The figure represents the distorted mirror at rest, but the reasoning is the same when it is in motion, save that all the images will be deflected in the direction of rotation.
Imperfection of the Lens.
It has also been suggested that, as the pencil goes through one-half of the lens and returns through the opposite half, if these two halves were not exactly similar, the return image would not coincide with the slit when the mirror was at rest. This would undoubtedly be true if we consider but one-half of the original pencil. It is evident, however, that the other half would pursue the contrary course, forming another image which falls on the other side of the slit, and that both these images would come into view, and the line midway between them would coincide with the true position. No such effect was observed, and would be very unlikely to occur. If the lens was imperfect, the faults would be all over the surface, and this would produce simply an indistinctness of the image.
Moreover, in the latter part of the observations the mirror was inverted, thus producing a positive rotation, whereas the rotation in the preceding sets was negative. This would correct the error mentioned if it existed, and shows also that no constant errors were introduced by having the rotation constantly in the same direction, the results in both cases being almost exactly the same.
Periodic Variations in Friction.
If the speed of rotation varied in the same manner in each revolution of the mirror, the chances would be that, at the particular time when the reflection took place, the speed would not be the same as the average speed found by the calculation. Such a periodic variation could only be caused by the influence of the frame or the pivots. For instance, the frame would be closer to the ring which holds the mirror twice in every revolution than at other times, and it would be more difficult for the mirror to turn here than at a position 90 from this. Or else there might be a certain position, due to want of trueness of shape of the sockets, which would cause a variation of friction at certain parts of the revolution.
To ascertain if there were any such variations, the position of the frame was changed in azimuth in several experiments. The results were unchanged showing that any such variation was too small to affect the result.
Change of Speed of Rotation.
In the last four sets of observations the speed was lowered from 256 turns to 192, 128, 96, and 64 turns per second. The results with these speeds were the same as with the greater speed within the limits of errors of experiment.
Bias.
Finally, to test the question if there were any bias in taking these observations, eight sets of observations were taken, in which the readings were made by another, the results being written down without divulging them. Five of these sets are given in the "specimen," pages 133-134.
It remains to notice the remarkable coincidence of the result of these experiments with that obtained by Cornu by the method of the "toothed wheel."
Cornu"s result was 300400 kilometers, or as interpreted by Helmert 299990 kilometers. That of these experiments is 299940 kilometers.
Postscript.
The comparison of the micrometer with two scales made by Mr. Rogers, of the Harvard Observatory, has been completed. The scales were both on the same piece of silver, marked "Scales No. 25, on silver. Half inch at 58 F., too short .000009 inch. Centimeter at 67 F., too short .00008 cm."
It was found that the ratio .3937079 could be obtained almost exactly, if, instead of the centimeter being too short, it were too _long_ by .00008 cm. at 67.