294
Dioptrics of the Eye
[253. G.
surface is a surface of revolution, the magnitudes denoted by W and S
(p" — p>) tan d
are given by the formulaeW = —-;-and *S = — (q—p)tanQ,
P'p"* 2
where d denotes the angle between the normal to the surface or the
chief ray of the bundle and a transversal plane perpendicular to the
axis.
If the bundle of rays is anastigmatic along a certain ray, the two
caustic surfaces are in contact with each other at the focal point.
The radius of curvature of the r-curve is the same as in the anastig¬
matic bundle of rays, and the curvature of the s-line is found from the
transverse asymmetry-value by means of the expression
R-2S
S2
In the case shown in Fig. 118 (R —2S) has therefore the same sign as R.
When the two asymmetry-values have the same sign, the focal points
f of a consecutive ray lie both on the
__ same side of Ühe focal plane at F, and
the intersection of the two caustic sur¬
faces with this focal plane is a curve
which has a cusp at the focal point com¬
mon to both surfaces. The tangents to the two branches of the curve
which come together in the cusp make with each other the angle
whose trigonometrical tangent is
2 y/RS
R-S '
Fig. 118.
Since in the cases that ordinarily occur the direct asymmetry-value
exceeds the transverse value, this angle whose bisector lies in the
plane of symmetry is an acute angle, and consequently the cross-
section of the bundle of rays is a characteristic figure similar to an
arrowhead.
If the calculation is carried a step further by another differentia¬
tion, the imagery-laws of third order are derived; which give, for
example, formulae that enable us to find in a system of revolution the
curvatures of the image-surfaces at the point of intersection with the
axis, together with the distortion-value of the primary magnification-
ratio and the coefficient of variation of the asymmetry-value. The
latter is the differential coefficient of the asymmetry-value for a
consecutive chief ray with respect to the distance of the object-point
from the axis, supposing this distance to become gradually less and
less; and accordingly the direct asymmetry-value is always three
times the limit of the transverse asymmetry-value. Again, the