45 



with Si* the 10 edges oonliiiiiing llie points of intersection of a^ and 

 Q^ ; the remaining (5 common edges (/ are singidar bisecants. For q. 

 is chord of the cm've </ passing tliroiigh P, and moreover of a q' 

 intersecting it on <'>\ l)iit in that case it mnst l)e ciioid of oc' 

 cnrves <j^ 'I'he surface f/''\ which may be laid tlu'ongh (/, o^ and (j^ 

 does lielong to tiie net ; (lie oilier surfaces of this net consequently 

 intersect this net in the pairs of a (piadralic in\ulution ; in oilier 

 words, q is a singidar bisecant. 



The six straight lines q lie apparently on //* ; this surface also 

 contains the five straight Vmes f\=: PF/c, which, as the above men- 

 tioned straight lines />, are ixirticular ^parabolic) sim/ular bisecants ; 

 through each point /' passes a ()^ which has its second point of 

 support in F, so that the involution of the points of support is 

 parabolic. The section of //" and S.* apparently consists of a^', 

 two straight lines t (which are nodal lines for both surfaces) five 

 straight lines ƒ and six straight lines y. ' . . 



For a point .S of the singular curve «Mhe surface /7' consists of two 

 parts : the inoiioid ^' and a ruf>ic cone formed by the singular bisecants 

 (/, which intersect o' in N. As a plane contains four points jS, 

 consequently 4x3 straight lines q, the singular bisecants form a 

 congruence of rays (fi, 12), belonging to the complex of secants 

 of ö^ which congruence of rays possesses in a* a singular curve of 

 order three. 



5. The singular trisecants / form, as has been próveóy a congru- 

 ence of rays of onlfi' tim. The latter has the five fundamental 

 points F as siiii/itinr jioiiils, tor e.ich of those points beai'S oc' 

 straight lines /, w Inch forui a cone -i. Willi the cone 5\ which 

 projects an arbiirary o' oul of /', .i has ihe four sti'aigiit lines to' 

 tlie remaining points in common and further the two straight lines' 

 t, passing through /•'. As these straight lines are nodal edges of 5% 

 Ï must be a quadric cone. The congruence [/] has therefore //t-ö 

 singular ijuiiits of ordei' tiini. 



The trisecants t of an elliptic jj" form '; a ruled surface ^v', with 

 nodal curve 9". The axial ruled surface 31 formed by the straig-ht 

 lines t which intersect a given straight line a, has in common 

 with an arbitrary p' in the tirst place 5x3 points, in which ^' is 

 intersected by the fi\e straight lines t resting on a. Moreovei' tliey 

 have in common the five points F, which, however, are nodes of 31. 

 Consequently 31 is a ruled surface of order five. As a is nodal line 



1) Vid. e.g. my paper in volume 11 (p. 374) of these Proceedings. 



