66 66 "gave nearly the same result, only a trifle less in the depression of high water "for an inch rise of mercury. The specific gravities of mercury and water 'being not far (if I recollect right) from this ratio of 14 or 15 to 1, it would "seem that the total weight of the compound column of air and water raised by the force which produces the tide, remains nearly unaffected by the "changes of atmospheric pressure. By introducing this new correction, therefore, a very considerable portion of our Residual Error is accounted for." The barometric observations which Mr. Bunt used for finding the effects of atmospheric pressure on the heights of high water at Bristol, were those contained in the register kept at the Bristol Institution, which extended back to a period earlier than the commencement of the tide observations. As it had appeared that all the other effects of external forces upon the height and time of high water corresponded not to the forces at the moment of observation, but to a state of the forces at an anterior period, it occurred as possible that this might also be the case with the effect of the atmospheric pressure upon the height of the tide; and that the correction corresponding to this effect might be most accurately obtained by taking the state of the barometer at some period anterior to the time of high water; for instance, twelve hours or twenty-four hours. If this were the case, we should be able to predict the effect of atmospheric pressure upon the tide, a day or half a day previous to the event. As this prospect gave an additional interest to the inquiry, I begged Mr. Bunt to try the comparative results of contemporaneous and anterior epochs of the barometric observation. This he proceeded to do, by arranging various portions of our observations according to the heights of the barometer. The following is the account of the result. "Bristol, Feb. 18, 1841. "I send you diagrams of the effects of atmospheric pressure on the heights of high water for every tenth of an inch height of the barometer, from 29.2 or 3 inches to 304 or 5 inches, for the years 1834, 35, 39; barometer and tide contemporaneous. Also for the year 1834, barometer heights being twenty-four hours anterior to high water. Also for 1839, barometer 29.2, 3, 4, 5 inches to 30°2, 3, 45 inches, twelve hours anterior to high water. Also the mean of the three years, giving about 14 inches depression of tide to one inch rise of barometer. "I have also taken the sums of the residues left after introducing the barometer correction, first, contemporaneously with high water, and secondly, at twenty-four hours anterior to high water, for the first six months of the year 1834, measuring the residue at about every high water. The total residues, in the two cases, were so nearly alike, as to leave it doubtful which epoch should be preferred. The diagram for 1834, made from observations of the barometer twenty-four hours anterior to high water, appears about as good as the one from contemporaneous observations of barometer and tide. The extreme groups, however, for 29.2, 3, 4 inches, 304, 5, 6 inches barometer, approximate slightly towards the mean line: the same tendency appears in the double groups 29.2, 3, 4, 5 inches, 30.2, 3, 4, 5 inches, for 1839, barometer observed contemporaneously with, and at twelve hours anterior to, high water. Hence I should be disposed to infer, that we do not improve the result by going back to an anterior epoch; for I take it for granted that the true epoch is that which shows the greatest amount of elevation and depression of tide corresponding with the least and greatest heights of the barometer; or that which makes the greatest angle of inclination between the line connecting the several points or groups, and the axis. "There is one peculiarity which I have noticed in these barometrical results, and in others which I obtained in my earlier trials, namely, that the effect produced on the heights when the barometer is at any point below 29-4 inches or 29.5 inches, is always greater than the proportion for the greater heights of the barometer. I imagine this arises from the effect of wind, which generally follows a great depression of the barometer, and generally, with us, comes from the S.W.; so that an additional elevating cause comes into operation. This, however, is mere conjecture." I then requested that, instead of arranging the observed heights according to the barometer, he would correct the observed heights for lunar and solar parallax and declination, and investigate the effect of atmospheric pressure on the residues; still comparing the contemporaneous and the anterior epochs. The following was the result. "March 17, 1841. "I send you the results of comparisons of the residues of height for 1834, 1835 and 1836, with the state of the barometer at different epochs. The heights were calculated carefully by numbers, using what I consider to be my best corrections for lunar and solar parallax and declination, and employing the same corrections in each of the three years. The only correction omitted was that for the diurnal inequality. The residues for 1834 were compared with the barometer contemporaneous, twelve hours anterior, twenty-four hours anterior, twenty-four hours posterior, and the extreme groups, with barometer, thirty-six hours anterior; in order to find what progressive changes of form the curves would thus be made to assume. The mean correction for one inch difference in height of barometer having been obtained, the proportional correction was applied to each observed height of high water, and the mean of all the errors (remaining after the barometrical correction) then taken for the whole year. In every instance the contemporaneous barometer gives the best correction. Thus in 1834 the mean error remaining, after applying the barometrical correction, is 5.817 inches contemporaneous barometer. 