E. R. HEDRICK
5. POSSIBLE EFFECT OF DIFFERENCE OF ANGLE ON THE POSITION OF THE INTERFERENCE BANDS
Figure 15 represents the network of wave fronts of the two interfering rays. The space between Fz and F2 represents the central bright fringe.
Let the ray s change its direction (relative to the ray t) by the amount Aa. If the new wave front f2 meets the old wave front fi near
the edge of the fringe at N, the center of the fringe will be shifted to the left from M to M', The amount of this shift, which is due wholly to the change of angle between the interfering rays, will depend on the distance of the point of intersection of the consecutive wave fronts from the edge of the fringe. As this point approaches the center of the fringe, the distance MM' becomes negligible. In
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.