189 



A Note on the Intersection op Osculating Planes 

 TO the Twisted Cubic Curve. 



By a. M. Kexyox. 



On page lo of his Differential Geometrj' Eisenhart proposes an exercise 

 to show that on any plane there is one and only one line through which can 

 be drawn two osculating planes to the twisted cubic. 



§1. 

 The twisted cubic 



ao + a, t + a2 t= + as t'' 



y = 



d„ + di t + d. f^ + d3 1^ 



ao ai a2 as 



bo bi b2 hi 



Co Cl C2 Cs 



do di d2 ds 



o (since the curve is twisted) 



is carried over by the nonsingular linear transformation: 

 aix' +a2y' + asz' + ao 



d,x' + dsy' + dsz' + do 



into the cubic 



x' = t y' = t- z' = t^; 



planes, straight linos, and points, go over into planes, straight lines, and 

 points, respectively, and in particular, osculating planes go into osculating 

 planes. 



The ec^uation of the osculating plane to the cubic 

 X = t y = t= z = t^ 



at the point whose parameter value is t, is 

 3t-x — Sty + z — t^ = 



