{
"id": "b7f308da10bfabae5fd929fb10130f568c3fcb98",
"text": "XV. Observations for ascertaining the length of the Pendulum at Madras in the East Indies, latitude $13^\\circ 4' 9''$, N. with the conclusions drawn from the same. By John Goldingham, Esq. F. R. S.\n\nRead January 31, 1822.\n\nThe object of the enquiry in this paper has been considered at all times one of interest and importance, and is particularly so at present, when investigations have been completed in Europe, by order of some of the governments there; such as, with reference to their accuracy, had never before been made in any quarter of the globe, so far as comes within my recollection. I had seen the details of Captain Kater's experiments in the Philosophical Transactions, and he also did me the favour to send me out a copy of his Paper. The simplicity and accuracy of the apparatus induced me to write to that Gentleman, requesting he would have the goodness to order a similar one to be sent to me. This request he not only most readily complied with, but made the experiments requisite for enabling me to draw the conclusions; and thence to form the comparison with the results obtained in Europe. The apparatus arrived in March, and I immediately set about fixing it; which, notwithstanding the little solid assistance to be obtained in an operation of this nature from workmen in this country, I was enabled to effect in a most satisfactory manner; and I am led to hope, these observations will not be deemed unworthy the attention of the learned in Europe.\nThe clock used in these experiments has a gridiron pendulum, the motion being given by a spring; the maker's name is Haswell, and the works are of the best description: it was fixed to the north wall of the Observatory, which is of solid masonry two feet in thickness: the rate was ascertained by comparisons with the transit clock each day, at the commencement and conclusion of the experiments: the transit of the noon before the comparison, and that after, were used in finding this rate; so that four results were obtained from the two comparisons: the transit clock, which is an excellent time keeper, was regulated by transits of the sun and stars; the weather fortunately having been clear, both at noon and at night, during the time the experiments were making.\n\nThe first operation performed was that of making the weight of the clock pendulum black, and fixing the disc on the centre. This having been done, five blocks of seasoned teak-wood, each 4 inches in diameter and 7 in length, were prepared; the place above the clock for the frame, which was to support the pendulum, was then marked. This I did with great care and precaution: intersecting lines were drawn upon the wall to show the exact position of the centres of the blocks and of the screws for fastening the frame; holes 4 inches in diameter and $10\\frac{1}{2}$ in depth, (it being necessary to let the outer part of the blocks $3\\frac{1}{2}$ inches within the surface of the wall, to bring the pendulum sufficiently near the clock case) were then made in the wall, and the blocks, coated with tar to preserve them from the white ants, were let in and firmly secured. The outer surfaces of the whole, which had previously been made smooth and level, being in\none plane. The frame, its two parts being firmly screwed together, was then placed, levelled by means of a spirit level, and fixed to the blocks in the firmest manner: the frame enclosing the agates was next put up, levelled, and screwed in its place, the Y's elevated, and the pendulum hung; the knife edges were then lowered upon the agates; when I had the satisfaction to find, from the precautions which I had taken, that the pendulum was most correctly in its place. In this distant part of the globe, there is an anxiety in handling and fixing any new apparatus which is not felt in England, where the maker of it is ready to give assistance, as well as to repair any damage that may chance to have been done: here, little or no assistance can be obtained; and if the use of any part of the apparatus should be mistaken, and the part forced into a wrong place, the injury may be fatal to the experiments, as it cannot be repaired here; it therefore affords no small gratification when an instrument is firmly secured, uninjured, in its proper position.\n\nThe pendulum is precisely the same, in all its parts, as that used by Captain Kater at the different stations of the Trigonometrical Survey of England, and which he has fully described in the Philosophical Transactions for 1819. Any farther description therefore of its construction, will here be unnecessary.\n\nThe next operation was to fix the arc for measuring the vibrations. The clock-case was of handsome mahogany enriched with projecting mouldings, with the door in front of plate glass. The mouldings kept the pendulum at too great a distance from the part of the case where the arc could otherwise have been fastened, and it became necessary to\nhave a support in front of the case. I therefore had a solid stand of teak wood made, similar to that for supporting the telescope, the inner part cut out to the form of the mouldings of the clock-case, so that it fitted perfectly close to it; in this position it was screwed to the floor; the ends for the supports of the arc were then let in, and secured to the top of the stand, and the arc fixed in its proper place, with reference to the extreme point of the pendulum. The floor outside of this apparatus was then separated from the part of the floor which supported it, to prevent any shake by persons moving about within the building.\n\nThe small telescope containing the diaphragm was now fixed upon its stand, and screwed to the floor at the proper distance from the pendulum: this was about $9\\frac{1}{2}$ feet. The telescope, and every other part of the apparatus, have been so fully described by Captain Kater in the paper published in the Philosophical Transactions, that I feel it unnecessary to be more particular here.*\n\nWhile making the holes in the wall above the clock for the insertion of the blocks for supporting the frame, a great deal of dust would necessarily fall upon the clock-case; every part of the case where dust could penetrate was therefore filled up with wax, and several folds of cloth were afterwards secured over the whole of the case, so that it was hardly possible that any dust could penetrate to the works of the clock. After the frame was fixed, the cloth and wax were removed, and fresh oil applied to the works.\n\n* The drawing [Pl. XIV] shows the inside of a part of the Observatory, the pendulum up, and the adjustment of the diaphragm making, preparatory to commencing the observations.\nThe clock was then set in motion. This was on the 22d of March, and the observations commenced two days afterwards.\n\nThe following is the mode pursued in making the observations.\n\nThe pendulum was lifted up from the Y's by myself and an assistant, and the knife edges wiped with a cloth saturated with oil. The pendulum was then replaced, and the Y's lowered, so that the knife edges rested upon the agates. The telescope was then adjusted (care being taken that the O on the arc of vibration coincided with the point of the slip), so that the edges of the slip were exactly embraced by the edges of the diaphragm. The height of the barometer, of the thermometer fixed near the middle of the pendulum, and that of the hygrometer, were taken and registered. The point of the slip at the end of the pendulum was then brought and kept by the hand to about $1^\\circ$,3 upon the arc; and an instant before the pendulum of the clock was at its highest point on the same side, the hand was withdrawn, and the pendulum thereby allowed to vibrate freely. I stationed the head Bramin assistant* to take down the time, and the youngest Bramin assistant to count the clock, which he does with the greatest correctness. Having placed myself at the telescope, I found there was a sensible portion of time, more or less, as the arc of vibration was greater or smaller, between the disappearance of the disc behind the slip, and its reappearance; I therefore noticed the seconds, and parts of a second, when the disc disappeared, and also the instant when it\n\n* The name of the head assistant is Senavassaehary, and that of the other Teroovencatachary.\nagain appeared, both which the Bramin put down; the mean of these I took as the true time of the coincidence, and registered it accordingly. These times I found could be accurately noted; and it is probable the mean of the two observations is generally correct to less than half a second. In this manner the times of the coincidences were observed. The thermometer often varying a good deal in a short time, I thought it right to take its height three times, at the third observation of each set, as well as the first and fifth. The barometer was observed at the end of each set, as well as at the beginning; and also the hygrometer, as mentioned above, at the beginning of the observations, and likewise at the end of those of each day; being desirous of seeing how much the atmosphere had changed in dryness, as well as in heat and weight; not that this was material, but it is satisfactory to know what change there actually was in the atmosphere during the time the observations were making. I now proceed to detail the observations.