On Systems of Rays. 91 
lates in all directions at the given final point to the hyperboloid (7): it is therefore 
sufficient to construct this osculatng hyperboloid, in order to deduce the sought para- 
boloid (D”). We might even employ the hyperboloid as a new guiding surface for 
the ray-lines from the oblique plane, instead of employing the paraboloid, since thes 
two osculating surfaces have the same deflexures and deflected lines, near their given 
point of osculation. 
Now to construct the osculating hyperboloid (J"*), by the oblique plane (B") or 
(F'”), and by the former elements of arrangement, that is, by the guiding paraboloid 
: ry : : 3 
(11), and by the coefficients = - ee , which determine the magnitude and plane of 
curvature of the final ray, we may compare the sought hyperboloid (J") with the 
following new paraboloid 
S=pL tq tare +s.ty + 3toy', (N*) 
which may be called the guiding paraboloid removed, since it is equal and similar to 
the guiding paraboloid (17°), and may be obtained by transporting that guiding para- 
boloid without rotation to a new position such that it touches the given oblique plane 
at the given point. ‘The intersection of the hyperboloid (7) and paraboloid (Vv), 
consists in general of an ellipse or hyperbola in the given plane 
z=0, (O") 
perpendicular to the given final ray, and of a parabola in the plane 
6 é 
x+y P=0, GP?) 
which contains the given final ray-line or ray-tangent, and is perpendicular to the 
final plane of curvature of the ray. If then, we make this final plane of curvature 
the plane of xz, so that its equation shall be 
y=9, (Q") 
and so that, by ( 7”), 
2 =0; (R”) 
we shall have the following equations for the two curves of intersection ; first, for the 
ellipse or hyperbola, 
, 2=0, pa+qy tyre +sry + 3ty'=035 (S*) 
and Sc for the parabola, 
a=0, z=qytaty’: ei) 
and these two curves may be considered as known, since they are the intersections of 
two known planes with the known guiding paraboloid removed to a known position. 
To examine now how far a surface of the second degree is restricted by the condition 
