38 
MR. T. Y. BAKER AND PROF. L. N. G. FILON: LONGITUDINAL 
infinite and afterwards changes sign. Thus the values a u = ±A correspond to poles of 
the longitudinal aberration. These poles, being the singularities of least modulus, 
govern the convergence of the Taylor’s series in this neighbourhood, and this radius 
of convergence tends to zero as the object approaches the front focus. It was a con¬ 
sideration of this difficulty which primarily led us to put the longitudinal aberration 
in a new form. 
This difficulty does not arise with the transverse spherical aberration. The poles of 
the longitudinal spherical aberration are due to the zeros of a u , and on multiplying 
by to get the transverse aberration, these poles disappear. 
4. Summary of Method and Results. 
The general principle of the method employed was suggested by an attempt to fit 
an empirical formula to the longitudinal aberration of a lens for a certain range of 
curvatures and object and image positions. This empirical formula was discussed by 
the authors in a paper recently read before the Optical Society* and was found to 
give, on the whole, a singularly good fit. Briefly stated, the formula is of the 
following type :— 
At' 2 
+ B t 2 . 
Ax — 
( 1 ) 
where t is the slope of the emergent “ Gaussian ” ray, so that t = tj M . A is the 
(known) theoretical constant of the first-order aberration, which is a quartic in the 
magnification, and B is a cubic in the magnification, the coefficients in which are 
determined empirically. This formula was found to give a good approximation, even 
when the magnification was high and we were working well outside the limits of con- 
vergency of the Taylor’s series for Ax. 
If we consider any given object point, the longitudinal spherical aberration will be 
a function of t 0 , that is, of t. Denoting it by fit), the reasoning of the preceding section 
shows that f{t) is always one-valued for a finite (and generally quite considerable) 
range of t, but it is not regular, having poles at t = ± r, where r = tan X/M and 
becomes rapidly small as the magnification increases numerically. 
If, however, we write (l —t' 2 /T 2 )f (t) = 0 ( t ), 0 (t) is now limited only by the original 
branch-points of f[t) and will, in general, have an adequate radius of convergence. 
We may therefore expand it in a Taylor’s series, and we get for f(t) the form 
Ar =f{t) = 
at? F Ijd -\- ct -\- 
1 -t 2 /r 2 
(2) 
* Baker and Filon, “On an Empirical Formula for the Longitudinal Spherical Aberrations in a Thick 
Lens,” ‘Proceedings of the Optical Society,’ December, 1918. 
