116 Dr Searle, The determination of the focal 
It is clear that the point is in this case at the centre of the 
spherical surface, and hence, if r be the radius of the sphere, 
r=U=v=2f. 
There is an important distinction between the two self-con- | 
jugate points, for at the pomt where w=v=0, the image of a | 
finite object is erect, but at the point where u=v = 2f, the image 
is wnverted. . 
>) 
§3. “Thick” mirrors. Many mirrors are formed of glass 
silvered at the back, the glass having two spherical surfaces which 
are generally not concentric. Although the front (unsilvered) 
surface acts as a spherical mirror, the images formed by it are 
very weak compared with those which are formed by two refrac- 
tions at the front surface and one reflexion at the silvered surface. 
The more general system is described in § 4. 
We see at once that we must not expect a thick mirror to 
act exactly as if the front surface of the glass were silvered and 
polished. Yet we can show that the positions of image and object 
are related in the same way as if the glass mirror were replaced 
by an ideal spherical mirror of appropriate radius placed in the 
proper position relative to the front surface of the actual system, 
under tke limitation that any object point is very close to the 
axis of the system and that the rays make only infinitesimal 
angles with that axis. 
_ §4. Theory of the thick mirror. The mirror may consist of a 
piece of glass bounded by two spherical surfaces and silvered at 
the back, or it may be a more complicated arrangement consisting 
of any number of thick or thin lenses arranged along an axis, the 
reflexion taking place at a plane or spherical silvered surface 
situated behind the last of the lenses. 
In Fig. 1, AC is the axis of the spherical surfaces. Let P’X 
be an incident ray parallel to the axis. When this ray emerges 
Fig. 1. 
from the system after suffering reflexion at the silvered surface 
M, it will not,in general, be parallel to the axis. It will, therefore, — 
intersect the incident ray in some point X and will cut the axis 
