Bauhaus-Universität Weimar

Helmholtz's treatise on physiological optics. Volume 1. Edited by James P. C. Southall. Translated from the 3rd German edition
Helmholtz, Hermann von
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 
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 
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 
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


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