378 E. R. HEDRICK
Therefore
_ r sin 0 _ rß sin 0
P~BM+MM' cos 2a~ EMß , '
1 MM’ß cos 2a
cos 2 a
rß sin 0 cos 2a
r—rß sin 0 sin 2a * But we have tan a = i — £; hence, to terms of the second order,
. £2
sin a=i--, cos 2a = H—.
2 2
Substituting these values in the expression for tan p, we have tan p = ß2 sin 0 (cos 0 —sin 0) , q.p.
Now if we denote the angle CAfT by <£ and the angle CArTr by \f/, we have (remembering that 0 and yf/ are negative angles)
<£+p=2a —p or $=2(p— a)
^ = 27 — 180° .
Thus the positive angle
T'A'T = (j) — \f/=2p— 2a— 27+180° .
To determine the tangent of this angle, we find
tan ( — 2a) = ^
tan (27 — 180°)
i-(i+£)2’
= 2(i+0
i — (i+£)2 ’
and therefore
*£2
tan ( — 2a — 27+180°) =-— q.p.
4 — f4
From this we obtain
* /, /\ £2+2ß2 sin 0 (cos 0 —sin 0)
i — 2ß2^2 sin 0 (cos 0 — sin 0) ?
since tan 2p = 2ß2 sin 0(cos 0 —sin 0) to terms of the second order. Substituting for £ and reducing, we have finally
tan (cj)—\p)=ß2 cos 20 .
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.