SPHERICAL ABERRATION FOR A SYMMETRICAL OPTICAL SYSTEM. 
39 
If the series in the numerator converges rapidly, it will be sufficient, provided a is not 
near zero, to stop at the first term, and we get as an approximate formula 
Ax = 
at 2 
l-t 2 /? 
which is of the same form as (l). 
We have necessarily a = A, and if the two formulae are to tally we should have in 
addition B = — 1/t 2 . 
The formula in the form (3), however, is not rigorously correct to the second order of 
aberrations inclusive, unless b happens to be small. If we wish to retain second order 
terms complete, we have to use 
Ax = 
at 2 + bt 4 
( 4 ) 
and this can be written, to the same order of algebraic approximation, in the form 
Ax = at 2 /{l—(\/r 2 + b/a) t 2 }, .(5) 
provided again a is not zero. 
If this form (5) is adopted, then the B of the empirical formula should be l/r 2 + b/a. 
But if this is done, the formula suffers from two defects : (i) it fails whenever a is 
near zero; (ii) it does not give exact compensation for the poles in the critical range 
for M large. 
The further cpiestion then arose : how far are formulae of type (4) or (5) suitable 
for dealing with combinations of surfaces or lenses ? An important guiding considera¬ 
tion, in all work of this kind, must be the relative simplicity of the formulae in 
passing from a single surface or lens to a combination, and whether these formulae are 
suitable for tracing the effect of individual surfaces or lenses upon the final result. 
We have ultimately been led to the conclusion that no single formula can satisfy 
completely the three ideal requirements, viz. : (i) exact agreement with development 
as far as the second order inclusive; (ii) simplicity in dealing with combinations ; 
(iii) exact compensation of the poles in the critical range of M. 
The method finally adopted satisfies conditions (i) and (ii). It only satisfies (iii) 
approximately. Numerical calculations show that numerically the approximation is 
adequate in the case of a lens or a simple surface. In the case of more complicated 
systems we have, as yet, no numerical data. 
The first part of the investigation deals with the single refraction. It is there shown 
that the longitudinal aberration can be put into the form (l), i.e., Ax — A£ 2 /(l+B£ 2 ), 
where the formula is correct to the second order inclusive. 
We also find, for the inclination of the emergent ray, the formula 
q = t{l+Bt 2 )/(l+Ct 2 ), 
G 2 
(6) 
