Tide

TIDE (O. Eng. tid, cf. Ger. Zeit, time or season, connected with root of Sanskrit a-diti, endless), a term used generally for the daily rising and falling of the water of the sea, but more specifically defined below.

I

General Account Of Tides And Tidal Theories §. 1. Definition of Tide. - When, as occasionally happens, a ship in the open sea meets a short succession of waves of unusual magnitude, we hear of tidal waves; and the large wave caused by an earthquake is commonly so described. But the use of the adjective " tidal " appears to us erroneous in this context, for the tide is a rising and falling of the water of the sea produced by the attraction of the sun and moon. A rise and fall of the sea produced by a regular alternation of day and night breezes, by regular rainfall and evaporation, or by any influence which the moon may have on the weather cannot strictly be called a tide. Such alterations may be inextricably involved with the rise and fall of the true astronomical tide, but we shall here distinguish them as meteorological tides. It is well known that there are strongly marked diurnal and semi-diurnal inequalities of the barometer due to the sun's heat, and they may be described as atmospheric meteorological tides.' These movements both in the case of the sea and in that of the atmosphere are the result of the action of the sun, as a radiating body, on the earth. True astronomical tides in the atmosphere would be shown by a ' Lord Kelvin shows that the attraction of the sun on these tides must produce an excessively small acceleration of the earth's rotation. See Societe de physique (September 1881), or Proc. Roy. Soc. Edin. (1881-1882), p. 396.

regular rise and fall in the barometer, but such tides are undoubtedly very minute, and we shall not discuss them in this article, merely '., referring the reader to the Mecanique celeste of Laplace, bks. i. and xiii. We shall in the present article extend the term " tide " to denote an elastic or viscous periodic deformation of a solid or viscous globe under the action of tide-generating forces.

§ 2. General Description of Tidal Phenomena. 2 - If we live by the sea or on an estuary, we see that the water rises and falls nearly twice a day; speaking more exactly, the average interval from high-water to high-water is about 12 h 25 m, so that the average retardation from day to day is about So m. The times of highwater are then found to bear an intimate relation with the moon's position. Thus at Ipswich high-water occurs when the moon is nearly south, at London Bridge when it is south-west, and at Bristol when it is east-south-east. For a very rough determination of the time of high-water it is sufficient to add the solar time of high-water on the days of new and full moon (called the " establishment of the port ") to the time of the moon's passage over the meridian, either visibly above or invisibly below the horizon. The interval between the moon's Variability passage over the meridian and high-water varies of sensibly with the moon's age. From new moon to first quarter, and from full moon to third quarter (or rather from and to a day later than each of these phases), the interval diminishes from its average to a minimum, and then increases again to the average; and in the other two quarters it increases from the average to a maximum, and then diminishes again to the average.

The range of the rise and fall of water is also subject to great variability. On the days after new and full moon the range of tide is at its maximum, and on the day after the first and third quarter at its minimum. The maximum neap. is called " spring tide " and the minimum " neap tide," and the range of spring tide is usually nearly three times as great as that of neap tide. At many ports, however, especially non-European ones, two successive high-waters are of unequal heights, and the law of variability of the difference is somewhat complex; a statement of that law will be easier when we come to consider tidal theories. In considering any single oscillation of water level we find, especially in estuaries, that the interval from high to low-water is longer than that from low to high-water, and the difference between these two intervals is greater at springs than at neaps.

In a river the current continues to run up stream for some considerable time after high-water is attained and to run down similarly after low-water. Much confusion has been River Tides. occasioned by the indiscriminate use of the term " tide " to denote a tidal current and a rise of water, and it has often been incorrectly inferred that high-water must have been attained at the moment of cessation of the upward current. The distinction between " rising and falling " and " flowing and ebbing " must be maintained in rivers, whilst it is unnecessary at the seaboard. If we examine the progress of the tide-wave up a river we find that highwater occurs at the sea earlier than higher up. If, for instance, on a certain day it is high-water at Margate at noon, it is high-water at Gravesend at a quarter past two, and at London Bridge a few minutes before three. The interval from low to high-water diminishes also as we go up the river; and at some distance up certain rivers - as, for example, the Severn - the rising water spreads over the flat sands in a roaring surf and travels up the river almost like a wall of water. This kind of sudden rise is called a "bore" 3 (q.v.). In other cases where the difference between the periods of rising and falling is considerable there are, in each high-water, two or three rises and falls. A double high-water exists at Southampton.

When an estuary contracts considerably, the range of tide becomes largely magnified as it narrows; for example, at the 2 Founded on G. B. Airy's " Tides and Waves," in Ency. Metro p. See a series of papers bearing on this kind of wave by Sir W. Thomson (Lord Kelvin) in Phil. Mag. (1886-1887).

entrance of the Bristol Channel the range of spring tides is about 18 ft., and at Chepstow about 50 ft. This augmentation of the height of the tide-wave is due to the concen- lion of tration of the energy of motion of a large mass of Height water into a narrow space. At oceanic ports the tidal phenomena are much less marked, the range of tide being usually only 2 or 3 ft., and the interval from high to low-water sensibly equal to that from low to high-water. The changes from spring to neap tide and the relation of the time of high-water to the moon's transit are, however, the same both on the open coast and in rivers.

In long and narrow seas, such as the English Channel, the tide in mid-channel follows the same law as at a station near the mouth of a river, rising and falling in equal times; the current cause departures from the predicted height of water, high barometer corresponding to decrease of height of water. Roughly speaking, an inch of the mercury column will correspond to about a foot of water, but the effect seems to vary much at different ports.2 Mariners and hydrographers make use of certain technical terms which we shall now define and explain.

Thp " establishment of the port," already referred to above, is the average interval which elapses between the moon's transit across the meridian, at full moon and at change of moon, and the occurrence of high-water. Since at these times the T e Sailors. moon crosses the meridian at twelve o'clock either of day or night, the establishment is the hour of the clock of high-water at full and change.

It has already been remarked that spring tide occurs at most places a day or a day and a half after full and change of moon. Now it is more important in the theory of the tides to know what occurs at spring tide than what occurs at full and change of moon. Thus the term " the corrected establishment of the port " is used to denote the interval in hours elapsing at spring tide between moon's transit and high-water. The difference between the ordinary and the corrected establishments is of small amount. At any other state of the moon, except full and change, the " interval " or " lunitidal interval " means the interval between the moon's upper or lower transit and high-water.

