SPHERICAL ABERRATION FOR A SYMMETRICAL OPTICAL SYSTEM. 
6 
o 
It should be noted carefully that all the above refer to expressions in terms of the 
tangent of the Gaussian inclination, this being the argument we have used in the 
numerical work. 
§ 12. Values of A., B, C, E for a Thin Lens. 
When the lens is thin, we have the relation 
J 1 = I 
fi- fz f ’ 
which enables the values of §11 to be considerably simplified. 
In this case it is useful to introduce a quantity K such that 
_ mean curvature of the lens _ f !\ 1 \ 
power of the lens 2 \r 1 rj 
When this is done the constants A, B, C, E take the following forms :— 
A = -{(l-M) 2 /2ft| {{n + 2) [(1—M)K —(1+M) (n+ l)/(ft + 2)] 2 
+ ft 3 (l —M) 2 /4 (ft — l) 2 —ft 2 (l +M) 2 /4 (n + 2)}. 
B = (l—M) 2 K 2 {ft—1 —(ft+2) M}/2n + (l-M) K (1 + M + 4M 2 ) [n+l)/±n 
+ (l—M) {M (l+M)/4ft + (l-M) [3ft-2ft 2 -3 + M (6ft-4ft 2 -3)]/8 (ft-l) 2 }. 
C = — 3(1 —M) 2 K 2 /2ft+ 3 (l — M 2 ) K (n+ l)/4ft—f (l — M 2 )—fft (l — M) 2 /(ft — l) 2 . 
(l+M 4 ) (— 4ft 5 +8ft 4 —ft 3 —4ft 2 +3ft—l) 
+ M (l + M 2 ) (8ft 6 — 16ft 5 + 4ft 4 + 4ft 3 — 12ft 2 + 12ft —4) 
+ M 2 ( — 16ft 7 + 32ft 6 —8ft 5 —8ft 4 + 10ft 3 —16ft 2 + 18ft —6) 
+ 8 (ft-1) (l—M) (1+M) K {(1+M) 2 
(— 2 n 5 + 5 ft 4 — 2 ft 3 — 3 n 2 + 2>n — 1) 
+ M (2n 6 — 8ft 4 + 4ft 3 + 2ft 2 )} 
+ 8 (ft —l) 2 (l — M) 2 K 2 {(1+M) 2 
( — 2ft 5 + 8ft 4 — 7 ft 3 —6ft 2 + 9n— 3) 
— 2M n 2 ( n 2 — 2n — 1)} 
+ 16 (ft — l) 3 (l+M) (l—M) 3 K 3 {2n 4 — 4ft 3 — 2ft 2 + 6n — 2} 
+ 16 (ft—l) 4 (l —M) 4 K 4 (—ft 3 + 3ft —l) 
We notice that when M = 1 (which gives one of the zeros of A) B, C and E all 
vanish with it, and also E/A remains finite. Hence in this case, the term E^ 4 4 will 
not rise in importance, even when A = 0. But in this case A may have two other 
real zeros, and these are not zeros of E, so that E plays an important part in the 
neighbourhood of such zeros. 
3 (1 — M) 2 
128 (ft-1) 4 ft 3 
k 2 
