1188 



ratois of' two cones, wliicli liavc as curves of' direction the (>'' passing 

 tlironi^li /', and tlie singidar curve o\ These cones pass throngli the 

 10 intersections of o^ and o^ ; of the 15 common generators 5 are 

 lying in lines q. As a plane contains 5 points S, consequently 10 

 lines ([, the singular hisecanU of the second kind form a congruence 

 (5, 10), which has o'" as singular curve of the second order. 



The cubic cone l^ wiiich projects a o^ out of one of its points 

 P, has a nodal line in the trisecant u of (>\ which trisecant passes 

 through P. The latter is at the same time nodal line of the sur- 

 face n\ To the section of W and l-^ belongs in the first place the 

 curve (>' ; further the singular bisecants hy^Ji^, which connect ^^ and 

 H^ with P, while u represents four common lines; the rest of the 

 section consists of the 5 lines q, whicti meet in P. 



As u with (y and o' determines a *?>^ it is cut by the net [0'] 

 in the triplets of an involution P. and is therefore singular trisecant 

 of the congruence (common bisecant of x^ curves (/). 



5. Let us now consider the qitadru/jle im^olution {Q*) in & pl^ne q , 

 which is determined by the congruence [q*]- It has Jive singular 

 points of the third order in the five intersections Si: of the singular 

 curve Q". The monoid 2^/^ cuts <f along tlie nodal curve o^i^, the 

 points of which are arranged in the triplets of an P, which form 

 witii S;^ quadruples of {Q^); <i/.: also contains the remaining points .S'(§ 3). 



If the point Q describes a line /, the remaining three points Q' 

 of its (piadruple describe a curve a, wliicli passes three times through 

 each of tlie points S/^. The curves / and /* belonging to / and /*, 

 have, besides tlie 45 intersections lying in the points Sl-, the three points 

 in common, which form a (piadruple with i^ ; moreover as many 

 pairs of points as the order of a indicates. For, if /* is cut by / in 

 L*, then / contains a point L of the quadruple determined by L*, 

 and the remaining two points belonging to it ai'e intersections of ). 

 and A*. The order .c of those curves is consequently found from 

 .t' = 2,1' -f 48 ; hence x = 8. 



The coincidences of the P on the singular curve (J^'- are at the 

 same time coincidences of the {Q^). Each point *S produces two 

 coincidences, the locus y of the coincidences has therefore in S 6 

 points in common with o^' ; further two in each of the remaining 4 

 points >b' and 4 in the coincidences of the P. From this it ensues, 

 that the curve of coincidences y is of order six. 



{Q*) consists of the quadruples of base-points of the pencils of 

 cubic curves belonging to a net with the fixed base-points Sk- Each 

 point of y^ is node of a curve belonging to the net. 



