α a=R2― R1 + 3".49 cos (2 z+ a + 30) sin From this it appears that the correction to (R2— R1) depends almost wholly on the first term of the periodic part. It appears also that when cos (2 z+a+3) sin a is a maximum, the range of the different values of (RR) should be 2 × 3.49= 6.98. The correction to the mean reading of the two microscopes is 1.74 sin (2 z+3°)+0°.37 sin (4 z+327°) To test the accuracy of the periodic function thus found, we construct the table given below from the results obtained on an angle not used in the previous calcuation. Each value of (R-R1) in this table is a mean of two combined results, so that errors of observation, twist, and collimation may be assumed to be well eliminated. The agreement between the corrected results shows that the remaining discrepancies are well accounted for by the hypothesis of periodic errors. APPENDIX F.-LEVELING OBSERVATIONS. REPORT OF MESSRS. L. L. WHEELER AND F. W. LEHNARTZ, ASSISTANT ENGINEERS. OFFICE UNITED STATES LAKE SURVEY, February 10, 1876. GENERAL: We have the honor to submit the following report on leveling operations during the summer of 1875. The object of this leveling was to determine with as much accuracy as possible the elevation of the great lakes above the level of the sea. For this purpose, water-level observations were taken at Oswego, Charlotte, Port Dalhousie, Port Colborne, Cleveland, Rockwood, Lakeport, Escanaba, and Marquette, during the months of May, June, July, and August, to determine the mean elevation of the lakes with reference to certain fixed bench-marks established at those places. The gauge at Rockwood was referred to bench-marks in Gibraltar, 2 miles distant. The bench-marks at Oswego were to be connected by duplicate lines of levels with certain bench-marks at Greenbush, N. Y., the elevation of which above mean tide at New York had previously been determined by the United States Coast Survey. The bench-marks at Port Dalhousie were to be connected with those at Port Colborne, those at Gibraltar with those at Lakeport, and those at Escanaba with those at Marquette, in a similar manner. The bench-marks at Oswego were to be connected with those at Charlotte and Port Dalhousie, those at Port Colborne with those at Cleveland and Gibraltar, and those at Lakeport with those at Escanaba, by a comparison of water-level observations, it being assumed that the mean water-surface of the lakes during the summer months is level from one end of the lake to the other. This report is, therefore, divided into two parts: I. Leveling by means of the spirit-level. II. Leveling by means of water-level observations. 1.-LEVELING BY MEANS OF THE SPIRIT-LEVEL. For this work, two parties were detailed; Assistant Lehnartz having charge of the first, and Assistant L. L. Wheeler having charge of the second party. 1st. Instruments used.-The instruments used in this work were: Stackpole level, No. 1496, 11 inches focal length, and object-glass of 14 inches diameter, and magnifying power of 24, and Würdemann level No. 2, 17 inches focal length, and object-glass of 11 inches diameter. The spirit-level in each instrument is attached to the lower side of the telescope, and the value of one division of the level tube 6."42 and 3."17 respectively. The leveling-rods were of the pattern known as New York rods, were graduated to hundredths of a foot, and read by vernier to thousandths of a foot. They were supported, while being used, on steel pins, eight inches long, driven into the ground. 2d. Method of leveling.-The method of leveling was as follows: At the end of each day's work the first party established two permanent bench-marks; also one at midday, if one could be obtained. Where it was not possible to establish permanent bench-marks at the end of a day's work, three stakes, at least one foot long, were driven into the ground, 20 feet apart, until their tops were even with the surface of the ground, and used as stopping points. A description of these bench-marks, with their elevations, was sent to the second party, who determined the elevation of the same bench-marks. The second party had instructions, that, when the difference of elevation of a bench-mark and the preceding one differed from the first party's determination by more than 0.1 foot X (distance in miles), the line should be run twice again to ascertain which party was in error. This was necessary only twice throughout the season. In working, the instrument was set up at a convenient distance from the benchmarks, and readings taken on the bench-marks. The rodman then drove the steel pin at the same distance from the instrument as the bench-marks, and a reading was taken on the pin. The instrument was then set up beyond the pin and the work continued in this manner. The length of sight, where possible, was 200 