Chapter 2

The eye-piece of the micrometer was then set approximately[3] and the revolving mirror started. If the image did not appear, the mirror was inclined forward or backward till it came in sight.

[Footnote 3: The deflection being measured by its tangent, it was necessary that the scale should be at right angles to the radius (the radius drawn from the mirror to one or the other end of that part of the scale which represents this tangent). This was done by setting the eye-piece approximately to the expected deflection, and turning the whole micrometer about a vertical axis till the cross-hair bisected the circular field of light reflected from the revolving mirror. The axis of the eye-piece being at right angles to the scale, the latter would be at right angles to radius drawn to the cross-hair.]

The cord connected with the valve was pulled right or left till the images of the revolving mirror, represented by the two bright round spots to the left of the cross-hair, came to rest. Then the screw was turned till the cross-hair bisected the deflected image of the slit. This was repeated till ten observations were taken, when the mirror was stopped, temperature noted, and beats counted. This was called a set of observations. Usually five such sets were taken morning and evening.

Fig. 13 represents the appearance of the image of the slit as seen in the eye-piece magnified about five times.

Determination of The Constants.

Comparison of the Steel Tape with the Standard Yard.

The steel tape used was one of Chesterman's, 100 feet long. It was compared with Wurdeman's copy of the standard yard, as follows:

Temperature was 55° Fahr.

The standard yard was brought under the microscopes of the comparator; the cross-hair of the unmarked microscope was made to bisect the division marked o, and the cross-hair of the microscope, marked I, was made to bisect the division marked 36. The reading of microscope I was taken, and the other microscope was not touched during the experiment. The standard was then removed and the steel tape brought under the microscopes and moved along till the division marked 0.1 (feet) was bisected by the cross-hair of the unmarked microscope. The screw of microscope I was then turned till its cross-hair bisected the division marked 3.1 (feet), and the reading of the screw taken. The difference between the original reading and that of each measurement was noted, care being taken to regard the direction in which the screw was turned, and this gave the difference in length between the standard and each succesive portion of the steel tape in terms of turns of the micrometer-screw.

To find the value of one turn, the cross-hair was moved over a millimeter scale, and the following were the values obtained:

Turns of screw of microscope I in 1mm--

7.68 7.73 7.60 7.67 7.68 7.62 7.65 7.57 7.72 7.70 7.64 7.69 7.65 7.59 7.63 7.64 7.55 7.65 7.61 7.63

Mean =7.65

Hence one turn = 0.1307mm.

or = 0.0051 inch.

The length of the steel tape from 0.1 to 99.1 was found to be greater than 33 yards, by 7.4 turns =.96mm +.003 feet. Correction for temperature +.003 feet. Length 100.000 feet. -------------- Corrected length 100.006 feet.

Determination of the Value of Micrometer.

Two pairs of lines were scratched on one slide of the slit, about 38mm apart, i.e., from the center of first pair to center of second pair. This distance was measured at intervals of 1mm through the whole length of the screw, by bisecting the interval between each two pairs by the vertical silk fiber at the end of the eye-piece. With these values a curve was constructed which gave the following values for this distance, which we shall call D′:

Turns of screw. At 0 of scale D′ =38.155 10 of scale D′ 38.155 20 of scale D′ 38.150 30 of scale D′ 38 150 40 of scale D′ 38.145 50 of scale D′ 38.140 60 of scale D′ 38.140 70 of scale D′ 38.130 80 of scale D′ 38.130 90 of scale D′ 38.125 100 of scale D′ 38.120 110 of scale D′ 38.110 120 of scale D′ 38.105 130 of scale D′ 38.100 140 of scale D′ 38.100

Changing the form of this table, we find that,--

For the _first_ 10 turns the _average_ value of D′ is 38.155 20 turns 38.153 30 turns 38.152 40 turns 38.151 50 turns 38.149 60 turns 38.148 70 turns 38.146 80 turns 38.144 90 turns 38.142 100 turns 38.140 110 turns 38.138 120 turns 38.135 130 turns 38.132 140 turns 38.130

On comparing the scale with the standard meter, the temperature being 16°.5 C., 140 divisions were found to = 139.462mm. This multiplied by (1 + .0000188 × 16.5) = 139.505mm.

One hundred and forty divisions were found to be equal to 140.022 turns of the screw, whence 140 turns of the screw = 139.483mm, or 1 turn of the screw = 0.996305mm.

This is the _average_ value of one turn in 140.

But the average value of D, for 140 turns is, from the preceding table, 38.130.

Therefore, the true value of D, is 38.130 × .996305mm, and the average value of one turn for 10, 20, 30, etc., turns, is found by dividing 38.130 × .996305 by the values of D;, given in the table.

