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XII. 



SOME THEOEEMS ON THE TWISTED CUBIC. 



By MATTHEW J. CONEAN, M.A, 



University College, Cork. 



Read Febiiuary 8. Ordered for Publication March 24. Published June 14, 1909. 



Inteoduction. 



1. This paper is the outcome of an attempt to find some metrical properties 

 of the twisted cubic. 



It is shown that the three diameters of the cubical hyperbola are the 

 medians of the triangle formed by the " points " in the " plane of centres." 



Moreover, the common point of intersection of the diameters is the centre 

 of the " locus of centres " of conic sections of the developable, and is also the 

 middle point of the chord joining the " points " (real in case of the cubical 

 ellipse), the osculating planes at which are parallel. 



These points are referred to in the paper as the points vji, w^ ; and the 

 centre of the " locus of centres " is referred to as the point 0. 



2. A theory of correspondence and a geometrical construction for corre- 

 sponding points are also given. 



3. Finally, the analytical forms for these theorems are stated for the 

 general equation of the cubic. 



I. 



4. It is convenient to state here three known theorems of which con- 

 siderable use is made in this paper. 



Theorem I. — " If a' be the line of intersection of the osculating plane at a, 

 with the plane through a touching the curve at j3, and if V be the line of 

 intersection of the osculating plane at /3, with the plane through j3 touching 



K, I. A. PROC, VOL. XXVII,, SECT. A. [36] 



