Lie's Theorem of the Torsion of a Complex Curve 71 



= oxi — avi . 



dp 



Whence 



(22) $1 = cy-bz+ a, 



62 = az — ex + /5, 



^3 = bx — ay -i-y, 



where a, /3, 7 are arbitrary functions of g. But these equations show 

 that the parametric curve 2 = const, belongs to the complex. 



Sa (y 8z) -]-Sadx = 0. 



Let us next consider the tangents to the curves ^ = const, along a curve 



g = const. If for 8x, 8y, dz we substitute — ^r-^ X2, — = 72 , — = 3'2 , 



VG VG VG 



we see that these tangents belong to the complex of (11) corresponding 



to the value of g considered. In a similar manner we find that the 



tangents to the curves p = const, along the curve g_-\- 8q belong to the 



complex 



{Sa(y8z-z8y)-\-Sa8x}-\-8q{Sa'(y8z-z8y)+Sa'8x}=0, 



where the primes indicate differentiation with respect to g. Thus the 

 tangents to the curves p = const, are all contained in the complex obtained 

 by eliminating q between the equations 



Sa (ydz—z8y)-}-Sa8x = 0, 

 Sa'(y8z-z8y) + Sa'8x = 0, 



that is, the enveloping complex of the family defined by (11). This result 

 was given by Lie in reference cited above. 



Finally we consider a surface (S) for which both families of para- 

 metric curves are complex curves. It follows from the results established 

 above that the solutions 61, 62, dz of (A) must satisfy the two relations. 



(23) a (g) di-\-h {-g) 02+c (g) ^3=1, 

 a{p)di-\-lb(p)d2-{-c(p)dz=l. 



The two families of complexes are then defined by the equations 



(24) Sa(y8z-z8y) + Sa8x=0, 

 Sa(y8z—zSy)-{-Sa8x = 0. 



Following the same line of reasoning as that employed to establish 

 equations (17), we establish the two sets of equations 



