CONFERENCE ON MICHELSON-MORLEY EXPERIMENT 371
Similarly, the square of the resultant displacement in the interfering beams below the dividing line is found to be
2#2 i-|-cos ^ (x+a)J cos2 co(/ — 8) .
The intensities, being proportional to the squares of the amplitudes, can be represented by
J^Æa^i+cos ^ (%— a)j
and
/2 = &a2£i+cos ^ (#+a)J .
Now co = 27tv where v = frequency of the light. Hence co/c=27r/X. Therefore
and
/i = &a2|^i+cos ^ (x— a) j J2 = &a2£i+cos —■ (#+a)j .
For values oi x = n\/4, where n is an integer,
7i = Æa2^i±cos >
the sign being positive for even values of n and negative for odd values. The same expression holds for Z2; hence, under these conditions,
Jx = /a.
To the observer, then, the field of view is equally intense on both sides of the dividing line when x = n\/4.
We have now to determine the least change in x from this value which will produce a perceptible difference in illumination in the two sides of the field. If x is given the variation dx while a is kept constant, the difference in intensity will be
372
ROY J. KENNEDY
Now
Similarly,
Therefore
dlx , 47rka2 . 47ra d# X X
dl2 —dirka2 . 4xa -r-= - sin .
dx X X
. I"87rka2 . 47ral .
a7=±b^sinVr’
the sign being of no importance.
The perceptibility of the variation is determined not by 37 alone, but by the ratio of 37 to the total intensity, 7t or I2. According to the Weber-Fechner law, if hi is taken to be the least perceptible variation in intensity, the foregoing ratio is nearly constant for a considerable range of intensities. With this meaning of 37, Sx becomes the least detectable change of position of M2.
If initially we have uniformity of illumination, we have from the equations above,
. 47ra
37 8tt
^ = ÔX
I X , 47ra
I ± COS -7—
or
47ra
, X 8/,±COS X
bx = — —
8w I . 4xa sin ——
If now 37/7 were a true constant, we should have for the case of negative sign, which corresponds to dim illumination of the field, the sensibility of the apparatus increasing indefinitely as the factor a was made smaller. 7 decreases with a, however, and the Fechner “constant” soon diminishes rapidly. Nevertheless, the conditions of illumination and contrast here are similar to those in the halfshade polariscope, and from the theory of the Lippich instrument it appears that 37/7 equals about 8X10“3. The lack of perfect planeness in the mirrors and of equality of intensity in the interfering beams is a further limiting factor; a little experimenting indicated