Bauhaus-Universität Weimar

Titel:
Helmholtz's treatise on physiological optics. Volume 1. Edited by James P. C. Southall. Translated from the 3rd German edition
Person:
Helmholtz, Hermann von
PURL:
https://digitalesammlungen.uni-weimar.de/viewer/image/lit39649/318/
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
        

Nutzerhinweis

Sehr geehrte Benutzer,

aufgrund der aktuellen Entwicklungen in der Webtechnologie, die im Goobi viewer verwendet wird, unterstützt die Software den von Ihnen verwendeten Browser nicht mehr.

Bitte benutzen Sie einen der folgenden Browser, um diese Seite korrekt darstellen zu können.

Vielen Dank für Ihr Verständnis.