202 president's address. 



would be by an ordinary adfected quadratic equation, neither 

 very long nor troublesome. If anyone goes through this they 

 will find that when the second is a converging lens, i.e., has a 

 positive focus, the roots are imaginary. One should judge 

 from this that it is impossible to construct an aplanatic doublet 

 of this form ; if, however, the second lens is allowed to have a 

 negative focus, the roots become real and two forms of lenses 

 are possible. 



In the first doublet the second lens is a biconcave, the 

 anterior curve being deeper than the posterior ; the other 

 doublet has a diverging meniscus for its second lens, with its 

 posterior curve deeper than its anterior; but both these 

 doublets have such steep curves that they are practically 

 useless. We now come to the last case, viz., that when the 

 equation has no real root it can be solved for a minimum value. 

 This is a very simple matter ; the curves of a doublet so calcu- 

 lated are given in the appendix. 



Having constructed our lens, we can compute the amount of 



y 2 



its aberration, and it will be found to be — *214 — or 27 per 



F 

 cent, better than that of Herschel's. To sum up our work 



1. We have seen that Herschel's doublet has an aberration 



of — -296 — 

 F. 



2. We have investigated aberration at a single spherical 

 surface, and have found that in our example it did not ex- 

 ceed + -4167 ?/. 



3. We found that no lens thus constructed could neutralise 

 the negative aberration of a crossed lens for parallel rays. 



4. We have investigated the construction of an aplanatic 

 front of a homogeneous immersion objective. 



5. We have seen that an aplanatic meniscus, when combined 

 with a crossed lens, yields a result somewhat similar to that of 

 Herschel's doublet. 



6. That the quadratic equation for aberration has no real 

 roots for a lens of positive focus with the given conditions. 



7. That two doublets of no aberration may be constructed 

 when the second lenses have negative foci. 



8. That when we are contented with a minimum of aberra- 



