d′
(1) tan φ = -----
r
2592000″ × D × n
(2) V = -----------------
φ″
φ = angle of deflection.
d′ = corrected displacement (linear).
r = radius of measurement.
D = twice the distance between the mirrors.
n = number of revolutions per second.
α = inclination of plane of rotation
d = deflection as read from micrometer.
B = number of beats per second between electric Vt₂ fork and
standard Vt₃
Cor = correction for temperature of standard Vt3.
V = velocity of light.
T = value of one turn of screw. (Table, page 126.)
Substituting for d, its value or d×T×sec α (log sec α = .00008), and
for D its value 3972.46, and reducing to kilometers, the formulæ become--
dT
(3) tan φ = c′ ----; log c′ = .51607
r
n
(4) V = c ---; log c = .49670
φ
D and r are expressed in feet and d′ in millimeters.
Vt₃ fork makes 256.070 vibrations per second at 65° Fahr.
D = 3972.46 feet.
tan α = tangent of angle of inclination of plane of rotation = 0.02
in all but the last twelve observations, in which it was 0.015.
log c′ = .51607 (.51603 in last twelve observations.).
log c = .49670.
The electric fork makes ½(256.070 + B + cor.) vibrations per second,
and n is a multiple, submultiple, or simple ratio of this.
Observations.
Specimen Observation.
June 17. sunset. Image good; best in column (4).
The columns are sets of readings of the micrometer for the deflected image
of slit.
112.81 112.80 112.83 112.74 112.79
81 81 81 76 78
79 78 78 74 74
80 75 74 76 74
79 77 74 76 77
82 79 72 78 81
82 73 76 78 77
76 78 81 79 75
83 79 74 83 82
73 73 76 78 82
------- ------- ------- ------- -------
Mean = 112.801 112.773 112.769 112.772 112.779
Zero = 0.260 0.260 0.260 0.260 0.260
------- ------- ------- ------- -------
d = 112.451 112.513 112.509 112.512 112.519
Temp = 77° 77° 77° 77° 77°
B = + 1.500
Corr = - .144
-------
+ 1.365
256.070
-------
n = 257.426 257.43 257.43 257.43 257.43
r = 28.157 28.157 28.157 28.157 28.157
The above specimen was selected because in it the readings were all taken
by another and noted down without divulging them till the whole five sets
were completed.
The following is the calculation for V:
2d, 3d,
1st set. and 4th sets. 5th set.
log c′ = 51607 51607 51607
" T = 99832 99832 99832
" d = 05131 05119 05123
------- ------- -------
56570 56558 56562
" r = 44958 44958 44958
------- ------- -------
" tan φ = 11612 11600 11604
φ = 2694″.7 2694″.1 2694″.3
" c = 49670 49670 49670
" n = 41066 41066 41066
------- ------- -------
90736 90736 90736
" φ = 43052 43042 43046
------- ------- -------
" V = 47684 47694 47690
V = 299800 299880 299850
In the following table, the numbers in the column headed "Distinctness of
Image" are thus translated: 3, good; 2, fair; 1, poor. These numbers do
not, however, show the relative weights of the observations.
The numbers contained in the columns headed "Position of Deflected Image,"
"Position of Slit," and displacement of image in divisions were obtained
as described in the paragraph headed "Micrometer," page 120.
The column headed "B" contains the number of "beats" per second between
the electric Vt₂ fork and the standard Vt₃ as explained in the paragraph
headed "Measurement of the Speed of Rotation." The column headed "Cor."
contains the correction of the rate of the standard fork for the
difference in temperature of experiment and 65° Fahr., for which
temperature the rate was found. The numbers in the column headed "Number
of revolutions per second" were found by applying the corrections in the
two preceding columns to the rate of the standard, as explained in the
same paragraph.
The "radius of measurement" is the distance between the front face of the
revolving mirror and the cross-hair of the micrometer.
The numbers in the column headed "Value of one turn of the screw" were
taken from the table, page 127.
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.
[Illustration: FIG. 14.]
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)