1375 
C(p.q, 8) are asymptotic. curves, but also for any ruled surface 
generated by these curves. - 
Proof: Let the ruled surface be represented by the equations 
x= (a + av) t? | 
y = (6 + Pv) 49 fart eae cee eae 
z= (c + yv)és | 
Let P,, P,, P,, P, be the four points of intersection of the four 
curves C(p, q,s) cerresponding to the four parameter values v,, v,, 
Vs, V‚, With the generatrix corresponding to the parameter ¢,. The 
cross ratio of these four points is equal to that of the four projec- 
tions of these points on the axis OX and in its turn this cross ratio 
is equal to that of the four points of OX for which the x coordinate 
has the values 
Wid AT 5 @-F adv, 5, aa- dU 
These four coordinates being independent of ¢., the cross ratio of 
the last group of four points does not vary with ¢,. So the cross 
ratio of the four points P,, P,, P,, P, is independent-of f,, i.e. this 
cross ratio is the same for any group of-four points determined by 
the four curves C'(p,q,s) corresponding to the parameter values 
Vis Vis Vas V, On any generatrix. 
Example. The curves of the system C(1, 2,3) intersecting a given 
right line lie on a ruled surface of order four, for which one of 
the twisted ecubies Cl, 2,3) is double curve (nodal curve, isolated 
curve or cuspidal curve). According to the theorem just proved any 
definite group of four curves of the system C (1, 2,3) cuts all the 
generatrices in four points with a constant cross ratio. 
§ 12. In the case of a rectangular system of coordinates we 
easily find for the first differential coefficient of the length of are o 
in the point P(w, y, z) of the curve Cp (p, q,s) corresponding to the 
parameter value ¢ the expression 
do 
ENE 
Le 
pa" EE gy” 4 sz? | 
Let A6 be the angle between the binormals of the curve C'(p‚, q, 5) 
in the points corresponding to the values ¢ and ¢+ Af; then we 
easily find : 
dO oe PQS § pix? + g?y? + 87272 
ie En ne 
2 a 2 a 2 
© y z°| 
So the radius of torsion @ becomes : 
(eye 
