This post relates to a previous post *here*.

The Michelson-Morley experiment is a famous “null” result that has been understood as leading to the Lorentz transformation. However, an elementary error has persisted so that the null result is fully consistent with classical physics.

The Michelson-Morley paper of 1887 [*Amer. Jour. Sci*.-*Third Series*, Vol. XXXIV, No. 203.–Nov., 1887] explains using the above figures:

Let

sa, fig. 1, be a ray of light which is partly reflected inab, and partly transmitted inac, being returned by the mirrorsbandc, alongbaandca.bais partly transmitted alongad, andcais partly reflected alongad. If then the pathsabandacare equal, the two rays interfere alongad. Suppose now, the ether being at rest, that the whole apparatus moves in the directionsc, with the velocity of the earth in its orbit, the directions and distances traversed by the rays will be altered thus:— The raysais reflected alongab, fig. 2; the anglebab, being equal to the aberration =a, is returned alongba_{1}, (aba_{1}=2a), and goes to the focus of the telescope, whose direction is unaltered. The transmitted ray goes alongac, is returned alongca_{1}, and is reflected ata_{1}, makingca_{1}eequal 90—a, and therefore still coinciding with the first ray. It may be remarked that the raysba_{1}andca_{1}, do not now meet exactly in the same pointa_{1}, though the difference is of the second order; this does not affect the validity of the reasoning. Let it now be required to find the difference in the two pathsaba_{1}, andaca_{1}.Let

V= velocity of light.

v = velocity of the earth in its orbit,

D= distanceaborac, fig. 1.

T= time light occupies to pass fromatoc.

T_{1}= time light occupies to return fromctoa_{1}(fig. 2.)

The paper then goes on to give expressions for *T* and *T*_{1} **incorrectly** as

These are wrong because they assume that time is the variable in common, but actually *distance* is the variable in common since all distances are pre-determined, such that “the paths *ab* and *ac* are equal”. *Time* is a dependent variable, and the velocities *V* and *v* are actually *inverse velocities*, and so are combined *reciprocally* [see *here*], not arithmetically.

The **correct** expressions are as follows:

Inverse velocity is the reciprocal of lenticity and combines with reciprocal addition and subtraction:

where ‘boxplus’ and ‘boxminus’ are the reciprocal versions of addition and subtraction. Another approach is instead of velocities to use *lenticities*, which are inverse velocities, with distance as the variable in common. Then the correct values for *T* and *T*_{1} are

Either approach leads to

So the elapsed time and the speed of light are independent of the earth in its orbit. Thus the Galilean transformation (actually the Euclidean transformation) may be used with a finite constant space mean speed of light. The Lorentz transformation is not needed.