SPHERICAL ABERRATION FOR A SYMMETRICAL OPTICAL SYSTEM. 
55 
or, dropping the factor 
= {3B 3 M j (U 1 +/ 3 B,//, 1 ) + 3U 3 (U,+/ ) B,//; j ) 
-M 3 2 M,B, dBJdM ,}. 
The term 
3U :! M j s M s (U 1 +/ j B 1 // 1 ,) 
is clearly of the form cubic x cubic and the terms leading to expressions of 7th degree 
reduce to 
M/M 3 {MaUi (3B 3 —MgdBa/cZMg) + M 3 Bj (3B 3 / 3 // 13 -M 13 dB 3 /dM 3 )}, 
and since dM v JdM-., — f Jf iM this can be written 
M/M, {M,U, (3B,—M ! <*B 3 / ( M 3 )UM3B 1 /,/4) (3B 3 -M 1s dBJdMJ}. 
But since B :i is a cubic, in either M 3 or M 13 , 3B 3 —M 3 c?B 3 /c£M 3 and 3B 3 —M 13 dB.JdM n 
are both quadratics in M 3 or M J3 . 
The above expression therefore reduces to M 3 xsum of two quantities each of form : 
cubic in M 13 x quadratic in M 13 , that is, to a sextic in M 13 . 
We see, therefore, that those terms in E 13 which do not involve Ej or E 3 are a 
polynomial of sixth degree in M 13 or M 1:s . 
It follows that for a lens E is necessarily a sextic in the magnification. 
Suppose now that E! and E 3 are both sextics in M 1? M. ; respectively. Then E 3 is a 
sextic in M 13 and E,M 3 4 M 3 “ will also be a sextic in M 13 , that is E 13 will again he a sextic 
in M 13 . 
Hence, since any system is built up of combinations of lenses or single refracting 
surfaces, we find that E is a sextic polynomial in M for any system. 
Examination of particular cases shows that E is not, in general, divisible by A, so 
that the vanishing of the latter does not usually involve the disappearance of the 
second order terms. 
§9. Invariant Relations. 
Certain relations exist between the coefficients A, B, C, E which remain the same 
in form, whatever the number of refracting surfaces. One of these we have already 
dealt with, namely the fact that 
A-MB 
reduces to an expression of the third degree, i.e., the coefficients of highest degree in 
M in A and B are the same. 
This we shall refer to as the first invariant relation (I.). 
A second invariant relation takes the form 
B-C = f (1 — M 2 ) + £ dA/dM, 
I 2 
(II) 
