( 664 ) 



If A is taken on the line of intersection of two planes h, two 

 pencils are separated from the cone. Three pencils are obtained 

 when A is point of intersection of three principal planes. 



If we take A on the curve c" -- which a plane h has in common 

 with Q" then (A) consists of p concentric pencils and a cone of 

 order (?i— 2) (p-'J)— 1. 



If yl is a point of intersection of q" with two principal planes 

 the number of pencils evidently becomes (/) + l). 



4. The curve of the complex («) is of order (p — 1) (2n-|-2' — 6) '). 

 It possesses ^ {p — 1 ) {p — 2) (m — 2) threefold tangents ") which are 

 transversals of as many triplets of right lines belonging to a group 

 of Ip. The cone of the complex (^4) jiossessing evidently as many 

 threefold edges, the scrolls eacli having three conjugate right lines / 

 as directrices form together a congruence y of order and class 

 k{p — l){p — 1){n-2). 



Each principal point H is for this congruence a singular point oi 

 order [p — 2); the singular cone is broken up into [p — 2) pencils. 



Each principal plane /* is a singular plane of order {p — 2) and 

 contains (p — 2) pencils of rays of congruence. 



5. The right lines resting on four lines / belonging to a group 

 of ƒ/, form a scroll enclosed in r, of which the order is going to 

 be determined. 



Each transversal t of three conjugate right lines 1^,1,, Ig and the 

 arbitrary right line a intersects still (n — 3) generatrices m of 9". 

 To each of these right lines m can be made to correspond the (2> — 3) 

 right lines /' forming with l^, 4, 4 a group of I^j. 



To each ray /' belong {p — 1), triplets /,, 4, 4, so 2{p — l)^ trans- 

 versals t and therefore 2 (« — 3)(2>--l)3 rays m. 



Tiie congruence (1 , 1) of the i-ight lines resting on m and a has 

 with the congruence y in common (/i — 2) (2^ — Ij [p — 2) rays t, 

 so that to ni are conjugate (n — 2) yp — 1) (p — 2) {p — 3) right 

 lines r. 



Now each transversal of four lines / belonging to a group of If, 

 evidently gives four coincidences of the correspondence (/' , m). 



1) The characteristic numbers of the curve of involution of an Ip on a rational 

 c" are found in the dissertation of Joh. A. Vreeswijk Jr. (Involulies op rationale 

 krommen, Utrecht 1905, page 38). 



2) See also my paper "Complexes of rays in relation to a rational skew curve" 

 (These Proceedings, VI, page 12). 



