CONFERENCE ON MICHELSON-MORLEY EXPERIMENT 379
This formula was obtained by Righi, who concluded from it that a rotation of the apparatus (in the ideal experiment) through 90° would produce absolutely no effect, since, although the distances traversed by the two rays are exchanged, yet at the same time their positions are also exchanged; that is, the ray having the longer path occupies the same relative position with respect to that one having the shorter path, after the rotation as before. It follows that the pattern of the interference fringes after the rotation cannot be distinguished from that before the rotation.1
4. THE USUAL THEORY
A careful computation of the difference in the length of path traversed by the two rays yields precisely the same result as is given in the usual theory, namely, ß2 cos 20. As a matter of fact, it was also known that, under ideal conditions, there exists a second-order difference in the directions of the final rays.2 The view has been, however, that this difference in direction could affect the difference in time, up to the telescope, and therefore the difference in phase, only by an amount of the third order in ß. Thus it was thought that this difference had no appreciable effect on the position of the interference fringes, although it might modify the width of the fringes.
In the next section we investigate, as far as we can, the legitimacy of this older view. As a basis for this investigation, we use a conception of the mechanism of interference phenomena which has been employed in other connections. Whether its application in the present instance is legitimate is perhaps a matter to be decided by experiment, but there does not seem to be any very evident reason why it cannot be employed with safety.
We may mention here that, quite apart from any special hypothesis concerning interference phenomena, the argument of Righi given at the conclusion of the preceding section proves absolutely that the second-order change in the angle between the final rays is by no means negligible, since in the ideal experiment the expected shift for a rotation through 90° is proportional to 2ß2 if that angle is not taken into account, but is zero when it is taken into account.
1 See, for example, Larmor, Aether and Matter, p. 53.
2 See Michelson and Morley, loc. cit.; also Larmor, p. 48.
380
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
Fig. 15
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