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
200, 201.] 
§16. Illumination of the Eye and the Ophthalmoscope 
where a, ß denote the angles made by the normals to a and b, respec¬ 
tively, with the straight line connecting these elements. Similarly, the 
quantity of light which would proceed from b to a would be the same 
as before, supposing the brightness of the element b were the same as 
that of a in the first case. 
Let 111,712 denote the indices of refraction of the first and last media, 
respectively, of a centered system of spherical refracting surfaces; and 
let a and ß designate twTo surface-elements near the axis and per¬ 
pendicular to it, in the first and last media, respectively. Assuming 
that the brightness of the surface-element a is nx2H and that of ß is n22H, 
the quantity of light that goes from ß to a is equal to that that goes from 
a to ß. 
Not to make the proof more complicated than is necessary for the 
applications we have here in view, the losses of light due to reflection at 
the refracting surfaces may be neglected; and it may be also assumed 
that the rays are all incident on the refracting surfaces at angles so 
small that the cosine of any such angle may be put equal to unity; 
although the proposition is true when this is not the case. 
1. Case when ß does not coincide with the image of a. 
In Fig. 100 the straight line AC represents the axis of the optical 
system, with its first and second principal points at F and G, respec¬ 
tively. The element of surface a in the first medium may be repre¬ 
sented by a point in the diagram, since it is infinitely small; and the 
£ & 
Fig. 100. 
image of a in the last medium is designated by 7. Consider the bundle 
of rays proceeding from a; its cross section in the first principal plane is 
represented by/i/2, and the corresponding cross section in the second 
principal plane by gxg2) so that gxg2 =/i/2 = say- The other surface- 
element ß is supposed to be in a plane perpendicular to the axis at the 
point designated by B; and 6162 represents the cross section of the 
bundle of rays in this plane. Let A and C designate the feet of the 
perpendiculars on the axis drawn from the conjugate points a and 7, 
trespectively. According to equation (1), the quantity of light coming 
from a that falls on/i/2 is 
nx2H- a. $ 
AF2 ’


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