SPHERICAL ABERRATION FOR A SYMMETRICAL OPTICAL SYSTEM. 
33 
All distances parallel to the axis will be denoted by x (the attribution of the 
symbol being indicated in each case) and will be measured positively from left to 
right. 
The longitudinal aberration will be denoted by Ax, with the suffix of the medium, 
and will be reckoned positive when the point of intersection of the ray with the axis 
is to the right of the geometrical image. Thus Ax 2 = J 2 I 2 . This is opposite to the 
usual convention which is based on the fact that for positive or convergent lenses, I 2 is 
generally to the left of J 2 ,; but, in the first place, this is not universally true, and, in 
the second place, the convention adopted by us was found more convenient in handling 
the algebra. 
It is to be noted that, with the notation used, the well-known formula for a lens 
- + - = \ becomes - — - = \, the distances u and v being measured in the same 
u v j v u j 
direction. 
The distances between successive refracting surfaces we denote by c, with the suffix 
of the medium. 
In dealing with a system, especially where the initial and final media are not the 
same, it is very convenient to use an “ equivalent" Gaussian system, in which lengths 
parallel to the axis are measured in each medium in terms of a unit proportional to its 
absolute refractive index. 
If we denote the corresponding points in the equivalent Gaussian system by accents, 
we find that 
A' V — A' V — Ai-L* A' A' — o' — -ffi 
W 0 — ) 2 — 5 1^ 3 — o 2 — , 
n n 
n. 
n. 
A' T' — A3J2 
n. 
A'Jh = } & c . 
n. 
If then we denote the quantities ———, -- - -- U2 , &c., by 4-, - 7 -...; 
n r 3 ./; y, 
the focal lengths of the successive refracting surfaces. The equations connecting 
image and object in the equivalent Gaussian system are 
... may be called 
1 1 1 
A\J’ 2 A'J',7.’ 
which is of the same form as the equation connecting image and object for a thin lens 
at A'j. 
Now bearing in mind that A 3 J 2 = A,J 2 —Aj A ;i and therefore A' 3 J' 2 = A.\3' 2 —A! X A! Z , 
it is easy to show that the effects of the successive refracting surfaces in the actual 
system can be obtained by compounding a corresponding set of thin lenses in the 
equivalent Gaussian system. By dealing with the latter, we get rid of the 
asymmetry introduced by the difference of initial and final index. Of course this 
applies only to the calculation of the geometrical images. We note that in the 
