Bauhaus-Universität Weimar

Titel:
Helmholtz's treatise on physiological optics. Volume 2. Edited by James P. C. Southall. Translated from the 3rd German edition
Person:
Helmholtz, Hermann von
PURL:
https://digitalesammlungen.uni-weimar.de/viewer/image/lit39650/96/
84 
The Sensations of Vision 
[69, 70. 
Let r0, rh . . . rm denote the lengths of the portions of the path of the 
ray in the various media, and let n0, ni, . . . nm denote the indices of 
refraction; then 
^ = n0r0 + niri +• • • + nmrm . 
In this equation r is a function of a, b and c; consequently, 
d'V drm a — xm 
— flm ~ — flm 
do, do? y fri 
d'V drm b — ym 
Mm Mm 
db db rm 
d'V drm c — zm 
= flm — = flm y 
de de T m 
and hence equations (2) become : 
dc 
Ça %m) “h (c %m) 0 
da 
dc 
(b ym) + Çc Zm) — 0 
db 
(2a) 
The interpretation of these equations is that the straight line drawn 
from the point ym, zm) to the point (a, b, c) is the normal to the 
surface ^ = C at the latter point. 
The easiest way to see this is by recalling that the distance measured 
along the normal to the surface is itself the longest or shortest distance 
from a given point to the surface. Now if the distance 
rm = V (*m - a)2 + (ym — b)2 + (zm - c)2 ; 
between (xm, ym, zm) and (a, b, c) is to be a maximum or minimum, then 
we must have: 
dvm, dvm, de a dc C Zm, 
0 = ---1----=---1— ■--, 
da dc da rm da rm 
dr m dr m de b y m dc c Zm 
0 =---1----=----1--‘-, 
db dc da rm db rm 
These conditions are the same as equations (2a). And so the ray that 
goes through (a, b, c) is normal to the surface ^ = C that passes through 
this same point. 
As the light traverses equal optical lengths in equal times, it 
reaches all points of the surface ^ = C at the same instant, and hence 
this surface is a wave-surface, that is, it is a surface which contains all
        

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