CONFERENCE ON MICHELSON-MORLEY EXPERIMENT 381
this case the effect is to widen the fringe without appreciably altering the position of its center.
The foregoing is based, of course, on the hypothesis that the distances traversed by the two rays are not changing. If the distance traversed by t changes, then the wave front LM takes a new position indicated by the dotted line.
Now actually, the two changes occur simultaneously; and as both are periodic it seems inevitable that the point of intersection of fz and f2 should at times come near enough to produce an appreciable displacement.
It is conceivable, of course, that the two effects might neutralize each other, as indicated at the bottom of the figure, where the point of intersection of consecutive rays is supposed to come outside of the central fringe.
6. FORMULA FOR THE SHIFT OF THE FRINGES
It seems to be impossible to obtain a formula for the amount of the shift of the fringes without making certain assumptions concerning the nature of interference phenomena.
The simplest procedure seems to be to study the network of parallelograms so drawn that each system of parallel sides represents the successive positions of some definite phase of the waves of the corresponding ray.
Let Figure 15 represent this network of parallelograms, and let a denote the distance of the middle of the central fringe to the right of some convenient origin. This distance will depend upon the initial adjustment between the distances traversed by the two rays.
If it is agreed that only the relative positions and lengths of paths of the two rays s and t are involved, we may suppose that one of the rays remains fixed in length while the other remains fixed in direction.
Let the ray t be supposed to rotate about a point in the neighborhood of its image. Then one of the lines / representing a certain phase of t may be supposed to envelop a circle. Let b denote the distance to the right of the origin of the point of contact of this circle with/in its initial position.
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E. R. HEDRICK
Use the following notations: a' equals the new value of a due to change of length of s, b' equals the new value of b due to change of direction of t, Mr denotes the middle of central fringe after s has changed length, and Mn denotes the final position of the central fringe.
After the apparatus has rotated through an angle 0, we have
af'=a—Dß2(i —cos 20) , b' = b+Dß2(i —cos 20) ,
MfMn _a—b — 2Dß2(i —cos 20) i —cos 20 r+cos 20 *
Adding M'M" to a', we have for the position of M", the new middle point of the central fringe,
//_q(r+i)+6 cos 20 — Dß2[r+2 — (r+3) cos 20+cos2 20]
r+cos 20 ‘
7. POSITION OF MAXIMUM SHIFT
The formula of the last section shows that the fringes have a periodic motion across the field of the telescope. The maximum and minimum positions of M, however, depend upon the values of the quantities a, b, and r. The values of a and r depend upon the initial adjustments, and the value of b would very likely be different for experiments performed at different times.
If, then, no effort is made to control the values of these quantities, we must suppose that the maximum and minimum positions for a series of experiments will have an entirely random distribution. It will not be legitimate, therefore, simply to average the readings of a series of observations, as was done in the Michelson-Morley experiment. In fact, there would seem to be a high degree of probability that this procedure would lead to a quite small result in case it is applied to a large number of observations.
[Professor Hedrick remarked at the end of his report that his results had been discussed by Professor Epstein from the physical point of view. This discussion has kindly been supplied for publication here.]