( 487 , 



of wliich it is ea»y to show that it consists of 3fi {?i — 2) lines; for, 

 throngii an arbitrary point of this section a prijicipal tangent of the 

 surface mnst pass resting on the nuiltiplis line, tiierefore lying entirel}' 

 in the phine. Tiie 3/^ (/^^2) lines are evidently tlie inflectional tangents 

 of the section of the plane nnder consideration whh the surface of 

 order n. 



An ordinary point of contact of a generatrix of £2 with <I* is a 

 single point of the curve of contact c' (§3); in each ot' ihcndi^— 4) 

 points /-* just now found, however, the generatrix A^P has with 

 a three point contact, with 77, a two point one, and therefoi-e 

 also with c'^ a two point one; so there are n{n^ — ^) generatrices of 

 i2 touching 6'*'\ 



§ 5. A generatrix of i^ touches 0, and lias thus, besides the 

 point of contact, still {n — 2) points in common with this surface; in a 

 plane A^r^ lie therefore n {n — V) in — 2) such points, namely on each 

 of the n{n — 1) generatrices in this plane every time n — 2. All these 

 points lie on a curve of order {n — 1) {n — 2), the satellite curve of 

 the first polar curve Pi"~"^ of A^ with respect to h\ If the plane 

 revolves around r^ , the satellite curve will generate a surface which 

 we shall call "the satellite surface" of r^ with respect to andr2, 

 and which will evidently cut out of 'T» the residual intersection of 

 52 with ^. 



The intersection of the satellite surface ^^ with a plane A^r^ consists 

 of a satellite curve s^^ of order (n — i) ()i — 2), and of the line r, ; 

 the question is how many different satellite curves pass through an 

 arbitrary point of ?•, . In order to answer this question we shall 

 consider again in particular a poijit of intersection S^ of r^ and <!>. 

 If the curve .s\ lying in a plane A^r^ is to pass through S^ , then 

 AiS^ must be a tangent to «T» without the point of contact coinciding 

 with S2 . Now the plane r^S, cuts <P in a curve of order n 

 containing the point S^ itself and to which ?i (n — I) — 2 tangents 

 can be drawn out of <S', , not touching in *S', itself; in the planes 

 through these tangents and r^ the curves ^j will pass through S, . 

 So lüe find for the satellite surface 2^ a surface of order 

 {n—\) (?i— 2) + n {n—l) — 2 = 2n (n— 2), ivith an [n (n—1 — 2\-fold 

 line r^. The satellite curve of c""^, the intersection of *P and 2^, is 

 thus a curve of order 2 n"" [n — 2), inith |;i(/z— 1) — 1]-f old points in 

 the n points of intersection S^ of <P arid r, . 



Now however it is clear, that just as there is only one curve of 

 contact c'^ immaterial whether we start from the polar surface of 

 7'i or of r, , there is also only one satellite curve; for the curve of 



33* 