\n\nEvery observation taken is here given. As although in this as in similar cases, one feels better pleased with some observations than with others, yet I do not recollect more than two of these observations, which I felt dissatisfied with at the time, and that not in a sufficient degree to induce me to think of rejecting them.\n\nThe results, I trust, will prove how unnecessary it would have been to have rejected any of the observations.\nthe length of the pendulum at Madras.\n\nOBSERVATIONS.\n\nFIRST SERIES.\n\nMarch 24th A. M.\n\nRate of the Clock — $0''$.39.\nHygrometer $11^\\circ$.5 dry.\n\nMean - $30,093$\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|----------------------|-------------------|-----------|---------------------|----------------------|---------------------------------|------------------|------------------|------------------------|\n| 81 | h. m. s. | | | | | | | | |\n| | 6 23 44 | | | | | | | | |\n| | 35 45.25 | | | | | | | | |\n| 81.3 | 47 48 | | | | | | | | |\n| | 59 52 | | | | | | | | |\n| 81.4 | 7 11 55.8 | | | | | | | | |\n| | | | | | | | | | |\n| 81.23 | Mean | | | | | | | | |\n\nRate of the Clock — $0''$.39\n\nBarometer $\\{30,105$ $30,123$\n\nMean - $30,114$\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|----------------------|-------------------|-----------|---------------------|----------------------|---------------------------------|------------------|------------------|------------------------|\n| 81.5 | h. m. s. | | | | | | | | |\n| | 7 24 50.5 | | | | | | | | |\n| | 36 52.9 | | | | | | | | |\n| 82 | 48 58.6 | | | | | | | | |\n| | 8 1 2.25 | | | | | | | | |\n| 82.7 | 13 6.0 | | | | | | | | |\n| | | | | | | | | | |\n| 81.73 | Mean | | | | | | | | |\n\nRate of the Clock — $0''$.39\n\nMean - $30,114$\nMr. Goldingham's observations for ascertaining\n\nBarometer \\( \\begin{cases} 30,137 \\\\ 30,134 \\end{cases} \\)\n\nMean - 30,135\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|---------------------|------------------|-----------|---------------------|----------------------|---------------------------------|-------------------|------------------|------------------------|\n| 82,9 | h. m. s. | 1,15 | | | | | | | |\n| | 8 29 26,5 | 1,15 | | | | | | | |\n| | 41 27,1 | 1,05 | | | | | | | |\n| 83,4 | 53 32,1 | 0,95 | | | | | | | |\n| | 9 5 34,75 | 0,87 | | | | | | | |\n| 84,3 | 17 37,4 | 0,79 | | | | | | | |\n| 83,53 | Mean | | | | | | | | |\n\nRate of the Clock\n\n\\( 86168,135 - 0,390 = 86167,745 \\)\n\nBarometer \\( \\begin{cases} 30,129 \\\\ 30,140 \\end{cases} \\)\n\nMean - 30,135\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|---------------------|------------------|-----------|---------------------|----------------------|---------------------------------|-------------------|------------------|------------------------|\n| 84,5 | h. m. s. | 1,11 | | | | | | | |\n| | 9 33 39,25 | 1,11 | | | | | | | |\n| | 45 38,6 | 1,03 | | | | | | | |\n| 85 | 57 36,75 | 0,95 | | | | | | | |\n| | 10 9 35,5 | 0,88 | | | | | | | |\n| 85 | 21 35,0 | 0,80 | | | | | | | |\n| 84,83 | Mean | | | | | | | | |\n\nRate of the Clock\n\n\\( 86167,437 - 0,390 = 86167,047 \\)\nthe length of the pendulum at Madras.\n\nHygrometer 13, dry\n\nBarometer \\( \\{ 30,140 \\\\ 30,143 \\} \\)\n\nMean - 30,141\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|---------------------|------------------|-----------|----------------------|---------------------|---------------------------------|-------------------|-----------------|------------------------|\n| 84,9 | h. m. s. | | | | | | | | |\n| | 10 30 16,6 | 1,20 | | | | | | | |\n| | 42 10,25 | 1,13 | | | | | | | |\n| 85 | 54 5,5 | 1,04 | | | | | | | |\n| | 11 6 2,5 | 0,99 | | | | | | | |\n| 85 | 17 58,15 | 0,90 | | | | | | | |\n\nMean\n\nRate of the Clock\n\nMarch 25, A. M.\n\nClock losing 1\",49\n\nHygrometer 12°. dry\n\nBarometer \\( \\{ 30,143 \\\\ 30,147 \\} \\)\n\nMean - 30,145\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|---------------------|------------------|-----------|----------------------|---------------------|---------------------------------|-------------------|-----------------|------------------------|\n| 80,3 | h. m. s. | | | | | | | | |\n| | 6 14 0,25 | 1,24 | | | | | | | |\n| | 25 57,75 | 1,17 | | | | | | | |\n| 81,0 | 37 58 | 1,09 | | | | | | | |\n| | 49 58,5 | 1,02 | | | | | | | |\n| 81,2 | 7 2 1,1 | 0,94 | | | | | | | |\n\nMean\n\nRate of the Clock\nMr. Goldingham's observations for ascertaining\n\nEnd of the Experiments.\n\nHygrometer $12^\\circ\\frac{1}{8}$ dry.\n\nBarometer $\\{30,147$ $30,161$\n\nMean - $30,154$\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|---------------------|------------------|-----------|----------------------|----------------------|---------------------------------|-------------------|------------------|------------------------|\n| 81,2 | h. m. s. | 1,28 | | | | | | | |\n| | 7 10 19,5 | | | | | | | | |\n| | 22 18,6 | 1,18 | | | | | | | |\n| 81,6 | 34 20,6 | 1,07 | | | | | | | |\n| | 46 23,6 | 0,97 | | | | | | | |\n| 82 | 58 28,1 | 0,85 | | | | | | | |\n| 81,6 | Mean | | | | | | | | |\n\nRate of the Clock\n\nMarch 26, P. M.\n\nClock losing $1''$,91\n\nHygrometer $15^\\circ$,6 dry\n\nBarometer $\\{30,132$ $30,133$\n\nMean - $30,133$\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|---------------------|------------------|-----------|----------------------|----------------------|---------------------------------|-------------------|------------------|------------------------|\n| 84 | h. m. s. | 1,20 | | | | | | | |\n| | 4 58 37,1 | | | | | | | | |\n| | 10 33,6 | 1,10 | | | | | | | |\n| 83,9 | 22 31,45 | 1,04 | | | | | | | |\n| | 34 30,5 | 0,98 | | | | | | | |\n| 83,5 | 46 30,5 | 0,90 | | | | | | | |\n| 83,8 | Mean | | | | | | | | |\n\nRate of the Clock\nMarch 26, A. M.\n\nClock losing \\(2''\\),08\n\nHygrometer \\(14^\\circ\\), dry.\n\nBarometer \\(\\{30,149\\)\n\nMean - \\(30,149\\)\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|---------------------|------------------|-----------|----------------------|----------------------|---------------------------------|-------------------|-----------------|------------------------|\n| 79,3 | h. m. s. | 5 58 58,1 | 1,19 | 1, 14 | 723,27 | 721,27 | 86161,08 | 2,129 | 3,955 | 86167,164 |\n| | | | | | | | | | | |\n| | | | | | | | | | | |\n| 79,5 | | | | | | | | | | |\n| | | | | | | | | | | |\n| 79,8 | | | | | | | | | | |\n\nMean\n\nRate of the Clock\n\n\\(86167,457\\)\n\n\\(-2,080\\)\n\n\\(86165,377\\)\n\nHygrometer \\(13^\\circ\\),6 dry.\n\nBarometer \\(\\{30,150\\)\n\nMean - \\(30,162\\)\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|---------------------|------------------|-----------|----------------------|----------------------|---------------------------------|-------------------|-----------------|------------------------|\n| 79,8 | h. m. s. | 6 56 30,62 | 1,29 | 1, 24 | 721,88 | 719,88 | 86161,63 | 2,519 | 4,264 | 86168,413 |\n| | | | | | | | | | | |\n| | | | | | | | | | | |\n| 80,9 | | | | | | | | | | |\n| | | | | | | | | | | |\n| 81,3 | | | | | | | | | | |\n\nMean\n\nRate of the Clock\n\n\\(86168,031\\)\n\n\\(-2,08\\)\n\n\\(86165,971\\)\nMr. Goldingham's observations for ascertaining\n\nMarch 27, P. M.\n\nClock losing 2\",06\nHygrometer 20° dry.\n\nBarometer\nMean -\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|---------------------|------------------|-----------|---------------------|----------------------|---------------------------------|-------------------|---------------------|------------------------|\n| 84.7 | h. m. s. | | | | | | | | |\n| | 4 13 44.25 | 1.24 | | | | | | | |\n| | 25 40.12 | 1.16 | | | | | | | |\n| 84.3 | 37 36.63 | 1.08 | | | | | | | |\n| | 49 34.5 | 0.99 | | | | | | | |\n| 84.1 | 5 1 32.37 | 0.92 | | | | | | | |\n| 84.37 | Mean | | | | | | | | |\n\nMean of the Clock\n\nHygrometer 19°,3 dry.\n\nBarometer\nMean -\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|---------------------|------------------|-----------|---------------------|----------------------|---------------------------------|-------------------|---------------------|------------------------|\n| 84.1 | h. m. s. | | | | | | | | |\n| | 5 10 14.75 | 1.28 | | | | | | | |\n| | 22 11.25 | 1.19 | | | | | | | |\n| 83.9 | 34 9.75 | 1.09 | | | | | | | |\n| | 46 8.62 | 1.00 | | | | | | | |\n| 83.5 | 58 9.25 | 0.92 | | | | | | | |\n| 83.83 | Mean | | | | | | | | |\n\nRate of the Clock\nthe length of the pendulum at Madras.