Pursuing her voyage the ship arrives at St. Michaels, and observations for the difference of longitude are made there on the 17th of June, nine days after the last observations at Plymouth. Chron. fast mean time at St. Michaels, on 17th June Chron. fast of Greenwich, M. T. on 8th June Oh. 1m. 10s. Gain in 9 days = 3.5s. X 9= +31.5 1h. 44m. 28.1s. Chron. fast M. T. at Greenwich, on 17th June 0 1 41.5 0 1 Again proceeding, the ship arrives at Halifax, and the 41.5 46.6 W. 4h. 17m. 11·4s. Oh. 1m. 10s. 10 Chron. fast M. T. at Greenwich on 28th June 0 2 20 0 2 20 Approximate longitude of Halifax dock-yard 4 14 51.4 W. But on the 3rd of July, a second set of observations are obtained at Halifax, which, by comparison with the first set obtained there on the 28th of June, give the mean rate for the five intervening days 5.58s. gaining. Let us now proceed to apply our rule to find the correction; first, in the case of St. Michaels. CASE I. v5.58s.3.50s. 2·08s. t 26. and = 0·08s. the ratio of acce t leration per day. n=7. p=9. therefore " + 2 p r = 8 × 9 × '08 5.76s. the correction to be substracted from the longitude according to the given rule. Approximate longitude of St. Michaels 1h. 42m. 46.6s. 5.76 p, in this case, = 20 all else are the same, therefore "+Px p r = 13·5 × The same hypothesis, that the rates of chronometers usually increase or decrease uniformly, will enable us to find directly the correction to the longitude of intermediate stations when the difference of longitude between the first and last stations is known, and when the rate at one of them only has been ascertained. For instance, let us suppose the longitudes of Plymouth and Halifax to be known; that the rate of the chronometer at the former place had been ascer tained to be 3-5 gaining, between the 1st and 8th of June; its error on the 8th Chron. fast M. T. on the 28th of June, at Plymouth at Halifax Oh. 17m. 50.8s. +1 10 0 19 00.8 4 17 11.4 Hence it appears, that the rate of the chronometer has increased, and if there be nothing in its previous rate of going, or in the comparisons with the other watches to prevent us, we may consider the acceleration of its rate to commence from the middle of the period (n), for which the first-rate was obtained. But if the comparisons indicated that the chronometer had changed its rate immediately, or soon after the ship put to sea, or if its rate immediately previous to the last seven days at Plymouth had been decreasing instead of increasing, and that there was no reason to think that it had ceased to do so before the ship sailed, then our hypothesis would be inadmissible, and we could not do better in such a case, perhaps, than take the mean sea-rate which would be found thus: : Chron. fast M. T. on the 8th of June, at Plymouth Oh. 17m. 50.8s. 3 57 49 66 Chron. fast M. T. on the 8th of June, at Halifax 4 15 39.8 In such a case, however, we should not by any means have the same faith in the results, as we should have in cases where there is no reason to suppose otherwise than that the change of rate had been uniform. For instance, let us suppose that the chronometer had begun to increase its rate before the seven days rating at Plymouth, and that the comparisons indicated that it had continued to do so since, we might then safely adopt our hypothesis and proceed as follows. The chronometer gives too much longitude west by 21-6s, which is the whole accelerated motion in p, 20 days; but we assume that the acceleration commenced from the middle of the period n = 7 days, therefore 21.6 half the ratio of acceleration per day. But we may get at r the ratio by a more simple process, by consider ing that the mean sea-rate found as above belongs to the middle of the period p, as the first rate does to the middle of the period n. Mean sea-rate between 8th and 28th answering to 18th June answering to midway 4.58s. 4.5 3.50 Having the ratio we may apply our rule for the correction to the intermediate longitudes as before. I have been induced to explain at some length the principle upon which a rule for correcting the difference of longitude for a subsequently discovered increase or decrease in the rate of the chronometer should be founded, because, as I have before remarked, a correct rule for the purpose does not appear to have been yet made public. But when we know the ratio of increase or decrease of the rate per day it is nearly as easy, (or at any rate not much more troublesome,) to repeat the cal culation for the difference of longitude with a mean rate during the interval between the observations at the two places, as to correct the result of the previous computation with the first rate. I will, therefore, proceed to give a rule, founded upon the principle