Bauhaus-Universität Weimar

Helmholtz's treatise on physiological optics. Volume 2. Edited by James P. C. Southall. Translated from the 3rd German edition
Helmholtz, Hermann von
The Sensations of Vision 
Now, as a matter of fact, this latter equation will be always satisfied, 
provided either Aa=0 or A6=0, that is, provided the intersecting 
normals lie either in the 2/2-plane or in the xs-plane. And, finally, if at 
the same time 
d2c d2c _ 
~db2~ ~d^~ 9 
the condition that adjacent rays shall intersect the ray A is satisfied 
for all arbitrary values of A a and Ab; that is, all the adjacent rays 
meet the ray A. Still supposing that = and then putting either 
Aa = 0 or Ab =0, we find, as was mentioned above, the distance z of the 
point where the adjacent rays meet the ray A by putting x=y = 0 in 
equations (3). 
For the rays in the xz-plane, A& =0; hence from equations (2) the 
distance of the point of intersection from the wave-surface is 
Z ~ C ~ H2c' 
The second of equations (2) becomes 0=0. 
For the rays in the 2/z-plane, A a = 0, and 
z — c 
Hfl * 
Finally, if 
— = —, then for all adjacent rays without dis 
db2 p 
z — c = p . 
Moreover, in this case the æz-plane and 2/2-plane are also the 
principal sections of the surface for which the curvature has its maxi¬ 
mum and minimum values; and the values of the corresponding radii 
of curvature are: 
Pb = 
da2 db2 
and hence the focal points of the bundle of rays are also at the centres 
of principal curvature of the wave-surface. 
Constitution of an infinitely narrow bundle of rays that meets the 
wave-surface in a circle. In order to get a clearer notion of the way the


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