1370 
we get 0=0, for the condition under which the curve C'(p',q', 3’) 
hes on Oe, is 
Pp' + Qq + Ss = 0. 
For this proof we have not made use of the fact that the two 
systems C'( p,q, s) and C'(p,, q,, 5,) are complementary ; so this theorem 
also holds for any surface generated by curves of the system Cp, q, 8) 
meeting a curve of a system C'(p,,q,, 52): 
§ 7. Let P,@,,y,,2,) be onee more an arbitrary point of Oe: 
Then the two ruled surfaces osculating O,,, along the curve through 
P, of the system C(p+ap,, g+q,, s+/s,) are the surfaces the 
generatrices of which are the principal tangents of Oe in the points 
of this curve. So these two osculating ruled surfaces are represented 
by the following two sets of equations : 
x“ == PTP: (1-}- pv), oS we iP (1+ 7,2), | 
y= ytrrn(L+gr)) A) yy inl +912), | (ZZ). 
zet eS ze tl gw): 
As soon as two surfaces are generated by curves of a same 
system C(p, q,s) the intersection of these surfaces consists exclusively 
of curves of this system, whether single or degenerated ones; for 
the curve C(p, q,s) passing through any common point of the two 
surfaces must lie on both. So, as the two surfaces / and // oscu- 
lating O.-, partake of the property of O,.., of being generated by 
curves of the system C(p+dp,, ¢+/q,, s+4s,), or CA) for short, the 
intersection of each of these two surfaces with O,,, and their mutual 
intersection must break up into curves of the system C(A). 
In the case A=0O, ie. if the system CA) coincides with the 
system Cp, q,s), the ruled surface / is developable. If C(A) coincides 
with the system C(p,, ,,5,) the ruled surface // is developable. 
For 4,==—A4, the cross ratio 4,:4,==— 1 (see $ 6) and the 
four tangents form a harmonic quadruple. Then the tangents to the 
curves of the systems Cà) and C(—A,) are two conjugate diameters 
of the indicatrix, the tangents to the curves of the systems C(p, q, 5) 
and C(p,,q,,5,) being the asymptotes of the indicatrix. So the two 
systems of curves corresponding to 4, and — 2, are conjugate on Ope. 
So the developable enveloping O,, according to a curve of the 
system C(4,) is represented by the equations 
=d ft ob v (p — A. ~,)) (Pp, 
& 
yn (telg dg) erin, j. . . 2 TZ) 
==, (1 + Vv (s aa A, s‚)) pets, 