already explained, for finding the mean rate for any required interval, and from it the correct difference of longitude by the chronometer. If it be required to know the rate at the termination of any number of days after the middle of the period n, to which the first rate belongs, it is only necessary to multiply the ratio by the given number of days, and add it to, or subtract it from, the first rate according as it may be increasing or decreasing. And half the sum of the ratio so found, for any two days will be the mean rate for the interval between them. But such a mean rate for any interval may be more directly and easily found by the following rule. Let n, p, and r, have the same signification as before. Let a be the first rate; and m, the mean rate for the given interval. Then will a+ (n + px } r) = m. and m ×p: the whole loss or gain of the chronometer during the internal p. Which rule may be thus expressed. To the number of days for which the first rate was obtained, add the number of days between the observations for the difference of longitude; multiply the sum by half the ratio; the product added to the first rate will give the mean rate, which multiplied by the number of days between the observations will give the correct gain or loss of the watch. Thus referring to Ex. I. for the longitude of St. Michaels; a + (n + p × } r) = m, that is, first rate 3.5s. + (16 × 0.04s.) 4.14s. the mean rate. Chron. fast M. T. at St. Michaels, on the 17th of June, at Greenwich, 8th June, Oh. 1m. 10s. Ditto 66 +37.26 1h. 44m. 28.1s. Gain in 9 days, mean rate 4-14 X 9= Chron. fast M. T. Greenwich 17th June 0 1 47.26 0 1 47.26 Longitude of St. Michaels as before 1 42 40.84 W. In further illustration of the subject I may add that, we might, if we wished, find the correction to the longitude deduced, from the first rate by means of the mean rate found as just explained. For (ma) px the correction to the longitude required.— That is, the difference between the mean rate for the interval, and the first rate multiplied by the interval will give the correction required. But this, requiring a double operation, that is one for the mean rate, and another for the correction, would be evidently more operose than the rule which I have previously given and endeavoured to illustrate by Ex. I. It will be perceived that I have used the same quantities in both examples. I have done so that the agreement in the results of different methods for finding the correct chronometric difference of longitude may be the more easily perceived. There is one more remark which it may be useful to add, namely, that when the periods for which the rates have been obtained at two stations are equal, then half the sum of those two rates will be the correct mean rate by which to deduce the difference of longitude between those two extreme stations only; and half the difference of those two rates multiplied by the number of days' interval between the observations at those two stations will give the correction to the longitude deduced from the first rate. But when the periods of rating are unequal, or when the longitude of intermediate stations is required to be found or corrected, then the rules which I have given may be resorted to with advantage. In conclusion I may remark, that when the rate of a chronometer is found to have greatly changed during a voyage, I should put little faith in the longitude deduced from it, whatever might be the hypothesis adopted to allow for such change. Nevertheless, there are good reasons, even apart from experience, for preferring in common cases the hypothesis which I have adopted, reasons founded upon a consideration of the nature of the instrument itself; and of the varying action, as to degree of natural causes or agents on the machinery, and consequently on the performance of chronometrs, subjected to such a gradual change of geographical position, and of temperature as often takes place during a voyage. However, in all cases, quick runs between the two places whose difference of longitude is required, (and which, if possible, should not be more than ten days' sail apart,) thrice repeated in quick succession, so as to have the sea-rates during the voyage backwards and forwards, are the best mode of proceeding to insure an accurate result. In hopes that what I have written may be useful to those who may not previously have had an opportunity of well considering the subject, which I have endeavoured to elucidate, To the Editor, &c. I remain, &c., HENRY H. BAYFIELD, Captain R.N., We have taken the opportunity of annexing the following table to Capt. Bayfield's paper, not as any illustration of it, but as somewhat ENLARGED SERIES.-NO. 4.-VOL. FOR 1843. 2 G |