( 301 ) 



JO. Among the particular sections of <S\' the conies of this surface 

 come into account. These conies have two points in common with 

 AV ; yo (3) to these must corres[)on(l in — ljyperhoh)i(lic systems of 

 focal rays of ./. These can be constructed in the following way : 



Let again a point .1 he taken on K\ its focal |)iaue a he deter- 

 mined, moreover the second point of intersection .4' of « witii A'" and 

 the focal plane u' of A' . If now a pencil of rays l)e dra^vn in a' 

 through A (vvhicii rays are not focal rays) and likewise through A' 

 in «, the pencils (.1, «'), (.1', «) consist of ctmjugate polars of A 

 between which a projective correspondence is establislied by means of 

 the focal rays. In connection with AV each pair of conjugate i)olars 

 causes a hyperboloidic system of focal rays to appear. Tiiese two 

 pencils generate them all, so their number is oo . 



Jl. Finally a few particular cases ask for our attention. 



a. The line of intersection x cuts the plane PRS in a point of 

 the tangent plane PR. The pencil, of focal rays in tlie plane PR 

 has as \'ertex this point of intersection; to this pencil corresponds a 

 pinchpoint on AV> but at the same time this pencil of rays has more- 

 oxQv a ray in common with the pencil of rays in the focal ])lane of 

 the point R; so the obtained pinchpoint is at the same time a point 

 of y^i; from this follows that in the point of intersection of A\- and 

 />! two pinchpoints have coincided; so through this point only a single 

 generator of >S'i'' can be draw^n. 



h. Application to the motion of an invariable system. In this case 

 K^ is imaginary (the imaginary circle in the plane at intinity) ; so 

 the congruence (2,2) consists entirely of imaginary rays. The pencil 

 of rays F/FRS, however, remains real; so the representation in ^ 

 becomes an imaginary ruled surface >S'i^ with real double curve con- 

 sisting of a straight line and a conic intersecting it. The same obsei'- 

 vation can be made for other cases where 'K'^ l>econies imaginary. 



c'. Another particular case occurs when the ray A'A' zE .v i« taken 

 in such a way that it cuts the conic K^'^; by doing so the character 

 of the congruence does not change, but its I'epresentation does. If 

 ^ve now consider a pencil of rays in a |)lane brought through .r, it 

 is apparent that always one of the two rays of congi-uence to A'-' coin- 

 cides with ,/". Of the two rays cutting in J!i\ the double conic A\- ojily 

 one is situated on S^\ the other one passes into a ray situated in $. ; 

 from this follows: 



"When the focal ray x cuts the conic K^ the surface .-S/' bj''-ak> 

 up into §1 and a cubic ruled surface >S'i' of which j)^ is a double 

 Une; so this gives a simpler representation of the congruence '2.2'! 



26^» 



