60 
MR. T. Y. BAKER AND PROF. L. N. G. FILON: LONGITUDINAL 
But, in the case of the ideally thin lenses, where the thickness is zero, the ideally 
thin positive and negative lenses, having the same mean curvature and numerical 
power, correspond in this way. 
Now I 4 and I 0 are also interchanged. If we consider I 4 as the initial point, then we 
are really considering a set of rays starting from I 4 in the last medium and travelling 
backwards through the original positive lens. In other words, the aberration constants 
for the corresponding negative lens are identical with those for the original positive 
lens reversed, and the equations (III), (64), (IV) and (V) are applicable to calculate 
them. 
This, again, will greatly diminish the work of calculation. In the case of ideally 
thin negative lenses, we see that A, B, C, E are directly obtained from the corre- 
sporftling thin positive lenses. In the case of thick negative lenses the corresponding- 
positive lens has a negative thickness. 
Now for various reasons it will probably be convenient, in calculating A, B, C, E 
for lenses, to express them in the form 
A c = A 0 + c (d A/dc)„, 
&c., where c, the thickness, is small, as it usually is in practice, and A 0 refers to an 
ideally thin lens. 
When the formulae are put in this form, it is perfectly simple to calculate A_ c , B_ c , 
&c., and then to obtain the corresponding results for the negative lens with a positive 
thickness. 
§ 11. Explicit Values of A, B, C, E for a Thick Lens (Tangent Formula). 
For the purposes of numerical calculation and comparison with correct trigonometri¬ 
cally found values, we have worked out explicitly the form of the expressions A, B, 
C, E for a thick lens, when we use tan /3 4 as the argument; the formulas are 
expressed in terms of the focal lengths of each surface and of the combination and the 
thickness does not appear explicitly. The initial and final media being the same 
n 0 =n± and we have written n — nfn 0 . 
The work of algebraic calculation has been straightforward but extremely heavy, 
and we therefore omit it here entirely, the object being to publish the results for 
reference, in case other workers desire to use them for tabulation purposes, but it is 
hardly to be expected that designing opticians should work direct from the algebraic 
expressions as they stand. 
For the purpose of this section we shall write 
A = A„ + A X M + A 2 M 2 + A 3 M 3 + A 4 M 4 . 
B = B 0 +Bj M+B 2 M 2 +B 3 M 3 . 
C = C 0 + CiM + CbM 2 . 
E = E 0 + E X M + E 2 M 2 + E 3 M 3 + E 4 M 4 + E 6 M 5 + E 6 M\ 