The average interval elapsing between full or change of moon and spring tide is called the " age of the tide "; as already remarked this interval is commonly about a day or a day and a half, but it may be twice as great in some places. The use of this term arises from the idea that spring tides are generated at some undefined place exactly at full or change of moon, and take an interval of time denoted the " age " to reach the place of observation. The term is not altogether satisfactory, since it implies a theory, but it must be referred to as in general use.

The average height at spring tide between high and low-water marks is called " the spring rise "; the similar height at neap tides is, however, called " the neap range." " Neap rise " is used to mean the average height between high-water of neap tides and low-water of spring tides. Thus both at springs and neaps the term " rise " refers to the rise above the level of low-water at spring tide. French hydrographers call half the spring rise " the unit of height." 1 Airy, " Tides and Waves." 2 Ibid. §§ 572-573.

The " diurnal inequality " of the tide denotes the fact that successive high-waters and successive low-waters are unequal to one another. In England the diurnal inequality scarcely exists. The practice of the British admiralty is to refer their soundings and tide tables to " mean low-water mark of ordinary spring tides." This datum is found by taking the mean of all the available observations of spring tides, excluding, however, from the mean any spring tides which may be considered abnormal. The admiralty datum is not, then, susceptible of exact scientific definition; but when it has once been fixed with reference to a bench-mark ashore it is expedient to adhere to it, by whatever process it was first fixed.' When new tidal stations are established in India the datum of reference has, since about 1885, been " Indian low-water mark," which is defined as being below mean sea-level by the sum of the semi-ranges of the tides M2, S2, K 1, 0 (see §§ 24, 25 on Harmonic Analysis below).

In ordinary parlance sailors very commonly use the term " tide " when they mean what may be more accurately described as a tidal current.

graduated staff fixed in the sea, with such allowance as may be possible for wave motion. It is, however, far preferable to sink a tube into the sea into which the water penetrates through small holes; and the wave motion is thus annulled. In the calm water inside the tube there lies a float, to which is attached a cord passing over a pulley and counterpoised at the end. The motion of the counterpoise against a scale is observed. In either case the observations may be made every hour, or the times and heights of high and low-water may be noted.

In more careful observations than those referred to above the tidal record is automatic and continuous and is derived by means of an instrument called a tide-gauge.

This gauge should be placed in a place where we may obtain a fair representation of the oscillation of the surrounding sea. In such a site a well or tank is built on the shore communi cating by a channel with the sea at about 10 ft. below lowest low-water mark. In some cases an artificially constructed well may be dispensed with, where some lagoon or pool exists so near to the sea as to permit junction with the sea by means of a channel below low-water mark. At any rate we suppose that water is provided rising and falling with the tide, without much wave-motion. A cylindrical float, usually a hollow metallic box or a block of greenheart wood, hangs and floats in the well, and is of such density as just to sink without support. The float hangs under very light tension by a platinum wire, or by a metallic ribbon, or by a chain. The suspension wire is wrapped round a wheel, and imparts to it rotation proportional to the rise and fall of tide. By a simple gearing this wheel drives another, by which the range is reduced to any convenient extent. A fine wire wound on the final wheel of the train drags a pencil or pen up and down or to and fro proportionately to the tidal oscillations. The pencil is lightly pressed against a drum, which is driven by clockwork so as to make one revolution per day. The pen leaves its trace or tide-curve on paper wrapped round the drum. The paper is fixed to the drum with the edges of the paper at the XII o'clock line, and the record of a fortnight may be taken without change of paper. An example of a tide-curve for Apollo Bunder, Bombay, from the 1st to the 15th of January 1884, is shown in fig. 1.

The curves are to be read from right to left, and when we reach the left-hand edge of the paper, we re-enter again at the same height on the right-hand edge. The numbers on the successive curves denote the days of the month.

We have chosen an example from a sub-tropical region because it illustrates the remarkable regularity of the tides in a region where the weather is equable. Further, if the reader will note the successive high-waters or low-waters which follow one another on any one day, he will see a strongly marked " diurnal inequality," which would have been barely perceptible in a European tide-curve.

3 See J. N. Shoolbred on datum levels, Brit. Assoc. Reports (1879).

§ 3. Tidal Observation: the Tide-gauge. - Tidal prediction is only possible when accurate observations have been made of the phenomena to be predicted; and the like is true of verification after prediction. It was formerly thought sufficient to note the heights of the water at high and low-water, together with the times of those events, and the larger part of the observations which exist are still of this character, but complete investigation of the law of tidal oscillations demands that the height of the water should be measured at other times than at high and low-water.

With whatever degree of thoroughness it is proposed to observe the tides the procedure is much the same. The simplest sort of observation is to note the height of i the water on a hours before and after high-water, and down stream for the same period before and after low-water. But near the sides of channels and near the mouths of bays the changes of the currents are very complex; and near the headlands separating two bays there is usually at certain times a very swift current, termed a " race." In inland seas, such as the Mediterranean, the tides are nearly insensible except at the ends of long inlets. Thus at Malta the tides are not noticed by the ordinary observer, whilst at Venice they are conspicuous.

The effect of a strong wind on the height of tide is generally supposed to be strongly marked, especially in estuaries. In the case of an exceptional gale, when the wind veered round appropriately, Airy states 1 that the water has been known to depart from its predicted height at London by as much as 5 ft. The effect of wind will certainly be different at each port. The discrepancy of opinion on this subject appears to be great - so much so that we hear of some observers concluding that the effect of the wind is i nsensible. Variations in barometric pressure also 'p ' Land- runs in the direction analogous to up stream for three §4. Tide-Tables and the Degree of Accuracy in Tidal Prediction.' - The connexion between the tides and the movements of the Empirical moon and sun is so obvious that tidal predictions Tide-tables. were regularly made and published long before mathematicians had devoted their attention to them; and these predictions attained considerable success, although they were founded on empirical methods. During the 18th century, and even in the earlier part of the 19th, the art of prediction was regarded as a valuable family secret to be jealously guarded from the public. The best example of this kind of tide-table was afforded by Holden's tables for Liverpool, founded on twenty years of observation by a harbour-master named Hutchinson.2 the heights and times are tabulated according to the hour of the clock at which the moon will cross the meridian at the place of observation, distinguishing between the visible and invisible transits. Certain simple corrections have also to be applied. A considerable degree of elaboration has to be given to the table, in order that it may give accurate results, and it would occupy some half-dozen to a dozen pages of a book, its extension varying according to the degree of accuracy aimed at. It might occupy about five minutes to extract a prediction from the more elaborate form of such a table. There are many ports of considerable commercial importance where, nevertheless, it would hardly be worth while to incur the great and repeated A. M.

