SPHERICAL ABERRATION FOR A SYMMETRICAL OPTICAL SYSTEM. 
35 
will involve for I 0 a positive ray magnification M exceeding some definite finite limit. 
The Gaussian image point J 2 will then lie a finite distance in front of F 0 . 
Consider first a nearly paraxial incident ray I 0 P. Such a ray will be refracted 
approximately according to the Gaussian law and will emerge at an inclination a 2 , 
where a 2 is nearly equal to «„/M and both a 0 and M being finite and positive, a 2 is 
also finite and positive. The ray emerges as QR, passing through a point I 2 , finitely 
different from J 3 . 
As a 0 increases, a 2 at first increases with it, but as a 0 reaches the value X, 
corresponding to the inclination of the ray I 0 L which touches the front focus caustic, 
a 2 is again zero. Hence between those two values a 2 has at least one maximum , and 
for a given value of a 2 there are at least two values of a 0 . 
Thus, within the range of values which are of practical importance, a 0 is a many¬ 
valued function of a 2 having one or more branch-points, of which the one of least 
modulus corresponds to the first maximum of a 3 . 
Now, within the same range of values, all the aberrations must be given as single 
valued functions of a 0 , since clearly there can only be one physical emergent ray, 
corresponding to one given physical incident ray. This statement, as we shall see, 
needs to be qualified when we are dealing with purely geometrical rays, but this need 
not affect the present stage of the discussion. 
In consequence, if any aberration be expressed in terms of a 2 —or of any trigono¬ 
metrical function of a 2 —that aberration must, in general, be a many-valued function 
of a 2 , having for its branch-point of least modulus the first maximum value of a 2 
mentioned above. It follows by a well-known result in theory of functions, that 
no Taylors series in a 2 , or in sin a 2 , or tan a 2 , can be valid for values of a 2 
exceeding this modulus numerically. For such values the series will be definitely 
divergent. 
It is interesting to consider what happens when I 0 is on the other side of F 0 , so 
that we are dealing with a large negative magnification. In this case no real 
tangent can be drawn from I 0 to the front focus caustic and the value of a 0 , for 
which a 2 = 0, is a pure imaginary. But here again, although we are now dealing 
with imaginary values, we get two values of a 0 for a given (pure imaginary) value 
of cl 2 , and, although no such maximum of a 0 occurs in the purely real values, the 
modulus of the imaginary branch-point limits the validity of Taylor’s series in a 2 as 
before. 
Thus there exists always a certain range, extending a finite distance (depending 
on the nature of the optical system) on either side of the front focus, within which 
no development of any aberration in powers of a 2 or of its trigonometrical functions 
(or, indeed, by similar reasoning, of any inclination of the ray, except in the 
original medium) is valid for the whole pencil of rays which actually traverse the 
system. 
Indeed, as the object point I 0 approaches the front focus, it is clear that both X and 
