a, a, section of brass tripod; b, b, section of outer axis; c, inner axis; d, d, nutsecuring axis; e, adjustable plate for raising or lowering inner axis. Let CA be the first position of the level; C F its horizontal projection; C B second position of level; C G its horizontal projection; a angle between first and second position; c= = deviation for first position; b= deviation for second position. There is one position in which the deviation of the level will be zero. Let D E be that position: sin e sin (180°-a) cos A CD― cos (180 — a) sin A CD=sin b sin A CD Dividing by sin A CD it becomes or, sin e sin (180° — a) cot A C D — cos (180° — a) = sin b cot A CD= sin b+cos (180° — a) a, b, and e being known, the angle A C D becomes known, which gives the position of DE, or the line of no deviation. The maximum deviation is at right angles to D E, or on the line of C H. To find maximum deviation, sin A C D : sin H C D or 90° :: sin c: sin of maximum deviation, or Usually b and c are so small that the angles themselves can be used instead of their sines. The level was read in third position, and the deviation thus obtained compared with the computed deviation for a check upon the work. Three separate determinations, requiring 464 level-readings, were made with the following results: The line on which the horizontal projection of the inner axis falls when the outer axis is vertical: The reason that the first result differs so much from the second and third is probably because its level-readings were taken with an interval of five hours between them, during which time the stone-post may have changed postiion. The angle which the outer vertical axis makes with the horizontal limb was found in the following manner: The deviation of the outer axis was found in the manner previously described. The inner axis was then removed and a steel rule laid on the horizontal limb. The level was then placed on the rule and readings taken as before, the rule being reversed in each position to eliminate the difference in thickness between its two ends. The computations were made as before. Sixty-four level-readings gave the following result: The angle between the outer axis and its projection on the horizontal limb 59' 48".1-. : 89° The horizontal projection of the outer axis falls on the line from the center of limb to 44°. The eccentricity of both limbs was determined by readings every 30° around the limb, each reading being repeated four times and a mean taken. The following are the results: The eccentricity was then computed by the formulæ given in Chauvenet. Eccentricity of horizontal limb.. 1".5 Eccentricity of vertical limb.. 1.9 The line joining centers of horizontal limb and inner axis passes through 56° The line joining centers of vertical limb and of alidade passes through 15°; angular distance between microscopes, 179° 59′ 54′′.4; angular distance between verniers on the vertical limb, 179° 59′ 58′′. The horizontal axis was tested by reading the striding-level at ten different elevations of the telescope's axis. The readings were repeated four times and a mean taken. The result shows the irregularities to be less than the errors of level-readings. The difference in the diameters of the ends of the horizontal axis was found by reversing the axis in its supports and reading the level in both positions. The mean of twelve trials gave a change of 6.2 divisions of the level, the illuminating end being the smallest. The angles of the level and axis V's are equal; therefore, dividing 6.2 divisions by 4 gives 1.55 divisions, or 1".627, which is the correction for inequality of ends of axis. When the instrument is used for an astronomical transit and the illuminating end of the axis is west, this correction should be added to the elevation of the west end of the axis as given by the level. The graduation of the horizontal limb has been tested in three ways: 1. By computing, with the eccentricity, the readings of opposite microscopes: (with Chauvenet's formula) and comparing them with the observed readings. The maximum difference found in this way was 2.5; the average difference being 1".1, which is about the same as the errors of reading. 2. By measuring the space between two consecutive divisions with the micrometer, the greatest difference found being 0".5. 3. By placing two collimators on the same post and repeating the angle between them. No satisfactory results were obtained by this method, owing to the unsteadiness of the collimator-posts. The focal length of the telescope is 18.7 inches. 10 inches, the power of the object-glass alone is found to be 1.87 diameters. Dividing by The power of the telescope, in combination with each of its eye-pieces, was measured with Ramsden's dynameter, and found to be, for the diagonal eye-piece, 35 diameters; for the direct eye-pieces, 35 and 24 diameters. The microscopes for reading the horizontal limb and the magnifiers for reading the vertical limb were used as eye-pieces in the telescope, and the power of the combination measured with the dynameter. The microscopes gave a power of 94 diameters, which, divided by the power of the object-glass, (1.87,) shows the power of the microscopes alone to be 50 diameters. The power of the magnifiers for the vertical circle was found in the same way to be 7 diameters. The power of the microscopes was again found by placing two similar scales so that one was in the focus of the microscope and the other 10 inches from the eye; then looking at both at the same time, one division of the magnified scale covered 55 divisions of the other scale, indicating a power of 55 diameters in the microscope. With high powers the error by this method may be 4 or 5 diameters. The first method is much more exact. The power of the magnifiers being low, both methods gave it the same. A number of five-minute divisions on different parts of the limb were measured with microscope B. The mean was 4 revolutions, 59.2 divisions. The readings of this microscope should, therefore, be multiplied by 1.0027, to reduce them to minutes and seconds. In a similar way it was found that the readings of microscope A should be multiplied by 1.0069. There are four vertical, two horizontal, and two oblique spider-lines in the telescope. The two oblique lines are nearly vertical, crossing each other in the center of the field. For transit-work their intersection can be used for the middle line. The intervals between the lines were found by bringing each line separately to coincide with the spider-line of the collimator, reading the angle from the horizontal limb of the theodolite. The vertical angle being zero, no corrections were required. The lines are numbered from the illuminating end of the axis. The mean of 60 observations gave the interval from 1 to 2. 