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FIG. I. - Tide-curve for Bombay from the beginning of the civil year 1884 to the midnight; ending Jan. 14, Zero of 1884, or from 12" Dec. 31, 1883, to 12 h Jan. 14, 1884, astronomical time. Gauge 8 rt. is rt. ca rt. 10 ft. 8 ft. About 1832 the researches of W. Whewell and of Sir John Lubbock (senior) pointed the way to improvement on the empirical tables prepared by secret methods, and since that time the preparation of tide-tables has become a branch of science.

A perfect tide-table would tell the height of the water at the place of observation at every moment of the day, but such a Prediction table would be cumbrous; it is therefore usual to for each predict only the times and heights of high-water and Day. of low-water. The best kind of tide-table contains definite forecasts for each day of a definite year, and we may describe it as a special table. Although the table is only made for one definite place, yet it is often possible to give fairly accurate predictions for neighbouring ports by the application of corrections both for time and height. Special tide-tables are published by all civilized countries for their most important harbours.

But there is another kind of table, which we may describe as a general one, where the heights and times are given by reference to the time at which the moon crosses the meridian. Prediction Reference A l t hough such a table is only applicable to a definite to Time of place, yet it holds good for all time. In this case it is Moon's necessary to refer to the Nautical Almanac for the Transit. 3' time of the moon's transit, and a simple calculation then gives the required result. In a general tide-table 1 References may be given to two papers by G. H. Darwin on this subject, viz. " Tidal Prediction," Phil. Trans., A. (1891) pp. 159-229; and " An apparatus for facilitating the reduction of tidal observations," Proc. Roy. Soc. (1892), vol. lii. For a general account without mathematics see Darwin's Tides, &c.; this section is founded on chs. xiii. and xiv. of that book. For mathematical methods see Maurice Levy, Theorie des marees (Paris, 1898).

Whewell, History of Inductive Sciences, ii. 248; Darwin's Tides, &c., ch. iv.

expenditure involved in the publication of special tables. But this kind of elaborate general table has been used in few cases,3 and the information furnished to mariners usually consists either of a full prediction for every day of a future year, or of a meagre statement as to the average rise and interval, which must generally be almost useless.

The success of tidal predictions varies much according to the place of observation. In stormy regions the errors are often considerable, and the utmost that can be expected of a tide-table is that it shall be correct with a steady barometer and in calm weather. But such conditions are practically non-existent, and therefore errors are inevitable.

Notwithstanding these perturbations, tide-tables are usually. of surprising accuracy even in northern latitudes; this may be seen from the following table showing the results of com parison between prediction and actuality at Portsmouth. Amount of The importance of the errors in height depends, of Error a t course, on the range of the tide; it is well, therefore, to Porismouth. note that the average ranges of the tide at springs and neaps are 13 ft. 9 in. and 7 ft. 9 in. respectively.

Prediction at such a place as Portsmouth is difficult, on account of the instability of the weather, but, on the other hand, the tides in themselves are remarkably simple in character. Let Amounf of us now turn to such a port as Aden, where the weather Error at is very uniform, but the tides very complex on account Aden. of the large diurnal inequality, which frequently obliterates one of two successive high-waters. The short series of comparisons between actuality and prediction which we give below may be taken as a fair example of what would hold good when a long series is examined. The results refer to the intervals Toth of March to the 9th of April and the 12th of November to the 12th of December 1884. In these two periods there should have been 118 high waters, but the tide-gauge failed to register on one occasion, Darwin, " Tidal Prediction," quoted above. This kind of table has been applied with some success at Cairns in North Queensland, where there is a large diurnal inequality.

ft. re. 2 rt. Meteorological Disturbance of Prediction. so that one comparison is lost. We thus have 117 cases to consider, but on one occasion the diurnal inequality obliterated a high-water, leaving 116 actual comparisons. The maximum range of the tide at Aden is 8 ft. 6 in., and this serves to give a standard of importance for the errors in height.

Table of Errors in the Prediction of High-Water at Portsmouth in the months of January, May and September 1897. Table of Errors in the Prediction of High-Water at Aden in March - April and November - December 1884. It would be natural to think that when a prediction is erroneous by as much as fifty-seven minutes it is a very bad one, but such a conclusion may be unjust. There was one case in which the highwater was completely obliterated by the diurnal inequality, but there were many others in which there was nearly complete obliteration, so that the water stood nearly stagnant for several hours. A measure of the degree of stagnation is afforded by the amount of rise from low to high-water. Now, on examining all the eleven cases where the error of time was equal to or over twenty minutes, we find five cases in which the range from low to high-water was less than 8 in., and these include the errors of fifty-six and of fifty-seven minutes. There is one case of a rise of 13 in. with an error of thirty-six minutes; one case of a rise of 17 in. with an error of twenty-two minutes; one of 19 in. rise with thirty-three minutes error. The remaining three cases have rises of 2 ft. 10 in., 3 ft. 9 in., 3 ft. I I in., and errors of twenty-two, twenty-three, twenty minutes. Thus all the very large errors of time correspond with approximate stagnation, and are unimportant. It is fair to conclude, therefore, that the predictions as to time are very good. The predictions as to height are obviously good, for more than half were within I in., and only eleven had an error of as much as 4 in.

When it is considered that the incessant variability of the tidal forces, the complex outlines of the coast, the depth of the sea, the earth's rotation and the perturbations by meteorological influences are all involved, it should be admitted that the success of tidal prediction is remarkable. If further evidence were needed, we might appeal to tidal prediction as a convincing proof of the truth of the theory of gravitation.

