311 



Let B* be a point of ii* , ii* the raj which the congruence (1,/;) 

 sends through that point. Any plane passing through u* contains 

 two straight lines /', which intei'sect in /i* ; B* is consequently a 

 double null-point. 



The surface (/'')/" + •* has consequently ^-i* as nodal curve ; \i fnvüiev 

 cointains the curve oi', the ('S p -\- \) singular straight lines .s- aiul 

 passes /> times through the singular sti-aight line a. 



The ruled surface (?;*) is of order 4/;, the ruled surface {n*} of 

 order (3/; -[-3). wddle the straiglit lines t'*, as bisecants of i^\ form 

 a ruled surface of the fourth order. 



If the congruence (!,/>) is also replaced by the eongi-uence (1,3) 

 of the bisecants of a curve 6r-, a null-system 9^(1,9, fi) arises. The 

 surface (Py has u^ and li' as nodal curves and contains 10 singular 

 straight lines s; (P)' and (Q)' have moreover a curve {iy^" in com- 

 mon. The ruled surfaces (u*) and (?;*) are of order 12. 



6. For }> = i, </ = 1 we have a bilinear null-system ^t (1,1,2), 

 in which the rays u rest on two straight lines a, a', the rays v on 

 two straight lines b,b'. 



The singular figure consists then of the straight lines n,a',b,b' 

 and their two transversals .v, .v'. For each singular point the null- 

 planes form a pencil ; the axes of those pencils form four quadratic 

 systems of generatrices. The surface (/'*)' has a triple tangent 

 plane ^) in the null-plane of P. 



') Cf. my paper "Un bilinear nuU-systemb" (These Proceedings, vol. XV, p. 1160). 



