length of a thick mirror 117 
in some point /. Draw a plane XH at right angles to the axis, 
cutting the axisin H. ‘Then, if we reverse the ray #X, we have 
the two incident rays P’X and FX giving rise to the emergent 
rays AF and XP’, and thus X is its own image. From the 
general properties of refractions and reflexions at spherical surfaces, 
it follows that every point near H in the plane XH coincides with 
its own image or, in other words, is self-conjugate. 
The point conjugate to / is at infinity, since the ray XP’ is 
parallel to the axis, and hence every ray which passes through F’ 
on incidence and makes a small angle with the axis is parallel to 
the axis on emergence. 
The point F' 1s called the Focus of the system and # is called 
the Principal or Unit Point. The plane HX is called the Principal 
or Unit Plane of the system. 
We can now find the image of the point P’. Let the incident 
ray P’F cut the principal plane in Y. Since Y coincides with its 
own image, the emergent ray passes through Y. This ray YQ’ is 
parallel to the axis, since the incident ray P’FY passed through 
the focus F. The image of P’ is therefore at Q’ where XF’ and 
YQ’ intersect. 
If P and Q be the feet of the perpendiculars from P’ and Q’ 
upon the axis, then P and Q are conjugate points. 
Let wu and v be the distances of P and @ from the principal 
plane; let HF =f and let wu, v and f be counted positive when in 
each case a point moving in the direction of the normal drawn from 
M reaches H before it reaches P, Q and F' respectively. Then 
fe lif SE 
oe Laos 
Hence, by addition, _ =1, 
Teel, Yili 
or x ap a Tih Peaster ietoverscecelnieisi ois Tavete’ tratovelare, os (2) 
Comparing (2) with (1), we see that the mirror system may be 
replaced by an ideal spherical mirror of focal length /, if the 
reflecting surface of the latter intersects the axis at right angles 
in H, We shall show presently how / and the position of H may 
be found by experiment. 
If we put w= 2/, we find from (2) that 
OHVs 2 
and thus the object point P and its image Q coincide. If C 
(Fig. 1) be the corresponding point, then the focus F is midway 
between C and H. 
VOL. XVIII. PT. III. 9 