§ 5. General Explanation of the Cause of Tides. - The moon attracts every particle of the earth and ocean, and by the law of - gravitation the force acting on any particle is directed towards the moon's centre, and is jointly proportional Forces. to the masses of the particle and of the moon, and inversely proportional to the square of the distance between the particle and the moon's centre. If we imagine the earth and ocean subdivided into a number of small portions or particles of equal mass, then the average, both as to direction and intensity of the forces acting on these particles is equal to the force acting on that particle which is at the earth's centre. For there is symmetry about the line joining the centres of the two bodies, and, if we divide the earth into two portions by an ideal spherical surface passing through the earth's centre and having its centre at the moon, the portion remote from the moon is a little larger than the portion towards the moon, but the nearer portion is under the action of forces which are a little stronger than those acting on the farther portion, and the resultant of the weaker forces on the larger portion is exactly equal to the resultant of the stronger forces on the smaller. If every particle of the earth and ocean were being urged by equal and parallel forces there would be no cause for relative motion between the ocean and the earth. Hence it is the departure of the force acting on any particle from the average which constitutes the tide-generating force. Now it is obvious that on the side of the earth towards the moon the departure from the average is a small force directed towards the moon; and on the side of the earth away. from the moon the departure is a small force directed away from the moon. Also these two departures are very nearly equal to one another, that on the near side being so little greater than that on the other that we may neglect the excess. All round the sides of the earth along a great circle perpendicular to the line joining the moon and earth the departure is a force directed inwards towards the earth's centre. Thus we see that the tidal forces tend to pull the water towards and away from the moon, and to depress the water at right angles to that direction.

D FIG. 2. - Tide-generating Force.

In fig. 2 this explanation is illustrated graphically. The relative magnitudes of the tidal forces are given by the numbers on the figure. M is the direction of the moon, V the centre of the hemisphere of the earth at which the man in the moon would look, I the centre of the hemisphere which would be invisible to him, DD are the sides of the earth where the tidal force is directed towards the earth's centre. The outward forces at V and I are exactly double the inward forces at D and D.

If it were permissible to neglect the earth's rotation and to consider the system as at rest, we should find that the water was in equilibrium when elongated into a prolate ellipsoidal or oval form with its longest axis directed towards and away from the moon.

But it must not be assumed that this would be the case when there is motion. For, suppose that the ocean consisted of a canal round the equator, and that an earthquake or any other cause were to generate a great wave in Equatorial the canal, this wave would travel along it with a velocity dependent on the depth. If the canal Earth' were about 13 miles deep the velocity of the wave would be about woo miles an hour, and with depth about equal to the depth of our seas the velocity of the wave would be about half as great. We may conceive the moon's tide-generating force as making a wave in the canal and continually outstripping the wave it generates, for the moon travels along the equator at the rate of about 1000 miles an hour, and the sea is less than 13 miles deep. The resultant oscillation of the ocean must therefore be the summation of a series of partial waves generated at each instant by the moon and always falling behind her, and the aggregate wave, being the same at each instant, must travel 1000 m. an hour so as to keep up with the moon.

Now it is a general law of frictionless oscillation that, if a slowly varying periodic force acts on a system which would oscillate quickly if left to itself, the maximum excursion on one side of the equilibrium position occurs simultaneously with the maximum force in the direction of the excursion; but, if a quickly varying periodic force acts on a system which would oscillate slowly if left to itself, the maximum excursion on one side of the equilibrium position occurs simultaneously with the maximum force in the direction opposite to that of the excursion. An example of the first is a ball hanging by a short string, which we push slowly to and fro; the ball will never quit contact with the hand, and will agree with its excursions. If, however, the ball is hanging by a long string we can play at battledore and shuttlecock with it, and it always meets our blows. The latter is the analogue of the tides, for a free wave in our shallow canal goes slowly, whilst the moon's tide-generating action goes quickly. Hence when the system is left !a to settle into steady oscillation it is low-water under and opposite to the moon, whilst the forces are such as to tend to make high-water at those times.

If in this case we consider the moon as revolving round the earth, the water assumes nearly the shape of an oblate spheroid or orange-shaped body with the shortest axis pointed to the moon. The rotation of the earth in the actual case introduces a complexity which it is not easy to unravel by general reasoning. We can see, however, that if water moves from a lower to a higher latitude it arrives at the higher latitude with more velocity from west to east than is appropriate to its latitude, and it will move accordingly on the earth's surface. Following out this conception, we see that an oscillation of the water to and fro between south and north must be accompanied by an eddy. The solution of the difficult problem involved in working out this idea will be given below.

The conclusion at which we have arrived about the tides of an equatorial canal is probably more nearly true of the tides of a globe partially covered with land than if we were to suppose the ocean at each moment to assume the prolate figure of equilibrium. In fact, observation shows that it is more nearly lowwater than high-water when the moon is on the meridian. If we consider how the oscillation of the water would appear to an observer carried round with the earth, we see that he will have low-water twice in the lunar day, somewhere about the time when the moon is on the meridian, either above or below the horizon, and high-water half-way between the low waters.

If the sun be now introduced we have another similar tide of about half the height, and this depends on solar time, giving low-water somewhere about noon and midnight.

The superposition of the two modified by friction Y and by the interference of land, gives the actually observed aggregate tide, and it is clear that about new and full moon we must have spring tides and at quarter moons neap tides, and that (the sum of the lunar and solar tide-generating forces being about three times their difference) the range of spring tide will be about three times that of neap tide.

So far we have supposed the luminaries to move on the equator; now let us consider the case where the moon is not on the equator. It is clear in this case that at any place the moon's zenith distance at the upper transit is different from her nadir distance at the lower transit. But the tide-generating force is greater the smaller the zenith or nadir distance, and therefore the forces are different at successive transits. This was not the case when the moon was deemed to move on the equator. Thus there is a tendency for two successive lunar tides to be of unequal heights, and the resulting inequality of height is called a " diurnal tide." This tendency vanishes when the moon is on the equator; and as this occurs each fortnight the lunar diurnal tide is evanescent once a fortnight. Similarly in summer and winter the successive solar tides are generally of unequal height, whilst in spring and autumn this difference is inconspicuous.

One of the most remarkable conclusions of Laplace's theory of the tides, on a globe covered with ocean to a uniform depth, is that the diurnal tide is everywhere non-existent. But this hypothesis differs much from the reality, in Ocean of and in fact at some ports, as for example Aden, the diurnal tide is so large that during two portions of each lunation there is only one great high-water and one great low-water in each twenty-four hours, whilst in other parts of the lunation the usual semi-diurnal tide is observed.

