AN APPLICATION OF LIE'S THEOREM ON THE TORSION OF A 



COMPLEX CURVE 



By Charles T. Sullivan, Ph.D., D.Sc; F.R.S.C. 



Read May Meeting, 1920. 



Introduction 



A curve is said to belong to a linear complex or, more briefly, to be a 

 complex curve, if all of its tangents are lines of a linear complex. All 

 com.plex curves through a point have a common osculating plane at the 

 point, viz., the polar plane of the point for the complex. It has been 

 shown by Lie that the torsion of a complex curve at a point is a function 

 of the constants of the complex and the co-ordinates of the point only; 

 and, therefore, the same for each complex curve through the point.*. 

 In the succeeding pages it is proposed to apply this theorem on the torsion 

 of a complex curve to show that: If one family of the parametric curves 

 on a surface determined by Lelieuvre's Integrals are complex curves, it is 

 necessary and sufficient that the Lelieuvre Integrals be constructed from 

 a Moutard equation of range three at most. 



We shall first develop certain formulae required in this discussion. If 

 di, 02, G3 be three linearly independent solutions of the Moutard equa- 

 tion 



(A) W^=^^P'^''' 



then the paramictric curves on the surface determined by the Lelieuvre 

 Integrals 



^-{(^■^)^^-(*-^>' 



are asymptotic curves on the surface (Eisejihardt — Differential Geo- 

 metry, p. 194). 



* Proceedings of the Society of Science at Christiania (1883). 



63 



