Lie's Theorem of the Torsion of a Complex Curve 



65 



From the realtions (2) and (3), we derive: 



( 



= s 



''t) (^^^) 



This expression reduces to 



(5) Tr=^e^+e.^+e,')(^e,^^y, 



by virtue of formula (4). 



"" V ~dp da J ' °y ^^ elementary property of determinants. 



In this case, the quantity T^ = LN—M^ reduces to — M^; and the 

 Gauss measure of curvature K has the value 



(7) 



^ ( F ) (Se^j^ 



In the customary notation, the direction cosines of the normal to 

 (5) at the point P (x, y, z) are: 



X, Y,Z 



xi yi 2i 

 X2 yi zi 



On using the relations (1) and (5), these become 



(8) 



X,Y,Z= -^^= , -^^ 



Vsr 



If now pg, cTg denote the radius of curvature and the radius of 

 torsion respectively at the point P(.r, y, z) on the curve g = const., then 



Pa 



= Zf X 



1 



Pa" Cg 



(X y z ), 



—5 



