1371 
Indeed this ruled surface proves to be developable, as it is possibe 
to determine v in such a way that the director cosines of the tangent 
to the curve of the system C(A,) situated on this surface /// and 
corresponding to this value of v become proportional to the director 
cosines of the generatrices. 
Indeed it is possible to find values v and w satisfying the equations 
Pp Je A, Pr Sa (p? <i A py) + w (p Dil A, P,) = 0, 
gg, de wg? Sh 4) + w(q— 4,0) = 9, 
s+a,s, + v(s? —A,’s,7) + w(s — A, s,) = 0, 
as the sum of the three first members, multiplied respectively by 
P, Q, and S disappears. 
This developable also cuts O,., according to curves of the system 
C(4,) to which also belongs the curve of contact. 
- s 
$ 8. By assuming for 2, the value — we find s—a, s, = 0 and 
all 
the system C{A) becomes the system C(p(s—p), g(s—q), 0). The con- 
jugated system, i.e. the system corresponding to the value A, —=—4,—=s:s,, 
is then the system C(pq, pq,s,). Then the first system consists of 
curves lying in the planes z= constant and the second of curves 
lying in planes through the axis OZ. 
The developable D circumscribed to O,,, along a curve of the 
system C(p(s—p), g(s—q),9) is generated by the tangents to the 
curves C(pq, pg, s,) and admits therefore the equations : 
aa, (Ì + pgv) trs—P), 
y =y, (Ll + pgo) t9(s—9, 
2=2,(1 + s,v), 
PY» 2, Satisfying the relation 
x Pk z Sk — By—€., 
As the system of curves conjugated to the curves of contact 
consists of curves situated in planes through the axis OZ, the 
developable must be a cone (according to the theorem of Korntes'), 
the vertex of which lies on .OZ. It is easily verified that all the 
s 
0,0,2,(1 2) 
Pq 
The developable D’ circumscribed to O,,, along a curve of the 
system C(pq, pq, s,) is represented by the equations : 
generatrices of D pass through the point 
1) G. DarBoux, Théorie gén. des surfaces T. 1 § 91. 
