# Bauhaus-Universität Weimar

### Volltext Helmholtz's treatise on physiological optics. Volume 2. Edited by James P. C. Southall. Translated from the 3rd German edition (2)

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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