\n\nMarch 29, P. M.\n\nClock losing $1''_{46}$\nHygrometer $17^\\circ_{7}$ dry.\n\nBarometer $\\{30,084$\n$\\{30,113$\n\nMean - $30,098$\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|---------------------|------------------|-----------|----------------------|----------------------|---------------------------------|-------------------|-------------------|------------------------|\n| 84,5 | h. m. s. | ° | 1,23 | \" | 719,12 | 86159,71 | + | + | 86168,142 |\n| | 2 59 8 | | | | 717,12 | | | | |\n| | 3 11 7,12 | 1,15 | | | 720,88 | 86160,29 | 2,019 | 6,070 | 86168,389 |\n| 84,3 | 23 8,0 | 1,07 | | | 723,0 | 86160,99 | 1,705 | 6,028 | 86168,723 |\n| | 35 11,0 | 0,97 | | | 723,63 | 86161,20 | 1,417 | 5,985 | 86168,602 |\n| 84,1 | 47 14,63 | 0,89 | | | 723,63 | | | | |\n| | Mean | | | | | | | | |\n\nRate of the Clock\n\n$86168,464$\n$- 1,460$\n$86167,004$\n\nHygrometer $17^\\circ_{7}$ dry.\n\nBarometer $\\{30,113$\n$\\{30,087$\n\nMean - $30,100$\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|---------------------|------------------|-----------|----------------------|----------------------|---------------------------------|-------------------|-------------------|------------------------|\n| 84,1 | h. m. s. | ° | 1,26 | \" | 721,0 | 86160,33 | 2,418 | 5,956 | 86168,704 |\n| | 3 56 44,75 | | | | 719,0 | | | | |\n| | 48 45,75 | 1,17 | | | 721,75 | 86160,58 | 2,037 | 5,930 | 86168,447 |\n| 84 | 20 47,50 | 1,06 | | | 723,62 | 86161,20 | 1,688 | 5,922 | 86168,810 |\n| | 32 51,12 | 0,97 | | | 723,13 | 86161,04 | 1,417 | 5,922 | 86168,379 |\n| 84 | 44 54,25 | 0,89 | | | 723,13 | | | | |\n| | Mean | | | | | | | | |\n\nRate of the Clock\n\n$86168,585$\n$- 1,46$\n$86167,121$\nMr. Goldingham's observations for ascertaining\n\nMarch 30, P. M.\n\nClock losing $1''$,96\nHygrometer $16°$,2 dry.\n\nBarometer $\\{30,136$\n$\\{30,117$\n\nMean - $30,126$\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibration in 24 hours. |\n|-------|---------------------|------------------|-----------|----------------------|----------------------|---------------------------------|-------------------|-------------------|---------------------|\n| 84,4 | h. m. s. | | | | | | | | |\n| | 2 57 | 1,5 | 1,27 | | | | | | |\n| | 3 9 | 2,5 | 1,18 | | | | | | |\n| 84,3 | 21 3,75 | 1,09 | 1,135 | | | | | | |\n| | 33 6,25 | 1,00 | 1,045 | | | | | | |\n| 84,3 | 45 9,62 | 0,92 | 0,96 | | | | | | |\n\nMean Rate of the Clock\n\n84,33\n\nBarometer $\\{30,117$\n$\\{30,115$\n\nMean - $30,116$\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibration in 24 hours. |\n|-------|---------------------|------------------|-----------|----------------------|----------------------|---------------------------------|-------------------|-------------------|---------------------|\n| 84,3 | h. m. s. | | | | | | | | |\n| | 3 52 18,25 | 1,24 | 0 | | | | | | |\n| | 4 4 18,87 | 1,14 | 1,19 | | | | | | |\n| 84,1 | 16 21 | 1,05 | 1,095 | | | | | | |\n| | 28 24,75 | 1,96 | 1,005 | | | | | | |\n| 84 | 40 38,28 | 0,89 | 0,925 | | | | | | |\n\nMean Rate of the Clock\n\n84,13\nthe length of the pendulum at Madras.\n\nMarch 30, A. M.\n\nClock losing 2'',23\nHygrometer 13°,7 dry.\n\nBarometer\n\nMean - 30,134\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|----------------------|------------------|-----------|---------------------|----------------------|---------------------------------|-------------------|-----------------|------------------------|\n| 80,1 | h. m. s. | 0 | 1,29 | | | | | | |\n| | 6 0 30 | | | 1,245 | 720, 5 | 718, 5 | 86160,17 | 2,539 | 4,327 |\n| | 12 30,5 | | | | | | | | |\n| | 1, 16 | | | 721,25 | 719,25 | 86160,41 | 2,205 | 4,433 | 86167,048 |\n| 80,6 | 24 31,75 | | | | | | | | |\n| | 1, 08 | | | 721,87 | 719,87 | 86160,62 | 1,911 | 4,518 | 86167,049 |\n| | 36 33,62 | | | | | | | | |\n| | 1, 0 | | | 723,50 | 721,50 | 86161,16 | 1,639 | 4,577 | 86167,376 |\n| 80,9 | 48 37,12 | | | | | | | | |\n| | 0,96 | | | | | | | | |\n| 80,53 | Mean | | | | | | | | |\n\nRate of the Clock\n\n86167,127\n— 2, 23\n86164,897\n\nHygrometer 13°,3 dry.\n\nBarometer\n\nMean - 30,153\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|----------------------|------------------|-----------|---------------------|----------------------|---------------------------------|-------------------|-----------------|------------------------|\n| 81 | h. m. s. | 0 | 1,19 | | | | | | |\n| | 6 59 7,12 | | | 1,145 | 722,75 | 720,75 | 86160,91 | 2,148 | 4,695 |\n| | 7 11 9,87 | | | | | | | | |\n| | 1, 05 | | | 723,63 | 721,63 | 86161,20 | 1,806 | 4,780 | 86167,786 |\n| 81,4 | 23 13, 5 | | | | | | | | |\n| | 0,965 | | | 724, 0 | 722, 0 | 86161,33 | 1,526 | 4,843 | 86167,699 |\n| | 35 17, 5 | | | | | | | | |\n| | 0,905 | | | 724, 5 | 722, 5 | 86161,49 | 1,341 | 4,886 | 86167,717 |\n| 81,6 | 47 22, 0 | | | | | | | | |\n| | 0,88 | | | | | | | | |\n| 81,33 | Mean | | | | | | | | |\n\nRate of the Clock\n\n86167,739\n— 2,230\n86165,509\nMr. Goldingham's observations for ascertaining\n\nMarch 31, P. M.\n\nClock losing \\(2''\\), 24\nHygrometer 16°, 2 dry.\n\nBarometer \\(\\{30,07\\)\n\\(30,066\\)\n\nMean - 30,068\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|---------------------|------------------|-----------|----------------------|----------------------|---------------------------------|-------------------|---------------------|------------------------|\n| 85.1 | h. m. s. | 1,24 | | | | | | | |\n| | 3 8 27,5 | | | | | | | | |\n| | 20 25,75 | 1,14 | | | | | | | |\n| 84.9 | 32 25,12 | 1,05 | | | | | | | |\n| | 44 25,5 | 0,96 | | | | | | | |\n| 84.6 | 56 25,25 | 0,88 | | | | | | | |\n| 84.87 | Mean | | | | | | | | |\n\nRate of the Clock\n\n86167,953\n— 2,240\n86165,713\n\nHygrometer 16°, 3 dry.\n\nBarometer \\(\\{30,066\\)\n\\(30,061\\)\n\nMean - 30,064\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|---------------------|------------------|-----------|----------------------|----------------------|---------------------------------|-------------------|---------------------|------------------------|\n| 84.6 | h. m. s. | 1,37 | | | | | | | |\n| | 4 6 55,5 | | | | | | | | |\n| | 18 52,12 | 1,27 | | | | | | | |\n| 84.4 | 30 49,87 | 1,17 | | | | | | | |\n| | 42 49,87 | 1,07 | | | | | | | |\n| 84.26 | 54 49,62 | 0,97 | | | | | | | |\n| 84.42 | Mean | | | | | | | | |\n\nRate of the Clock\n\n86167,806\n— 2,240\n86165,566\nMarch 31, P. M.\n\nClock losing $2''10$\nHygrometer $14^\\circ,6$ dry.\n\nBarometer $\\{30,098$\n$\\{30,114$\n\nMean - $30,106$\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc- | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Aic. | For Temperature. | Vibrations in 24 hours. |\n|-------|----------------------|------------------|-----------|---------------------|----------------------|---------------------------------|-------------------|-----------------|------------------------|\n| 81,3 | h. m. s. | 0 | 0 | \" | 721,5 | 86160,50 | + | + | 86167,506 |\n| | 5 59 27, 5 | 1,21 | 1,17 | 721,5 | 719,5 | 86161,03 | 1,964 | 4,733 | 86167,727 |\n| | 6 11 28, 5 | 1,13 | 1,095 | 723,12 | 721,12 | 86161,29 | 1,721 | 4,725 | 86167,736 |\n| 81,15 | 23 31,62 | 1,06 | 1,025 | 723,88 | 721,88 | 86161,61 | 1,494 | 4,750 | 86167,854 |\n| | 35 35, 5 | 0,99 | 0,955 | 724,87 | 722,87 | 86161,61 | | | |\n| 81,25 | 47 40,37 | 0,92 | 0,955 | 724,87 | 722,87 | 86161,61 | | | |\n\nMean\n\nRate of the Clock\n\n86167,706\n— 2,100\n86165,606\n\nHygrometer $13^\\circ,5$ dry.\n\nBarometer $\\{30,114$\n$\\{30,130$\n\nMean - $30,122$\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc- | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Aic. | For Temperature. | Vibrations in 24 hours. |\n|-------|----------------------|------------------|-----------|---------------------|----------------------|---------------------------------|-------------------|-----------------|------------------------|\n| 81,35 | h. m. s. | 0 | 0 | \" | 722,75 | 86160,91 | 2,205 | 4,822 | 86167,937 |\n| | 6 59 59, 5 | 1,20 | 1,16 | 722,75 | 720,75 | 86161,45 | 1,911 | 4,864 | 86168,225 |\n| | 12 2,25 | 1,12 | 1,08 | 724,37 | 722,37 | 86161,90 | 1,622 | 4,941 | 86168,463 |\n| 81,55 | 24 6,62 | 1,04 | 0,995 | 725,75 | 723,75 | 86162,19 | 1,357 | 5,021 | 86168,568 |\n| | 36 12,37 | 0,95 | 0,91 | 726,63 | 724,63 | 86162,19 | | | |\n| 81,95 | 48 19,00 | 0,87 | 0,91 | 726,63 | 724,63 | 86162,19 | | | |\n\nMean\n\nRate of the Clock\n\n86168,298\n— 2,100\n86166,198\nMr. Goldingham's observations for ascertaining\n\nApril 1, P. M.\n\nClock losing \\(1''85\\)\nHygrometer \\(16^\\circ8\\) dry.