§ 6. Progress of the Tide-wave over the Ocean and in the British Seas. - Sufficient tidal data would give the state of the tide at every part of the world at the same instant of time, and if the tide wave is a progressive one, like such wave as we may observe travelling along a canal, we should be able to picture mentally the motion of the tide-wave over the ocean and the successive changes in the height of water at any one place. But we are not even sure that the wave is progressive, for in some oceans, such as perhaps the Atlantic, the motion may be only a see-saw about some line in mid-ocean - up on one side and down on the other; or it may more probably be partly a progressive wave and partly a see-saw or stationary oscillation. In contracted seas the wave is undoubtedly predominantly progressive in character, but too little is known to enable us to speak with any confidence as to wider seas.

Whewell and Airy, while acknowledging the uncertainty of their data, made the attempt to exhibit graphically the progress of the tide-wave over a large portion of the oceans of the world. In the first edition of this article (Ency. Brit., 9th ed.) we reproduced their chart. But, since doubts as to its correctness have gradually accumulated, we think it more prudent to refrain from reproducing it again.' As we have already indicated, the tide in British seas has mainly a progressive character, and the general march of the wave may be exhibited on a chart by what are called cotidal lines. If at the full and change of moon we draw lines on the sea through all the places which have high-water simultaneously, and if we mark such lines successively XII, I, II, &c., being the Greenwich time of high-water along each line, we shall have a succession of lines which show the progress of the wave from hour to hour.

For phases of the moon, other than full and change, the numbers may be taken to represent the interval in hours after the moon's transit, either visible or invisible, until the occurrence of high water. But for these other phases of the moon the interval varies by as much as one hour in excess or defect of the number written on any of the lines. Thus when the moon is about five days old, or five days past full, the numbers must all be reduced by about one hour so that I, II, III, &c., will then be replaced by XII, I, II, &c.; and when the moon is about ten days old, or ten days past full, the numbers must all be augmented by about one hour, and will read II, III, IV, &c. However, for a rough comprehension of the tides in these seas it is unnecessary to pay attention to this variation of the intervals.

Airy in his " Tides and Waves " gives such a chart for Great Britain and the North Sea, and he attempts to complete the cotidal lines conjecturally across the North Sea to Norway, Denmark and the German coast. In this case, as in the more ambitious attempt referred to above, further knowledge has led to further doubt. We therefore give in fig. 3 Berghaus's modification of Airy's Chart,' abandoning the attempt to draw complete%cotidal lines. In this chart we can watch, as it were, the tide-wave running in from the Atlantic, passing up the Bristol Channel and Irish Sea, travelling round the north of Scotland and southward along the east coasts of Scotland and England. Another branch comes up the Channel, and meets the wave from the north off the Dutch coast. The Straits of Dover are so narrow, however, that it may be doubted whether ' Portion of Airy's Chart (Encycl. Metrop., art. " Tides and Waves ") is given in Darwin, Tides and Kindred Phenomena in the Solar System. 2 Berghaus's Physical Atlas (1891),(1891), pt. ii., " Hydrography." the tides on the English coasts would be profoundly modified if the Straits were completely closed.

It will be noticed that between Yarmouth and Holland the cotidal lines cross one another. Such an intersection of lines is in general impossible; it is indeed only possible if there is a region in which the water neither rises nor falls, because at such a place the cotidal line ceases to have adefinite meaning. A set of observations by Captain Hewitt, R.N., made in 1840, appears to prove the existence of a region of this kind at the part of the chart referred to.

(From Berghaus's Atlas.) FIG. 3. - Cotidal Lines in British Seas.

§ 7. Historical Sketch.' - The writings of various Chinese, Arabic and Icelandic authors show that some attention was paid by them to the tides, but the several theories advanced are fantastic. It is natural that the writings of the classical authors of antiquity should contain but few references to the tides, for the Greeks and Romans lived on the shores of an almost tideless sea. Nevertheless, Strabo quotes from Posidonius a clear account of the tides on the Atlantic coast of Spain, and connects the tides correctly with the motion of the moon. He also gives the law of the tide in the Indian Ocean as observed by Seleucus the Babylonian, and the passage shows that Seleucus had unravelled the law which governs the diurnal inequality of the tide in that sea.

We shall not give any details as to the medieval speculations on the tides, but pass on at once to Newton, who in 1687 laid the foundation for all that has since been added to the theory of the tides when he brought his grand generalization of universal gravitation to bear on the subject. Johann Kepler had indeed. at an early date recognized the tendency of the water of the ocean to move towards the centres of the sun and moon, but he was unable to submit his theory to calculation. Galileo expresses regret that so acute a man as Kepler should have produced a theory which appeared to him to reintroduce the occult qualities of the ancient philosophers. His own explanation referred the phenomenon to the rotation and orbital motion of the earth, and he considered that it afforded a principal proof of the Copernican system. .

In the 19th corollary of the 66th proposition of bk. i. of the Principia, Sir Isaac Newton introduces the conception of a canal. circling the earth, and he considers the influence of a satellite on the water in the canal. He remarks that the movement of each molecule of fluid must be accelerated in the conjunction and opposition of the satellite with the 1 The account from the time of Newton to that of Laplace is founded on Laplace's Mecanique celeste, bk. xiii. ch. i.

molecule, that is to say when the molecule, the earth's centre and the satellite are in a straight line, and retarded in the quadratures, that is to say when the line joining the molecule and the earth's centre is at right angles to the line joining the earth's centre and the satellite. Accordingly the fluid must undergo a tidal oscillation. It is, however, in propositions 26 and 27 of bk. iii. that he first determines the tidal force due to the sun and moon. The sea is here supposed to cover the whole earth and to assume at each instant a figure of equilibrium, and the tide-generating bodies are supposed to move in the equator. Considering only the action of the sun, he assumes that the figure is an ellipsoid of revolution with its major axis directed towards the sun, and he determines the ellipticity of such an ellipsoid. High solar tide then occurs at noon and midnight, and low-tide at sunrise and sunset. The action of the moon produces a similar ellipsoid, but of greater ellipticity. The superposition of these ellipsoids gives the principal variations of the tide. He then proceeds to consider the influence of latitude on the height of tide, and to discuss other peculiarities of the phenomenon. Observation shows, however, that spring tides occur a day and a half after full and change of moon, and Newton falsely attributed this to the fact that the oscillations would last for some time if the attractions of the two bodies were to cease.

