SPHERICAL ABERRATION FOR A SYMMETRICAL OPTICAL SYSTEM. 
41 
component of the combination and the other factor to the second. In addition the E 
for the combination involves linear terms in the E’s of the components. 
These results are found to hold good in the more general case of the combination 
of three or more systems. It will follow that if the constants A, B, C, E are 
tabulated for lenses of various curvatures, the effect of the combination can be traced 
relatively easily and the aberrations corrected, so far as possible, by suitably bending 
the lenses, while keeping the general arrangement and the magnifications the same. 
Explicit values of the constants for the single refracting surface and a single thick 
or thin lens have been obtained and are tabulated for reference, so as to he available 
for eventual computation of the required tables. We have also given some numerical 
values for a single lens, and a numerical test of the accuracy in this case, which works 
out at about -g^jo of the total aberration for the range of cases taken. 
The corresponding formulae with sin y instead of tan /3 as argument are discussed, 
and it is shown that the equations of combination are of the same form as before. 
Certain invariant relations between the coefficients in A, B, C, E are developed, 
which enable various calculations to be simplified and in particular to determine these 
constants for a system reversed, when they are known for the direct system. This 
will generally halve the work of tabulation. 
§ 5. The Single Refracting Surface. 
Using the general notation described in § 2 , consider refractions at a single refracting 
surface. 
Let \fr o, f., denote the angles of incidence and refraction, so that fo = CjPJo, = 
C,P,L (fig. 1 ). 
Let CJ 0 = X 0 = x 0 , CjL = X 2 = x 2 + Ax 2 , C x J 2 = x 2 we then have the set of refraction 
equations 
sin f 0 = pft\ = x 0 sin a 0 /r x .( 8 ) 
sin — pjr } = X 2 sin a fr A .(9) 
n 2 p 2 = n 0 p 0 ..(10) 
.( 11 ) 
Let A = A J 0 , g 2 = A'jJ'g in the “ equivalent ” Gaussian system (see § 2 ). Then 
nf 0 = x 0 + r 1} nf 2 = x 3 -+r u . ( 12 ) 
and we find, using the first approximation when a 0 , &c., are small 
1/6-1/6 = i//.- • 
(13) 
