( -279 ) 



is the paraniclei-. So (lie radius of ciirxjitiire /■„ of (he eonie c/, in 



2 a^ Cj 

 the origin is ^r-,^-. 



So, 



From (he viUues 



IXr. = ;r-T and r„ = 



now follows : 



72„ :>■„=: 3 : 4. Q. E. D. 



§ 2. The theorem can be easily expanded to a iieneral t\vis(ed 

 cnrve C. 



Let P l)e an ordinar\ poijd of (_\ the (ajii>en( and the oscnlatino- 

 plane in P showing no pai-tieulai'ilies. 'Jlirongh poini I* and live 

 conseenti\e points of (' a twisted cnhic ('" can al\va\s he laid. 

 The radins of cni'vatnre /i, in (he [)oin( /' is the same for the enrves 

 6' and 6'"\ havijig six eonseeutive })()ints in eommon. The oscnlatiiif»- 

 planes of the enrves C and C^ in the |)oinl P will eoineide too. 

 This common osenlating i)lane < > intersects the developables belonging 

 to 6' and 6'"' according to the tangent in 1* counting donble and 

 moreover according to two plane cnrxes d and d.^. 



If the cnrves (' and ('■' had but a (liree-point contact in /-*, the 

 curves <l and (/., would have a commoji tangent in the common 

 point P, so that (he curves d and d, would have in /^ at least two 

 consecutive i)oints in connnon. If (he curves f' and ^ ' ' were to 

 have a live-point contact, a common generatrix of the two develop- 

 ables not lying in the common osculating plane (f would meet (he 

 osculating [)lane (f in a (bird common poiid of (he cnrxes ^/andf/j. 

 Now that the curves (' and T' ha\e a six-[)oijit contact in 1* the 

 curves d and (/,, \vill ha\ e a( leas( four consecn(i\e points in common. 

 These two sections d and d., have (hus in /-' (he same i-adius of 

 curvature y\,. Conse(piently in the ordinary j)oin( P of the twisted 

 curve 6' we have: 



R, : /•„ = 3:4. 



§ 3. When (wo arbi(rary twis(ed cnrves ha\e in a point /-* a 

 three-poinl coidact, they have in (lia( |)oiii( (lie same radius of cur- 

 vature //. If now (he common osculabng plane (> in /* of the (wo 

 curxes cu(s (lie (wo developables belonging to the cur\ es in (he j)lane 

 cni-ves d and d' (hen (he radii of cnr\a(ure in P of these sections 



4 

 d and d are bodi R and (lierefoi-e e(pial. The cnr\es d and d' 



o 



have (hus in P also a thi-oe-poiid eoidac(. Fi-om (his follows (he 

 theorem : 



19 



Proceedings Royal Acad. Amsterdam. Vol. VII. 



