( 486 ) 



is shown that these two surfaces have in each point of r, the same 

 tangential plane. 



The equation of ^ can not only be written in the foi'm -2'j:"i -—=iO, 

 but also in the symbolic form -S'.r, r — = 0. Let us put in it 



/ OXi \ 



x^ = i}\ = in order to determine the n points of intersection *S, 

 with i\, then exactly' the equation K*=^Q remains, from which 

 follows that the n points S^ lie at the same time on K* and therefore 

 jalso on K^ ; it is even easy to show that each of these points counts 

 double among the number of points of intersection of the three 

 surfaces 0, TJ^ , K*. In a plane S-^i\ lie namely, as intersection with 

 fP, a curve k\ as intersection with /Zj the first polar curve of these, 

 ^^/'-^ and these curves touch each other in aS'i. Now however the 

 curve 5'i"~' is again the first polar curve of S^ with respect to the 

 curve of order n, consisting of pi"~' and i\; so <7,"~' touches in S^ 

 the two other curves. The tangential planes in S^ to the three mentioned 

 suifaces intersect each other according to the same line, namely 

 tlie torsal line of i2 through S^ (§ 2); each of these points counts 

 thus indeed for two points of intersection of the three surfaces. Now 

 outside i\ (see above) lie n{7i^ — 2) of these points; if moreover we 

 subtract still the 27? points S^ then n (?i' — 4) points remain, lying 

 neither on i\ nor on i\. If we suppose a plane through such a 

 point P and i\, which is intersected in A^ with i\, then the curves 

 ^•», ƒ)/'—', ^/'—i lying in this plane (and therefore also the second 

 polar curve /)i"~- of A^) all pass through P, from which ensues that 

 P is for I'' an inflectional point and therefore A^P one of the two 

 pi-iucipal tangents (osculating tangents) of <P in ]\ With this we 

 have shown, that in the cotic/ruence of the principal tangents of the 

 cjeneral surface of the n''' order n{n'' — 4) of these lines rest on two 

 arbitrary lines, or in other words, that the principal tangents intersecting 

 an arbitrary line form a scroll of order n {n^ — 4). 



Through an arbitrary point of space pass n [n — 1) {n — 2) of those 

 lines ^) ; for we have but to take the points of intersection of the 

 surface itself with the first and the second polar surface of the chosen 

 point ; the surface just found has thus the right line 07i which all 

 generatrices rest, as an n {n — 1) {n — 2)-fold line. 



A plane through this line contains, besides the 7i{n — l){n — 2)- fold line, 

 a curve of intersection of order n {n- — 4) — ?i {n — 1} {n — 2) = 'èn {n—2), 



1) Cremona— GuRTZE : "Grundziige einer allgemeinen Theorie der Oberflachen", 

 p. 64, or Salmo^^ — Fiedler: "Anal. Geom. des Raumes", II. Theil, S. 24. 



