SPHERICAL ABERRATION FOR A SYMMETRICAL OPTICAL SYSTEM. 
57 
itself reversed, with the difference that, in the second set of refractions, the measure¬ 
ment of length parallel to the axis is reversed in direction. An examination of the 
equations (41) et seq., § 7, on which the formulae of combination are based, shows that 
this is analytically equivalent to changes in the sign of the focal length in the second 
set of refractions. 
We have therefore 
/, = /, fs = -/, M, = M, M s = I/M, M, = M, M : , = l/M, M 13 = M 1;) = I, 
and we also find that f 13 = oo. But f 13 A 13 , and f n E 13 have definite limiting values, 
and as f z does not otherwise explicitly enter into the equations of combination, no 
difliculty arises on that account. 
Now, after retracing our steps in this way, we necessarily arrive at a perfect image, 
so that A;r 4 = 0 and tan a i = tan /3 4 , leading to 
and 
/A — 0’ 
= 0 , 
B 13 -C 13 EE 0. 
These lead to the following identical relations 
A (M)/M 2 M 2 -A' (M- 1 ) EE 0,.(58) 
E (M)/M 2 M 4 —E' (M _1 ) +A (M) M~ 2 |B' (M -1 ) M _2 +3A' (M _1 ) M-'+dA' (M-^/dM} 
-A / (M- 1 )M- 2 {3B(M)-2C(M)} = 0,.(59) 
B'(M- 1 )-C'(M- 1 ) + M- 2 {B(M)-C(M)}-A(M)M- 1 M- 2 = 0. . . (60) 
Equation (58) may be written in either of the two forms 
A (M) = M 2 M 2 A'(M- 1 ) 
A (M )/n 0 2 = WA' (M- 1 )/^ 2 .(Ill) 
This we shall refer to as the third invariant relation. It shows that, if we divide 
A by the square of the initial refractive index, the coefficients of powers of M 
equidistant from the beginning and end of the development are interchanged by 
reversing the system. Equation (59) becomes on multiplying up by M 2 M 4 , using (58) 
and simplifying 
E —M 2 M 4 E' + A{M 2 B'—A/M—3B + 2C + dA/dM} = 0, . . . (61) 
omitting the arguments M, l/M of A, B, B', &c., since no confusion can occur. 
Use now the second invariant relation 
dA/dM = 4B-4C —Rr(l—M-), 
