( 388 ) 



Ld the (liivclioii of \\\o M'locilv (f ol' a poiiil A iiilcrsccl llic 

 plane of coiiicidiMicc /VAS' in /, ■ \u)\\ the loejil [)laiie of J iiilersecis 

 this |tlaiie iii llie polai' /> of A with i-especl (o llie conic A''. 'J'lie 

 i'avs of llie complex, at the same time ravs of the cojijii,i>ale foc.il 

 svslem, ai-e situated iii the focal plane a of A: from this ensues that 

 these ravs intersect the plane J^RS in two coincident |)oints, al llie 

 same lime c()njn<>'ale with respect to A"': so these ravs \\iJ! intersect 

 X'^ and now ensues the liieorem : 



"The i-avs of the conni-nence (2,2), which is the section of the 

 complex with the focal s\stem, ha\-e a ]»oi]it in common \\\\\\ the 

 conic A'-'; so they are found as rays of A cutting- A -." 



So the c()ni>-ruence {'2,'2) arisinu' from this belongs to those con- 

 i'-ruences, not possessinu' a focal surfac(\ hut a singular or double 

 cui-\(''), U'eomelrical locus of the lii'st series of foci of the conurnence. 



() The coniirueuce can be constructed as a whole out of |)oints 

 of the conic A": for these points lia\"e the properly of Ixmuu' the 

 f»oinls of intersection not only of two but of a \\ hole pencil of ra\s 

 of the conurueiice (2,2). situated in the focal planes l)elon,i>in,u' lo 

 each of the points. 'IMiese focal planes enxelop a (piadi'alic cone /'"-, 

 with the vertex /^ so llie conui-nence must touch the cone. From 

 this ensues the h)llow inii' construction : 



"Let a point .1 be taUeii on A', the focal ray /'A be di-awn, 

 ciittinu' A' for the second time in J'. Let the two lanuenlial planes 

 to /'• be brought throuLih /\\: each of these planes coidains a pencil 

 of |-ays of the con,i;i'uence, the xcrtex of one pencil Ixmuu' J, of the 

 Other' A'." 



7. \\'e ]iow' pi'oceed by ,iii\ inii' some \isible properties of the con- 

 U'l-iience (2,2). 



((. 'I'lie two foci of each ray ai'c the points of iidersection with 

 A'' and the point of contact wilh /'". The focal surface of poiids 

 becomes /*": the focal sui'face of laniiential plaJies consists of the 

 lanuenlial planes of K'. 



I). All rays of Ihe conu-rnence (2,2) bek)n^U'in,ü' h) a con,i>i-uencc of 

 rays (1,1) of A cut two conjugate polai-s of A, and cntlinu- at the 

 same time A' lliey form a ruled surface of order foui' with a, simple 

 conic and two doid»le lines. 



c. The rays of the conuiMience (2,2), lyinu' on a hypei'boloid of 



^) Cungruences of this type are ranged in the '"Index du répertoire bibliogra- 

 phique des sciences niatliémali(iues'" under X '~ 1 e x and placed by R. Sturm in 

 a separate division ; see "Liuiengeomelrie", II, p. 3:^3. 