The Newtonian hypothesis, although it fails in the form which he gave to it, may still be made to represent the tides if the lunar and solar ellipsoids have their major axes always directed toward a fictitious moon and sun, 3', Fictifs. which are respectively at constant distances from the true bodies; these distances are such that the full and change of the fictitious moon as illuminated by the fictitious sun occur about a day or a day and a half later than the true full and change of moon. In fact, the actual tides may be supposed to be generated directly by the action of the real sun and moon, and the wave may be imagined to take a day and a half to arrive at the port of observation. This period has accordingly been called " the age of the tide." In what precedes the sun and moon have been supposed Age of Tide. to move in the equator; but the theory of the two ellipsoids cannot be reconciled with the truth when they move, as in actuality, in orbits inclined to the equator. At equatorial ports the theory of the ellipsoids would at spring tides give morning and evening high waters of nearly equal height, whatever the declinations of the bodies. But at a port in any other latitude these high waters would be of very different heights, and at Brest, for example, when the declinations of the bodies are equal to the obliquity of the elliptic, the evening tide would be eight times as great as the morning tide. Now observation shows that at this port the two tides are nearly equal to one another, and that their greatest difference is not a thirtieth of their sum. Newton here also offered an erroneous explanation of the phenomenon.

In 1738 the Academy of Sciences of Paris offered, as a subject for a prize, the theory of the tides. The authors of four essays received prizes, viz. Daniel Bernoulli, Leonhard Euler, Cohn Maclaurin and Antoine Cavalleri. The first three adopted not only the theory of gravitation, but also Newton's method of the superposition of the two ellipsoids. Bernoulli's essay contained an extended development of the conception of the two ellipsoids, and, under the name of the equilibrium theory, it is commonly associated with his name. Laplace gives an account and critique of the essays of Bernoulli and Euler in the Mecanique celeste. The essay of Maclaurin presented little that was new in tidal theory, but is notable as containing certain important theorems concerning the attraction of ellipsoids. In 17 4 6 Jean-le-Rond D'Alembert wrote a paper in which he treated the tides of the atmosphere; but this work, like Maclaurin's, is chiefly remarkable for the importance of collateral points.

The theory of the tidal movements of an ocean was therefore, as Laplace remarks, almost untouched when in 1774 he first undertook the subject. In the Mecanique celeste he gives an interesting account of the manner in which he was led to attack the problem. We shall give below the investigation of the tides Laplace. of an ocean covering the whole earth; the theory is Lap substantially Laplace's, although presented in a different form, and embodying an important extension of Laplace's work by S. S. Hough. This theory, although very wide, is far from representing the tides of our ports. Observation shows, in fact, that the irregular distribution of land and water and the various depths of the ocean in various places produce irregularities in the oscillations of the sea of such complexity that the rigorous solution of the problem is altogether beyond the power of analysis. Laplace, however, rested his discussion of tidal observation on this principle - The state of oscillation of a system of bodies in which the primitive conditions of movement have disappeared through friction is coperiodic with the forces acting on the system. Hence if the sea is acted on by forces which vary periodically according to the law of simple oscilla Forced tions (a simple time-harmonic), the oscillation of osciila- the sea will have exactly the same period, but the tions. moment at which high-water will occur at any place and the amplitude of the oscillation can only be derived from observation. Now the tidal forces due to the moon and sun may be analysed into a number of constituent periodic parts of accurately determinable periods, and each of these will generate a corresponding oscillation of the sea of unknown amplitude and phase. These amplitudes and phases may be found from observation. But Laplace also used another principle, by which he was enabled to effect a synthesis of the various oscillations, so that he does not discuss a very large number of these constituent oscillations. As, however, it is impossible to give a full account of Laplace's methods without recourse to technical language, it must suffice to state here that this procedure enabled him to discuss the tides at any port by means of a combination of theory with observation. After the time of Laplace down to 1870, the most important workers in this field were Sir John Lubbock (senior), William Whewell Lubbock, and Sir G. B. Airy. The work of Lubbock and Whewell Whewell (see § 33 below) is chiefly remarkable for and Airy. the co-ordination and analysis of enormous masses of data at various ports, and the construction of trustworthy tide-tables and the attempt to construct cotidal maps. Airy contributed an important review of the whole tidal theory. He also studied profoundly the theory of waves in canals, and explained the effects of frictional resistance on the progress of tidal and other waves.

The comparison between tidal theory and tidal observations has been carried out in two ways which we may describe as the synthetic and the analytic methods. Nature is herself synthetic, since at any one time and place we only observe one single tide-wave. All the great investigators from Newton down to Airy have also been synthetic in their treatment, for they have sought to represent the oscillation of the sea by a single mathematical expression, as will appear more fully in chapter V. below. It is true that a presupposed analysis lay behind and afforded the basis of the synthesis. But when at length tide-gauges, giving continuous records, were set up in many places the amount of data to be co-ordinated was enormously increased, and it was found that the simple formulae previously in use had to be overloaded with a multitude of corrections, so that the simplicity became altogether Kelvin. fictitious. This state of matters at length led Lord Kelvin (then Sir William Thomson) to suggest, about 1870, the analytic method, in which the attempt at mathematical synthesis is frankly abandoned and the complex whole is represented as the sum of a large number of separate parts, each being a perfectly simple wave or harmonic oscillation. All the best modern tidal work is carried on by the analytic method, of which we give an account below in chapter IV.

Lord Kelvin's other contributions to tidal theory are also of profound importance; in particular we may mention that he established the correctness of Laplace's procedure in discussing the dynamical theory of the tides of an ocean covering the whole earth, which had been impugned by Airy and by William Ferrel. We shall have frequent occasion to refer to his name hereafter in the technical part of this article.

Amongst all the grand work which has been bestowed on the theory of this difficult subject, Newton, notwithstanding his errors, stands out first, and next to him we must rank Laplace. However original any future contribution to the science of the tides may be, it would seem as though it must perforce be based on the work of these two.

§ 8. The Tide-Predicting Instrument

In the field of the practical application of theory Lord Kelvin also made another contribution of the greatest interest, when in 1872 he Tide_ suggested that the laborious task of constructing a Predicting tide-table might be effected mechanically. Edward Instrument. Roberts bore a very important part in the first practical realization of such a machine, and a tide-predictor now in regular use at the National Physical Laboratory for the Indian government was constructed by Lege under his direction. We refer the reader to Sir William Thomson's (Lord Kelvin's) paper on "Tidal Instruments" in Inst. C.E., vol. lxv., and to the subsequent discussion, for a full account and for details of the share borne by the various persons concerned in the realization of the idea.

