is before the association, were made, (Pl. 3, or H No. 2.) The hours of the day are the ordinates, and the days of the month the abscissa. The signs H and L show in their proper place the hour of occurrence of high and low water for each day. The time of the moon's superior transit is marked, and the periods of greatest declination, and of crossing the equator. The result is easily generalized, that there is ordinarily but one high and one low water at Cat island in twenty-four (lunar) hours, and that when there are two tides they occur about the time of the moon's crossing the equator, and are usually most regular and strongly marked when in syzigies, with declination nearly zero. Following one set of high and low waters, it will be found that they occur later and later as the lunar day gains on the solar, with very remarkable differences, of which the explanation will be given towards the period of small declinations. The interval from high to low water is generally less by some hours than that from low to high. That as the moon approaches the equator, there are a few days of singular double tides, or of single tides, in which the times from low to high water are very much increased. That when the declination changes its name, a high tide takes nearly the place of a low in time, and vice versa, with an interval of irregularity; or, in other words, the tides are displaced by nearly twelve hours. 2. There is, as Mr. Whewell has remarked, no proper establishment to be derived from such tides; yet, we may obtain a desirable datum by throwing the results into the form of tables, in which the luni-tidal intervals are arranged according to the days from the zero of declination and the corresponding superior and inferior transits, and for north and south declinations. This will be made more clear by subsequent explanation. These afford a test of the theory of these tides by showing the displacement of the ordinate of high and low water, and might be used for the inverse purpose of forming prediction tables. Such tables of luni-tidal intervals for three months I now submit. They show considerable steadiness and similarity of intervals towards the maximum of declinations and great variations near the zero, and greater discrepancies than is usual in ordinary tides. These are from a series of tables computed by Mr. R. M. Bache for the year, and containing the times of high and low water, deduced from the daily curves, the readings of the gauge, the rise and fall of the tides, the times of the moon's superior and inferior transit, and the moon's declination. The intervals serve to show that the high water belongs alternately to the superior and inferior transits of the moon, according as the moon's declination is north or south, with a few cases only which admit of doubt. Two sets of luni-tidal intervals were computed (see tables) for three months, to ascertain the proper epoch of reduction, or age of the tide. In one case, the intervals were referred to the superior transit of one day before; and in the other, to the superior transit of two days before. The square of the discrepancy of the mean in the latter case was greater than in the former. An establishment deduced from these numbers for high-water, without correction, would have a probable error, as tried by discrepancy from the mean, of nearly eightyfour minutes. I have little doubt of being able to reduce this error, by computation, much within the limits of observation, so as to give useful prediction tables. The foregoing results point distinctly to a ruling cause depending upon the moon's declination. 3. The hourly observations for the year were thrown into the form of curves-the abscissas representing the hours, and the ordinates the heights. Of these I present, as characteristic, the months of January and March, (Pl. 4, 5, or H, Nos. 3, 4.) In January the tides are single throughout the month, the rise and fall diminishing towards the zero of declination; and in March, two periods of marked double tides occur. The times of new and full moon coincide nearly with the zero of declination of March; in January the syzigies occur at times of greatest declination. A series of diagrams, prepared for periods of declination zero, show irregularities, or double tides, near these times. Before disappearing, the tide which is lost appears rather as an irregularity than as a real tide, puzzling to the observer, and a severe test of his faithfulness. A similar set of diagrams for the periods of greatest declination show uniformly single tides and the greatest comparative rise and fall at the same periods, whether coinciding with syzigies or with first and last quarters. In computing the heights of spring and neap tides by the common methods, four months gave zero or negative differences. To discuss the epochs of the phenomena, as compared with greatest and least declinations, I prepared two sets of tables, which require revision. They show sometimes an actual coincidence in the epoch of least tides and zero of declination-sometimes a precedence and sometimes a subsequence-which, when not caused by irregularity of winds, I believe will find a satisfactory explanation; at a mean, there was little advantage in the discussion found from displacing the epoch. The average rise and fall for the second day before the greatest declination was 1.68 feet; for the day next preceding the greatest declination 1.78; for the day of greatest declination 1.81; for the next day 1.86; and for the next 1.77. Tracing a curve from these would give the epoch of greatest rise and fall about 0.75 days after the greatest declination. The average rise and fall on the corresponding days, in reference to declination zero, were 0.96 feet, 0.75, 0.60, (dec. zero,) 0.63, 0.73, the curve giving the epoch about one-sixth of a day after the zero of declination. The numbers, as stated, require revision; and there are causes for apparent displacement, which require further examination. 