The value of V depends on three quantities D, n, and φ. These will now be considered in detail.

The Distance.

The distance between the two mirrors may be in error, either by an erroneous determination of the length of the steel tape used, or by a mistake in the measurement of the distance by the tape.

The first may be caused by an error in the copy of the standard yard, or in the comparison between the standard and the tape. An error in this copy, of .00036 inch, which, for such a copy, would be considered large, would produce an error of only .00001 in the final result. Supposing that the bisections of the divisions are correct to .0005 inch, which is a liberal estimate, the error caused by supposing the error in each yard to be in the same direction would be only .000014; or the total error of the tape, if both errors were in the same direction, would be 000024 of the whole length.

The calculated probable error of the five measurements of the distance was ±.000015; hence the total error due to D would be at most .00004. The tape has been sent to Professor Rogers, of Cambridge, for comparison, to confirm the result.

The Speed of Rotation.

This quantity depends on three conditions. It is affected, first, by an error in the rate of the standard; second, by an error in the count of the sound beats between the forks; and third, by a false estimate of the moment when the image of the revolving mirror is at rest, at which moment the deflection is measured.

The calculated probable error of the rate is .000016. If this rate should be questioned, the fork can be again rated and a simple correction applied. The fork is carefully kept at the Stevens Institute, Hoboken, and comparisons were made with two other forks, in case it was lost or injured.

In counting the sound beats, experiments were tried to find if the vibrations of the standard were affected by the other fork, but no such effect could be detected. In each case the number of beats was counted correctly to .02, or less than .0001 part, and in the great number of comparisons made this source of error could be neglected.

The error due to an incorrect estimate of the exact time when the images of the revolving mirror came to rest was eliminated by making the measurement sometimes when the speed was slowly increasing, and sometimes when slowly decreasing. Further, this error would form part of the probable error deduced from the results of observations.

We may then conclude that the error, in the measurement of n, was less than .00002.

The Deflection.

The angle of deflection φ was measured by its tangent, tan φ = d/r; d was measured by the steel screw and brass scale, and r by the steel tape.

The value of one turn of the screw was found by comparison with the standard meter for all parts of the screw. This measurement, including the possible error of the copy of the standard meter, I estimate to be correct to .00005 part. The instrument is at the Stevens Institute, where it is to be compared with a millimeter scale made by Professor Rogers, of Cambridge.

The deflection was read to within three or four hundredths of a turn at each observation, and this error appears in the probable error of the result.

The deflection is also affected by the inclination of the plane of rotation to the horizon. This inclination was small, and its secant varies slowly, so that any slight error in this angle would not appreciably affect the result.

The measurement of r is affected in the same way as D, so that we may call the greatest error of this measurement .00004. It would probably be less than this, as the mistakes in the individual measurements would also appear in the probable error of the result.

The measurement of φ was not corrected for temperature. As the corrections would be small they may be applied to the final result. For an increase of 1° F. the correction to be applied to the screw for unit length would be -.0000066. The correction for the brass scale would be +.0000105, or the whole correction for the micrometer would be +.000004. The correction for the steel tape used to measure r would be +.0000066. Hence the correction for tan. φ would be -.000003 t. The average temperature of the experiments is 75°.6 F. 75.6-62.5 = 13.1. -.000003×13.1 = -.00004

Hence φ should be divided by 1.00004, or the final result should be multiplied by 1.00004. This would correspond to a correction of +12 kilometers.

The greatest error, excluding the one just mentioned, would probably be less than .00009 in the measurement of φ.

Summing up the various errors, we find, then, that the total constant error, in the most unfavorable case, where the errors are all in the same direction, would be .00015. Adding to this the probable error of the result, .00002, we have for the limiting value of the error of the final result ±.00017. This corresponds to an error of ±51 kilometers.

The correction for the velocity of light in vacuo is found by multiplying the speed in air by the index of refraction of air, at the temperature of the experiments. The error due to neglecting the barometric height is exceedingly small. This correction, in kilometers, is +80.

Final Result.

The final value of the velocity of light from these experiments is then—299940 kilometers per second, or 186380 miles per second.