( 362 ) 



For p = — 1 these are twisted ciibics. If we bring these through 

 a point C''i,y],^i) the oo^ curves all lie on the cone of the complex 

 of this point. This holding for each point, the bisecants (and not 

 only the tangents) are rays of the complex. 



Indeed, all the twisted cubics pass through the vertices of our 

 tetrahedron and the four planes passing through a bisecant and these 

 four points have thus a constant auharmonic ratio. From this ensues 

 that the bisecants intersect the four planes of coordinates in the same 

 auharmonic ratio. 



For j^ = 1 we have the rays of the complex themselves. 



For p ^ 2 we have conies which can be nothing but conies of 

 the complex, e.g. for s ^= ^^ I the curve touches the plane YOZ, etc. 



For J) = 3 we have twisted cubics whose bisecants are not rays 

 of the complex, etc. 



In general tlie tangents to the "curves of a complex" lie always 

 in linear congruences belonging to the tetrahedral complex. For such 

 a tangent namely we have 



d,v dii dz 



{I ^ s)- = {m -\- s)-^ ={n ^ .^-. 

 .V y z 



From this ensues among others : 



dz {I -\- s) dx -\- k{m -\- s) dx 

 {n -\- s) — = . {k an arbitrary constant.) 



z X + ky 



This is evidently always satisfied by rays of the complex, satisfying 

 at the same time : 



xdz — zdx ■=. k (zdy — yds) and kdy = — Rdx, 

 for which we can write in coordinates of lines : 

 Ih = k\ e.Ji — kp, = Rp,. 

 These satisfy the equations of the tetrahedral complex and lie in 

 congruences ; the two linear complexes determining such a congruence, 

 are themselves special, and the position of their axes is evident from 

 their equation. 



§ 7. Finally it proves to be simple to bring in equation the 

 curves which are drawn on an arbitrary surface in such a way that 

 the cone of the complex touches the surface in each'point of the curve. 



Let the surface be f{x,y,z) = and the ray of the complex 



^ =~ — ^ --Z L^ passing through the point x^,y^,z^ of the 



« /? y 



surface. 



A ray of the complex in the tangential plane must satisfy 