6'085 inches barometer twelve hours anterior, "These two latter epochs, the one anterior the other posterior, producing nearly equal errors, seem to show (like equal altitudes) that the truth lies midway between them. "In like manner, the mean residual error for 1835 is 5-277 inches, barometer contemporaneous, 5.421 inches, barometer twelve hours anterior, and for 1836 is 6.450 inches, barometer contemporaneous, 6.535 inches, barometer twenty-four hours anterior. "The introduction of the correction for the contemporaneous barometer reduces the mean error, previously remaining, about one-fourth, being as 1 : 0.753, for the whole of the year 1834; and as 1 : 0·705 for the year 1835. "The mean effect on the tide corresponding to a change of one inch in the mercurial column was carefully obtained, by taking into account the number of observations in each parcel, so as to get the true average. The contemporaneous barometer gives, in every instance (as shown in the diagrams), the greatest result: and in this case also equal differences from the maximum attend the anterior and posterior epochs for 1834-viz. 11 inches tide (instead of 13.4 inches) to one inch of mercury. "The mean depression of tide corresponding to 1 inch rise of barometer, is Thus this last investigation appears to have put a negative upon the supposition that the barometric correction of the height of high water corresponds to an anterior epoch; for we cannot doubt the justice of the remark made by Mr. Bunt, that since not only the contemporaneous barometer gives the greatest result, but since also equal differences from the maximum attend the epochs anterior and posterior by twenty-four hours, the contemporaneous epoch must be the true one. And thus it appears that the effect of atmospheric pressure on the height of the tide is something local and immediate, not an effect transmitted in a finite time from some other place. I next wished Mr. Bunt to try how far the correction curves of height for lunar and solar parallax and declination would have been different if the barometric correction had been made first, before the heights were arranged for the other corrections. This also he undertook. The following is his communication on the subject. "April, 1841. "I send you new correction curves made from the observed heights in 1839, after having first cleared them of the effects of the changes of atmospheric pressure, allowing 13 inches of water to one inch difference in the barometric column. The greatest difference is in the solar declination curve at the hour 6 of transit. I hardly think this can be entirely owing to the atmospheric correction, but most likely to some difference in the working out of the new lunar corrections, especially that for declinations, with which the solar declination is almost inseparably mixed up in any short series of observations. Indeed I can scarcely see how the effects, of the two kinds of declination can be separated, with any certainty, about the hours of Oh and 6h transit, except by taking two sets of observations, the one having the moon's declination a maximum, and the other a minimum." The very small differences between these correction curves in their former shape, and as modified by allowing for the barometric correction, might have been expected, since the barometric correction will, on the whole, compensate itself. The smallness of the differences is, however, evidence of the care and consistency with which our results were formerly obtained. W. WHEWELL. Trinity College, Cambridge, July 1, 1841. Report on the Discussion of Leith Tide Observations, executed by Mr. D. Ross, of the Hydrographer's Office, Admiralty, under the direction of the Rev. W. WHEWELL. ALTHOUGH tables of the corrections of the heights and time of high water, due to lunar parallax and declination, have already been obtained for several places, (London, Liverpool, Plymouth, and Bristol) it is still desirable to correct and confirm these results by the discussion of observations made at other places, especially if continued for a considerable series of years. Our methods of discussion and tabulation may admit of improvement, and new features may appear in the new results; with these views I applied at the last meeting 1841. D of the Association for the sum of 50l. to enable Mr. Ross to complete the discussion of a series of tide observations made at Leith, extending from 1827 to 1839 inclusive, and now including 1840. This long series of years is advantageous for the purpose of obtaining the declination correction; since, in consequence of the motion of the moon's nodes, the range of lunar declination and the mean declination is very