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′_.
In other words, besides the image coinciding with _a_, there would be two
images, one on either side of _a_, and in case there were more than two
portions having different inclinations there would be formed as many
images to correspond. If the surfaces are not plane, the only effect is to
produce a distortion of the image.
As no multiplication of images was observed, and no distortion of the one
image, it follows that the distortion of the mirror was too small to be
noticed, and that even if it were larger it could not affect the
deflection.
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°.
On this supposition the following tables were obtained. They represent the
value of one turn of the micrometer in millimeters.
Table 1 is the result from centimeter scale.
Table 2 is the result from half-inch scale.
Table 3 is the result from page 31.
It is seen from the correspondence in these results, that the previous
work is correct.
(1) (2) (3)
From 0 to 13 .99563 .99562 .99570
25 .99562 .99564 .99571
38 .99560 .99572 .99576
51 .99567 .99578 .99580
64 .99577 .99586 .99585
76 .99582 .99590 .99592
89 .99590 .99598 .99601
102 .99596 .99608 .99605
115 .99606 .99614 .99615
128 .99618 .99622 .99623
140 .99629 .99633 .99630