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
G. 252, 253.] 
I. Optical Imagery 
cuspidal edge, as is evident from the fact that the meridian plane is a 
plane of symmetry. Hence, the curve in which the secondary caustic 
surface is cut by the secondary principal plane has a cusp at the 
secondary focal point, where the two branches of the curve are tangent 
to each other and to the ray OFF". 
Picturing these relations as deviations or aberrations of the separate 
rays, we can say that in an astigmatic bundle of rays the distance of 
the point of intersection of a ray with the primary focal plane, belong¬ 
ing to the chief ray, from the corresponding primary focal line repre¬ 
sents the primary lateral deviation of the ray. The secondary lateral 
deviation is defined similarly with respect to the secondary focal line. 
Thus, the primary and secondary lateral deviations are equal to 
— ^ R-l-S and —uvS, 
respectively, where v denotes the angle which the ray makes with the 
meridian plane. However, these formulae are approximately correct 
only for infinitely small angles and give merely the deviations that 
depend on the second power of the angle of inclination. Consequently, 
they are inserted here not to be used, but simply in order to show the 
connection between the asymmetry-values that represent exact 
magnitudes and the ordinary conception of deviations. 
Let us employ here the symbol s to denote the length of an arc of 
the curve made by the intersection of the meridian plane with the 
wave-surface, and at the same time also introduce the following other 
symbols, namely, 
T ■ 
then generally: 
dDr dD" 
n‘u' ~w’ 
The magnitudes U, W, called the direct and transverse curvature- 
asymmetries, respectively, which accordingly are the. rates of change 
qf the principal curvatures of the surface from one point to a con¬ 
tiguous point, are not to be employed for the. wave-surface but for 
the various refracting or reflecting surfaces of.the system. Fig. 
117 the straight line O.F>F>< is, supposed to represent the normal to a 
refracting surface of a system 0f revolution, the s-curve would have to 
be a straight line coinciding with the axis, since in a*surface of revolu¬ 
tion the centre of curvature in the secondary principal section lies on 
the axis of revolution itself. Therefore for a surface of revolution, and 
for the bundle of chief rays of a system of, revolution whose wave-


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