Part 3

Date. | Distinctness of image. | | Temperature, Fahr. | | | Position of deflected image. | | | | Position of slit. | | | | | Displacement of image in divisions. | | | | | | Difference between greatest and least values. | | | | | | | B. | | | | | | | | Cor. | | | | | | | | | Number of revolutions per second. | | | | | | | | | | Radius of measurement, in feet. | | | | | | | | | | | Value of one turn of the screw. | | | | | | | | | | | | Velocity of light in air, in kilometers. | | | | | | | | | | | | | Remarks. | | | | | | | | | | | | | | June 5|3|76|114.85| 0.300|114.55|0.17|1.423|-0.132|257.36|28.672|0.99614|299850|Electric light. June 7|2|72|114.64| 0.074|114.56|0.10|1.533|-0.084|257.52|28.655|0.99614|299740|P.M. Frame inclined at various angles June 7|2|72|114.58| 0.074|114.50|0.08|1.533|-0.084|257.52|28.647|0.99614|299900|P.M. Frame inclined at various angles June 7|2|72| 85.91| 0.074| 85.84|0.12|1.533|-0.084|193.14|28.647|0.99598|300070|P.M. Frame inclined at various angles June 7|2|72| 85.97| 0.074| 85.89|O.07|1.533|-0.084|193.14|28.650|0.99598|299930|P.M. Frame inclined at various angles June 7|2|72|114.61| 0.074|114-53|0.07|1.533|-0.084|257.42|28.650|0.99614|299850|P.M. Frame inclined at various angles June 9|3|83|114.54| 0.074|114.47|0.07|1.533|-0.216|257.39|28.658|0.99614|299950|P.M. Frame inclined at various angles June 9|3|83|114.54| 0.074|114.46|0.10|1.533|-0.216|257.39|28.658|0.99614|299980|P.M. Frame inclined at various angles June 9|3|83|114.57| 0.074|114.47|0.08|1.533|-0.216|257.39|28.662|0.99614|299980|P.M. Frame inclined at various angles June 9|3|83|114.57| 0.074|114.50|0.06|1.533|-0.216|257.39|28.660|0.99614|299880|P.M. Frame inclined at various angles June 9|2|83|114.61| 0.074|114.53|0.13|1.533|-0.216|257.39|28.678|0.99614|300000|P.M. Frame inclined at various angles June 10|2|90|114.60| 0.074|114.52|0.11|1.517|-0.300|257.29|28.685|0.99614|299980|P.M. June 10|2|90|114.62| 0.074|114.54|0.08|1.517|-0.300|257.29|28.685|0.99614|299930|P.M. June 12|2|71|114.81| 0.074|114.74|0.09|1.450|-0.072|257.45|28.690|0.99614|299650|A.M. June 12|2|71|114.78| 0.074|114.70|0.05|1.450|-0.072|257.45|28.690|0.99614|299760|A.M. June 12|1|71|114.76| 0.074|114.68|0.09|1.450|-0.072|257.45|28.690|0.99614|299810|A.M. June 13|3|72|112.64| 0.074|112.56|0.09|1.500|-0.084|257.49|28.172|0.99614|300000|A.M. June 13|3|72|112.63| 0.074|112.56|0.10|1.500|-0.084|257.49|28.172|0.99614|300000|A.M. June 13|2|72|112.65| 0.074|112.57|0.08|1.500|-0.084|257.49|28.172|0.99614|299960|A.M. June 13|3|79|112.82| 0.260|112.56|0.06|1.517|-0.168|257.42|28.178|0.99614|299960|P.M. June 13|3|79|112.82| 0.260|112.56|0.13|1.517|-0.168|257.42|28.178|0.99614|299960|P.M. June 13|3|79|112.83| 0.260|112.57|0.07|1.517|-0.168|257.42|28.178|0.99614|299940|P.M. June 13|3|79|112.82| 0.260|112.56|0.06|1.517|-0.168|257.42|28.178|0.99614|299960|P.M. June 13|3|79|112.83| 0.260|112.57|0.11|1.517|-0.168|257.42|28.178|0.99614|299940|P.M. June 13|3|79|113.41| 0.260|113.15|11 |1.517|-0.168|258.70|28.152|0.99614|299880|P.M. Set micrometer and counted oscillations. June 13|3|79|112.14| 0.260|111.88|6 |1.517|-0.168|255.69|28.152|0.99614|299800|Oscillations of image of revolving mirror. June 14|1|64|112.83| 0.260|112.57|0.12|1.500|+0.012|257.58|28.152|0.99614|299850|A.M. June 14|1|64|112.83| 