\n\nBarometer \\(\\{30,100\\)\\)\nMean - \\(30,094\\)\n\n| Temp. | Time of coincidence. | Arc vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|---------------------|---------------|-----------|----------------------|----------------------|---------------------------------|-------------------|------------------|------------------------|\n| 85,3 | h. m. s. | | | | | | | | |\n| 2 | 23 30,62 | 1,18 | | | | | | | |\n| | | | | | | | | | |\n| 35 | 28,12 | 1,095 | | | | | | | |\n| | | | | | | | | | |\n| 85,3 | 47 28,25 | 1,000 | | | | | | | |\n| | | | | | | | | | |\n| 59 | 29 | 0,96 | | | | | | | |\n| | | | | | | | | | |\n| 85,15 | 3 11 30,5 | 0,845 | | | | | | | |\n| | | | | | | | | | |\n| 85,25 | Mean | Rate of the Clock | | | | | | | |\n\nHygrometer \\(16^\\circ8\\) dry.\n\nBarometer \\(\\{30,087\\)\\)\nMean - \\(30,085\\)\n\n| Temp. | Time of coincidence. | Arc vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|---------------------|---------------|-----------|----------------------|----------------------|---------------------------------|-------------------|------------------|------------------------|\n| 85,15 | h. m. s. | | | | | | | | |\n| 3 | 19 39,75 | 1,27 | | | | | | | |\n| | | | | | | | | | |\n| 31 | 39 | 1,15 | | | | | | | |\n| | | | | | | | | | |\n| 85,10 | 43 38,62 | 1,05 | | | | | | | |\n| | | | | | | | | | |\n| 55 | 39,25 | 0,97 | | | | | | | |\n| | | | | | | | | | |\n| 85,06 | 4 7 41,25 | 0,89 | | | | | | | |\n| | | | | | | | | | |\n| 85,10 | Mean | Rate of the Clock | | | | | | | |\n\n\\(86168,379 - 1,850 = 86166,529\\)\nthe length of the pendulum at Madras.\n\nApril 1, P. M.\n\nClock losing 1\",86\nHygrometer 48° dry.\n\nBarometer\n\n\\[\n\\begin{align*}\n&30,126 \\\\\n&30,134 \\\\\n\\text{Mean} & - 30,130\n\\end{align*}\n\\]\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Contraction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|---------------------|------------------|-----------|----------------------|----------------------|-------------------------------|-------------------|-----------------|------------------------|\n| 81,95 | h. m. s. | 0 | 1, 19 | \" | 722,25 | 86160,83 | 2,148 | 5,114 | 86168,092 |\n| | | | 1, 10 | | 723,13 | 86161,04 | 1,824 | 5,233 | 86168,097 |\n| 82,5 | 40 55,13 | 1, 01 | 1,055 | | 723,37 | 86161,12 | 1,567 | 5,300 | 86167,987 |\n| | 52 58,5 | 0,945 | 0,978 | | 723,62 | 86161,20 | 1,363 | 5,321 | 86167,884 |\n| 82,6 | 7 5 2,12 | 0, 88 | 0,912 | | 723,62 | 86161,20 | 1,363 | 5,321 | 86168,015 |\n| | Mean | | | | | | | | 86166,155 |\n\nRate of the Clock\n\nHygrometer 13°,8 dry.\n\nBarometer\n\n\\[\n\\begin{align*}\n&30,134 \\\\\n&30,172 \\\\\n\\text{Mean} & - 30,153\n\\end{align*}\n\\]\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Contraction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|---------------------|------------------|-----------|----------------------|----------------------|-------------------------------|-------------------|-----------------|------------------------|\n| 82,6 | h. m. s. | 0 | 1,195 | \" | 720 | 86161,00 | 2,159 | 5,355 | 86168,514 |\n| | | | 1, 10 | | 723,88 | 86161,28 | 1,841 | 5,410 | 86168,531 |\n| 82,85 | 37 18,5 | 1, 02 | 1,06 | | 723,12 | 86161,69 | 1,574 | 5,444 | 86868,708 |\n| | 49 23,62 | 0, 94 | 0,98 | | 725,63 | 86161,86 | 1,321 | 5,469 | 86168,650 |\n| 82,95 | 8 1 29,25 | 0, 855 | 0,89 | | 725,63 | 86161,86 | 1,321 | 5,469 | 86168,650 |\n| 82,80 | Mean | | | | | | | | 86168,601 |\n| | Rate of the Clock | | | | | | | | 86166,741 |\n\nMDCCCXXII.\nMr. Goldingham's observations for ascertaining\n\nApril 2, P. M.\n\nClock losing 2\"09\nHygrometer 16°7 dry.\n\nBarometer \\{30,185 \\\\\n30,127\\}\n\nMean - 30,131\n\n| Temp | Time of coincidence | Arc of vibration | Mean Arc. | Interval in seconds | Number of vibrations | Observed vibrations in 24 hours | Correction for Arc. | For Temperature | Vibrations in 24 hours |\n|------|-------------------|-----------------|-----------|---------------------|---------------------|-------------------------------|-------------------|------------------|---------------------|\n| 85.75 | h. m. s. 2 52 1,5 | 0 1, 23 | \" | 719,62 717,62 | 86159,87 | 2,281 | + | + | 86168,771 |\n| 3 4 1,12 | 1, 13 | 1,085 | 719,88 717,88 | 86159,96 | 1,929 | 6,535 | 86168,424 |\n| 85.35 | 16 1,0 | 1, 04 | 0,993 | 721, 0 719, 0 | 86160,33 | 1,616 | 6,455 | 86168,401 |\n| 28 2 | 0,945 | 0, 91 | 720,12 718,12 | 86160,04 | 1,357 | 6,383 | 86167,780 |\n| 85.0 | 40 2,12 | 0,875 | | | | | | |\n| 85.37 | Mean | Rate of the Clock | 86168,344 — 2,090 | 86166,254 |\n\nHygrometer 16°7 dry.\n\nBarometer \\{30,127 \\\\\n30,118\\}\n\nMean - 30,123\n\n| Temp | Time of coincidence | Arc of vibration | Mean Arc. | Interval in seconds | Number of vibrations | Observed vibrations in 24 hours | Correction for Arc. | For Temperature | Vibrations in 24 hours |\n|------|-------------------|-----------------|-----------|---------------------|---------------------|-------------------------------|-------------------|------------------|---------------------|\n| 85 | h. m. s. 3 46 27,87 | 1,195 | \" | 719,88 717,88 | 86159,96 | 2,178 | + | + | 86168,462 |\n| 58 27,75 | 1,110 | 1,052 | 720,25 718,25 | 86160,08 | 1,813 | 6,282 | 86168,175 |\n| 84.8 | 4 10 28,0 | 0,995 | 0,958 | 722,12 720,12 | 86160,70 | 1,504 | 6,260 | 86168,464 |\n| 22 30,12 | 0, 92 | 0,885 | 722,03 720,03 | 86160,68 | 1,283 | 6,260 | 86168,223 |\n| 84.8 | 34 32,15 | 0, 85 | | | | | | |\n| 84.87 | Mean | Rate of the Clock | 86168,331 — 2,090 | 86166,241 |\nThe correction for the arc of vibration was ascertained by multiplying the square of the mean arc by $1,6385$. The correction for temperature was found as follows: the mean of the thermometer at the beginning and middle of the observations was taken, and that of the middle and end; which gave five heights, one for each observation; the mean of the first and second, of the second and third, and so on in succession was taken, which gave four mean heights; the difference between each of these and $70^\\circ$ was multiplied by $0.423$, the part of a vibration due to each degree of the thermometer, as furnished by Captain Kater, and the required correction was obtained.\n\nThe rate of the clock was found as before mentioned. The following shows the daily rate of the transit clock, in the interval during which the observations were taken; and furnishes a satisfactory example of the good performance of this standard for finding the rate of the other clock.\n\n**Rate of the Transit Clock.**\n\n| March 22 | - | - | - | 0.25 |\n|----------|---|---|---|------|\n| 23 | + | 0.13 |\n| 24 | - | - | - | 0.05 |\n| 25 | + | 0.23 |\n| 26 | + | 0.10 |\n| 27 | + | 0.15 |\n| 28 | + | 0.20 |\n\n| March 29 | - | - | + | 0.09 |\n|----------|---|---|---|------|\n| 30 | + | 0.05 |\n| 31 | - | - | - | 0.03 |\n| April 1 | - | - | + | 0.02 |\n| 2 | - | - | - | 0.08 |\n| 3 | - | - | + | 0.05 |\n| 4 | - | - | - | 0.04 |\nMr. Goldingham's observations for ascertaining\n\nTable of the Results of the foregoing Experiments.\n\n| Day. 1821 | Time of the Experiment | Mean Height of the | Number of Vibrations in 24 hours, at the temperature of 70° of Fahrenheit |\n|-----------|------------------------|--------------------|---------------------------------------------------------------------|\n| | | Thermometer. | Barometer. | Hygrometer. | dry. | |\n| March 24 | A. M. | 81°23 | 30°093 | | | 86166,908 |\n| | | 81°73 | 30°114 | | | 86167,339 |\n| | | 83°53 | 30°135 | | | 86166,275 |\n| | | 84°83 | 30°135 | 12°25 | | 86167,047 |\n| | | 84°97 | 30°141 | | | 86166,218 |\n| | A. M. | 80°83 | 30°145 | | | 86165,111 |\n| | | 81°6 | 30°154 | 12°22 | | 86166,083 |\n| | P. M. | 83°8 | 30°133 | | | 86165,176 |\n| | A. M. | 79°53 | 30°149 | 14°4 | | 86165,377 |\n| | | 80°67 | 30°162 | | | 86165,971 |\n| | P. M. | 84°37 | 30°123 | | | 86164,930 |\n| | | 83°83 | 30°109 | 19°6 | | 86165,320 |\n| | P. M. | 84°3 | 30°098 | | | 86167,004 |\n| | | 84°03 | 30°100 | 17°7 | | 86167,121 |\n| | P. M. | 84°33 | 30°126 | | | 86166,742 |\n| | | 84°13 | 30°116 | | | 86166,715 |\n| | A. M. | 80°53 | 30°134 | 14°92 | | 86164,897 |\n| | | 81°33 | 30°153 | | | 86165,509 |\n| | P. M. | 84°87 | 30°068 | | | 86165,713 |\n| | | 84°42 | 30°064 | 15°15 | | 86165,566 |\n| | A. M. | 81°23 | 30°106 | | | 86165,606 |\n| | | 81°62 | 30°122 | | | 86166,198 |\n| April | P. M. | 85°25 | 30°094 | | | 86166,273 |\n| | | 85°10 | 30°085 | | | 86166,529 |\n| | A. M. | 82°35 | 30°130 | 15°65 | | 86166,155 |\n| | | 82°80 | 30°153 | | | 86166,741 |\n| | P. M. | 85°37 | 30°131 | | | 86166,254 |\n| | | 84°87 | 30°123 | 16°7 | | 86166,241 |\n\nMean | 83°48 | 30°121 | 15°38 | 86166,108 |\n\nSecond Series of Experiments for ascertaining the length of the Pendulum at Madras.\n\nThinking it possible that these Observations might be referred to by future observers in other parts of the world, and wishing to have as accurate results as I could obtain, I deter-\nmined to take a second series; having made what I considered some improvement in detaching the clock and apparatus from the floor of the building. In this series, besides comparisons for the rate of the clock used in the experiments, with the transit clock at the time of making the experiments, transits of stars were taken with this clock for the purpose. The result of this series, however, seems to prove, that every necessary precaution had been used in the first, the difference of the two being only 0.06 of a vibration in 24 hours.