This gives the value of a turn--

mm. For the first 10 turns 0.99570 20 turns 0.99570 30 turns 0.99573 40 turns 0.99577 50 turns 0.99580 60 turns 0.99583 70 turns 0.99589 80 turns 0.99596 90 turns 0.99601 100 turns 0.99606 110 turns 0.99612 120 turns 0.99618 130 turns 0.99625 140 turns 0.99630

NOTE.--The micrometer has been sent to Professor Mayer, of Hoboken, to test the screw again, and to find its value. The steel tape has been sent to Professor Rogers, of Cambridge, to find its length again. (See page 145.)

Measurement of the Distance between the Mirrors.

Square lead weights were placed along the line, and measurements taken from the forward side of one to forward side of the next. The tape rested on the ground (which was very nearly level), and was stretched by a constant force of 10 pounds.

The correction for length of the tape (100.006) was +0.12 of a foot.

To correct for the stretch of the tape, the latter was stretched with a force of 15 pounds, and the stretch at intervals of 20 feet measured by a millimeter scale.

mm. At 100 feet the stretch was 8.0 80 feet the stretch was 5.0 60 feet the stretch was 5.0 40 feet the stretch was 3.5 20 feet the stretch was 1.5 --- --- 300 23.00

Weighted mean = 7.7 mm. For 10 pounds, stretch = 5.1 mm. = 0.0167 feet. Correction for whole distance = +0.33 feet.

The following are the values obtained from five separate measurements of the distance between the caps of the piers supporting the revolving mirror and the distant reflector; allowance made in each case for effect of temperature:

1985.13 feet. 1985.17 feet. 1984.93 feet. 1985.09 feet. 1985.09 feet. ------- Mean = 1985.082 feet.

+.70. Cap of pier to revolving mirror. +.33. Correction for stretch of tape. +.12. Correction for length of tape. -------- 1986.23. True distance between mirrors.

Rate of Standard Ut₃ Fork.

The rate of the standard Ut₃ fork was found at the Naval Academy, but as so much depended on its accuracy, another series of determinations of its rate was made, together with Professor Mayer, at the Hoboken Institute of Technology.

_Set of determinations made at Naval Academy._

The fork was armed with a tip of copper foil, which was lost during the experiments and replaced by one of platinum having the same weight, 4.6 mgr. The fork, on its resonator, was placed horizontally, the platinum tip just touching the lampblacked cylinder of a Schultze chronoscope. The time was given either by a sidereal break-circuit chronometer or by the break-circuit pendulum of a mean-time clock. In the former case the break-circuit worked a relay which interrupted the current from three Grove cells. The spark from the secondary coil of an inductorium was delivered from a wire near the tip of the fork. Frequently two sparks near together were given, in which case the first alone was used. The rate of the chronometer, the record of which was kept at the Observatory, was very regular, and was found by observations of transits of stars during the week to be +1.3 seconds per day, which is the same as the recorded rate.

Specimen of a Determination of Rate of Ut₃ Fork.

Temp.=27° C. Column 1 gives the number of the spark or the number of the second. Column 2 gives the number of sinuosities or vibrations at the corresponding second. Column 3 gives the difference between 1 and 11, 2 and 12, 3 and 13, etc.

July 4, 1879. 1 0.1 2552.0 2 255.3 2551.7 3 510.5 2551.9 4 765.6 2551.9 5 1020.7 2552.1 6 1275.7 2552.0 7 1530.7 2551.8 8 1786.5 2551.4 9 2041.6 2551.7 10 2297.0 2551.5 ------- 11 2552.1 255.180 = mean ÷ 10. 12 2807.0 + .699 = reduction for mean time. 13 3062.4 + .003 = correction for rate. 14 3317.5 + .187 = correction for temperature. ------- 15 3572.8 256.069 = number of vibrations per second at 65° Fahr. 16 3827.7 17 4082.5 18 4335.9 19 4593.3 20 4848.5

The correction for temperature was found by Professor Mayer by counting the sound-beats between the standard and another Ut₃ fork, at different temperatures. His result is +.012 vibrations per second for a diminution of 1° Fahr. Using the same method, I arrived at the result +.0125. Adopted +.012.

_Résumé of determinations made at Naval Academy._

In the following table the first column gives the date, the second gives the total number of seconds, the third gives the result uncorrected for temperature, the fourth gives the temperature (centigrade), the fifth gives the final result, and the sixth the difference between the greatest and least values obtained in the several determinations for intervals of ten seconds:

July 4 20 255.882 27.0 256.069 0.07 5 19 255.915 26.4 256.089 0.05 5 18 255.911 26.0 256.077 0.02 6 21 255.874 24.7 256.012 0.13 6 9 255.948 24.8 256.087 0.24 7 22 255.938 24.6 256.074 0.05 7 21 255.911 25.3 256.061 0.04 8 20 255.921 26.6 256.100 0.02 8 20 255.905 26.6 256.084 0.06 8 20 255.887 26.6 256.066 0.03 ------- Mean = 256.072

In one of the preceding experiments, I compared the two Vt₃ forks while the standard was tracing its record on the cylinder, and also when it was in position as for use in the observations. The difference, if any, was less than .01 vibration per second.

