SPHERICAL ABERRATIONS. 23 
(b) APLANATIC POINTS OP A SINGLE REFRACTING SPHERICAL SURFACE. 
The wave-front developed in Fig. 10 by waves emerging from the axial 
point P and entering the glass surface is not spherical but is a warped surface 
whose normals or ray directions intersect the principal axis at different 
points. This deviation of the refracted wave-front from the spherical shape 
is called spherical aberration and its effect is to produce a wave-front whose 
rays appear to come from very different points and not the common point P, 
as was the case while the waves were still propagated in air. The shape of 
the refracted wave-front is directly dependent on the position of P and 
changes its shape as P is moved nearer to, or farther away from, the spheri- 
cal surface. It can be shown, moreover, that for waves converging to one 
particular position of P, the refracted wave-front is still spherical and con- 
verges to a definite point P'. In Fig. 12, let a spherical wave converging in 
FIG. 12. 
air toward the point P be incident at the glass spherical surface BM. Let 
n n' 
the distance CP = .R (as in Fig. n) and the distance CP' = R, where R 
n n 
is the radius of the spherical surface BM. Then the Weierstrass construc- 
tion shows that any ray DBP is refracted to the point P', the optical path 
BP' in glass being equal to the optical path BP in air. As the ray DB is any 
ray converging toward P, the construction is valid for the entire area of the 
spherical surface and not alone for paraxial rays. The refracted wave-front 
is accordingly strictly spherical with A' as the center. The points P and P' 
are conjugate points and for them the relation is readily obtainable from 
Fig. ii. 
CP n' sin u' 
= constant (n) 
CP' 
n sin M 
which is valid for all possible angles u ; these two points are free, therefore, 
from the spherical aberration resulting from the use of other than central 