0.260|112.57|0.05|1.517|+0.012|257.60|28.152|0.99614|299880|A.M. June 14|1|65|112.81| 0.260|112.55|0.11|1.517| 0.000|257.59|28.152|0.99614|299900|A.M. June 14|1|66|112.83| 0.260|112.57|0.09|1.517|-0.012|257.57|28.152|0.99614|299840|A.M. June 14|1|67|112.83| 0.260|112.57|0.12|1.517|-0.024|257.56|28.152|0.99614|299830|A.M. June 14|1|84|112.78| 0.260|112.52|0.06|1.517|-0.228|257.36|28.159|0.99614|299790|P.M. Readings taken by Lieut. Nazro. June 14|1|85|112.76| 0.260|112.50|0.08|1.500|-0.240|257.33|28.159|0.99614|299810|P.M. Readings taken by Lieut. Nazro. June 14|1|84|112.72| 0.260|112.46|0.08|1.483|-0.228|257.32|28.159|0.99614|299880|P.M. Readings taken by Lieut. Nazro. June 14|1|84|112.73| 0.260|112.47|0.09|1.483|-0.228|257.32|28.159|0.99614|299880|P.M. June 14|1|84|112.75| 0.260|112.49|0.09|1.483|-0.228|257.32|28.129|0.99614|299830|P.M. June 17|2|62|112.85| 0.260|112.59|0.09|1.517|+0.036|257.62|28.149|0.99614|299800|A.M. June 17|2|63|112.84| 0.260|112.58|0.06|1.500|+0.024|257.59|28.149|0.99614|299790|A.M. June 17|1|64|112.85| 0.260|112.59|0.07|1.500|+0.012|257.58|28.149|0.99614|299760|A.M. June 17|3|77|112.80| 0.260|112.54|0.07|1.500|-0.144|257-43|28.157|0.99614|299800|P.M. Readings taken by Mr. Clason. June 17|3|77|112.77| 0.260|112.51|0.08|1.500|-0.144|257.43|28.157|0.99614|299880|P.M. Readings taken by Mr. Clason. June 17|3|77|112.77| 0.260|112.51|0.11|1.500|-0.144|257.43|28.157|0.99614|299880|P.M. Readings taken by Mr. Clason. June 17|3|77|112.77| 0.260|112.51|0.09|1.500|-0.144|257.43|28.157|0.99614|299880|P.M. Readings taken by Mr. Clason. June 17|3|77|112.78| 0.260|112.52|0.08|1.500|-0.144|257 43|28.157|0.99614|299860|P.M. Readings taken by Mr. Clason. June 18|1|58|112.90| 0.265|112.64|0.07|1.500|+0.084|257.65|28.150|0.99614|299720|A.M. June 18|1|58|112.90| 0.265|112.64|0.10|1.500|+0.084|257.65|28.150|0.99614|299720|A.M. June 18|1|59|112.92| 0.265|112.66|0.07|1.483|+0.072|257.62|28.150|0.99614|299620|A.M. June 18|2|75|112.79| 0.265|112.52|0.09|1.483|-0.120|257-43|28.158|0.99614|299860|P.M. June 18|2|75|112.75| 0.265|112.48|0.10|1.483|-0.120|257-43|28.158|0.99614|299970|P.M. June 18|2|75|112.76| 0.265|112.49|0.08|1.483|-0.120|257-43|28.158|0.99614|299950|P.M. June 20|3|60|112.94| 0.265|112.67|0.07|1.517|+0.063|257.65|28.172|0.99614|299880|A.M. June 20|3|61|112.92| 0.265|112.65|0.09|1.517|+0.048|257.63|28.172|0.99614|299910|A.M. June 20|2|62|112.94| 0.265|112.67|0.07|1.517|+0.036|257.62|28.172|0.99614|299850|A.M. June 20|2|63|112.93| 0.265|112.66|0.03|1.517|+0.024|257.61|28.172|0.99614|299870|A.M. June 20|2|78|133.48| 0.265|133.21|0.13|1.450|-0.156|257.36|33.345|0.99627|299840|P.M. June 20|2|79|133.49| 0.265|133.23|0.09|1.500|-0.168|257.40|33.345|0.99627|299840|P.M. June 20|2|80|133.49| 0.265|133.22|0.07|1.500|-0.180|257.39|33.345|0.99627|299850|P.M. June 20|2|79|133.50| 0.265|133.24|0.13|1.483|-0.168|257.39|33.345|0.99627|299840|P.M. June 20|2|79|133.49| 0.265|133.22|0.06|1.483|-0.168|257.38|33.345|0.99627|299840|P.M. June 20|2|79|133.49| 0.265|133.22|0.10|1.483|-0.168|257.38|33.345|0.99627|299840|P.M. June 21|2|61|133.56| 0.265|133.29|0.12|1.533|+0.048|257.65|33.332|0.99627|299890|A.M. June 21|2|62|133.58| 0.265|133.31|0.08|1.533|+0.036|257.64|33.332|0.99627|299810|A.M. June 21|2|63|133.57| 0.265|133.31|0.09|1.533|+0.024|257.63|33.332|0.99627|299810|A.M. June 21|2|64|133.57| 