14.67 These wire intervals are not satisfactorily determined. Before being adopted they should be redetermined by some other method. The error of reading a single microscope was found by reading the microscope upon a division of the limb, then turning it off, bringing it back, and reading a second time. Twenty-one trials showed the average difference of a single observation, from the mean of all, to be 0".3. The error of a single pointing was found by bringing the spider-line of the telescope upon that of the collimator and reading both microscopes; the telescope was then turned slightly in azimuth and both micrometers turned part of a revolution. The telescope and micrometers were then brought back as near as possible to the original position and read a second time. The difference in the two readings is the combined error of pointing and reading. Twenty trials showed the average difference of a single pointing, from the mean of all, to be 0.4. The repeating powers of the telescope were tested by repeating the angle between two collimators standing on the same post. It was found that passing street-cars and heavily loaded wagons jarred the collimators so as to make permanent changes in the angles. This test was therefore valueless. A large astronomical transit was next used for a collimator, and the angle between the outside spider-lines was measured. The mean difference of a single pointing from the mean of all was 1".1. The mean difference of a set of five repetitions from the mean of all was 0.3. These quantities, of course, vary with the observers, and the conditions of light, steadiness, &c. They show approximately how much of the error in actual work should be attributed to the instrument and observer and how much to phase, horizontal refraction, twist of post, &c. During the examination of this instrument there have been no defects of any magnitude discovered. The errors of eccentricity, ellipticity, and graduation are small The material and workmanship of the construction are of the best quality. Wire intervals from 3 to 4. Wire intervals from 4 to 5 Angle between outer axis and its projection on horizontal limb Projection of outer axis on horizontal limb falls on the line from cen ter of limb to................. Angle between two vertical axes. Projection of inner axes on horizontal limb, when outer axis is verti cal, falls on the line from center of limb to..... Eccentricity of horizontal limb Line through centers of limb and inner axis passes through. Angular distance between microscopes.. Eccentricity of vertical limb..... Line through centers of limb and alidade passes through. Co-efficient for correction of micrometer A. Mean error of single reading of microscope. Mean error of single pointing at some object. Very respectfully, your obedient servant, General C. B. COMSTOCK, Major of Engineers. 1".05 2".24 148.19 158.03 14.83 148.44 89° 59' 48".1 440 6".5 2050 1.5 56° 179° 59' 54".4 1.9 150 179° 59' 58" 1.0069 1.0027 1".627 35 diameters. 35 diameters. 24 diameters. 50 diameters. 7 diameters. 24.5 0.3 0.4 1".1 0".3 E. S. WHEELER, Assistant United States Lake Survey. APPENDIX F. UNITED STATES NAVAL OBSERVATORY, SIR: I herewith inclose a description of the instruments used in the observations to determine the clock-corrections employed in the determination of the difference of longitude between Detroit and the United States Naval Observatory, together with the deduced clock-corrections and rates. The observations were made with the transit-instrument, which is mounted in the east wing of the Observatory, 42.9 feet (0.036) east of the center of the dome. This instrument has a focal length of 84.4 inches, and an aperture of 5.33 inches. The system of transit-threads is divided into five sets, known as sets A, B, C, D, and E, and the following table, derived from observations by Professor M. Yarnall, United States Navy, in 1871, shows the reduction for each wire to the mean of sets B, C, and D, when the clamp of the instrument is east: The observations were recorded on the cylinder-chronograph, which has been in use at the Observatory for many years by means of make-circuit signals in the usual way. The clock (a Charles Frodsham) records each beat on the chronograph except the sixtieth second of each minute, which, for the convenience of fixing the position of the beginning of each minute on the sheet, is omitted by means of a small arm, which, revolving on the shaft of the escapement-wheel of the clock, prevents the closing of the circuit at that instant by the clock. The corrections to the observed transit of a star were derived from the observed errors represented by c', n', and m'; c' represents the error of collimation, a' the equatorial value of the distance between the plane of collimation and the true meridian at the pole, and m' the distance between the plane of collimation and the true meridian at the equator. The quantities represented by n' and m' are used instead of errors of azimuth and level. By means of a collimating eye-piece, the error of collimation and level was obtained by reversing the instrument over a basin of mercury and measuring with the right-ascension micrometer the distance between the central thread and its image reflected from the mercury. Letting c, n, and m represent the corrections obtained from the observed and computed errors c', n', and m'; 2 A the distance of the central thread west of its image when the clamp-end of the axis is east; 2 the distance of the central thread west of its image when the clamp-end of the axis is west; p the correction for the excess of the radius of the clamp pivot = 0.008; the reduction of the mean of sets B, C, and D to the middle thread = 0.014; a the correction for diurnal aberration —— b the level correction; -0.016; a the American Nautical Almanac's place of the star; a' the observed place of the star; c' the approximate clock correction; c the clock correction for each star; and we have the following formula, which have been employed in determining the clock-corrections from the observations of each star: c = 4 (^—0')—p−7+a, clamp east. c = ·† (^— ^') +p+7+a, clamp west. b = − (A+)-p, clamp east. b = − } (^+4)+p, clamp west. n =α · (a' + c'+c sec 8) mn tan + b seco c = α- - (a'+m+n tan d+csec 8) , representing the latitude of the observing station = 38° 53′.65. One revolution of the right-ascension micrometer-screw = 1a. 5865. |