118 Dr Searle, The determination of the focal { 
There are thus two points on the axis which are self-conj ugate, 
viz. H and C, and if we were to use merely a point as an object 
we could not determine by experiment which of the two points | 
was H and which was C. But the image of a finite object in the 
transverse plane through C lies in that plane and is inverted. 
This follows from the fact that, if the transverse plane through 0 
cut P’X in D and XF in D’, D and D’ are on opposite sides of C. 
Since HF'= FC and since XD is parallel to the axis, CD’ = CD. 
The point C corresponds to the centre of the ideal spherical 
mirror and may be spoken of as the centre of the optical system. 
On the other hand the image of an object in the principal | 
plane HX is erect*. 
Since CF =f, the focal length could be found by measuring | 
CF. The point C is easily found by adjusting a pin so that its tip | 
coincides with the tip of its own inverted image, and the point 
F could be found by means of a small screen of ground glass, 
for distant objects will be focused at #'; illumination troubles — 
would, however, make this plan difficult. 
Since the position of the centre C can be readily determined, 
we shall transform (2) so that we can use C as an origin of 
measurement. Let p and q be the distances of P and Q from C, 
and let p and q be positive when P and Q lie on the same side of — 
Cas Ff. Then 
u=2f—p, v=2f—¢q. 
1 1 1 
Hence, by (2), ~—_ + => — =a. 
Ee) Y—p* fg F 
This equation leads to 
1 ee te lee 
ee Oe I 6 ois so Soc ac < 3 
Denon a ®) 
Thus, if we find two conjugate points P and Q and measure their 
distances from the centre C, we can find f, The positions of F and 
H follow at once, since CF =f and CH = 2f. 
§ 5. Gauss’s method. The following method of finding f may — 
be called Gauss’s method, as it corresponds for mirrors to Gauss’s 
method of finding the focal length of lenses. The centre C (Fig. 1) 
is found by adjusting a pin so that its tip coincides with the tip 
* Mr R. E. Baynes has pointed out to me that, in the case of any general 
mirror system, the vertex M (Fig. 1) of the reflecting surface and the centre of — 
curvature of that surface are the images of H and C respectively formed by the — 
lens system in front of WM in the medium in contact with M. 3 
t If a telescope with cross-wires or a goniometer (§ 9) is available, the position — 
of the focal plane is easily found. The instrument is first focused for infinity and 
is then turned to view the image of an object seen by reflexion in the mirror 
system. A grain of lycopodium on that face of a glass plate which is turned 
towards the mirror would serve excellently. The object is adjusted so that an 
image of it is focused (without parallax) upon the cross-wire of the telescope or 
goniometer. i 
