1378 
§ 14. We now suppose s=p + q; then the numbers p, q, s are 
mutually prime two by two. We then find p, =q, = 2 pq,s,=0; 
so the complementary system C'(p, ,q,,5,) is a system of right lines 
resting on the axis OZ and on Xy Yo. The surface Oc, is a ruled 
surface with two right director lines. 
Furthermore we find: 
ASS a 
q+p 
so the lowest common multiple of the denominators of P, Q,S is 
either q+ p or (q + p): 2 according to the numbers q and p being 
either one even and the other odd, or both odd. 
We suppose in the first place that one of the numbers p, q is 
even (see § 15, examples I and III). 
Then the equation of the ruled surface QO... is: 
we9tPz7—P = ByitF; 
so the ruled surface is of order 2g. The enveloping cone is of order 
2Pk = 2(q+p), see § 9, and of class 2g.. 
If p and q are both odd and therefore p—- g and p+ q both 
even (see § 15, examples II, IV and V), the equation of OQ... is | 
x\P+9):2 2P—9):2 — ByP+9):2, 
so Occ, is a ruled surface of order g. The enveloping cone with 
arbitrary vertex A is of order q+ p, see $ 9, and of class g. 
The ruled surface osculating O.,. along a generatrix /is generated 
by the principal tangents of Oe, in the points of / which do not 
coincide with /, ie. by the tangents of the curves of the system 
C(p. q. p+q)- So this osculating ruled surface is represented by ‘the 
equations : | 
x=, (1-+pr)t 
y= Ate) t, 
2 = 2, fl +(p+q) 
or by the equation 
LY, = Ys, 2 
pay — qr (pd) 
§ 15. Evample [. Suppose p=1, yg=?, s=3; then we have 
1 
s= pa"? dik PS = OA. Da So: the 
equation of the ruled surface with the twisted cubics of the system 
Cd, 2, 3) as asymptotic lines is 
2 Die. 
