( 489 ) 



double generatrices, i.e. double tangents of <P, cutting i\ and i\) 

 we determine their number by determining the order of the scroll 

 formed by all the double tangents of <f» which intersect i\. A plane 

 through i\ cuts in a k' and this possesses ^ 7i{7i — 2) (7^* — 9) double 

 tangents, and through an arbitrary point of r, pass 4 Mfi — ^){n — 2){n~'S) 

 double tangents;^) the surface to be found is therefore of order 

 è 7^"— 2) Or— 9) + }, n {n—l) {n—2) {n—3) = (/i+l) (?i) («—2) {ii~'S), 

 and it has }\ as an h 7i{n—i) {n— 2) {n — 3)-fold line. The number of 

 double generatrices of il is equal to the number of poi?its of inter- 

 section of this surface ivith r.^, so equal to {:n-\-l) (n) {n — 2) (/i — 3). 



With the aid of the points of contact of the double generatrices 

 with <P, likewise of the n {u^ - 4) points found in § 4 on principal 

 tangents of *P, we can now entirely survey the mutual j)Osition of 

 the four surfaces 52, fp, TJ^, ^^, likewise of their intersections. We 

 fix our attention in particular on the curve of contact c"'' and the 

 corresponding satellite curve. According to § 4 there are n{n'* —4) 

 generatrices of ii touching c "; if F is one of the points of contact, 

 A^ the point of intersection with ;\, then P is an inflectional point 

 for the section /.' with <P lying in the plane At^i\, A^^P the corre- 

 sponding inflectional tangent, and it counts for two of the n{n — 1) 

 tangents which can be drawn out of A^ to k\ so that besides the 

 inflectional tangent only n[;n — l) — 2 tangents pass through A. 

 Each of these intersects k' in n — 2 points, altogether thus in 

 \n{n — 1) — 2 I (/i — 2), whilst the com})lele number of i)oints of 

 intersection of the satellite curve of yVj" " ^ with k^ amounts to 

 n{ii — l){n — 2); the missing 2 [ii — 2) must thus be furnished by 

 the inflectional tangent. Now it is easy to see, that by a slight 

 change of position of /I, the inflectional tangent would break up 

 into two se[)arale tangents; by attending in this position to the 

 satellite curve and then b)- returning to the inflectional tangejit we 

 convince ourselves that the satellite curve of 7>i"~' touches k' in 

 the n — 3 points of intersection of the inflectional tangent. 



Now but two points are missing and these can lie nowhere else 

 but in P ; so the satellite curve of /^i""' touches in P the curved", 

 Now this satellite curve lies on the satellite surface 2J,, which inter- 

 sects 4> according to the satellite curve of c"*; so this one too must 

 touch in P the line A^^P, just as c"% so that Xhonin" — 4) points 7-* 

 mentioned above represent 2 n {n"^ — 4) points of intersection of c'*^ 

 with its satellite curve. 



Let us further consider one of the {u-\-'\.) {n) [n — 2) (n — 3) double 

 generatiices of 52 with the points of contact P^, P.^, and the point 



1) Cremona — Guutze, 1. c. p. 64. Salmon — Fiedler, 1. c. p. 25. 