*\n\nThe following are the Observations of the Second Series.\n\n**OBSERVATIONS.**\n\n**SECOND SERIES.**\n\nApril 18th A. M.\n\nRate of Clock + 0\",97\n\nHygrometer 12°,6 dry.\n\nBarometer {30,018\n 30,029\n\nMean - 30,025\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For temperature. | Vibrations in 24 hours. |\n|-------|---------------------|------------------|-----------|----------------------|----------------------|---------------------------------|-------------------|------------------|------------------------|\n| 82.7 | h. m. s. | 0° 1,275 | 1,122 | 711,12 | 709,12 | 86157,00 | 2,447 | 5,402 | 86164,849 |\n| | | | | | | | | | |\n| | | | | | | | | | |\n| 83 | | | | | | | | | |\n| | | | | | | | | | |\n| 83.45 | | | | | | | | | |\n| | | | | | | | | | |\n| 83.05 | Mean | | | | | | | | |\n\n* By rejecting the 4 in each series, which differs most from the mean, we obtain a mean 0.03 of a vibration less than that from which the conclusions have been drawn.\nMr. Goldingham's observations for ascertaining\n\nBarometer \\( \\begin{cases} 30,029 \\\\ 30,044 \\end{cases} \\)\n\nMean - 30,037\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Are. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|---------------------|------------------|-----------|----------------------|----------------------|-------------------------------|------------------|-----------------|------------------------|\n| 83.45 | h. m. s. | 1° 30' | 1° 25' | 708.63 | 706.63 | 86156.15 | 2.560 | 5.748 | 86164.478 |\n| | 23 15.75 | 1° 20' | 1° 155' | 711.38 | 709.38 | 86157.09 | 2.186 | 5.863 | 86165.539 |\n| 84.0 | 35 7.13 | 1° 11' | 1° 067' | 711.12 | 709.12 | 86157.00 | 1.865 | 5.985 | 86164.850 |\n| | 46 58.25 | 1° 025' | 0° 990' | 712.12 | 710.12 | 86157.34 | 1.606 | 6.112 | 86165.058 |\n| 84.6 | 58 50.37 | 0° 955' | | | | | | | |\n| | Mean | | | | | | | | |\n\nRate of the Clock 86164.981 + .970 = 86165.951\n\nHygrometer 15° dry.\n\nBarometer \\( \\begin{cases} 30,044 \\\\ 30,045 \\end{cases} \\)\n\nMean - 30,044\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Are. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|---------------------|------------------|-----------|----------------------|----------------------|-------------------------------|------------------|-----------------|------------------------|\n| 84.6 | h. m. s. | 1° 39' | 1° 337' | 709.0 | 707.0 | 86156.28 | 2.929 | 6.218 | 86165.424 |\n| | 24 50.5 | 1° 285' | 1° 233' | 708.75 | 706.75 | 86156.19 | 2.491 | 6.303 | 86164.984 |\n| 85 | 36 39.25 | 1° 18' | 1° 132' | 710.0 | 708 | 86156.66 | 2.100 | 6.396 | 86165.156 |\n| | 48 29.25 | 1° 085' | 1° 043' | 709.37 | 707.37 | 86156.40 | 1.782 | 6.502 | 86164.684 |\n| 85.5 | 21 0 18.62 | 1° 00' | | | | | | | |\n| | Mean | | | | | | | | |\n\nRate of the Clock 86165.062 + .970 = 86166.032\nthe length of the pendulum at Madras.\n\nApril 19, P. M.\n\nRate of Clock o\"88\nHygrometer 16°5 dry.\n\nBarometer {29,983\n 29,961\n\nMean - 29,972\n\n| Temp. | Time of coincidence | Arc of vibration | Mean Arc. | Interval in seconds | Number of vibrations | Observed vibrations in 24 hours | Correction for Arc. | For Temperature. | Vibrations in 24 hours |\n|-------|---------------------|-----------------|-----------|--------------------|----------------------|-------------------------------|-------------------|------------------|----------------------|\n| 87,5 | h. m. s. | | | | | | | | |\n| | 2 21 17,62 | 0 | 1,275 | | | | | | |\n| | | | | | | | | | |\n| | 33 5,12 | 1,170 | | | | | | | |\n| 87,35 | 44 53,62 | 1,055 | | | | | | | |\n| | 56 44,5 | 0,975 | | | | | | | |\n| | 87,15 | 0,933 | | | | | | | |\n| | 3 8 34,12 | 0,890 | | | | | | | |\n\nMean\n\nRate of the Clock\n\nBarometer {29,961\n 29,952\n\nMean - 29,956\n\n| Temp. | Time of coincidence | Arc of vibration | Mean Arc. | Interval in seconds | Number of vibrations | Observed vibrations in 24 hours | Correction for Arc. | For Temperature. | Vibrations in 24 hours |\n|-------|---------------------|-----------------|-----------|--------------------|----------------------|-------------------------------|-------------------|------------------|----------------------|\n| 87,1 | h. m. s. | | | | | | | | |\n| | 3 17 0,12 | 0 | 1,26 | | | | | | |\n| | | | | | | | | | |\n| | 28 48,62 | 1,14 | | | | | | | |\n| 86,9 | 40 36,5 | 1,045 | | | | | | | |\n| | 52 27,12 | 0,98 | | | | | | | |\n| | 86,8 | 0,935 | | | | | | | |\n| | 4 4 17,62 | 0,89 | | | | | | | |\n\nMean\n\nRate of the Clock\nMr. Goldingham's observations for ascertaining\n\nHygrometer $16^\\circ 5$ dry.\n\nBarometer $\\begin{cases} 29.952 \\\\ 29.948 \\end{cases}$\n\nMean - $29.950$\n\n| Temp. | Time of coincidence | Arc of vibration | Mean Arc. | Interval in seconds | Number of vibrations | Observed vibrations in 24 hours | Contraction for Arc. | For Temperature | Vibrations in 24 hours |\n|-------|---------------------|-----------------|-----------|--------------------|---------------------|-------------------------------|-------------------|------------------|----------------------|\n| 86.8 | h. m. s. | 0° 1,19 | \" | 709,22 | 707,22 | 86156,35 | 2,129 | 7,064 | 86165,543 |\n| | | | 1° 14 | 710,0 | 708,0 | 86156,62 | 1,789 | 6,679 | 86165,388 |\n| 86.4 | 35 36,62 | 1,00 | 1,045 | 711,75 | 709,75 | 86157,22 | 1,491 | 6,908 | 86165,619 |\n| | 47 28,37 | 0,91 | 0,955 | 712,25 | 710,25 | 86157,39 | 1,226 | 6,840 | 86165,456 |\n| 86.1 | 59 20,62 | 0,82 | 0,865 | 713,63 | 711,63 | 86157,843 | | | |\n\n86.43 Mean Rate of the Clock $86165,5^2 + 880$\n\n86166,382\n\nApril 19, A. M. $18^h 20$ Civil.\n\nRate of Clock $+ 0''70$\n\nHygrometer $13^\\circ 4$ dry.\n\nBarometer $\\begin{cases} 30.046 \\\\ 30.041 \\end{cases}$\n\nMean - $30.044$\n\n| Temp. | Time of coincidence | Arc of vibration | Mean Arc. | Interval in seconds | Number of vibrations | Observed vibrations in 24 hours | Contraction for Arc. | For Temperature | Vibrations in 24 hours |\n|-------|---------------------|-----------------|-----------|--------------------|---------------------|-------------------------------|-------------------|------------------|----------------------|\n| 83 | h. m. s. | 0° 1,29 | \" | 710,25 | 708,25 | 86156,705 | 2,532 | 5,562 | 86164,799 |\n| | | | 1° 1,243 | 711,50 | 709,5 | 86157,132 | 2,167 | 5,689 | 86164,988 |\n| 83.6 | 18 29,37 | 1,105 | 1,150 | 711,50 | 709,50 | 86157,132 | 1,865 | 5,753 | 86164,750 |\n| | 30 20,87 | 1,030 | 1,067 | 713,63 | 711,63 | 86157,843 | 1,622 | 5,753 | 86165,218 |\n| 83.6 | 42 14,5 | 0,96 | 0,995 | 713,63 | 711,63 | 86157,843 | | | |\n\n83.4 Mean Rate of the Clock $86164,939 + 0.700$\n\n86165,639\nthe length of the pendulum at Madras.\n\nBarometer \\( \\left\\{ \\begin{array}{c} 30,041 \\\\ 30,050 \\end{array} \\right. \\)\n\nMean - 30,045\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|---------------------|------------------|-----------|----------------------|----------------------|---------------------------------|-------------------|-----------------|------------------------|\n| 83.6 | h. m. s. | 0 | 1,305 | 1,257 | 709.75 | 707.75 | 86156,533 | 2,589 | 5,795 | 86164,917 |\n| | | | | | | | | | | |\n| 84 | | 1, 21 | 1, 16 | 712.37 | 710.37 | 86157,429 | 2,205 | 5,880 | 86165,514 | |\n| | | | | | | | | | | |\n| 84.25 | | 1, 07 | 1, 03 | 711.38 | 709.38 | 86157,091 | 1,876 | 5,951 | 86164,918 | |\n| | | | | | | | | | | |\n| 83.95 | Mean | | | | | | | | | |\n\nRate of the Clock\n\nHygrometer 13°.6.\n\nBarometer \\( \\left\\{ \\begin{array}{c} 30,050 \\\\ 30,036 \\end{array} \\right. \\)\n\nMean - 30,043\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|---------------------|------------------|-----------|----------------------|----------------------|---------------------------------|-------------------|-----------------|------------------------|\n| 84.25 | h. m. s. | 0 | 1, 25 | 1, 21 | 710.25 | 708.25 | 86156,705 | 2,399 | 6,104 | 86165,208 |\n| | | | | | | | | | | |\n| 84.95 | | 1, 17 | 1, 125 | 711.37 | 709.37 | 86157,088 | 2,074 | 6,252 | 86165,414 | |\n| | | | | | | | | | | |\n| 85.1 | | 1, 08 | 1, 038 | 711.50 | 709.50 | 86157,132 | 1,765 | 6,341 | 86165,238 | |\n| | | | | | | | | | | |\n| 84.77 | Mean | | | | | | | | | |\n\nRate of the Clock\n\nMDCCCXXII.\noccasions the final equations, by using a formula for the computation of V, different from that given by M. Laplace.\n\nPreviously to the computation, I re-calculated the observations of April 8, 11, 14, 21, and May 3, using the places of the stars as given by M. Piazzi. This perhaps was unnecessary.\n\nThe results were\n\n| AR | Declination S. |\n|-----|----------------|\n| | h. m. s. |\n| April 8 | 2 34 15.0 | 7 51 52 |\n| 11 | 2 46 28.0 | 7 12 4 |\n| 14 | 2 57 14.2 | 6 33 51 |\n| May 21 | 3 17 46.1 | 5 13 35 |\n| 3 | 3 44 19.7 | 3 25 55 |\n\nBy M. Laplace's method, \"Determination approchée, &c.