Fig. 4 illustrates diagrammatically the nature of the instrument. A cord passes over and under a succession of pulleys, every other pulley being fixed or rather balanced and the alternate ones being movable; the cord is fixed at one end and carries a pen or pencil at the other end. In the diagram there are two balanced pulleys and one movable one; a second unit would require one more movable pulley and one more balanced one. If, in our diagram, the lowest or movable pulley were made to oscillate up and down (with a simple harmonic motion), the pencil would execute the same motion on half the linear scale. If the instrument possessed two units and the second movable pulley also rocked up and down, the pencil would add to its previous motion that of this second oscillation, again on half scale. So also if there were any number of additional units, each consisting of one movable and one balanced pulley, the pencil would add together all the separate simple oscillations, and would draw a curve upon a drum, which is supposed to be kept revolving uniformly at an appropriate rate.

The rocking motion is communicated to each movable pulley by means of a pin attached to a wheel C sliding in a slot attached to the pulley frame. All the wheels C and the drum are geared together so that, as the drum turns, all the movable pulleys rock up and down. The gearing is of such a nature that if one revolution of the drum represents a single day, the rocking motion of each movable pulley corresponds to one of the simple constituent oscillations or tides into which the aggregate tide-wave is analysed. The nature of the gearing is determined by theoretical considerations derived from the motions of the sun and moon and earth, but the throw of each crank, and the angle at which it has to be set at the start are derived from observation at the particular port for which the tide-curve is required. When the tide-predictor has been set appropriately, it will run off a complete tide-curve for a whole year; the curve is subsequently measured and the heights and times of high and low-water are tabulated and published for a year or two in advance.

The Indian instrument possesses about 20 units, so that the tidecurve is regarded as being the sum of 20 different simple tides; and tide-tables are published for 40 Indian and Oriental ports. A tidepredictor has been constructed for the French government under the supervision of Lord Kelvin and is in use at Paris; another has been made by the United States Coast Survey at Washington;; in 1910 one was under construction for the Brazilian government. These instruments, although differing considerably in detail from the Indian predictor, are essentially the same in principle.

§ 9. Tidal Friction

All solid bodies yield more or less to stress; if they are perfectly elastic they regain their shapes after the stresses are removed, if imperfectly elastic or viscous they yield to the stresses. We may thus feel certain that the earth yields to tide-generating force, either with perfect or imperfect elasticity. Chapter VIII. will contain some discussion of this FIG. 4. - Tide-Predicting Instrument.

subject, and it must suffice to say here that the measurement of the minute elastic tides of the solid earth has at length been achieved. The results recently obtained by Dr 0. Hecker at Potsdam constitute a conspicuous advance on all the previous attempts.

The tides of an imperfectly elastic or viscous globe are obviously subject to frictional resistance, and the like is true of the tides of an actual ocean. In either case it is clear that the system must be losing energy, and this leads to results of so much general interest that we propose to give a short sketch of the subject, deferring to chapter VIII. a more rigorous investigation. It is unfortunately impossible to give even an outline of the principles involved without the use of some technical terms.

In fig. 5 the paper is supposed to be the plane of the orbit of a satellite M revolving in the direction of the arrow about the planet C, which rotates in the direction of the arrow about an axis perpendicular to the paper. The rotation of the E Tidal planet is supposed to be more rapid than that of the satellite, so that the day is shorter than the month. Let Friction us suppose that the planet is either entirely fluid, or has an ocean of such depth that it is high-water under or nearly under the satellite. When there is no friction, with the satellite at m, the planet is elongated into the ellipsoidal shape shown, cutting the mean sphere, which is dotted. The tidal protuberances are drawn with much exaggeration and the satellite is shown as very close to the planet in order to illustrate the principle more clearly. Now, when there is friction in the fluid motion, the tide is retarded, and high-tide occurs after the satellite has passed the meridian. Then, if we keep the same figure to represent the tidal deformation, the satellite must be at ' M, instead of at m. If we number the four quadrants as shown, the satellite must FIG. 5. be in quadrant t. The protuberance P is nearer to the satellite than P', and the deficiency Q is farther away than the deficiency Q'. Hence the resultant action of the planet on the satellite must be in some such direction as MN. The action of the satellite on the planet is equal and opposite, and the force in NM, not being through the planet's centre, must produce a retarding couple on the planet's rotation, the magnitude of which depends on the length of the arm CN. This tidal frictional couple varies as the height of the tide, and as the satellite's distance. The magnitude of the tidal pro tuberances varies inversely as the cube of the distance of the satellite, and the difference between the attrac tions of the satellite on the nearer and farther pro tuberances also varies inversely as the cube of the distance. Accordingly the tidal frictional couple varies as the inverse sixth power of the satellite's distance. Let us now consider its effect on the satellite. If the force acting on M be resolved along and perpendicular to the direction CM, the perpendicular component tends to accelerate the satellite's velocity. It alone would carry the satellite farther from C than it would be dragged back by the central force towards C. The satellite would describe a spiral, the coils of which would be very nearly circular and very nearly coincident. If now we resolve the central component force along CM tangentially and perpendicular to the spiral, the tangential component tends to retard the velocity of the satellite, whereas the disturbing force, already considered, tends to accelerate it. With the gravitational law of force between the two bodies the retardation must prevail over the acceleration.' The action of Velocity tidal friction may appear somewhat paradoxical, but it is the exact converse of the acceleration of the linear and angular velocity and the diminution of distance of a satellite moving through a resisting medium. The latter result is generally more familiar than the action of tidal friction, and it may help the reader to realize the result in the present case. Tidal friction then diminishes planetary rotation, increases the satellite's distance and diminishes the orbital angular velocity. The comparative rate of diminution of the two angular velocities is generally very different. If the satellite be close to the planet the rate of increase of the satellite's periodic time or month is large compared with the rate of increase of the period of planetary rotation or day; but if the satellite is far off the converse is true. Hence, if the satellite starts very near the planet, with the month a little longer than the day,'as the satellite 1 This way of presenting the action of tidal friction is due to Sir George G. Stokes.

recedes, the month soon increases so that it contains many days. The number of days in the month attains a maximum and then diminishes. Finally the two angular velocities subside to a second identity, the day and month being identical and both very long.

