Let:
V = velocity of light.
v = velocity of the earth in its orbit.
D = distance ab or ac, Fig. 72.
T = time light occupies to pass from a to c.
T1 = time light occupies to return from c to
a1 (Fig. 73).
Then T = D / (V - v) and T1 = D / (V + v)
The whole time going and coming is
T + T1 = 2D [V / (V2 - v2)],
and the distance traveled in this time is
2D [V2 / (V2 - v2)] = 2D [1 + (v2 / V2)]
neglecting the terms of the fourth order. The length of the other path is evidently
2D [1 + (v2 / V2)]1/2,
or to the same degree of accuracy,
2D [1 + (v2 / 2V2)].
The difference is therefore D (v2 / V2).
If now the whole apparatus be turned through 900, the difference will be in the opposite direction, hence the displacement of the interference fringes should be
2D (v2 / V2).
Fitzgerald and Lorentz
Fitzgerald and Lorentz independently suggested that motion might cause a change of length in the direction of motion. The precise change can be shown very simply by making one distance equal to the other multiplied by a factor shown here as "X":
2DX [1 + (v2 / V2)] = 2D [1 + (v2 / V2)]1/2
Simplifying:
X [1 + (v2 / V2)] = [1 + (v2 / V2)]1/2
Solving for X:
X = [1 + (v2 / V2)]1/2 / [1 + (v2 / V2)]
Which comes to:
X = 1 / [1 + (v2 / V2)]1/2
If we substitute the modern "c" for "V", we have the Lorentz factor as it is commonly seen today.
1 / [1 + (v2 / c2)]1/2
Bear in mind that this was not the reason for the low results of the
Michelson-Morley experiment or of low results of the ones of a
similar nature
which followed. The detail can be found be reading Is There
a Dynamic Ether on this website.