68 
MR. T. Y. BAKER AND PROF. L. N. G. FILON: LONGITUDINAL 
If we apply this corrective factor to the first two entries of Table III., which'give 
a large percentage error—these correspond to cases approaching the failure of 
convergency and are therefore critical, we find, for lens (l) 
M. 
True 
aberration. 
Formula 
uncorrected. 
Percentage 
error. 
Formula 
corrected. 
Percentage 
error. 
3 
2 
-7-87422 
- 1-52334 
- 7•9S683 
- 1-55287 
1-41 
1-94 
-7-90811 
- 1-54089 
0-43 
1-16 
which shows a very sensible improvement. 
The significance of this alteration is brought out more clearly when we consider 
the limiting case M = co that is, rays actually issuing from the front focus (the 
case of an eye-piece). In this case, the geometrical image being at infinity, it is 
inconvenient to define the emergent ray by means of either longitudinal or transverse 
aberrations. 
Let us consider the intercept of the ray on the back focal plane. 
This = ( — n 2 /M + Ax 2 ) tan a_, 
= n 2 f\_ (A tan 2 /3 2 + e tan 4 /L)/(l + b tan 2 /3 2 ) — M] tan a 2 . 
Also 
tan cl 2 = (tan ajM) (l + 6M -2 tan 2 a u )/(l +cM 2 tan 2 a 0 ), 
where 
b = B— E (j M 2 /A 4 , c = C-E 6 M 3 /A 4 , e - E—E 6 AM a /A 4 , 
and it is clear that, in the limit, where M (and IVl) = o°, tan a 2 must he finite. 
This requires that b and c shall be of order M 3 and M 2 respectively, which is right, 
and leads to 
tan a 2 = nb :i tan 3 ajn (l +c 2 tan 2 a 0 ), 
b 3 and c 2 being the coefficients of M 3 and M 2 in b and c respectively. 
On the other hand, it is equally obvious that the intercept on the back focal plane 
must also approach a definite limit. Hence the factor 
(A tan 2 /3 2 + e tan 4 /L)/( 1 + b tan 2 /3 2 ) — M, 
i.e., 
— 1 + (A — M/>) tan 2 /3.,/M + e tan 4 fi.J M 
1 /M + b tail 2 /T/M 
or 
-1 +(A-M&) tan 2 a„/MM 3 + e tan 4 «„/MIT 
l/M + b tan 2 a,/MM 2 
must tend to a finite limit as M approaches oo. 
This necessarily involves that (l) A — M6 has M 3 in its leading term—a result 
already established, and (2) that e involves M 5 (and not M 6 ) in its leading term. 