different in different years; as was stated in my Report on the subject, presented to the Association last year. The present Report will refer to a new mode of presenting the corrections of the height of high water for lunar parallax and declination. It has been shown by me in various memoirs that the correction of the height both for lunar parallax and declination is nearly the same for all the hours of moon's transit. This being the case, the greater part of this correction may be expressed by means of a table of double entry; the two arguments being the moon's parallax and declination, Mr. Ross suggested to me the advantage of such a table, and has constructed it from the Leith observations, and it is laid before the Association along with the present Report. It appears by this table, when separated into the two parts dependent upon parallax and declination, that the parallax correction varies very exactly as the parallax; and that the declination correction applicable to declination 0°, varies very nearly as the square of the declination; results agreeing both with those obtained from the tide observations made at other places, and with the consequences of the equilibrium theory modified, as I have previously shown that it must be, in order to express the results of observation. As I have stated, the principal part of the correction of the height of high water for lunar parallax is constant for all hours of moon's transit. But there is a further term of this correction, though a small one, which goes through a cycle of positive and negative values in the course of a semilunation. This has already appeared in the results of the London, Liverpool, Plymouth, and Bristol observations, and also agrees with the theory above referred to. A like result appears in the results of Leith tides by the discussions now reported, but at first sight with a remarkable difference. At Plymouth it appeared (Ninth Series of Tide Researches, Phil. Trans. 1838), that the correction for parallax is least when the hour of moon's transit is 10h, and greatest when the hour of moon's transit is 4 or 5h; the mean parallax correction when the part depending on the hour of transit disappears, occurs at transit 1 and 7. At Leith, on the contrary, the effect of the parallax is greatest when the transit is about 6", least when the transit is Oh, and the mean value obtains when the transit is about 3h and 9h. But this great difference in the results, which at first appears to make the course of this correction nearly opposite at the different places, is, in fact, the result of the difference of the time which the original tide-wave employs in reaching Plymouth and Leith. This correction varies nearly as the sine of the double angle of the moon from the sun, minus a certain epoch. Or to be more exact, instead of the sine we may substitute a circular function, which vanishes, and is positive and negative when the sine is, but which does not exactly follow the law of the sine. If this function be called s, the term of which we are now speaking is, in the Plymouth tables, as s, 2 p 14; in the Leith tables it is as s, 2 - 18h. The difference of the epochs, 14 and 18", depends on the time of transmission of the tide from Plymouth to Leith. This is further illustrated by remarking that in the results of London observations this term is also represented by s, 24-18h; while the Bristol observations give the term s, 2 — 15h. The agreement of these results cannot but be considered as decisive evidence of the correctness of the tables which we have obtained, as to their form and general law. And this is the more remarkable when we consider how small are the results in which this coincidence is found. The coefficient of the term now spoken of, is at London 3 inches; at Plymouth it is 1 inch; at Bristol, when the rise and fall is very great, this coefficient is 6 inches; at Leith, by the present discussion, its amount is found to be little more than 1 inch. The smallness of this term also leads us to this inference, that Mr. Ross's table of double entry may be used to obtain the corrections of heights for parallax and declination, almost without a sensible error. The table being obtained from Leith observations, will require a constant multiplier to adapt it to other places. TABLES. (1.) Mr. Ross's table of the correction of height for parallax and declination. (2.) Mr. Ross's table of the difference of the parallax correction from the mean for each hour of transit, (3.) The mean value of this difference for each hour of transit. (4.) The mean value of this difference for each 3° of declination showing that the declination correction is nearly as the square of the declination. (5.) The semimenstrual inequality of height for Leith. in. 5.2 5.6 6.4 6.7 7.6 9-2 55. 56. 57. 58. 59. 601 in. in. in. 1.6+2·3 +64 +8·7+114+17·5+ 19-3 0.8+3·0+64 + 10·2 + 12·4+16·3 6.2-3·8-0·3+4·1+ 7·8 +11·2+14·3 11.0 9.0 - 3·4 — 1·3 + 3·4+ 5·8+ 9·1+10.5 - 5·7 — 0·6+2-1+ 7·2+10·1+9.0 2.2+03+47 + 8·7+ 7·0 4.8+0·7+ 2·3+ 6·4+ 8·4 .11.5 8.8 27 29 15.0 - 12.4 0 - 30 1-30 1.7 2.01 230+ 0.3 0.7-1.6 ·30 + 1·0+ 1·1+0·1 0+ 0.5 30+11+ 1·3+0·9+0·9+0·6+ 0·3| 30+11+ 1·5 + 1·0+1·8+0·8+ 0·2 0·2+0·6 +0.9 8-30 0·2+ 0·3 0+ 0.6 0.6 9-30 11 0-3 0.8+0·1+0.1 1.8 1 0.7-1.5+0.2 2.5 0.4 1.1 0+ 0.2 0.8+0.1 |