feet, determined by pacg, and back-sights and fore-sights were always taken of equal length. The instrument was carefully adjusted each morning before commencing work, and also at any time during the day, when, for any reason, it was supposed that it might be out of adjustment. Each time the instrument was set up, it was carefully leveled, so as to turn in azimuth with little displacement of the bubble, but instead of trying to have it perfectly level at the instant of pointing, the scale readings of the ends of the bubble were noted and recorded. The value of the divisions of the levels were determined before leaving the office, and a table prepared, showing the correction to be applied to the readings of the rod for a given distance, (never to exceed 200 feet,) and a given inclination of the level-tube, (never to exceed 5 level divisions.) The instrument was sheltered from the sun by an umbrella. The value of the divisions of the levels was redetermined after returning to the office. It was found that the value of one division of Stackpole level changed very materially with the temperature. The leveling-rods were compared with the standard neter, both before leaving and after returning to the office, and the proper corrections have been made to elevations determined with them. 3. History of summer's work.-The following is a brief history of operations during the summer: The party under charge of Mr. Lehnartz left Detroit May 4, established the watergauge stations at Port Colborne, Port Dalhousie, and Oswego, and commenced work at Greenbush May 13. Oswego was reached by this party August 15. Work was commenced at Port Dalhousie August 16, and Port Colborne reached September 2. Work was commenced at Rockwood September 7, and Lakeport reached October 28. The party under charge of Mr. Wheeler left Detroit May 6, established water-gauge stations at Gibraltar, Rockwood, and Lakeport, examined water-gauge station at Sacket's Harbor, and commenced work at Greenbush May 28. Oswego was reached October 15 and Port Colborne October 29. This party was delayed during the summer by needed repairs of instrument and by releveling of lines. The route followed by both parties was along Erie Canal to Higginsville, along wagon-roads to Fish Creek, and along the New York and Oswego Midland Railroad to Oswego. The Welland Railway was followed from Port Dalhousie to Port Colborne. The average distance run by the first party per day, including days on which no work was done, was 1.97 miles, and by the second party 1.89 miles. Throughout the season, E. S. Davis and J. B. Johnson, recorders, acted as rodmen, and deserve much credit for the care with which they performed their duties and the interest manifested in the work. 4. Levels from Greenbush to Oswego.-The bench-marks at Greenbush were established by the United States Coast Survey. The following extract from letter of J. E. Hilgard contains the data for elevation of bench-marks at Greenbush: "UNITED STATES COAST SURVEY OFFICE, "March 20, 1875. "Lerels between New York and Albany.-The two levelings of Mr. Vose in 1857-'58 give the following results: "Greenbush bench-mark at Boston and Albany Railroad bridge.. Feet. Feet. 0.000 0.000 * 11.066 11.040 "New York, Hudson River, Eighteeenth-street bench-mark "Zero of tide-staff at New York, Eighteenth street, below bench-mark. "Zero of tide-staff below bench-mark at Albany. 11.053 18.665 29.718 3.78 25.94 "J. E. H. “Bench-marks near Albany.—The bench-mark to which the preceding levels are referred is on the stone abutment of the Boston and Albany Railroad bridge at Greenbush, a copper bolt in the northwest side of the northeast corner, on the third tier of stones. (See accompanying sketch.) "There is another bench-mark at Greenbush, on the southwest upper corner of the lower tier of stones, at the northwest corner of the culvert, a few rods south of the one on the railroad-bridge. It is 9.316 feet lower than the latter. "Another bench-mark is on the steam grist-mill at Greenbush, a cross-cut on the northwest side of the northeast corner of stone foundation, (see sketch.) It is 10.564 feet lower than the one on the railroad-bridge. "The bench-mark on the railroad-bridge was established in 1856, and was then found to be 29.081 feet above miter-sill of southwest or lower lock in basin in Albany; also 28.995 feet above miter-sill of northeast lock." (See letter of J. E. Hilgard, March 24, 1875.) In consequence of the rebuilding (1861 or '62) of the bridge and culvert on which the first and second of these bench-marks were placed, those bench-marks were lost, and only the third remained. The Coast Survey data give this bench-mark 10.564 feet below the bench-marks on the railroad-bridge. Hence the bench-mark on the grist-mill is 15.374 feet above mean tide at Governer's Island. Since the two bench-marks on the railroad structures had been lost, there was no check on the bench-mark on the grist-mill, except the difference in elevation between it and the lower miter-sill of lock No. 1 of the Erie Canal. According to the data Combining the results of the two methods of leveling, we have the following results: Mean surface of water (May 11 to August 31) above zero.. 1.71 Elevation of mean surface of Lake Ontario from May 11 to August 31, inclusive. 