- Those of p^ depend on the position 

 of P AN'itli respect to 7v-. 



a. Let /-' be outside K'. When a ray through P cuts K' in two 

 points, we get two pencils of rays of the congruence, to which two 

 real generators of >SV correspond, concurring in a point of p.^. For 

 the tangential lines out of P to lO these two generators coincide, 

 so the point of S^\ from ^^■hich they are drawn is a pinchpoint ; so 

 for this position there are two real pinchpoints on p^ ; from this ensues : 



"If P lies outside A'% ^>i has one part appearing as double line and 

 anothei- which is isolated; two j)inchi)oints separate these two parts." 



h. Let P lie within A'". All focal rays through P cut A'" : there 

 are no tangents Jo A'-, so there are no pinchpoints on p^. So the 

 double line p^ is in its whole leugth reall}' double line. 



Besides the pinchpoints on p-^ the surface ^SV has also pinchpoints 

 on AV- To tlnd these we .must keep in view that the points on ATi" 

 correspond to the pencils of rays whose \'ertices lie on XX' = x, 

 which are thus situated in planes through x. Let y be a plane throngli .1; 

 and C its focal point; the pencil of rays (6'y) has two rays cutting 

 A^ viz. the two rays coiniecting C and the points of intersection 

 B and B of y and A'-. These two rays are represented in ^^ 

 by a single poini />\ of X^\ Now CB i)elongs still to another 

 pencil of focal rays, \iz. to the pencil \\'hose vertex is J5 and whose 

 plane is the i)lane CBI*Ez.^- The latter pencil belongs to the con- 

 gruence (2,2) and is thus represented by a straight line through B^ 

 lying on S^\ In a similar w ay it appears that also a second straight 

 line of >S\^ passes through B^, namely the one which is represented 

 by the pencil of rays {B'i^') lying in plane CB'P. Now again two 

 principal cases may occur: 



a. ,v cuts the plane BBS in a point P outside K\ The pencil 

 of rays T lying in this plane has rays cutting K^ in two points, 

 touching A'" or having two imaginary points in common with A' "^ 

 In this case these are parts of AV' through ^^ Inch two generators 

 of /SV pass, which have thus to be regarded as points of a double 

 curve, and parts ^N'hich are isolated ; the transition is formed by two 

 pinchpoints, through which two coinciding generators pass ; and these 

 last correspond to the pencils of rays, having their vertices on the 

 tangents draAvn from T to A"'. 



h. The above mentioned i)oint of intersection 7' lies within X^*. 

 All rays through T cut A"'; through each point of A^i' two generators 

 pass, so the whole conic AV is a double curve. 



