( 278 ) 



The coordinates of the poiiit.s of the curve C^ satisfy tlie conditions : 

 tC = z, t' B=z, t' A = z, 



whence 



«4 ^1 ^' "4 t (^"2 — ^2 ^—^^3 ^'-2 t') 

 " = -1^' " = N • 



Now the radius of curvature R^, of the twisted curve C' in the 

 point P is the same as tlie radius of curvature of its orthogonal 

 projection on its osculating plane in F, the curve with its projection 

 in P having three consecutive points in common. The parameter 

 expressions for the coordinates of this projection are 



y ■=. — —_ — and ./; = 



N N 



dt 



From the value of y we find -r=0 for ^ = 0; so for the general 



formula 



E — 



d.c d^y — dy d^x 



giving the radius of curvature of a plane curxe, can be substituted 

 the simi)ler expression : 



i h^ c^ X «4 <^2 j ' 



The equation of the surface 0^ enveloped by the plane 



Af — 3Be -\-3 Ct — D=zO 

 is 



A' D' — QAB CD + 4:AC' + iB' D- 3 B' C' — 0. 



The curve of intersection with the osculating plane Z) := ^ = is : 



C' (4 .1 C — 3 B') — 0. 



So the equation of the conic d^ lying in the osculating plane is: 



4 (a^ .c + a, y + aj 0, ^ — 3 {h, .v -\- /^, y)' = 0. 



The equation of the parabola osculating this conic d.^ in the 

 origin is: 



4a,c,y-3b,-^.v' = 0. 



This parabola has in the origin the same radius of cur\ature r„ 

 as the conic d^. The radius of curvature in the vertex of the parabola 



