382
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.]
CONFERENCE ON MICHELSON-MORLEY EXPERIMENT 383
V. PROFESSOR PAUL S. EPSTEIN (CALIFORNIA INSTITUTE OF technology)
The result of Professor Hedrick’s analysis is that the two beams of light acquire a difference of phase
d—8r = hß2 cos 2#
and a difference in direction
Aa = ß2 cos 2#
Fig. 16
in which terms of the fourth order are neglected.
Let us now choose the plane in which we observe the fringes as x = o of a cartesian system (Fig. 16). We can then represent the two waves by the formulae (s = light-vector)
I 27r I
s = A cos — (x cos a+y sin a+c/)+5 ,
s' = A cos cos a'+^ sin a'+c/)+5'j .
The illumination of the screen is then (x = o, sin a = a)
(s-\-sr)2 = ^A2 c°s2 |^- y(a— a')+5 — ô'j cos2 ;y(a+a') + 2c/+5+<5/j == 2A.2 cos2 |^- y(a— a')+ô — 5'j .
We have maxima, where the argument of the cosine is a multiple of 7r.
The position of the central fringe is therefore given by