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′:

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.

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.