Uncertainty of Laws of Reflection and Refraction in Media in Rapid Rotation.

What is probably hinted at under the above heading is that there may be a possibility that the rapid rotation of the mirror throws the reflected pencil in the direction of rotation. Granting that this is the case, an inspection of Fig. 14 shows that the deflection will not be affected.

In this figure let m m be the position of the mirror when the light first falls on it from the slit at a, and m′ m′ the position when the light returns.

fig 14

From the axis o draw op op, perpendicular to m m and to m′ m′, respectively. Then, supposing there is no such effect, the course of the axis of the pencil of light would be a o c mirror c o a′. That is, the angle of deflection would be a o a′, double the angle p o p′. If now the mirror be supposed to carry the pencil with it, let o c′ be the direction of the pencil on leaving the mirror m m; i.e., the motion of the mirror has changed the direction of the reflected ray through the angle c o c′. The course would then be a o c, mirror c′ o. From o the reflection would take place in the direction a″, making the angles c′ o p, and p′ o a″ equal. But the angle c o c′ must be added to p o a″, in consequence of the motion of the mirror, or the angle of deviation will be a o a″ + c o c′; or a o a″ + c o c′ = d. (1)

By construction—

c o p′ = p′ o a′ (2)

c′ o p′ = p′ o a″ (3)

Subtracting (3) from (2) we have—

c o p′ - c′ o p′ = p′ o a′ - p′ o a″, or

c o c′ = a′ o a″

Substituting a′ o a″ for c o c′ in (1) we have— a o a″ + a′ o a″ = a o a′ = d.

Or the deflection has remained unaltered.