SPHERICAL ABERRATION FOR A SYMMETRICAL OPTICAL SYSTEM. 
59 
It gives, however, on using (IV) to eliminate O', 
M“B' + B = A/M + 2C + 4<r (1 — M L ).(64) 
which is a convenient form for calculating B'. 
If now A, B, C, E are known for any system, the corresponding quantities are 
immediately obtainable for the reversed system, A', B', C', E' being given by 
equations (III), (64), (IV) and (V) respectively. 
This will generally halve the labour of calculations, if it is found desirable to 
tabulate these constants for a complete set of lenses. It will then be sufficient to 
start from the equi-convex lens and vary the curvatures in one sense only. 
Incidentally we note also that the second invariant relation enables us to find C, 
so soon as A and B are known, so that only A, B and E require to be calculated. 
The aberration constants for a reversed system have a further important application 
in the case of lenses. Consider a positive lens (fig. 4), the initial ray converging to 
I 0 and the final ray to I 4 . If now we interchange the full and dotted portions of the 
initial and final rays in fig. 4, we obtain, since here the initial and final media are the 
Fig. 4. 
same, the case of a ray going through a lens in which the front and back character of 
the two surfaces have been interchanged. In fact n and r 3 have been interchanged 
and the sign of the thickness c 2 has been reversed. This leads to a negative lens, of 
the same numerical power as the original positive lens, and with the same mean 
curvature, but a negative thickness. Such a lens, of course, is not physically 
realisable, although a part of it can be physically obtained by rotating the wedge 
beyond the intersection V of the two surfaces. 