\" the observations of April 8, 11, and 14, give perihelion dist. \\((p) = 0.0865\\), and time of perihelion, March \\(19^d 14^h 4^m\\).\n\nLet the true perihelion dist. \\(= p + dp\\), and time of perihelion \\(= March 19^d 14^h 4^m - dt\\), so that \\(t\\) being the interval between March \\(19^d 14^h 4^m\\), and the observation of April 8, the true value of \\(t = t + dt\\).\n\nLet also \\(T =\\) the time in the table of the comet of 109 days, when the anomaly \\(= v\\).\n\n\\(\\Delta =\\) variation of anomaly in that table in one day at time \\(T\\), and \\(r =\\) comet's distance from the sun\n\nThen \\(dv = \\left( \\frac{dt}{p^2} - \\frac{3tdp}{p^3} \\right) \\Delta \\)\n\nand \\((x) \\frac{d \\log r}{\\sin 1''} = dv \\tan \\frac{1}{2} v + \\frac{dp}{p \\sin 1''}\\)\n\nConceive the triangle in which \\(S, T, C\\), represent the sun, earth, and comet respectively, and let \\(P\\) represent the projection of the comet on the plane of the earth's orbit.\nthe length of the pendulum at Madras.\n\nHygrometer $17^\\circ$,5 dry. Barometer $\\{30,030 \\\\ 30,032$\n\nMean - $30,031$\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|----------------------|------------------|-----------|---------------------|----------------------|---------------------------------|-------------------|------------------|------------------------|\n| 86,9 | h. m. s. | | | | | | | | |\n| | 4 2 37,87 | 1,19 | 0 | | | | + | + | |\n| | 14 27,62 | 1,10 | 1,145 | 709,75 | 707,75 | 86156,533 | 2,148 | 7,140 | 86165,821 |\n| 86,8 | 26 18,5 | 1,0 | 0,955 | 710,88 | 708,88 | 86157,061 | 1,806 | 7,115 | 86165,982 |\n| | 38 9,87 | 0,91 | 0,875 | 711,37 | 709,37 | 86157,088 | 1,491 | 7,085 | 86165,664 |\n| 86,6 | 50 2,37 | 0,84 | | | | | | | |\n| | Mean | | | | | | | | |\n\nRate of Clock $0''$,63\n\nApril 21, A. M. Barometer $\\{30,064 \\\\ 30,062$\n\nHygrometer $14^\\circ$ dry.\n\nMean - $30,063$\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|----------------------|------------------|-----------|---------------------|----------------------|---------------------------------|-------------------|------------------|------------------------|\n| 83,4 | h. m. s. | | | | | | | | |\n| | 17 51 37 | 1,305 | 0 | | | | + | + | |\n| | 18 3 25,62 | 1,205 | 1,255 | 708,62 | 706,62 | 86156,489 | 2,581 | 5,681 | 86164,751 |\n| 83,5 | 15 15,75 | 1,114 | 1,159 | 710,13 | 708,13 | 86156,664 | 2,201 | 5,702 | 86164,567 |\n| | 27 6,75 | 1,060 | 1,082 | 711,00 | 709,00 | 86156,692 | 1,918 | 5,732 | 86164,342 |\n| 83,7 | 38 58,87 | 0,985 | 1,023 | 712,12 | 710,12 | 86157,344 | 1,715 | 5,773 | 86164,832 |\n| | Mean | | | | | | | | |\n\nRate of the Clock $+ ,630$\n\n86164,623\n\n86165,253\nMr. Goldingham's observations for ascertaining\n\nBarometer \\( \\begin{cases} 30,062 \\\\ 30,068 \\end{cases} \\)\n\nMean - 30,065\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|----------------------|------------------|-----------|---------------------|----------------------|---------------------------------|-------------------|-----------------|------------------------|\n| 83,7 | h.m.s. | | | | | | | | |\n| | 18 45 38,5 | 1,205 | 0 | \" | 709,87 | 707,87 | 86156,575 | 2,193 | 5,808 |\n| | | | | | | | | | 86164,576 |\n| | 57 28,37 | 1,110 | 1,157 | | 711,26 | 709,26 | 86157,050 | 1,922 | 5,829 |\n| | | | | | | | | | 86164,801 |\n| 83,8 | 19 9 19,63 | 1,055 | 1,083 | | 711,74 | 709,74 | 86157,214 | 1,711 | 5,858 |\n| | | | | | | | | | 86164,783 |\n| | 21 11,37 | 0,990 | 1,022 | | 712,50 | 710,50 | 86157,473 | 1,488 | 5,901 |\n| | | | | | | | | | 86164,862 |\n| 84 | 33 3,87 | 0,915 | 0,953 | | 712,50 | 710,50 | 86157,473 | 1,488 | 5,901 |\n| | | | | | | | | | 86164,862 |\n\nMean Rate of the Clock\n\nHygrometer 14° dry.\n\nBarometer \\( \\begin{cases} 30,068 \\\\ 30,072 \\end{cases} \\)\n\nMean - 30,070\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|----------------------|------------------|-----------|---------------------|----------------------|---------------------------------|-------------------|-----------------|------------------------|\n| 84,1 | h.m.s. | | | | | | | | |\n| | 19 40 10 | 0,19 | 0 | \" | 709,63 | 707,63 | 86156,492 | 2,167 | 5,985 |\n| | | | | | | | | | 86164,644 |\n| | 51 59,63 | 1,11 | 1,15 | | 713,12 | 711,12 | 86157,684 | 1,858 | 6,028 |\n| | | | | | | | | | 86165,570 |\n| 84,3 | 20 3 52,75 | 1,02 | 1,065 | | 712,0 | 710,0 | 86157,303 | 1,574 | 6,104 |\n| | | | | | | | | | 86164,981 |\n| | 15 44,75 | 0,94 | 0,98 | | 713,88 | 711,88 | 86157,942 | 1,342 | 6,209 |\n| | | | | | | | | | 86165,493 |\n| 84,8 | 27 38,63 | 0,87 | 0,905 | | 713,88 | 711,88 | 86157,942 | 1,342 | 6,209 |\n| | | | | | | | | | 86165,493 |\n\nMean Rate of the Clock\nthe length of the pendulum at Madras.\n\nApril 22, P. M.\n\nRate of Clock + 0\",76\nHygrometer 18°,4 dry.\n\nBarometer {30,025\n 30,017\n\nMean - 30,021\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc- | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|----------------------|------------------|-----------|---------------------|----------------------|---------------------------------|-------------------|-------------------|------------------------|\n| 87,6 | h. m. s. | 0 | 1,24 | \" | 708,13 | 706,13 | 86155,977 | 2,301 | 7,411 | 86165,689 |\n| | | | | | 708,37 | 706,37 | 86156,059 | 1,929 | 7,352 | 86165,340 |\n| 87,3 | | 1,04 | | | 709,63 | 707,63 | 86156,492 | 1,638 | 7,309 | 86165,439 |\n| | | | | | 710,87 | 708,87 | 86156,917 | 1,558 | 7,288 | 86165,763 |\n| 87,2 | | 0,96 | | | | | | | |\n| | | | | | | | | | |\n| 87,37 | Mean | Rate of the Clock| | | | | | | |\n\nBarometer {30,025\n 30,017\n\nMean - 30,021\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc- | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|----------------------|------------------|-----------|---------------------|----------------------|---------------------------------|-------------------|-------------------|------------------------|\n| 87,2 | h. m. s. | 0 | 1,23 | \" | 708,25 | 706,25 | 86156,018 | 2,243 | 7,268 | 86165,529 |\n| | | | | | 709,01 | 707,01 | 86156,279 | 1,858 | 7,246 | 86165,383 |\n| 87,1 | | 1,02 | | | 710,75 | 708,75 | 86156,876 | 1,574 | 7,221 | 86165,671 |\n| | | | | | 711,49 | 709,49 | 86157,122 | 1,312 | 7,204 | 86165,638 |\n| 87 | | | | | | | | | |\n| | | | | | | | | | |\n| 87,1 | Mean | Rate of the Clock| | | | | | | |\n\nBarometer {30,025\n 30,017\n\nMean - 30,021\nMr. Goldingham's observations for ascertaining\n\nHygrometer $16^\\circ,2$ dry. Barometer $\\{29.977$\n\nMean - $29.977$\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|---------------------|------------------|-----------|----------------------|----------------------|---------------------------------|-------------------|-----------------|------------------------|\n| 87 | h. m. s. | 0° 30\" | \" | 707,87 | 705,87 | 86155,887 | 2,560 | 7,161 | 86165,608 |\n| | | | 1° 25\" | 708 | 706 | 86155,932 | 2,186 | 7,094 | 86165,212 |\n| 86.7 | 23 39:37 | 1° 11\" | 1° 067 | 709,13 | 707,13 | 86156,321 | 1,865 | 7,043 | 86165,229 |\n| | 35 28:5 | 1° 025 | 0° 993 | 709,87 | 707,87 | 86156,575 | 1,516 | 7,000 | 86165,091 |\n| 86.5 | 47 18:37 | 0° 96 | | | | | | | |\n| 86.73 | Mean | | | | | | | | |\n\nRate of the Clock\n\nApril 20, A. M.\n\nBarometer $\\{30.047$\n\nRate of Clock $0''$,22 + $\\}$\n\nHygrometer $13^\\circ,4$ Mean - $30.059$\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|---------------------|------------------|-----------|----------------------|----------------------|---------------------------------|-------------------|-----------------|------------------------|\n| 83.3 | h. m. s. | 0° 275 | \" | 711,37 | 709,37 | 86157,088 | 2,491 | 5,647 | 86165,226 |\n| | | | 1° 233 | 710,75 | 708,75 | 86156,876 | 2,167 | 5,689 | 86164,732 |\n| 83.5 | 16 53:62 | 1° 11 | 1° 065 | 712,13 | 710,13 | 86157,347 | 1,858 | 5,732 | 86164,937 |\n| | 28 45:75 | 1° 020 | 0° 990 | 711,00 | 709,00 | 86156,962 | 1,606 | 5,774 | 86164,342 |\n| 83.7 | 40 36:75 | 0° 96 | | | | | | | |\n| 83.5 | Mean | | | | | | | | |\n\nRate of the Clock\nthe length of the pendulum at Madras.\n\nBarometer \\( \\left\\{ \\begin{array}{c} 30,071 \\\\ 30,088 \\end{array} \\right. \\)\n\nMean \\( 30,079 \\)\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|---------------------|------------------|-----------|----------------------|----------------------|---------------------------------|-------------------|------------------|------------------------|\n| 83,7 | h. m. s. | 1°, 36 | 1°, 31 | 709,37 | 707,37 | 86156,403 | 2,812 | 5,829 | 86165,044 |\n| | | | | | | | | | |\n| | | | | | | | | | |\n| 84 | | | | | | | | | |\n| | | | | | | | | | |\n| 84,3 | | | | | | | | | |\n\nMean Rate of the Clock\n\nHygrometer 13°,4 dry.