64 
MR. T. Y. BAKER AND PROF. L. N. G. FILON: LONGITUDINAL 
§ 13. Numerical Test for a Single Lens. 
To test the formulae, a number of longitudinal aberrations were calculated trigono¬ 
metrically for five positive lenses of refractive index 1*52, unit focal length and 
thickness fg-. The first was a meniscus-shaped lens for which r, = 0'349418, r 3 = 1. 
The second was a plano-convex lens, of which the convex side is towards the 
incoming light. The third was an equi-convex lens. The fourth and fifth were the 
second and first reversed. If Iv has the meaning defined in § 12, the values of K for 
these five lenses are 1‘93094, 0'96154, 0, — 0‘96154 and — 1'93094 respectively, so 
that they proceed by approximately equal steps of K. 
The constants A, B, E were calculated for these five lenses and the longitudinal 
aberrations computed from the formula. The rays selected met the first principal 
plane at a distance 0‘15 from the axis, corresponding to an aperture f/ 3'4, nearly. 
The results are shown in Table III. 
Table III.—Longitudinal Aberrations of Five Selected Lenses. 
Lens 1. 
Lens 2. 
Lens 3. 
M. 
Per- 
Per- 
Per- 
Formula. True. 
centage 
Formula. 
True. 
centage 
Formula. 
True. 
centage 
error. 
error. 
error. 
3 
-7-98683 - 7-87422 
1-41 
- 1-12450 
-1-12814 
0-32 
-0-32378 
-0-32439 
0-19 
2 
- 1-55287 - 1-52334 
1-94 
-0-40167 
-0-40283 
0-29 
-0-14240 
-0-14282 
0-29 
0-5 
-0-003527 -0-003530 
0-09 
-0-011697 
-0-011705 
0-07 
-0-033008 
-0-033096 
0-27 
0 
-0-06028 -0-06043 
0-24 
-0-024728 
-0-024749 
0-08 
-0-036194 
-0-036213 
0-05 
-0-5 
-0-20902 1-0-20946 
0-21 
-0-083863 
-0-083950 
o-io 
-0-057067 
-0-057101 
0-06 
- 1 
-0-42692 -0-42772 
0-19 
-0-18428 
-0-18448 
0-11 
-0-094736 
-0-094787 
0-05 
-2 
- 1-00383 -1-00531 
0-15 
-0-48874 
- 0•48930 
0-12 
-0-21707 
-0-21724 
0-08 
Lens 4. 
Lens 5. 
M. 
Formula. 
True. 
Percentage 
error. 
Formula. 
True. 
Percentage 
error. 
3 
-0-098969 
-0-099086 
0-12 
-0-10401 
-0-10444 
0-41 
2 
-0-046089 
-0-046128 
0-30 
-0-013671 
-0-013686 
0-11 
0-5 
-0-073959 
-0-074535 
0-77 
-0-14762 
-0-15068 
2-03 
0 
-0-099843 
-0-100044 
0-20 
-0-22172 
-0-22324 
0-68 
-0-5 
-0-13967 
-0-13984 
0-12 
-0-32828 
-0-32966 
0-42 
- 1 
-0-19084 
-0-19101 
0-09 
-0-45753 
-0-45887 
0-29 
_ 2 
-0-32454 
-0-32476 
0-07 
-0-77417 
-0-77511 
0-12 
The mean percentage error of these results is 0‘34, so that the formula determines, 
on the average, the longitudinal aberration correct to 1 part in 300. The bulk of 
