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,
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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
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E. A. LORENTZ
beams. This effect has been observed by Professors Michelson and Gale.
In the following there will be no question of the rotation of the earth; the annual aberration only will be considered. For the explanation of this the foregoing considerations suffice. If, at a point at some distance from the earth, the direction of the rays coming from a star is given in a system of co-ordinates in which the earth is moving, one can deduce from that the direction of the rays in a system of co-ordinates fixed to the earth, and the further course of these relative rays is determined by the ordinary laws of optics.
We proceed with the discussion of some special theories. In Fresnel’s theory the ether is supposed to be at rest; its motion relative to the earth may be considered as a uniform translation, which, obviously, is irrotational. It is necessary to introduce the dragging coefficient because the ether moves through the ponderable bodies (lenses) contained in our instruments of observation.
Stokes proposed a theory in which the ether was supposed to have an irrotational motion, such that at all points of the earth’s surface its velocity is equal to that of the earth. By this latter assumption he could avoid the introduction of Fresnel’s coefficient.
However, at least when the ether is supposed to be incompressible, Stokes’s assumptions contradict each other. If a sphere moves with a constant velocity in an incompressible medium, the motion of the medium is completely determined by the condition that it is irrotational and that, in the direction of the normal to the surface, a point of the sphere and the adjacent medium have the same velocity. In a tangential direction the two velocities will necessarily be different.
So far as aberration is concerned, a modification of Fresnel’s theory is certainly admissible. When we admit his value of the dragging coefficient, we may assume the existence of any motion of the ether, provided that it be irrotational. In fact, this is a necessary condition. Suppose, for instance, that over a part of the earth’s surface which may be considered as plane the ether flows in a horizontal direction x with a velocity wx increasing with the height y above the earth. This motion would not be irrotational and would not lead to the observed aberration. Since the existence of a velocity