200 president's address. 



Important as this may be, it is nevertheless a digression, so 

 we must return to the example before ns. We found that 

 when the focus of the converging incident rays was an inch, and 

 the radius of the convex surface four-tenths of an inch, that 

 the aplanatic focus of the refracted rays was two- thirds of an 

 inch. 



Now, if we further increase the convexity of the surface, we 

 shall find that the refracted rays are still brought nearer to the 

 surface, but the aberration will now be no longer positive, but 

 negative, and will increase rapidly; that is to say, the spherical 

 aberration will bring the refracted rays nearer to the surface 

 than the geometrical focus. Thus, with a radius of two-tenths, 

 it amounts to no less than — '544 y 2 , a quantity greater than 

 any we have yet had. As it is so important, let me sum up 

 the results concisely. Let us call the focus of the incident rays 



P 



P, then if the radius is less than , the aberration is 



ju + 1 



P 



negative and considerable. If the radius is equal to , 



/*+ 1 

 there is no aberration, but there are two aplanatic foci. If the 

 P 



radius is greater than , the aberration is positive : there is, 



^ + 1 

 however, one exception to this last rule, viz., when the radius 

 is equal to P, then there is no aberration. Finally, in our 

 example, where P = 1, the positive aberration never exceeds 

 + -4167 f. 



The next step is this. Having obtained positive aberration in 

 a block of glass with a convex surface, how is it to be preserved 

 when the block of glass is made into a lens ? All that is neces- 

 sary is to make the back curve of the lens a radius measured 

 from the focal point of the refracted rays ; the rays will then 

 pass out of the glass normally to the surface, so there will be 

 neither refraction nor aberration. 



We see, therefore, that a meniscus constructed on this plan 

 cannot have more positive aberration than '4167 y 2 , and, more- 

 over, the lens will be a diverging meniscus, because the convex 

 front surface will have a much flatter curve than that of the 

 concave back. We must now consider the first lens of the 

 doublet upon which the parallel rays impinge. The best form 



