On Systems of Rays. 93 
the three curves (S"*) (7""’) (5"), in the three rectangular co-ordinate planes, we 
shall thereby oblige it to become the sought hyperboloid (2). It is not necessary, 
however, though it is sufficient, to assign the hyperbola (6"), as a third curve upon 
this hyperboloid. or, in general, if we know the intersections of a surface of the 
second degree with two known planes, there remains only one unknown quantity in 
the equation of that surface, and the intersection with a-third known plane is more 
than sufficient to determine it. Thus, in the present question, if the intersection 
(C") be distinct from the following parabola 
y=0, z=p2+4r2, (D") 
that is, if the surface (Y"™), containing the two known curves (S™) (7""), be dis- 
tinct from the known guiding paraboloid removed, which also contains the same two 
curves, the intersection (C'') with the plane of curvature of the ray is in general a 
hyperbola, which touches the known parabola (D") at the known origin of co- 
ordinates, and meets this parabola again in another known point on the axis of 2, 
that is on the radius of curvature of the known final ray, namely, in the point 
2p 
r= — q=O,42—0); (Ee) 
Tr. ’ 
the hyperbola (€") has also one asymptote parallel to the known final ray-line or axis 
of z, namely, the asymptote having for equations 
1 
os y=9, ce”) 
and it will be entirely determined, if, in addition to the foregoing properties, we know 
also a line parallel to its other asymptote, namely, to that which has for equations 
Ap 2s 1 2p 
w= —2(—)2-5->, P=VE (GS) 
it will therefore be obliged to coincide with the hyperbola (B"), if only we oblige its 
second asymptote (G@") to be parallel to the following known right line, 
= te, 22 4 
Diet igen aa es 
in which the coefficient 
ope yew COTTER ORIOLE TENE (ry 
r. dz deflecure of guiding paraboloid ” 
° 
the plane of the deflexure 7, being the plane of curvature of the ray. We see, then, 
that this last condition, respecting the direction of the second asymptote (G"*) of the 
hyperbolic section (C™), is sufficient, when combined with the conditions of contain- 
ing the two known curves (S") (Z"), to determine completely the sought hyperbo- 
loid (J). Even the conditions of containing the two curves (S*) (7"") are not 
perfectly distinct and independent ; nor would their coexistence be possible, in the 
VOL. XVII. 2B 
