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 
[79, 80. 
and equation (7b) by 
z — Ç = ri cos v ; 
and add the three equations thus obtained. All the terms that are 
multiplied by — will vanish, and we shall get : 
r i 
{(1 — n2 cos2 m) Ay — n2 cos m cos v Az) 
* - r 
H--{ — n2 cos m cos v Ay + (1 — n1 cos2 v) Az\ 
r o 
H--{(1 — n2 cos2 ju) Au + w2 cos ji cos v Af} =0 
• • (8d) 
If the values of Au and A^ in terms of Ay and Az as obtained from 
equations (8a) and (8b) are substituted in equations (8c) and (8d), 
two equations will be obtained containing the unknown quantities 
^ and When one of them is eliminated, the other is given by a 
quadratic equation which has two roots. Thus, for any arbitrary com¬ 
bination of values of the angles m, y, v, we get at least one definite 
numerical value of the ratio j. Consequently, for a given direction of 
the bundle of rays r2 is proportional to r0, supposing that the latter 
varies. If r0 is infinite, so also is r2. It is not worth while actually to 
give the elimination equations here. We shall merely investigate 
certain special cases that are of interest to us. 
First, let us inquire in what cases a homocentric bundle of incident 
rays will issue from the prism as a homocentric bundle of emergent rays.1 
If all the rays emanating from the luminous point are to intersect each 
other, the conditions (8c) and (8d) must be satisfied, no matter what 
values we take for Ay and Az. Each of these magnitudes, therefore, 
may be put equal to zero, and thus the following conditions are 
1. If we put Ay =0 in equations (8c), which, according to equations 
(8a) and (8b), means also that Au =0 and A£ = Az, then 
(--J--\ (1 — ^2 cos2 j,) = o......(9a) 
Vfo r2 / 
1 IfThis whole subject has been beautifully treated synthetically by L. Burmester, 
Homocentrische Brechung des Lichtes durch das Prisma. Zft. f. Math. u. Phys., XL (1895), 
65-90. See also: J. P. C. Southall, The 'principles and methods of geometrical optics, 1910, 
pp. 97-105. (J. P. C. S.)


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