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light issued to the centre of curvature, should be to the radius, as the 
refractive index was to unity. When those conditions were obtained, 
then all the light passed without aberration to another point. It is 
this theorem which makes it possible to construct an aplanatic 
meniscus. For if, in a converging meniscus, the more convex curve 
satisfies the above conditions, the shallower curve may be made any 
radius from the focal point, and therefore the light will pass through 
that surface without refraction. Fig. 32 makes this abundantly 
Fig. 32. 
evident. Let the- ray E fall on the convex curve A, whose radius is 
A r, and let B, which is technically known as the focus of E, be the 
point where E produced will cut the axis, and let yit the refractive 
index of A D = 1*5. Then, if A r : r B : : 1 : 1 *5, all rays 
falling on A which have their focus at B will be refracted without 
aberration to the conjugate focus C. All that is necessary now is, 
from the centre C to describe the curve D, and the meniscus A D is 
constructed, for it is quite evident that all the rays refracted by A 
to C are radii of D, and so pass through without refraction. 
The converse problem is also true. If a light is placed at C, all 
rays falling on the meniscus A D will be refracted aplanatically as if 
they came from B. So far the books help one, but as r and B are 
both unknown, it is a tedious business to construct an aplanatic 
meniscus for purposes of a bull’s-eye, &c. I have investigated the 
problem, and have devised some very simple formulae which make the 
construction of an aplanatic meniscus perfectly easy. Let us see 
what we have given us. We have two things, viz. the refractive 
index of the medium, and C D, the back focus. Now, we first 
require the distance A C, or the back focus + the thickness of the 
lens. The best way of determining the thickness of the lens is by 
drawing it. Extreme accuracy in this point is not necessary. Having 
found A C call it P ; and A B,F; A r, r ; and let /a = the re- 
fractive index. Then 
and P' = yit P. 
Having found r draw it. It is only necessary to lay off an angle at 