_Second determination_.

(Joint work with Professor A.M. Mayer, Stevens Institute, Hoboken.)

The fork was wedged into a wooden support, and the platinum tip allowed to rest on lampblacked paper, wound about a metal cylinder, which was rotated by hand Time was given by a break-circuit clock, the rate of which was ascertained, by comparisons with Western Union time-ball, to be 9.87 seconds. The spark from secondary coil of the inductorium passed from the platinum tip, piercing the paper. The size of the spark was regulated by resistances in primary circuit.

The following is a specimen determination:

Column 1 gives the number of the spark or the number of seconds. Column 2 gives the corresponding number of sinuosities or vibrations. Column 3 gives the difference between the 1st and 7th ÷ 6, 2nd and 8th ÷ 6, etc.

1 0.3 255.83 2 256.1 255.90 3 511.7 255.90 4 767.9 255.93 5 1023.5 255.92 6 1289.2 256.01 7 1535.3 255.95 ------- 8 1791.5 255.920 = mean. 9 2047.1 - .028 = correction for rate. ------- 10 2303.5 255.892 11 2559.0 + .180 = correction for temperature. ------- 12 2825.3 256.072 = number of vibrations per second at 65° Fahr. 13 3071.0

In the following _résumé_, column 1 gives the number of the experiments. Column 2 gives the total number of seconds. Column 3 gives the result not corrected for temperature. Column 4 gives the temperature Fahrenheit. Column 5 gives the final result. Column 6 gives the difference between the greatest and least values:

1 13 255.892 80 256.072 0.18 2 11 255.934 81 256.126 0.17 3 13 255.899 81 256.091 0.12 4 13 255.988 75 256.108 0.13 5 11 255.948 75 256.068 0.05 6 12 255.970 75 256.090 0.05 7 12 255.992 75 256.112 0.20 8 11 255.992 76 256.124 0.03 9 11 255.888 81 256.080 0.13 10 13 255.878 81 256.070 0.13 ------- Mean = 256.094

Effect of Support and of Scraping.

The standard Vt₃ fork held in its wooden support was compared with another fork on a resonator loaded with wax and making with standard about five beats per second. The standard was free from the cylinder. The beats were counted by coincidences with the ⅕ second beats of a watch.

_Specimen._

Coincidences were marked--

At 32 seconds. 37 seconds. 43.5 seconds. 49 seconds. 54.5 seconds. 61.5 seconds. 61.5 - 32 = 29.5. 29.5 ÷ 5 = 5.9 = time of one interval.

_Résumé._

1 5.9 2 6.2 3 6.2 4 6.2 ---- Mean = 6.13 = time of one interval between coincidences.

In this time the watch makes 6.13×5 = 30.65 beats, and the forks make 30.65 + 1 = 31.65 beats.

Hence the number of beats per second is 31.65 ÷ 6.13 = 5.163.

_Specimen._

Circumstances the same as in last case, except that standard Vt₃ fork was allowed to trace its record on the lampblacked paper, as in finding its rate of vibration.

Coincidences were marked at--

59 seconds. 04 seconds. 10.5 seconds. 17 seconds.

77 - 59 = 18. 18 ÷ 3 = 6.0 = time of one interval.

_Résumé._

No. 1 6.0 seconds. 6.31 × 5 = 31.55 2 6.0 seconds. + 1.00 3 6.7 seconds. ---- 4 6.3 seconds. 5 6.5 seconds. 32.55 6 6.7 seconds. 32.55 ÷ 6.31 = 5.159 7 6.0 seconds. With fork free 5.163 ---- ----- Mean = 6.31 seconds Effect of scrape = - .044

_Specimen._

Circumstances as in first case, except that both forks were on their resonators.

Coincidences were observed at--

21 seconds. 28 seconds. 36 seconds. 44 seconds. 51 seconds. 60 seconds. 60 - 21 = 39 39 ÷ 5 = 7.8 = time of one interval.

_Résumé_.

No. 1 7.8 seconds. 7.42 × 5 = 37.10 2 7.1 seconds. + 1.00 3 7.6 seconds. ----- 4 7.4 seconds. 38.10 5 7.2 seconds. 38.10 ÷ 7.42 = 5.133 ---- (Above) 5.159 ----- Mean = 7.42 seconds. Effect of support and scrape = - .026

Mean of second determination was 256.094 Applying correction (scrape, etc.) - .026 ------- Corrected mean 256.068 Result of first determination 256.072 ------- Final value 256.070

NOTE--The result of first determination excludes all work except the series commencing July 4. If previous work is included, and also the result first obtained by Professor Mayer, the result would be 256.089.

256.180 256.036 256.072 256.068 ------- Mean = 256.089

The previous work was omitted on account of various inaccuracies and want of practice, which made the separate results differ widely from each other.

The Formulæ.

The formulæ employed are--

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.