346
H. A. LORENTZ
case. One has simply to follow the elementary waves and to construct the successive wave fronts. As to the coefficient of entrain-ment, I mention that Fresnel found it at first on a mechanical basis from his elastic theory of light. This was a very remarkable performance at that time.
If we neglect terms in w2 we find
v = u-\-kw COS # ,
II kw Q
- =---- COS V .
■ V u u2
The course of a ray of light between given points is determined by the condition (Fermat’s principle)
or
-/( f-
cos â ds ) . (2)
u
Suppose now
* f = const.,
u*
that is, k is inversely proportional to /x2. For fi=i, there is necessarily k = i. Thus it follows that
*=-a.
p
The second term in (2) becomes then
— I w cos û ds c2
Now let the motion of the ether in our diagram be irrotational, so that w depends on a velocity potential <p,
w = grad <p .
Then the integral
jw cos û ds
for a path between two given points P and Pf becomes
rd* /jc
CONFERENCE ON MICHELSON-MORLEY EXPERIMENT 347
This has the same value for all paths, and the condition (2) becomes simply
a Cds
8 I — = 0 ,
J u
as if there were no ether motion. Thus we conclude that the course of the ray is not altered by the motion of the ether.
The considerations given above also include cases of reflection and diffraction.
Now let there be two paths, 1 and 2, for a ray of light from a given point P to another given point Pr. The time required for light going over them is, for the first path,
058
and, for the second path,
Cds Cds i r
I — = I----I w cos v ds .
J2v J2u c2J2
The last terms in these expressions are equal. Therefore, the difference between the two times is not altered by the motion of the ether. This motion, then, has no influence on phenomena either of interference or of diffraction.
It may be remarked that the difference between the times just considered will be altered by the motion of the ether if this motion is not irrotational. The change is given by the difference of the two integrals
w cos â ds and £ w cos ê ds
taken for the two paths between P and P\ For this difference one can write the line integral of the velocity w taken over the closed circuit formed by the two lines.
Let us consider, for instance, the earth’s rotation. If the ether is stationary, its motion relative to the earth will be a rotation in the opposite direction. If now a large horizontal circuit fixed to the earth, e.g., a rectangular one, is traveled over in opposite directions by two beams of light, the relative motion of the ether will change the position of the fringes produced by the interference of these