president's address. 201 



for this lens is that known as a crossed lens, and for the kind 

 of glass we are using the radii will be in the proportion of 1 : 6 

 for minimum aberration. The lens must be of one inch focus, 

 so that the rays may converge on the second lens to a focus one 

 inch from the surface ; the lenses are assumed to be in contact. 

 If we now calculate the spherical aberration of this crossed 

 lens, we shall find that it is — T0714 y 2 , but we have seen 

 above that the utmost amount of positive aberration from one 

 spherical surface that we can bring to oppose this is something 

 less than + '4167 y 2 , a quantity insufficient to balance even 

 half the negative aberration of the crossed lens, therefore a 

 doublet of no aberration cannot be constructed on this plan. 



Being baffled in our attempt to neutralise the negative 

 aberration of the crossed lens by the positive aberration of a 

 single surface, let us examine the conditions we shall obtain by 

 placing an aplanatic meniscus behind a crossed lens, so that the 

 longer focus of the meniscus shall be coincident with the focus of 

 the crossed lens. Applying the law we have just investigated, 

 and making the focus of the crossed lens unity, the front curve of 

 our meniscus will be, as we have seen, -§, and the back curve 

 will be the same as the conjugate focus we found for the 

 refracted rays, viz., f. The meniscus by itself will have a 

 focus of two inches, and as that of the crossed lens is one inch, 

 the focus of the combination will be f inch. It is usually more 

 convenient to transpose this into a combination of one inch 

 focus, then we shall have the focus of the crossed lens I and 

 that of the meniscus 3 inches. 



Now, as the aberration of the second lens is 0, the aberration 

 of the combination is that of the first lens in terms of the focus 

 of the combination, thus the aberration of the first lens, 



2 9 



y y- 



— 1 0714 — , becomes — 317 — , which is the aberration of the 



/ F 



doublet, a result a shade more than that of Herschel's. 



You will probably by this time have had enough of the 

 aberration of a single spherical surface, but the problem is 

 important for two reasons : First, because it is the point on 

 which those rely who say that an aplanatic doublet can be com- 

 posed of two converging lenses ; secondly, the calculation is so 

 long that few attempt it. 



The usual method of calculating the form of such a lens 



