Non-homocentric Pencils. 705 
It will be seen that the object-wire may be moved into the 
position of any generator of the second set, except the two 
focal lines, without altering the form of the shadow. 
If the plane on which the shadow is taken is not per- 
pendicular to the axis of z, the section is still a conic. If it 
is turned so as to contain one of the generators of the second 
set, the section will become two straight lines; the other 
being the ray which lies wholly in that plane and comes from 
the point in which the plane is cut by the object-line. 
This is an example of a two-part shadow, of which one part 
is due to a single point of the object. 
It is to be noted that although the complete shadow is an 
hyperbola, it does not follow that both branches will be seen 
with a pencil limited by any aperture. In fact only one 
branch will be seen, unless some of the rays which meet it 
have and some have not passed through one of the focal lines. 
The positions of the focal lines produced by reflexion or 
refraction of a small pencil may be often found by use of the 
method of sagitte due to Prof. Thompson. The case of 
refraction through the centre of a thin lens is taken as an 
example. 
Let r, and rv, be the radii of curvature of a small thin 
convex lens of diameter 2a, and refractive index p. 
Then the thickness (¢) of the lens is given by 
poe a i a deal 
ra ie To) 
Let a small pencil inclined at an angle ¢ to the axis of the 
lens fall centrally upon it. 
Then if a plane wave-front fall upon the glass, the centre 
t 
Os dy’ 
sin gee pide bel 
where sin pe a But this is in a direction making an 
of it travels (see fig. 2) through glass for a distance 

angle ¢—d, with the direction of the pencil ; so it is only 
cos(¢—¢,). Meanwhile 
cos dy 
the edges of the wave-front travelling in air have advanced a 
advanced thereby a distance 

distance ee so that the relative retardation of the 
cos py’ 
* t es ae . 
centre is —-— +4—cos (6—¢;)}> which reduces to 
cos dy 
od OS ae a 
“(5 +5, (#208 gi 008 p) = A. 3: 
Phil. Mag. 8. 6. Vol. 7. No. 42. June 1904. 3B 
