SPHERICAL ABERRATION FOR A SYMMETRICAL OPTICAL SYSTEM. 
51 
§ 7. Combination of Two Systems. 
Call the systems 1 and 3, and the initial, intermediate and final media 0 , 2 , 4. 
f, f 3 are the focal lengths of the systems, as defined in § 2 . M 1; M, are the trans¬ 
verse and ray magnifications in the first system, M 3 , M 3 in the second system. 
M3+AM3, M 3 + AM 3 refer to the transverse and ray magnifications in the second 
system when I 2 , the true intersection of ray 2 with the axis, is taken as the object 
point for the second refraction (instead of J 2 , which refers to transverse and ray 
magnifications M 3 , M :! ). 
Using the notation of §§ 2 , 5, we assume 
Ax 2 — n 2 f (Ai^ 2 2 + E^ 2 4 )/(l+ B^ 2 2 ).(41) 
T — C (1 + B 1 C 2 )/(1 + QC 2 ) 
(42) 
where q 2 = {a 2 }> t 2 = ^ j, and B l5 C 4 have suitable forms according as sines or 
tangents are considered. The constant E 4 is zero if the systems reduce to single 
refracting surfaces. Its form in the more general case will be discussed later. 
If we denote by A,ic 4 that part of Acc 4 which is due to Ax 2 and by A 3 ;r 4 that which 
is introduced by the aberrations proper to the system 3, 
Ai « 4 — n i f 3 AM 3 
where AM 3 is obtained from Ax 2 by means of 
leading to 
Thus 
Again 
Ax 2 = n 2 fz {l/(M 3 + AM 3 ) — l/M 3 }, 
AM 3 = — M 3 2 Axf {n,f, + M :i Ax 2 ). 
A,x 4 = nj x M 3 2 (A^ + EjC 4 )/!! +tf (B 4 + MaAj/j/Za)}. 
• • (43) 
A 3 x 4 = 
_ n^fs {(A3+AA3) q 2 (M 3 +AM 3 ) 2 + (E 3 + AE 3 ) q 2 (M 3 + AM 3 ) 4 | 
1 + (B 3 + AB 3 ) g 2 2 (M :! +AM :{ )- 2 
nj z [{ A 3 M 3 - 2 + AM 3 d ( A 3 M 3 - 2 )/cM 3 } q 2 2 + E^ 4 M,-‘]/( 1 + B 3 * 2 2 M 3 - 2 ), ( 44 ) 
retaining only terms of second order in t 2 . Writing now for q 2 in the above its “ first 
order” equivalent £ 2 2 + 2 (Bj — C 4 ) t 2 , we have 
Axjn i = A l x i /n i + A^xjri^ 
:/iM 3 2 A 1 + / 3 A 3 /M 3 2 } t»+t* [/1 {B :( AjM/M.r 2 +E 4 M 3 2 —AjM 3 2 d 
+ A 1 A 3 M 3 m 3 - 2 }+/ 3 {B 1 A 3 M 3 - 2 +2A 3 (B 1 -C 1 )M 3 - 2 + E 3 M3- 4 }l 
{1 + (B 1 + A 1 M 3 / 1 M )}{1 + B 3 ^/M 3 2 } 
(45) 
