ZOOLOGY AND BOTANY, MICROSCOPY, ETO. 
257 
error, a result sufficiently correct for practical purposes might he 
arrived at. 
By solving a cubic equation,* the details of which possess no special 
interest, the ratio of the radii is found ; then by means of equation (ii.) 
we obtain this final result : 
r — — 1*165 F and s = — 1*608 F. 
There are, however, some quantities which, for the sake of simplifying 
the calculation, have been neglected, and one of these is the thickness of 
the lens. If a lens mirror is of large size the thickness must be con- 
siderable, in order that it may stand the strain of grinding and polishing, 
and as thickness influences the aplanatism of the system we cannot 
disregard it. 
Further, one of the great advantages of the lens mirror is, that it 
permits a very wide-angled system to be usefully employed, because no 
ray makes an excessive angle with its normal ; the incident ray makes 
the largest angle, and that must always be less than half the angle of 
aperture. The effect of thickness and large aperture can be very well 
seen by making a drawing of a lens mirror on a large scale, and by 
tracing out wide, medium, and narrow angled rays. 
If the lens is drawn on the above formula, and with no thickness, it 
will be seen that the medium and narrow rays have both been spherically 
F 
aberrated to the extent of — — , but the extreme ray will be further dis- 
oo 
F 
placed to — — . If in a drawing of a lens mirror, on the same formula, 
which possesses sufficient thickness for practical construction, say -, w« 
F 
have three similar rays, we shall obtain — — for the values of the aber- 
ration of the medium and narrow-angled rays, and — — for that of the 
o 
extreme ray. From this we learn that the results obtained by the solu- 
tion of the cubic equation are not suitable for the construction of an 
aplanatic lens mirror. A glance at the drawing of the thick lens will 
show that the curvature ratio is not large enough, and on trial a ratio 
of 2 : 3 will be found to yield far better results. 
No great difficulty will be experienced in determining curves that 
will bring an extreme and a central ray to an identical focal point ; but 
it is by no means easy to find the precise curves that will render a lens 
mirror strictly aplanatic for all zones. I thought the subject sufficiently 
important to justify the trouble of working out the following curves for 
three different thicknesses of lens mirrors. These curves, when drawn 
on a large scale and tested by tracing three rays for apertures of 100°, 
60°, and 30°, show perfect aplanatism. A slight deviation from these 
curvatures will destroy the aplanatism. It is probable that errors arising 
from the thickness of the lenses were the cause of the fluffiness which 
Sir G. Airy observed in his telescopes, as he distinctly states that no 
* The equation is given in some of the text-books, but not its solution. 