\n\nBarometer \\( \\left\\{ \\begin{array}{c} 30,088 \\\\ 30,082 \\end{array} \\right. \\)\n\nMean \\( - 30,085 \\)\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|---------------------|------------------|-----------|----------------------|----------------------|---------------------------------|-------------------|------------------|------------------------|\n| 84,35 | h. m. s. | 1°, 39 | 1°, 337 | 710, 0 | 708 | 86156,619 | 2,929 | 6,112 | 86165,660 |\n| | | | | | | | | | |\n| | | | | | | | | | |\n| 84,75 | | | | | | | | | |\n| | | | | | | | | | |\n| 85,15 | | | | | | | | | |\n\nMean Rate of the Clock\nMr. Goldingham's observations for ascertaining\n\nApril 21, P. M.\n\nRate of Clock + \\( \\sigma''_{37} \\)\nHygrometer 17° dry.\n\nBarometer \\( \\{ 30,045 \\\\ 30,038 \\} \\)\n\nMean - 30,042\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|---------------------|------------------|-----------|----------------------|----------------------|---------------------------------|-------------------|-------------------|------------------------|\n| 87.5 | h. m. s. | | | | | | | | |\n| | 7 33, 5 | 1,275 | 1,220 | 708, 5 | 706, 5 | 86156,104 | 2,439 | 7,373 | 86165,916 |\n| | 19 22 | 1,165 | 1,123 | 709,37 | 707,37 | 86156,403 | 2,066 | 7,305 | 86165,774 |\n| 87.2 | 31 11,37 | 1,080 | 1,038 | 710,25 | 708,25 | 86156,705 | 1,765 | 7,254 | 86165,724 |\n| | 43 1,62 | 0,995 | 0,953 | 711,01 | 709,01 | 86156,965 | 1,488 | 7,212 | 86165,665 |\n| 87 | 54 52,63 | 0,910 | | | | | | | |\n| | Mean | | | | | | | | |\n\nRate of the Clock \\( + 0,370 \\)\n\nBarometer \\( \\{ 30,038 \\\\ 30,030 \\} \\)\n\nMean - 30,034\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|---------------------|------------------|-----------|----------------------|----------------------|---------------------------------|-------------------|-------------------|------------------------|\n| 87 | h. m. s. | | | | | | | | |\n| | 3 43,13 | 0, 29 | 1, 23 | 708,49 | 706,49 | 86155,959 | 2,479 | 7,191 | 86165,629 |\n| | 13 31,62 | 1, 17 | 1,127 | 709,88 | 707,88 | 86156,584 | 2,081 | 7,191 | 86165,856 |\n| 87 | 25 21,50 | 1,085 | 1,043 | 711,0 | 709,0 | 86156,962 | 1,782 | 7,183 | 86165,927 |\n| | 37 12, 5 | 1, 00 | 0,955 | 710,88 | 708,88 | 86157,061 | 1,491 | 7,161 | 86165,713 |\n| 86.9 | 49 3,38 | 0, 91 | | | | | | | |\n| | Mean | | | | | | | | |\n\nRate of the Clock \\( + 0,370 \\)\nthe length of the pendulum at Madras.\n\nHygrometer $17^\\circ$, 5.\n\nBarometer $\\{30,017$ \n$30,007$\n\nMean - $30,012$\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|----------------------|-------------------|-----------|---------------------|----------------------|---------------------------------|------------------|------------------|------------------------|\n| 87 | h. m. s. | | | | | | | | |\n| | 3 56 31,37 | 1,14 | 0 | \" | 709,88 | 707,88 | 86156,584 | 1,929 | 7,170 |\n| | | | | | | | | | 86165,683 |\n| | 4 8 21,25 | 1,03 | 0,99 | \" | 711,37 | 709,37 | 86157,088 | 1,606 | 7,127 |\n| | | | | | | | | | 86165,821 |\n| 86,8 | 20 12,62 | 0,95 | 0,91 | \" | 711,51 | 709,51 | 86157,136 | 1,357 | 7,106 |\n| | | | | | | | | | 86165,599 |\n| | 32 4,13 | 0,87 | 0,83 | \" | 712,75 | 710,75 | 86157,558 | 1,128 | 7,106 |\n| | | | | | | | | | 86165,792 |\n| 86,8 | 43 56,88 | 0,79 | 0,83 | \" | 712,75 | 710,75 | 86157,558 | 1,128 | 7,106 |\n| | | | | | | | | | 86165,792 |\n\nMean\n\nRate of the Clock\n\n$86165,724$\n\n$+ ,760$\n\n$86166,484$\n\nApril 22, A. M.\n\nRate of Clock $+ 0''98$\n\nHygrometer $15^\\circ$ dry (beginning.)\n\nBarometer $\\{30,022$ \n$30,016$\n\nMean - $30,019$\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|----------------------|-------------------|-----------|---------------------|----------------------|---------------------------------|------------------|------------------|------------------------|\n| 83,2 | h. m. s. | | | | | | | | |\n| | 17 45 46 | 1,26 | 0 | \" | 714 | 712 | 86157,983 | 2,439 | 5,596 |\n| | | | | | | | | | 86166,018 |\n| | 57 40 | 1,18 | 1,22 | \" | 715,62 | 713,62 | 86158,531 | 2,110 | 5,617 |\n| | | | | | | | | | 86166,258 |\n| 83,3 | 18 9 35,62 | 1,09 | 1,045 | \" | 716,88 | 714,88 | 86158,955 | 1,789 | 5,626 |\n| | | | | | | | | | 86166,370 |\n| | 21 32, 5 | 1,0 | 0,965 | \" | 718, 5 | 716, 5 | 86159,498 | 1,526 | 5,626 |\n| | | | | | | | | | 86166,650 |\n| 83,3 | 33 31 | 0,93 | 0,965 | \" | 718, 5 | 716, 5 | 86159,498 | 1,526 | 5,626 |\n| | | | | | | | | | 86166,650 |\n\nMean\n\nRate of the Clock\n\n$86166,324$\n\n$+ ,980$\n\n$86167,304$\n\nMDCCCXXII.\nMr. Goldingham's observations for ascertaining\n\nHygrometer $14^\\circ$, dry (end.) Barometer $\\{30,016$ $30,040$\n\nMean - $30,028$\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|---------------------|------------------|-----------|----------------------|----------------------|---------------------------------|-------------------|-----------------|------------------------|\n| 83.4 | h. m. s. | 0° | | | | | | | |\n| | 18 43 | 3.87 | 1, 21 | | | | | | |\n| | 54 59 | 5 | 1, 111 | | | | | | |\n| 83.7 | 19 6 | 56.75 | 1, 04 | | | | | | |\n| | 18 54 | 5 | 0, 96 | | | | | | |\n| 84 | 30 53 | 0 | 0, 88 | | | | | | |\n| 83.7 | Mean | | | | | | | | |\n\nRate of the Clock $86166,667$ $+ 0.980$\n\n$86167,647$\n\nApril 23, P. M.\n\nRate of Clock $+ 0''.99$\n\nHygrometer $18^\\circ$, dry.\n\nBarometer $\\{30,017$ $30,008$\n\nMean - $30,012$\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|---------------------|------------------|-----------|----------------------|----------------------|---------------------------------|-------------------|-----------------|------------------------|\n| 87.6 | h. m. s. | 0° | | | | | | | |\n| | 2 26 10,12 | 1,25 | 1, 20 | | | | | | |\n| | 37 57,62 | 1,15 | 1, 10 | | | | | | |\n| 87.4 | 49 47,38 | 1,05 | 1,015 | | | | | | |\n| | 3 1 36,5 | 0,98 | 0, 94 | | | | | | |\n| 87.3 | 13 27 | 0,90 | 0, 94 | | | | | | |\n| 87.43 | Mean | | | | | | | | |\n\nRate of the Clock $86165,588$ $+ 0.99$\n\n$86166,578$\nthe length of the pendulum at Madras.\n\nHygrometer $18^\\circ,6$. Barometer $\\{30,008 \\\\ 30,000$\n\nMean $- 30,004$\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations | Observed vibrations in 24 hours. | Correction for Arc. | For Temperature. | Vibrations in 24 hours. |\n|-------|----------------------|-------------------|-----------|---------------------|---------------------|---------------------------------|------------------|------------------|------------------------|\n| 87,3 | h. m. s. | | | | | | | | |\n| 3 | 20 31 | 1,21 | 1,16 | 717,37 | 715,37 | 86155,714 | 2,205 | 7,297 | 86165,216 |\n| | 32 18,37 | 1,11 | 1,07 | 710,00 | 708,00 | 86156,619 | 1,876 | 7,254 | 86165,749 |\n| 87,1 | 44 08,37 | 1,03 | 0,98 | 709,38 | 707,38 | 86156,407 | 1,574 | 7,225 | 86166,143 |\n| | 55 57,75 | 0,93 | 0,895 | 712,12 | 710,12 | 86157,344 | 1,312 | 7,216 | 86165,872 |\n| 87,05 | 4 7 49,87 | 0,86 | | | | | | | |\n| 87,15 | Mean | | | | | | | | Rate of the Clock |\n| | | | | | | | | | + 3,990 |\n| | | | | | | | | | 86166,735 |\nMr. Goldingham's observations for ascertaining\n\nTable of the Results of the foregoing Experiments.\n\nSECOND SERIES.\n\n| Day. 1821 | Time of the Experiment | Mean Height of the | Number of Vibrations in 24 hours, at the temperature of 70° of Farenheit. |\n|-----------|-----------------------|--------------------|---------------------------------------------------------------------|\n| | | Thermometer. | Barometer. | Hygrometer. | Inch. | dry. | |\n| April 18 | A. M. | 83.05 | 30.023 | 13.8 | 86165.857 |\n| | | 84.02 | 30.037 | | 86165.951 |\n| | | 85.04 | 30.044 | | 86166.032 |\n| | P. M. | 87.33 | 29.972 | | 86166.442 |\n| | | 86.93 | 29.956 | | 86166.246 |\n| | | 86.43 | 29.950 | | 86166.382 |\n| | A. M. | 83.4 | 30.044 | 13.5 | 86165.639 |\n| | | 83.95 | 30.045 | | 86165.900 |\n| | | 84.77 | 30.043 | | 86165.970 |\n| | P. M. | 87.23 | 29.995 | | 86165.473 |\n| | | 87.0 | 29.979 | 17.1 | 86165.598 |\n| | | 86.73 | 29.977 | | 86165.745 |\n| | A. M. | 83.5 | 30.059 | 13.4 | 86165.029 |\n| | | 84.0 | 30.079 | | 86165.150 |\n| | | 84.75 | 30.085 | | 86165.626 |\n| | P. M. | 87.23 | 30.042 | | 86166.140 |\n| | | 86.93 | 30.034 | 14 | 86166.151 |\n| | | 86.73 | 30.031 | | 86166.179 |\n| | A. M. | 83.53 | 30.063 | | 86165.253 |\n| | | 83.83 | 30.065 | 17.4 | 86165.385 |\n| | | 84.4 | 30.070 | | 86165.805 |\n| | P. M. | 87.37 | 30.032 | | 86166.318 |\n| | | 87.1 | 30.021 | 17.3 | 86166.315 |\n| | | 86.87 | 30.012 | | 86166.484 |\n| | A. M. | 83.27 | 30.019 | 14.6 | 86167.304 |\n| | | 83.7 | 30.028 | | 86167.647 |\n| | P. M. | 87.43 | 30.012 | | 86166.578 |\n| | | 87.15 | 30.004 | 18.4 | 86166.735 |\n| Mean | | 85.49 | 30.258 | 15.6 | 86166.048 |\nThe height of the pendulum above the level of the sea was 27 feet; the distance in a direct line to the sea being about 4900 yards, or 2.784 miles. The country is flat; the nearest elevation being St. Thomas's Mount, which is 9950 yards, or 5.654 miles off, and rises but little above the ordinary level.