Non-homocentric Peneils. 703 
As this surface is ruled by rays passing through three 
straight lines, it is a ruled quadric ; further, since these 
three lines lie in parallel planes, it‘is a hyperbolic paraboloid. 
One set of generators is the rays. If p, g, 7 be the 
direction-cosines of the ray through %, yo, we get from its 
equations 
P q Te 
She, Mose. ae 
b a 
Since yp=mery+c, we have, eliminating a’ and yw, 
ag—mbp+cr = 0; 
that is, the rays are parallel to the plane 
ay—mbe+cz = 0. 
If c=0, that is if the object-line meets the axis, this becomes 
) ; 
7 = ‘a Mw. 
This case is shown in Model 5, which consists of the 
shadow-surface and the plane y= Mk. 
The other set of generators are the shadow-lines in planes 
parallel to z=0. 
The section by any plane not perpendicular to the axis 
will be an hyperbola whose equation can be obtained from the: 
equations of the plane and the surface. 
If the object-line does not lie in a plane perpendicular to 
the axis, the ray-surface is an hyperboloid of one sheet, and 
the shadow on a plane at right angles to the axis becomes an 
hyperbola. But since a generator of each set passes through 
every point on the surface, the shadow can always be reduced 
to a straight line by tilting the plane upon which it is 
received. 
The equation of the ray-surface in this case can be ob-- 
tained as follows :— 
The equations to a ray through the point (#9, yo, 2) are 
z—b ‘| 
tt =z X& 
2y—b Q9 | 
Lo 4 | 
Aan Oy 
Let the object-line be defined by the equations 

J = 
2=merh ) 
y=nitk i 