4. This general examination tends to point to the diurnal irregularity, as Mr. Whewell has stated, as the cause of the occurrence of these singleday tides; a view which is confirmed by such examinations as I have been able to make of the hourly tidal observations at Fort Morgan, at the entrance of Mobile bay. The interference in this case would be between the diurnal tide-wave, which represents the diurnal inequality, and the ordinary semi-diurnal wave; whether this wave has a regular progress along the coast, independently of the semi-diurnal wave, as was at first supposed by Mr. Whewell, or whether its phenomena are local, as he has since been led, from his investigations, to believe. If the observed wave is produced by its interference with a semi-diurnal wave, we can only study the phenomena to advantage after the observed wave has been separated into its components. 5. As a first approximation, I assumed the two waves to be governed by the law of sines, and then determined the curve which would result from the superposition of two such waves, having the same or different origins. The mean of the regular double tides, about the zero of declination, would present a first approximate value of the rise and fall of the semi-diurnal tides, and the mean of double and single tides, at the maximum of declination, would, especially when near the quadratures, give a first approximation to the height of the diurnal tide. The comparisons with the forms of curves already traced, addressing the eye, are easily made. I present, herewith, diagrams (Pl. 6, or H No. 5) for the case, in which the maximum of the diurnal tide coincides with that of the semi-diurnal; is three hours in advance, (or coincides with mean water falling,) six hours (or coincides with low water,) and nine hours, (or coincides with the second mean, or mean water rising,) using the approximate quantities referred to above for the greatest height of two component curves. It requires little examination to see that neither of the first three forms represents the case, and that the fourth does so remarkably, even in what appear to be small irregularities in the daily curves. This will be seen in the results for October, of which a diagram on a large scale is presented, giving the tidal curves near the zero, and thence up to the maximum of declination, for the first half of the month. In the singleday tides there was the same slow rise compared with fall; sharp rise and fall near high and low water, with the tendency to a stand during the rise; the same excess in the interval of time from low to high water, over that from high to low water. This hypothesis as to the position of the two waves may perhaps be slightly improved by further discussion. It is obvious, from the equation of the curve, (which I have already referred to, as given by Mr. Whewell,) that the form and position of remarkable points will vary with the constants in the component curves, as well as with the position of the origin of each in reference to that of the other. To carry out the representation graphically, I have drawn the curves for four values of the constants of the diurnal and semi-diurnal, formed from the observations with the same displacement of nine hours in the time of high water of the diurnal curve, and corresponding to the epochs of the maximum declination, two, four, and six days before or after the maximum. These show the general features of the curve sufficiently, and the variations in the times and heights, the passage from single to double tides, and the reverse; and the coincidence with observations is such as to warrant a close numerical discussion. 6. The equation of the curve shows how much the time of high and low water depends on the constants in the diurnal and semi-diurnal curve. The equivalent of the equation given by Mr. Whewell is— C. cos 2t+D cos (t—E)—y=0, in which t is the time in hours from the place of the maximum ordinate of the semi-diurnal curve as an origin; C is the constant of that curve of sines; E is the distance of the maximum ordinate of the diurnal curve for the former, and D the constant for the curve of sines; y is the ordiuate of the complex curve. By an easy transformation, this takes the form 2 C. cos 2t+D cos t. cos E+D. sin t. sin E-C=y. and y=2 C cos 2t+D sin E (sin t—cos t)—C. The differential co-efficient of which for the case of the maximum or minimum is dy dy =-4 C cos t. sin t+D sin E (sin t+cos. t)=0 or, since the second term is negative when t>6 hours, 4 C cosec. t-sec t= Ꭰ Ꮰ Applying this to the four cases shown in the diagrams E=9 