0.265|133.30|0.11|1.533|+0.012|257.61|33.332|0.99627|299820|A.M. June 21|2|65|133.56| 0.265|133.30|0.13|1.533| 0.000|257.60|33.332|0.99627|299800|A.M. June 21|3|80|133.48| 0.265|133.21|0.06|1.533|-0.180|257.42|33.330|0.99627|299770|P.M. June 21|3|81|133.46| 0.265|133.19|0.10|1.500|-0.192|257.38|33.330|0.99627|299760|P.M. June 21|3|82|133.46| 0.265|133.20|0.05|1.500|-0.204|257.37|33.330|0.99627|299740|P.M. June 21|3|82|133.46| 0.265|133.20|0.08|1.517|-0.204|257.38|33.330|0.99627|299750|P.M. June 21|3|81|133.46| 0.265|133.19|0.08|1.500|-0.192|257.38|33.330|0.99627|299760|P.M. June 23|3|89|133.43| 0.265|133.16|0.08|1.542|-0.288|257.32|33.345|0.99627|299910|P.M. June 23|3|89|133.42| 0.265|133.15|0.06|1.550|-0.288|257.33|33.345|0.99627|299920|P.M. June 23|3|90|133.43| 0.265|133.17|0.09|1.550|-0.300|257.32|33.345|0.99627|299890|P.M. June 23|3|90|133.43| 0.265|133.16|0.07|1.533|-0.300|257.30|33.345|0.99627|299860|P.M. June 23|3|90|133.42| 0.265|133.16|0.07|1.517|-0.300|257.29|33.345|0.99627|299880|P.M. June 24|3|72|133.47| 0.265|133.20|0.15|1.517|-0.084|257.50|33.319|0.99627|299720|A.M. June 24|3|73|133.44| 0.265|133.17|0.04|1.517|-0.096|257.49|33.319|0.99627|299840|A.M. June 24|3|74|133.42| 0.265|133.16|0.11|1.517|-0.108|257.48|33.319|0.99627|299850|A.M. June 24|3|75|133.42| 0.265|133.16|0.06|1.517|-0.120|257.47|33.319|0.99627|299850|A.M. June 24|3|76|133.44| 0.265|133.18|0.10|1.517|-0.132|257.45|33.319|0.99627|299780|A.M. June 26|2|86|133.42| 0.265|133.15|0.05|1.508|-0.252|257.33|33.339|0.99627|299890|P.M. June 26|2|86|133.44| 0.265|133.17|0.08|1.508|-0.252|257.33|33.339|0.99627|299840|P.M. June 27|3|73|133.49| 0.265|133.22|0.11|1.483|-0.096|257.46|33.328|0.99627|299780|A.M. June 27|3|74|133.47| 0.265|133.20|0.06|1.483|-0.108|257.44|33.328|0.99627|299810|A.M. June 27|3|75|133.47| 0.265|133.21|0.09|1.483|-0.120|257.43|33.328|0.99627|299760|A.M. June 27|3|75|133.45| 0.265|133.19|0.09|1.467|-0.120|257.42|33.328|0.99627|299810|A.M. June 27|3|76|133.47| 0.265|133.20|0.08|1.483|-0.132|257.42|33.328|0.99627|299790|A.M. June 27|3|76|133.45| 0.265|133.19|0.10|1.483|-0.132|257.42|33.328|0.99627|299810|A.M. June 30|2|85| 35.32|135.00 | 99.68|0.05|1.500|-0.240|193.00|33.274|0.99645|299820|P.M. Mirror inverted. June 30|2|86| 35.34|135.00 | 99.67|0.06|1.508|-0.252|193.00|33.274|0.99645|299850|P.M. Mirror inverted. June 30|2|86| 35.34|135.00 | 99.66|0.10|1.508|-0.252|193.00|33.274|0.99645|299870|P.M. Mirror inverted. June 30|2|86| 35.34|135.00 | 99.66|0.09|1.517|-0.252|193.00|33.274|0.99645|299870|P.M. Mirror inverted. July 1|2|83| 02.17|135.145|132.98|0.07|1.500|-0.216|257.35|33.282|0.99627|299810|P.M. Mirror inverted. July 1|2|84| 02.15|135.145|133.00|0.09|1.500|-0.228|257.34|33.282|0.99627|299740|P.M. Mirror inverted. July 1|2|86| 02.14|135.145|133.01|0.06|1.467|-0.252|257.28|33.311|0.99627|299810|P.M. Mirror inverted. July 1|2|86| 02.14|135.145|133.00|0.08|1.467|-0.252|257.28|33.311|0.99627|299940|P.M. Mirror inverted. July 2|3|86| 99.85| 0.400| 99.45|0.05|1.450|-0.252|192.95|33.205|0.99606|299950|P.M. Mirror erect. July 2|3|86| 66.74| 0.400| 66.34|0.03|1.450|-0.252|128.63|33.205|0.99586|299800|P.M. Mirror erect. July 2|3|86| 50.16| 0.400| 47.96|0.07|1.467|-0.252| 96.48|33.205|0.99580|299810|P.M. Mirror erect. July 2|3|85| 33.57| 0.400| 33.17|0.06|1.450|-0.240| 64.32|33.205|0.99574|299870|P.M. Mirror erect.

