( 387 ) 



.r l)('iii,U' all lioiiiolouoiis rav of holli pencils. The ra\s of ,/ are tlit- 

 lraiis\-('rsals of Iwo lioiiiolo.iioiis ravs of (.V^') and (A'^;). 



\a'\ lis now take Iwo slicaN'cs ol' ravs in llic space ^l', with llic 

 vertices A, and A', and estaldisli a pr(»Jecli\ e c(irre>p()iideiice helwcen 

 these sheaves and ihe poinlliehls ^ and >;', in such a wa\ thai Hie 

 pencil of' planes Ihronuh ihe axis A, A', is lioinolo^ons lo llie |)encils 

 (A'^') and (A'^). Lei / l>e a rav of J, cntlin;Li' Iwo honKiloiioiis ra\s 

 of* (A^') and (A'^j, lo which in Ihe hoinoloLioiis plane /, a ra\ /, 

 (Mil of A, and a rav /', oiil of A', correspond; /, and /', inlersecl 

 each oilier in a point /.^. 'I'liis point is hoinolo,i»oiis to the nw I. 

 S(> a ]H'()jocti\(' correspondence is established helweeji the points of 

 the space 2i,\ and the ravs of the focal svstein ./. 



As is the case with every representation, also here the l<nowle<l,ii-e 

 of its j)riiici|Kd curve cannot he dispensed with, it is a conic A','^ 

 ihroiiuh the points A\ and X\ situated in a jdane 5;^. its |)oiiils are 

 hoinoloii'ous to the pencils of rays of' ./ situated in planes tliroii;j;li ./'. 

 TlïG plane ^^ (princi|)al plane) itself is hoinolo_i>()iis to ,/'. 



To an arhitrary pencil of rays of ;/ a riiilit line corresponds ciitlin;^- 

 Ai% to ;i hyperl)oloidic system of focal rays a conic lia\ ini;- two 

 points in common with X^'\ to a linear congruence l)elon,i;in,i»- to J a 

 (piadratic snrtace throiigii X^\ 



4. TiCt a pi'ojectively variahh^ iiio\ijiii,' spacial system he <x\\v\\ ; 

 let as l)efore PQliS he the tetrahedron of coincidence of two suc- 

 cessive positions and let the corres|)ondin,ii' focal system ./ he deter- 

 mijied hy /V^ and /AS' as conjugale polars and the conic A ' toucliiiiLi- 

 /Vi and /N in Ii and S. Aceordini;- to the indicated method the 

 focal system can he repres<^nted in lli(> s|)ace -^\ : lor the tetrahedral 

 complex of the directions ol' the \(docilies, howcwer, we need an- 

 other re|)resentation, which can he taken in such a way that the 

 same |)rincipal cnrxe is retained; we shall succeed in this if wc do 

 not re|)reseiit the complex itself, iait its section with tli(> focal system . /. 

 Tills gives I'ise to a congruence (2,2) which we shall lirsl iii\estigate 

 more closely. 



5. TiCt A he an arhitrary point, ft its focal plane; at tlu^ same 

 time .1 is the ^•ertex of a (piadratic cone, geometrical locus of the 

 directions of the \(docities through J, hut of which only one is the 

 direclion ol' xclocily of A itself. 'Hiis cone will cut in genera Uf into 

 two rays helonging to the congruence (2,2); in this way we can 

 construct the whole congruence. \\\ this we have determined the 

 construction, hut not the ge(MnetricaJ character ol' tiic congriK.'iice ; 

 tliis can he done in the follow ini;- manner; 