hours, C=0.175, D=0.700, we find maximum at.......10 25.4 =0.615 hrs. min. 10 33.3 10 51.1 11 56.8 =0.400 and for the intervals between high and low water in lunar hours, 9h. 09.2m., 8h. 53.4m., Sh. 17.8m., and 6h. 06.4m. We might apply this mode to test the hypothesis, using for the values of C the half difference of the ordinates of six and twelve hours from the mean, and of eighteen and twenty-four hours with the signs changed; and for D, the average of the ordinates of six and eighteen hours from the first mean. The means present the best criterion, because not displaced in this combination, as the equation shows. This mode of proceeding, however, throws the test too much on the weak part of the results-the times of occurrence of high and low water, or of mean water-and does not take in all the points of the curve; and I have, therefore, preferred a different form of discussion. 7. Placing the maximum of the semi-diurnal curve at O hours, in the hypothesis that the high water of the diurnal curve is nine hours in advance of that of the semi-diurnal curve, the two curves cross the line of mean water at three hours, the diurnal curve rising and the semidiurnal falling; at six hours, the semi-diurnal curve has reached its maximum, and rises again at nine hours to its intersection with the mean water-line, at which time the diurnal curve has reached its maximum; the semi-diurnal curve attains its greatest rise at twelve hours, and the mean level at fifteen; the diurnal curve also descending to the same point at that time. Within these two intervals from mean level to mean level, the combinations of the ordinates forming the actual tidal curve are exhausted; the part of the curve below the mean level being symmetrical with the above from three to nine hours, the ordinates of the semi-diurnal curve as subtractive; from nine to fifteen hours additive. The mean is the average between high and low water. The tides of each day will give the forms of the component curves, beginning with the mean, and ending with it, considering as symmetrical the parts above and below the axis of X. In tabulating, the branch above the axis should be referred to the mean of the preceding and succeeding low water 缨 衛+} and of the high-water which it includes, and that below to the mean of the two high and of one low-water. From three to nine hours, the difference of the ordinates giving the actual curve, and from fifteen to nine in the reverse order, the sum of the same ordinates, half the sum of the two series of ordinates gives the value of the ordinates of the semidiurnal curve. The same being repeated with the second branch of the curve, the average will give two results for each day's observation. The case given in the table on the board, for March 5, will serve to illustrate the simple nature of this method of proceeding. The mean ordinate for the first and second branches of the curve having been obtained, and the hourly observation which coincides most nearly with it having been found before and after high-water, the hourly observations are arranged from it forwards for seven hours (m,) and backwards for seven (n.) The same is done for low-water (m' and n'.) The half sums and half differences are taken in each case, and then the means. The computation of the diurnal curve is made in the upper part of the table, and that of the semi-diurnal curve in the lower part. The number representing the mean level is eliminated by the mode of taking the means in each table, and the ordinates below the axis are treated as if having the same sign as those above. The semi-diurnal curve is turned over on its maximum ordinate, and the mean value of a single branch of it found. Then each curve is reduced to zero, in the mean level of the period. The last two columns of the upper and lower part of the table contain, respectively, the curves of sines corresponding to the diurnal and semi-diurnal curves. In the case shown in the first diagram, the ordinates of the semi-diurnal curve from mean water to high-water, and corresponding nearly to a minimum of declination, and new moon, are 0.00 feet,+ 0.02, +0.03, 0.05, +0.04, 0.02, +0.02. The moon's declination during the period being about from 2° 54' S., to 1° 45′ S., this curve obviously contains a residual of the semi-diurnal curve, not taken out; but supposing it to be deduced from a just mean, the corresponding ordinates of a semi-diurnal curve, calculated with 0.04 feet as the maximum, would be 0.00 feet, 0.01, 0.02, 0.03, 0.03, 0.04, 0.04, differing, at the most, 0.06 of a foot, or about three-quarters of an inch, and, in a single instance, the sum of all the six differences being .03 feet, and the average .004. The ordinates of the semi-diurnal curve are 0.00 feet, 0.14, 0.28, 0.32. The curve of sines computed with the greatest ordinate has, in this case, for its corresponding ordinates, 0.00 feet, 0.16, 0.28, 0.32, differing but .02 feet at the greatest. At the next period of declination, nearly zero and full moon in the month of March, the ordinates of the diurnal curve deduced are 0.00 feet, 0.05, 0.06, 0.06, 0.08, 0.06, 0.09, and the corresponding computed ordinates 0.00 feet, 0.02, 0.04, 0.06, 0.07, 0.09, 0.09, differing at the greatest 0.03 feet, and on the average, 0.004 feet, the observed ordi |