( 389 ) 



y1, |)as.-( lliroiiLiii the poiiils (tf iiilcrscctioii of llic lallcr with A'"', so 

 tlicy ni'c loiii' in uiiiiilK'r. 



(/. Lel A'' l)(' i"(>al and lel /' lic ^itiialcd williin A'"-': all local 

 ravs liii'onuli /\ llic focal poiiil ol' plane /'/('S, now cut A'-; so all 

 pencils of ravs are I'cal. It" /' lies ontside A' two laiiLicnls onl of 

 /' can l)e drawn to A''; these lajiji'cnls are the lines ol' intersection 

 ol" the (.-one /"*■ with [)lane /*/iS. 'Vhr planes toiichiiiLi /'■ accordiiiji; 

 to llicsc lines of interscTlion are focal planes, in which two |)eiK'ils 

 of ravs have eoincide(l; ravs thronuh /'. nol ciillin,u- A', uive rise to 

 inia,U'inarv pencils of ravs of the conu'riience (2,2). Fni-tliei' eiisnes 

 from this: 



■'If A'' is real and all the \t-rtices of the tetraliedi-on of coinci- 

 dence likewise ai'e real, the con,u-iMieiice (2.2) is hnilt up of real 

 and iinaiiinarv pencils of ravs, \\here as a Iransilioii t wo are doulile 

 ones; if A''' is i-eal hnl the \-erlices /' ajid >S' are iniauinarv. all the 

 pencils are real." 



r. 'J'lie cases in which A' is iinaiiijiarv, or also those in \\ liich 

 all the Nci'tices of the tetrahedron of coincidence are iinau'inarv, do jiot 

 li'ive real conu'rnences; so tlie>' are not nnder consideration. 



!S. AVe no\\ pass to the re[)i'esentation of the coji,u-rnence (2,2) 

 Itv ^^ hieh the iina,ue is obtained of the eonnerlion of foeal svslen» 

 and tetrahedi'al complex. 



tf. The eonu'i'nence containinu- x pejicils of rays which are 

 represented in -^\ hy siraiuht lines havin_i>' a |>oint in eonmion with 

 A'i% the whole cojiLirnence is i-epreseii(ed hy a rnled snrface passinu' 

 tliron^-h A'/'- 'I'o :i sti-aiuht line /^ in 2i, a hyperholoidic .system of 

 focal I'ays coi-responds, \\ Inch has fonr points in connnoii w itli A ' : 

 so it contains fonr rays of the conurneiice and the representing' snr- 

 lace /\^ of the congruence (2,2) is a rnled surface of order fonr. 



h. An arhitrai-y pencil of focal rays of ./ contains two ra\s of 

 the conurnence ; the strai.uht line in 2:i\ c(U'respoiidinu' to them cnttinL!,- 

 A,''' has another two points in common with .\ ' : so A," isad(ud»le 

 conic of .S'l'. 



c. To the pencil of I'ays in J^" with /' as \-ei'te\ and /'/t'S as 

 plane a strai.uht line ƒ>, in -Il\ corresponds. cutliiiLi' Ai'. I'lacli i-a\ of 

 the pencil /* /'/IS heloiiLiiiiLi- to two pencils of rays whose \-ei-lices are 

 points of intersection with A', in all points of y/, two u'l'iierators 

 (»l S.' com-ur; from this follows that .S', ' is a iided surface havin.U' 

 as douoie curve a conic with a straight line cnttinu' it: with this the 

 type of .S'l ' has lieeii estahlished. 



2U 



I'rucet'diiius lloval A<'ad. Auislerdaiii. Vol. V, 