246. 56 Mean surface of water at Charlotte from May 11 to August 31, inclusive, below zero of gauge, (new) Elevation of zero.. Bench-mark on light-house at Charlotte above zero, (office 219). Elevation of bench-mark on light-house... 2.78 249.34 34.53 233 87 36.62 Mean surface of Lake Ontario from January 1, 1860, to December 31, 1875, below bench-mark on light-house, (office 486) Elevation of mean surface of Lake Ontario from January 1, 1860, to December 31, 1875.... 247 25 2d. Lake Erie.-The following tables show the monthly mean readings of the gauges at the stations on Lake Erie. Since the readings in every case are taken by measuring down from a fixed point to the surface of the water, the water-level readings at any two stations on Lake Erie should show a constant difference. In the means given in the tables, the mean for each month has been given weight according to the number of days during which observations were taken: B. M. "C."-Cross cut into stone of foundation of Collector's Office, third course of ozes from top, north side 1.4 feet from northwest corner. It is 0.177 foot above benchLark B. The following bench-marks were established at Port Colborne: B. M. on Baptist Church.-Top of point of iron bolt in east end of window-sill in sement of steeple, south side, of Baptist Church. It is 4.202 feet below the benchark on the custom-house. B. M. on Church of England.—Top of point of iron bolt in stone foundation, lower r of stones in south side of Church of England, (street front, east side of entrance.) is 5.653 feet below bench-mark on Custom-House. The zero of tide-gauge is highest point of iron bolt in top-course of masonry in south cess of south extension of west wall of canal-lock, just above swing-bridge. It is feet below bench-mark on custom-house. Table B contains the elevation of bench-marks between Port Dalhousie and Port lborne, above bench-mark B at Port Dalhousie. The explanation already given with regard to Table A applies to Table B. The following bench-marks were established in connection with the gauge at Rock od. B. M. 1.-A small cross on the stone window-sill of the south window on the west le of the brick house of Mr. Craig, corner of Farnsworth and Adams streets, Gibtar, Michigan. It is 1.976 feet above bench-mark 2. B. M. 2.-The southeast corner of the store door-sill of the door in the southeast gle of the light-house at Gibraltar. The zero of the gauge at Rockwood and the her bench-marks have been referred to this bench-mark. B. M. 3.-The highest point of a large rock 120 feet southeast of the southeast corner log-house on the Story farm. It is 10.677 feet below bench-mark 2. The zero of the is the highest point of a spike driven into a post supporting the runway of the -house on the Story farm. It is 8,740 feet below bench-mark 2. II.-LEVELING BY MEANS OF WATER-LEVEL OBSERVATIONS. At each station where water-level observations were taken three permanent bencharks were established and their elevation with reference to the zero of the gauge termined. Readings were taken at 7 a. m., 1 p. m., and 7 p. m. each day. At Oso, the tide-gauge consists of a strap of iron spiked to a post and graduated to aths of a foot, the graduation commencing at a point below the surface of the water dextending upward. Readings at this station, therefore, show the height of the sure of the water above the zero of the gauge. At the other stations measurements ere taken by measuring down from a fixed point to the surface of the water with a I graduated to hundredths of a foot. These measurements, therefore, show the ght of the fixed point (zero of gauge) above the surface of the water. Ist. Lake Ontario.-The following tables show the monthly mean readings of the ges at the stations on Lake Ontario. Since the readings at Oswego show how much the surface of the water is above the to of the gauge, and at the other stations how much it is below the zero of the gauge, readings at Oswego and Charlotte, or Oswego and Port Dalhousie should show constant sum, and the readings at Charlotte and Port Dalhousie, should show a stant difference. In the means given in the tables, the mean for each month has en given weight according to the number of days during which observations were Observations during May extend from May 11 to May 31, inclusive. |