* There is a range of low hills a short distance beyond St. Thomas's Mount; and the Pulicat Mountains, which are of considerable elevation, are 39 miles off. The soil about Madras is composed of sand and blue mud, and this to as great depths as the wells have been sunk. I do not recollect any rock having been found. I have therefore used 0.66 as a multiplier to 0.095, the correction for 27 feet, which gives 0.06 to be added to the number of beats in 24 hours.\n\nThe last correction required was for the buoyancy of the atmosphere. Having no information relative to the specific gravity of the pendulum, I was obliged to determine it in the best way the limited means in this country afforded. This was done with a balance at a dispensary, and with the aid of Mr. Bruce, the proprietor of the establishment. The Madras water drawn from wells in the Black town here, and conducted into the cisterns in the fort, is considered among the purest in the world. This was boiled, and strained into a tin trough prepared for the purpose; the pendulum also was securely and properly slung by means of brass wire, with the assistance of Mr. Gordon, jeweller, of this place. The water was at the same temperature with the atmosphere, and the experiments were made with every care. It may be unnecessary to detail them here; I shall therefore proceed to the result, which was as follows:\n\n* About 150 feet above the level of the sea.\nThermometer $88^\\circ$, barometer $30,064$ inches, specific gravity of the pendulum $8,1085$. Hence the specific gravity of the pendulum for the mean of the first series of observations, the thermometer being $83^\\circ,48$, and barometer $30,121$, was $8,02096$, and the correction for the buoyancy of the atmosphere is $+6,2075$ vibrations. For the second series, the thermometer having been $85^\\circ,49$, and barometer $30,258$ inches, this correction is $6,220$ vibrations. These corrections being applied to the number of vibrations before found, will give the true number of vibrations of the pendulum in $24$ hours in vacuo at the level of the sea, the thermometer being $70^\\circ$, and are as follow:\n\nBy the first series of observations, $86172,3755$. By the second series, $86172,328$. The mean being $86172,352$.\n\nThe length of the seconds pendulum in London, (latitude $51^\\circ 31' 8'',4$ N.) at the temperature of $70^\\circ$, according to Captain Kater, is $39,142213$ inches. Now, the pendulum of experiment used at Madras, made $86293,44$ vibrations in $24$ hours in London, latitude as before, and $83$ feet above the level of the sea, the mean height of the thermometer being $67^\\circ,6$, of the barometer $29,97$ inches (vide Appendix). The correction for the height above the sea is $0,22$, and that for the buoyancy of the atmosphere $6,566$, both to be added: these corrections being applied, will give $86300,226$ for the number of vibrations of the pendulum of experiment in $24$ hours in vacuo at the level of the sea, the temperature being $70^\\circ$. Now, $86300,226^2 : 86400^2 :: 39,142213 : 39,232772$ the length of the pendulum of experiment.\n\nThen $86172,375^2 : 86400^2 :: 39,232772 : 39,026323087$, the length of the seconds pendulum at Madras by the first series of observations.\nAlso, $86172.328^2 : 86400^2 :: 39,232772 : 39,026280447$, the length of the seconds pendulum at Madras by the second series.\n\nThe mean of both is $39,026302$ inches, being, according to Sir George Shuckburgh's scale, the length of the seconds pendulum by these experiments at Madras in lat. $13^\\circ 4' 9''$, N. at the level of the sea, in vacuo, and at a temperature of $70^\\circ$ of Fahrenheit.\n\nThen comparing this length with $39,142213$ inches, the length in latitude $51^\\circ 31' 8,''4$ N. as before stated, the diminution of gravity from the pole to the equator will be $.0052894$, and the ellipticity $\\frac{1}{297,56}$ nearly.\n\nJ. GOLDINGHAM.\n\nMadras,\nMay, 1821.\nAPPENDIX.\n\nThe following are the Observations made by Captain KATER in England, before the Pendulum was sent out.\n\nJuly 25, 1820, in lat. $51^\\circ 31' 8''$,4.\nClock losing $1^s,20$ in a mean solar day. Barometer 29,83 Inches.\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | Vibrations in 24 hours. |\n|-------|----------------------|------------------|-----------|---------------------|----------------------|---------------------------------|-------------------|------------------------|\n| 67,4 | h. m. s. | | | | | | | |\n| | 1 5 25 | 1,17 | 1,70 | 1629 | | | 1,88 | |\n| | 32 34 | 0,97 | 0,90 | 1634 | | | 1,33 | |\n| | 59 48 | 0,84 | 0,77 | 1639 | | | 0,97 | |\n| | 2 27 7 | 0,71 | 0,65 | 1642 | | | 0,69 | |\n| 67,8 | 54 29 | 0,60 | | | | | | |\n| 67,6 | Mean | | | | | | | |\n\nJuly 26.\nClock losing $1^s,22$. Barometer 30,01 inch.\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | Vibrations in 24 hours. |\n|-------|----------------------|------------------|-----------|---------------------|----------------------|---------------------------------|-------------------|------------------------|\n| 66,2 | h. m. s. | | | | | | | |\n| | 1 37 28 | 1,28 | 1,18 | 1621 | | | 2,28 | |\n| | 2 4 29 | 1,08 | 0,99 | 1632 | | | 1,60 | |\n| | 31 41 | 0,91 | 0,84 | 1637 | | | 1,16 | |\n| | 58 58 | 0,77 | 0,71 | 1646 | | | 0,83 | |\n| 66,8 | 3 26 24 | 0,65 | | | | | | |\n| 66,5 | Mean | | | | | | | |\n### July 27.\n\n**Clock losing 1°,15.**\n\n**Barometer 30,01 inches.**\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | Vibrations in 24 hours. |\n|-------|----------------------|-------------------|-----------|---------------------|----------------------|---------------------------------|------------------|------------------------|\n| 67,2 | h. m. s. | | | | | | | |\n| | 1 2 27 | 1,22 | 1,12 | 1622 | | | | |\n| | 29 29 | 1,02 | 0,94 | 1630 | | | | |\n| | 56 39 | 0,86 | 0,79 | 1633 | | | | |\n| | 2 33 52 | 0,73 | 0,67 | 1638 | | | | |\n| 67,8 | 51 10 | 0,62 | | | | | | |\n| 67,5 | Mean | | | | | | | |\n\n### July 28.\n\n**Clock losing 1°,05.**\n\n**Barometer 30,01 inches.**\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | Vibrations in 24 hours. |\n|-------|----------------------|-------------------|-----------|---------------------|----------------------|---------------------------------|------------------|------------------------|\n| 67,8 | h. m. s. | | | | | | | |\n| | 0 50 41 | 1,19 | 1,09 | 1614 | | | | |\n| | 1 17 35 | 1,00 | 0,92 | 1624 | | | | |\n| | 44 39 | 0,84 | 0,78 | 1630 | | | | |\n| | 2 11 49 | 0,73 | 0,67 | 1627 | | | | |\n| 68,4 | 38 56 | 0,62 | | | | | | |\n| 68,1 | Mean | | | | | | | |\n\n### July 29.\n\n**Clock losing 1°,07.**\n\n**Barometer 30,01 inches.**\n\n| Temp. | Time of coincidence. | Arc of vibration. | Mean Arc. | Interval in seconds. | Number of vibrations. | Observed vibrations in 24 hours. | Correction for Arc. | Vibrations in 24 hours. |\n|-------|----------------------|-------------------|-----------|---------------------|----------------------|---------------------------------|------------------|------------------------|\n| 67,9 | h. m. s. | | | | | | | |\n| | 0 47 1 | 1,21 | 1,11 | 1616 | | | | |\n| | 1 13 57 | 1,02 | 0,94 | 1620 | | | | |\n| | 40 57 | 0,86 | 0,79 | 1625 | | | | |\n| | 2 8 2 | 0,73 | 0,68 | 1630 | | | | |\n| 68,8 | 35 12 | 0,63 | | | | | | |\n| 68,3 | Mean | | | | | | | |\n\nMDCCXXII.\n### Vibrations of the Pendulum at London.\n\n| Date | Barometer | Thermometer | Vibrations in 24 hours | Correction for Temperature | Correct vibrations in a mean solar day at 70° |\n|--------|-----------|-------------|------------------------|----------------------------|---------------------------------------------|\n| July 25 | Inch. 29.83 | 67.6 | 86294.40 | 1.02 | 86293.38 |\n| 26 | 30.01 | 66.5 | 86294.50 | 1.48 | 86293.02 |\n| 27 | 30.01 | 67.5 | 86294.21 | 1.06 | 86293.15 |\n| 28 | 30.01 | 68.1 | 86293.80 | 0.70 | 86293.10 |\n| 29 | 30.01 | 68.3 | 86293.76 | 0.72 | 86293.04 |\n| Mean | 29.97 | 67.6 | | | 86293.14 |\n\nFrom the above table it appears, that the pendulum makes $86293.14$ vibrations in a mean solar day, in latitude $51° 31' 8''$, the temperature being $70°$, and the height above the level of the sea $83$ feet. The correction employed for temperature is $0.423$ of a vibration for one degree. In computing the correction for the buoyancy of the atmosphere during the experiments, the temperature of $67°.6$ must be used, the barometer being at $29.97$ inches.\n\nJ. GOLDINGHAM.\n",
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"identifier": "jstor-107679",
"title": "Observations for Ascertaining the Length of the Pendulum at Madras in the East Indies, Latitude 13 degrees 4′ 9′ ′,1 N. with the Conclusions Drawn from the Same",
"authors": "Captain Kater, John Goldingham",
"year": 1822,
"volume": "112",
"journal": "Philosophical Transactions of the Royal Society of London",
"page_count": 49,
"jstor_url": "https://www.jstor.org/stable/107679"
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