( 280 ) 



[f t/ro tiristeil curres luire 'm P three cunsecutlve poii/ts in common 

 this lüill be also the case with the plane curves forming part of the 

 sections of the common osculatint/ plane unth the developahles belonging 

 to the twisted curves. 



The mdins of ('iir\alui'e of llie .section (/ in Uie i)oiiit P being 

 lour tliii'ds ot' ilie radius of curvature of the cuspidal curve C in 

 this same point, tlie curves d and C have in P but two points in 

 common. 



From the theorem proxed here, follows once again Ihc theorem 

 communicated by me before, concerning the situation of the three 

 points which a twisted curve has in common with its osculating 

 plane, (see These Proc, Febr. 27^'', 19Ü4). 



§ 4. By expansion of the coordinates of an arbitrary algebraic or 

 transcendent twisted curve in the proximity of an ordinaiy point 

 L* into convergent power series of a parameter t, the theorem ot 

 § 1 can be proved also directly for such a twisted curve without 

 using the twisted cubic. 



Let P be an ordinary point of the curve C\ if the tangent, the 

 j)rincipal normal and the binomial in P are taken respectively as 

 A'-axis, y-axis and Z-axis, then the coordinates of the twisted curve 

 C become: 



X =z a^ t -f- a.^ ^'^ -|- . . . . , 



y = b.^ej^b,e + 



Z=C,f +C^t' -i- 



The point P corresponds to the value zero of the parameter t. 

 If P is an ordinary point the coeflicients a^, b^ and c., cannot be zero. 

 Let Ro be the radius of curvature of C in point J\ thus the value 

 obtained by the radius of curvaliu-e R for / ^ 0. The radius of 

 curvature in P of the projection of C' on the osculating plane : = 

 is also Ro, this projection having in P three consecutive points in 

 common with 6'. 



The coordinates of the points of this projection are : 



./ = />, f + ^>3«« + 



As — is equal to for t := the general formula for the radius 

 dt 



of curvature 



E = 



da; d^y — dy d^x 



