410 Mr. T. Smith on the 



the glasses being the last step to be taken instead of being 

 assumed initially. 



The selection of" glasses for this purpose is greatly facili- 

 tated by drawing on tracing-paper a chart containing curves 

 corresponding to constant values of a . This is used in 

 conjunction* with a diagram on which the available kinds of 

 glass are plotted, the variables employed being the refractive 

 indices for a definite wave-length — usually the D line — and 

 the logarithm of the v\. The chart on the tracing-paper is 

 moved over the glass diagram in slide-rule fashion, and 

 suitable pairs of glasses are selected by noticing that when 

 the zero of the chart lies on the point representing one of the 

 glasses the other glass lies on the line corresponding to the 

 required value of a . This is one of the few directions in 

 which I have found graphical methods of distinct value 

 in lens calculations. I do not regard Mr. Allen's graphical 

 suggestion as a useful one for the practical computer, because 

 it is not only much easier to solve a quadratic equation 

 directly than by graphical methods which involve the con- 

 struction of a templet, but also, as will be explained later, 

 there is a very good reason for solving the spherical 

 aberration equation with greater accuracy than a graphical 

 method will generally afford. 



Before leaving the discussion of the detailed formulae it 

 may be pointed out that Problem (2) is not stated in a satis- 

 factory form, for the result obtained will depend upon the 

 interpretation given to " least aberration" as the object point 

 is varied. The boundary condition may be that the lens 

 aperture subtends a definite angle at the object, or at the 

 theoretical image point. The most natural assumption in 

 the absence of any statement would be that the linear 

 aperture of the lens is kept constant. In all cases, however, 

 it is to be remembered that expressions such as (1) are 

 multiplied by other factors which involve m, or the position 

 of the object, and these factors must be taken into account 

 when the expression for the aberration is differentiated to 

 find the stationary values. The result obtained will vary 

 with the criterion adopted for the measurement of the 

 aberration. For instance, the position of the object which 

 gives minimum longitudinal aberration will differ from that 

 for which the latitudinal aberration is least, and both will 

 be distinct from the one for which the difference of path 

 between axial and marginal rays is a minimum. 



* Proc. Phys, Soc. vol. xxviii. p. 220. 



f When the ordinary type of colour correction is not desired a modified 

 quantity is substituted for v. 



