67 
SPHERICAL ABERRATION FOR A SYMMETRICAL OPTICAL SYSTEM. 
or, to the same approximation which we have been using 
1 + tan 2 f3 2 ( B 3 M 3 + B,M 2 ).(70) 
We see, therefore, that the development of the correct convergency factor in descending 
powers of M will give a result which always agrees with our B, so far as the highest 
term in M is concerned, but makes the term in M 2 in general different. 
In the case of a lens E 6 /A 4 is in general small, compared with B 2 , so that this 
discrepancy makes little difference, but it may well be that, when we come to deal 
with more complicated systems, this will not be the case. 
A little consideration, however, shows that, when this is so, our formula is very 
readily corrected so as to take this difficulty into account, without involving any 
lengthy numerical computation. 
If we consider the formula 
Ax 2 = n 2 f{At 2 2 + (E-AE 6 M 2 /A 4 ) + (B-E 6 M 2 /A 4 ) t 2 2 } 
it is clear that it leaves the development of Ax 2 in powers of t 2 unaltered as far as 
the second order inclusive. It alters the coefficient B 2 of B so as to make the two 
leading terms agree with (69). It also alters E in such a way as to remove the term 
in M 6 and reduce E to a quintic. In fact it gives for the new E the remainder obtained 
after the first step in the division of E by A, according to the usual process. ■ 
In practice, the terms in (E, ; /A 4 ) M 2 are very readily added as follows :— 
a.,. = „ -■ At/ + EO / (l-E,My /A,) 
3 J 1+Be,7(l-E 6 M 2 «//‘A 4 ) 
and this amounts to applying the same corrective factor l/(l — E B M% 2 /A 4 ) or 
l/{l-En 0 % a /n 2 2 A 4 ) to the second terms in both numerator and denominator. This 
factor, expressed in terms of the inclination of the incident ray, is independent of the 
magnification, and a short table will enable it to be found in any given case without 
difficulty. 
A similar correction has then to be made in C; in order to keep the development 
of tan a 2 the same we must have 
tan a 2 = t 2 { 1 + (B-E u M 2 /A 4 ) ti) / {1 + (C-E 6 M 2 /A 4 ) 1 2 2 }, 
and writing this as 
Ul+Bf 2 2 /(1-E 6 MV/A 4 )} 
1 + C£ 2 2 /(1— e 6 mv/a 4 ) 
we see that the same corrective factor has to be applied to all the second terms in the 
formulae. 
In the above we have used tan /3 2 as our argument, but the formulae and the 
correction take precisely the same form if sin y 2 is the argument. 