In the last two sets of June 13, the micrometer was fixed at 113.41 and 112.14 respectively. The image was bisected by the cross-hair, and kept as nearly as possible in this place, meantime counting the number of seconds required for the image of the revolving mirror to complete 60 oscillations. In other words, instead of measuring the deflection, the speed of rotation was measured. In column 7 for these two sets, the numbers 11 and 6 are the differences between the greatest and the smallest number of seconds observed.

In finding the mean value of V from the table, the sets are all given the same weight. The difference between the result thus obtained and that from any system of weights is small, and may be neglected.

The following table gives the result of different groupings of sets of observations. Necessarily some of the groups include others:

Electric light (1 set) 299850 Set micrometer counting oscillations (2) 299840 Readings taken by Lieutenant Nazro (3) 299830 Readings taken by Mr. Clason (5) 299860 Mirror inverted (8) 299840 Speed of rotation, 192 (7) 299990 Speed of rotation, 128 (1) 299800 Speed of rotation, 96 (1) 299810 Speed of rotation, 64 (1) 299870 Radius, 28.5 feet (54) 299870 Radius, 33.3 feet (46) 299830 Highest temperature, 90° Fahr. (5) 299910 Mean of lowest temperatures, 60° Fahr. (7) 299800 Image, good (46) 299860 Image, fair (39) 299860 Image, poor (15) 299810 Frame, inclined (5) 299960 Greatest value 300070 Least value 299650 Mean value 299852 Average difference from mean 60 Value found for π 3.26 Probable error ± 5

Discussion of Errors.

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 mean value of V from the tables is 299852 Correction for temperature +12 ------------ Velocity of light in air 299864 Correction for vacuo 80 ------------ Velocity of light in vacuo 299944±51

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

Objections Considered.

Measurement of the Deflection.

The chief objection, namely, that in the method of the revolving mirror the deflection is small, has already been sufficiently answered. The same objection, in another form, is that the image is more or less indistinct. This is answered by a glance at the tables. These show that in each individual observation the average error was only three ten-thousandths of the whole deflection.

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.

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.

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.