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. Retardation Caused by Reflection. Cornu, in answering the objection that there may be an unknown retardation by reflection from the distant mirror, says that if such existed the error it would introduce in his own work would be only 1/7000 that of Foucault, on account of the great distance used, and on account of there being in his own experiments but one reflection instead of twelve. In my own experiments the same reasoning shows that if this possible error made a difference of 1 per cent. in Foucault's work (and his result is correct within that amount), then the error would be but .00003 part. Distortion of the Revolving Mirror. It, has been suggested that the distortion of the revolving mirror, either by twisting or by the effect of centrifugal force, might cause an error in the deflection. [Illustration: FIG. 15] The only plane in which the deflection might be affected is the plane of rotation. Distortions in a vertical plane would have simply the effect of raising, lowering, or extending the slit. Again, if the _mean_ surface is plane there will be no effect on the deflection, but simply a blurring of the image. Even if there be a distortion of any kind, there would be no effect on the deflection if the rays returned to the same portion whence they were reflected. The only case which remains to be considered, then, is that given in Fig. 15, where the light from the slit _a_, falls upon a distorted mirror, and the return light upon a different portion of the same. The one pencil takes the course _a b c d e f a′_, while the other follows the path _a f g h i b a′_.