( G25 ) 



§ 5. Tt follows iVom wluxt was proved above for a twisled cuhic 

 that also a general tAvisted curve C' has not more than two points i]i 

 common with the section d of its developal)le O and the osculating 

 plane V. If C and (/ had three points in common, then d and 

 the projection p of C' out of any arbitrary point A on the i)lan(^ I' 

 of (I would have three points in common. Let T', be a twisted cubic 

 having in /-* six successive points in common with (\ Now I^ is 

 also the osculating plane of C.^ in I* , let <l' be the trace of the 

 deveIoi)able ()^ belonging to (3 and let y/ be the j)rqjection of Tg on 

 V out of .4. As the developables (J and O^ have five successive 

 generating lines in common, so d and d' have at least three succes- 

 sixe jtoints in common, whilst p aiid p \\i\.YQ, six successive points 

 in common. Now, if (/ and p had three successiv^e points -n common 

 at P, this would also be the case with d' and p' . According to the 

 preceiling § the latter is not true, so (' and d Iuiac neither three 

 points in common. 



§ 6. The theorem can also be |)roved by searching for the [)()ints 

 of intersection of the cuspidal curve r or of the nodal curve ^ with 

 a second polar surface AV>, just as Crkimona did (Crkmona — Curtzk; 

 p. 87 — 90). To simplify the matter I shall tirst apply this proof to 

 a twisted cubic, after which the |»roof for the general case can be 

 more easily followed. 



Let us take for deveh)pable (/ the surface consisting of a develo|»- 

 able (>4 and of a quadratic cone A' with vertex T, passing thi'ough 

 the conic d.^ situated i]i the osculating plane V of P and on ()^. 

 The cuspidal curve of this surface of order six is tiie cuspidal curve 

 63 of ()^^ so r = 3. The nodal curve ii consists of d.^, and of a 

 curve of order six .y. This cur\e s intersects the plane T" of the 

 conic d^ six times, thi'ee times of which in the points where d.^ 

 meets a generating line of (J^, for Avhich the tangent plane [)asses 

 through T, these three points being points of contact of double 

 tangent planes of (/. Consequenth' .v has in P three successive 

 points in common with J.^. Let A'' (/'be the second polar surface 

 for a point A lying in V. The order of Z\V/ is four. 



In the formula of Cremona for § ((> — 2) I have oidy to kee[) the 

 llrst and the third term, none of the singularities appearing on the 

 surface under consitleration which are furnishing the other terms; 

 but a term P must be added for the particular [)oint P where 

 another third sheet (jf the surface O' passes through a i)lane curve 

 d., lying on 0^. This singularity has not been considered by Crkmoxa j 



