president's address. 199 



was 4- "4167 y 2 , gets gradually smaller until it vanishes. If we 

 further deepen the curvature the focus of the refracted rays 

 gets nearer the surface than that of the incident rays. But 

 what about the aberration? you will naturally ask. As the 

 calculation for the aberration at a spherical surface depends on 

 the solution of an equation of the third degree, the subject can 

 hardly be put in a popular form ; all that can be said is that 

 if you solve this cubic equation for our own particular example 

 you will find that the aberration is still positive, and that it 

 increases slowly until the radius is *526 inch, when it arrives at 

 a maximum of + 0213 y 2 ; it then rapidly declines until it 

 vanishes for the second time. You w r ill notice that the aberra- 

 tion at this second maximum is only a twentieth of that at a 

 plane surface. 



We now come to a very important point, viz., when the 

 spherical surface has two aplanatic foci. You will remember 

 that at our former aplanatic point both the foci coalesced at the 

 centre of curvature ; in this instance, however, they are 

 separated. When the radius is equal to the focus of the in- 

 cident light, divided by the refractive index plus one, then 

 those incident rays will be aplanatically refracted to another 

 point, whose focus w r ill be equal to the same distance (i.e., the 

 focus of the incident rays), divided by the refractive index. 



In simple words, in order to find the radius, we must divide 

 the focus of the incident rays by the refractive index plus one ; 

 and to find the focus of the refracted rays, we must divide the 

 focus of the incident rays by the refractive index. Therefore, 

 in our example, the radius is one divided by two-and-a-half and 

 equals -| inch, and the focus of the refracted rays is one divided 

 by one-and-a-half and equals § inch. 



The reason why this rule is so important in microscopy is 

 because it is the basis for the construction of homogeneous 

 immersion objectives.* The immersion front is made of such 

 a radius that it will refract the rays proceeding from the 

 object aplanatically to some other and longer focus, this longer 

 focus being the focal point of the second lens. The radius is 

 found, therefore, by dividing the distance from the apex of the 

 front lens to the object by (//, -f 1), and the longer focus by 

 dividing the same distance by /x. 



* " Jouni. E.M.S.," 1878, p. U2, fig. 2. 



