SPHERICAL ABERRATION FOR A SYMMETRICAL OPTICAL SYSTEM. 
37 
If I 0 be a source of light placed in front of a plate of thickness c 2 and refractive 
index n, the perpendicular from I 0 on the plate being the axis of the system, it is 
easily verified that the longitudinal spherical aberration 
n tan aj 
where sin a 0 = n sin a 2 , so that 
JJ4 = 
(l 
sill 2 a„ 
n 
J l -JEhilL 0 
v n 
The branch-points here correspond to = \ir or a 2 = i.e., to grazing incidence at 
the first or second surface respectively. 
Clearly if n >1, then, since sin 2 a 0 < 1, the second grazing incidence can never occur 
for real values of a 0 . 
But if we take as our argument t 0 = tan a 0 , which removes the first branch-point 
to infinity, we find 
m 
and this has imaginary branch-points where t 0 — ± 
v\ n ~ 1) 
vergence of the Taylor’s series in t 0 is therefore given by t 0 
n 
The radius of con- 
a value which 
V(n 2 - iy 
does not correspond to any physical limitation of the rays. This applies to both the 
longitudinal and the transverse spherical aberrations in this case. 
The above example also brings out another important point ; for if in it sin a 0 is 
taken as'the argument, the branch-points are ±1, + n ; both of which correspond to 
definite physical limitations, viz., grazing incidence and total internal reflection, so 
that in this case the limitations of the Taylor’s series are also the limitations of the 
problem. 
We see then that the validity even of the expansion in a 0 may be limited by the 
existence of branch-points, and that the choice of the particular trigonometrical 
function in which we expand may exercise a considerable influence on the result. 
The limitation of the a 0 developments due to branch-points will not, however, as in 
the case of the a 2 developments, lead to vanishing radii of convergence. There is 
always a finite region within which these developments may be used. In what 
follows, therefore, we have exclusively used as argument. 
In dealing with the longitudinal spherical aberration another limitation presents 
itself. We have seen that if a 0 = X (fig. 2), a 2 = 0. It follows that the intersection 
of the emergent ray with the. axis is then at infinity, or the longitudinal aberration is 
VOL. CCXX.I-A. G 
