1379 
isample I, “Bor pl gmt we finds a, 
Qi == Pye) spe Os Pa ST, S= 
So the surface admitting as asymptotic lines the twisted quartics 
of the system C'(1, 3,4) with two stationary tangents, is the cubic 
surface 
te aoa BY 
The section of O., by a plane «= constant breaks up into the 
line at infinity of this plane and a curve of the system C(O, 1, 2). 
The ruled surface osculating O,, along this section is represented 
by the equations : 
=e, (1+), 
y = y, 1+) t, 
z= 2,(1+4v)?. 
The equation of this oseulating ruled surface is 
yz, (4a—3e,) = zy,” (Br 2e). 
The intersection of this cubic surface and O,,, consists of the 
conic of contact counted thrice and of the two right directors of Oc. 
Example Ll. Suppose p = 2, ¢g=3, s=p+q=—5. Then Oe 
is a ruled surface of order 2g = 6, the equation of which is 
2" 2 Bij. 
So this ruled surface admits a system of asymptotic lines of 
order five. 
Hzample EW Suppose. p == 1, =d, s == p -- ¢ = 6. _Then 0... 
is a ruled surface of order q = 5 with the equation 
, Wi By, 
_So this ruled surface of order five admits a system of asymptotic 
lines of order six. 
Baamples Vo Suppose p= 3, ¢—= 5, sp +g == 8. Here Oi, 
is a ruled surface with the equation 
Fe eh sf 
Example VI. If the first system of asymptotic lines is formed 
by curves of the system C'(1,3,6), then the asymptotic lines of the 
second system belong to the system C'(2, 3, —3). So both systems 
are curves of order six. 
The equation of O0, is 
Tia by. 
Example VII. If the first system of asymptotic lines belongs to 
the system C(1,2,4) the second system belongs to the system 
C5, 6, —4), 
Then the equation of 0, is 
