SPHERICAL ABERRATION FOR A SYMMETRICAL OPTICAL SYSTEM. 
69 
Hence, whenever we deal with incident rays actually passing through the front 
focus—and this necessarily occurs as soon as the results of the present paper are 
applied to aberrations off the axis—the modified B, C and E have to be used. 
§ 15. Combination Formulae for More than Two Systems. 
The combination formulae (47), (48), (51) are capable of explicit generalisation for 
any number of systems. 
In the case of A and C successive applications of equations (47) and (51) lead at 
once to the results 
«/l35 ■■■ 2n+l"^ 135... 2n + l = ifl-^l) M 2 35 ... 2n + 1 M 2 3s2n+ 1 
+ (/ 3 A 3 )M 2 5 ... 2 n+ 1 M 2 5 ... 2n+1 + ... 
+ (/k—iA 2n —i) M 2 2n+1 M 2 2n+ ,+ i /’ 27i+ iA 2n+1 . . (71) 
Ci3 5 ...2n+1 — C 1 M 3 3 5 ... 2n + 1 4-C 3 M 2 5 __ 2n+1 +... 
+ C 2n _ 1 M 2 2n+1 + C 2n+1 .(72) 
in which we have reverted to the notation of §§ 2, 7. 
In the case of B, we have for three systems 
Bi % — B,<5 + BjM.j 2 + AfiLMfifijf^ 
= B 8 + B 3 M 5 2 + B 1 M 3 s 2 +(/ 1 A 1 ) M 35 M 35 2 //3 5 + (/ 3 A 3 ) MsM 5 2 // 5 , 
and the general law of combination is at once obvious. 
We have 
B 
135 ••• 2n+l 
BiM“ 35 _o n+1 +B 3 M“ 5 . 2n+1 + ... + B 2n _!M 2 2n+1 + B 2n+1 
+ (/iAx) (MM 2 //) 35 ... 2B+1 + (/ 3 A 3 )(MM 2 //),., n+1+ ... +(/ 2 n _ 1 A 2n _ 1 ) (MM 2 //) 2n+1 . (73) 
The case of E is more difficult and we have not been able to obtain a form in which 
it can be written down for the combination of n systems. 
But we can deal with it as follows, by determining the contribution of any one 
component to the whole :— 
Consider three systems 1, 3, 5. It will be convenient to look upon 3 as a single 
lens, forming part of a larger system. 1 will then be the system of lenses preceding 3, 
and 5 the system following. 
Applying equation (49), simplified by the use of the second invariant relation, we 
find 
/,35Ei35 = / 5 E 5 +/ 13 Ei 3 M 5 2 M 5 4 
+/ 13 A 13 M 5 2 M 5 2 (3A 5 /M 5 - 3B 5 +4C 5 + 4«r {1 - M 5 3 }) 
+ 3B 13 / 5 A 5 M 5 2 —2C 13 / 5 A 5 M 5 2 . 
YOL. CCXXI.-A. L 
