Lie's Theoeem of the Torsion of a Complex Curve 67 

 and, therefore, 



Hence along the curve g = const., 



1 1 



= f^Q2) ^ + V—K, by virtue of equation (7), 



if we choose the positive value of the root. With this convention as 

 to signs, we find in a precisely similar manner that 



1 



^--^-^^' 



along the curve ^ = const. 

 Thus 



11 11 



7 = r = — K^ and = — , 



Which is Enneper's theorem on the torsion of the asymptotic curves. 



We next consider the one parameter family of linear complexes 

 defined by the equation 



(11) Sa(ybz-zby)-\-S(ibx = 0, 



where 5 signifies the cyclic sum over the three axes and a, b, c; a, (3, y 

 are functions of one parameter q. We may, without loss of generality, 

 assume the realtion 



(12) aa + blS-hcy = 1. 



Consider now a point P {x,y, z) on the surface (S). Corresponding to 

 this point P (p, g) on (S) there is a definite complex of the family (11); 

 let X, Y, Z be the direction cosines of the polar plane of P in this complex. 

 Then from the theory of the linear complex, it follows that 



(13) VTX = a-\-bz-cy, 



y/ k Y = ^ -\- ex — as , 



\/kZ = y-\-ay — bx, 



where V ife is a factor of proportionality. By squaring and adding these 

 equations, we find 



(14) k = S ia-\-bz-cyy* 



But the expression on the right hand side of this equation is that found 

 by Lie for the torsion of a complex curve at the point P (x, y, z). Thus 



*Lie and Sheffers' Beriihrungs transformationen, p. 233. 