We have supposed that the ocean is of such depth that the tides are direct; if, however, they are inverted, with low-water under or nearly under the satellite, friction, instead of retarding, accelerates the tide; and it would be easy by drawing another figure to see that the whole of the above conclusions would hold equally true with inverted tides.

Attempts have been made to estimate the actual amount of the retardation of the earth's rotation, but without much success. It must be clear from the sketch just given that the effect of tidal friction is that the angular motion of the moon round the earth is retarded, but not to so great an extent as the earth's rotation. Thus a terrestrial observer, who regards the earth as a perfect timekeeper, would look on the real retardation of the moon's angular motion as being an acceleration. Now there is a true acceleration of the moon's angular motion which depends on a slow change in the eccentricity of the earth's orbit round the sun. After many thousands of years this acceleration will be reversed and it will become a retardation, but it will continue for a long time from now into the future; thus it is indistinguishable to us at present from a permanent acceleration. The amount of this true acceleration may be derived from the theories of the motions of the moon and of the earth when correctly developed. Laplace conceived that its observed amount was fully explained in this way, but John Couch Adams showed that Laplace had made a mistake and had only accounted for half of it. It thus appeared that there was an unexplained portion which might be only apparent and might be attributed to the effects of tidal friction.

The time and place of an eclipse of the sun depend on the motions of the moon and earth. Accordingly the records of ancient eclipses, which occurred centuries before the Christian era, afford exceedingly delicate tests of the motions of the moon and earth. At the time when Thomson and Tait's Natural Philosophy' was first published it was thought that all the numerical data were known with sufficient precision to render it possible to give a numerical estimate of the retardation of the earth's rotation. But the various revisions of the lunar theory which have been made since that date throw the whole matter into doubt. It seems probable that there is some portion of the acceleration of the moon's motion which is unexplained by gravitation, and may therefore be attributed to tidal friction, but its amount is uncertain. We can only say that the amount is very small. It is, however, not impossible that this smallness may be due to counteracting influences which tend to augment the speed of the earth's rotation; such an augmentation would result from shrinkage of the earth's mass through cooling. However this matter may stand, it does not follow that, because the changes produced by tidal friction in a man's lifetime or in many generations of man are almost insensible, the same must be true when we deal with millions of years. It follows that it is desirable to trace the effects of tidal friction back to their beginnings.

We have seen above that this cause will explain the repulsion of a satellite from a position close to the planet to a more remote distance. Now when we apply these considerations to the moon and earth we find that the moon must once have been nearly in contact with the earth. This very remarkable initial configuration of the two bodies seems to point to the origin of the moon by detachment from the earth.

Further details concerning this speculation in cosmogony are given below in chapter VIII.3 § t o. Bibliography. - Many works on popular astronomy contain a few paragraphs on the tides, but the treatment is generally so meagre as to afford no adequate idea of the whole subject. A complete list of works both general and technical bearing on the theory of the tides, from the time of Newton down to 1881, is contained in vol. ii. of the Bibliographie de l'astronomie by J. C. Houzeau and A. Lancaster (1882). This list does not contain papers on the tides of particular ports, and we are not aware of the existence of any catalogue of works on practical observation, reduction of observations, prediction and tidal instruments. The only general work on the tides, without mathematics, is George Darwin's Tides and Kindred Phenomena in the Solar System.' This book treats of all the subjects considered in the present article (with references to original sources), and also others such as seiches (q.v.) and the bore (q.v.).

The most extensive monograph on the tides is A Manual of Tides by Mr Rollin A. Harris, published by the United States Coast Survey in a series of parts, of which pt. i. appeared in 1897, and pt. iv.

' See that work (ed. 1883), § 830; P. H. Cowell, M. N. R. Ast. Soc. (1905), lxv. 861.

For a discussion of the subject without mathematics, see G. H. Darwin's Tides. 4 London (1898) and with important changes (1901, 191 1); (Boston, 1898); translations: German, by A. Pockels (Leipzig, 1902, 1911); Italian, by G. Magrini (Turin, 1905), with appendices by translator; Magyar, byRado von Kovesligethy (Budapest,1904),with appendices by translator.

B in 1904. This work contains an enormous mass of useful work, and gives not only complete technical developments both on the theoretical and practical sides but also has chapters of general interest. The present writer feels it his duty, however, to dissent from Mr Harris's courageous attempt to construct the cotidal lines of the various oceans.

This work contains the most complete account of the history of tidal theories of which we know. Laplace's admirable history of the subject down to his own time has been summarized in § 7. Dr Giovanni Magrini has an appendix to his translation of Darwin's book, entitled La Conoscenza della marea nell'antichita, founded on the researches of Dr Roberto Almagia. Dr Almagia. himself gives the results of his researches more fully in a memoir, presented to the Accademia dei Lincei of Rome (5th series, vol. v. fascic. x., 1905, 1 37 PP).

Another monograph on tides, treating especially the mathematical developments, is Maurice Levy's La The'orie des marees (Paris, 1898). Colonel Baird's Manual of Tidal Observation (1886) contains instructions for the installation of tide-gauges, and auxiliary tables for harmonic analysis. Airy's article on " Tides and Waves " in the Ency. Metrop., although superseded in many respects, still remains important. Harris's Manual contains a great collection of results of tidal observations made at ports all over the world.

The article " Die Bewegung der Hydrosphare " in the Encyklopeidie der mathematischen Wissenschaften (vi. I, 1908) gives a technical account of the subject, with copious references. The same article is given in English in vol. iv. (1911) of G. H. Darwin's collected Scientific Papers; and vols. i. and ii. contain reprints of the several papers by the same author referred to in the present article.

Since the date of the 9th edition of the Ency. Brit. some technical discussion of the tides has appeared in textbooks, such as H. Lamb's Hydrodynamics.' That work also reproduces in more modern form Airy's investigation of the effects of friction on the tides of rivers. We are thus able to abridge the present article, but we shall present the extension by Hough of Laplace's theory of the tides of an oceancovered planet, which is still only to be found in the original memoirs.

II

[[Tide-Generating Forces § I T]]. Investigations of Tide-Generating Potential and Forces. - We have already given a general explanation of the nature of tide-generating forces; we now proceed to a rigorous investigation. If a planet is attended by a single Forces. satellite, the motion of any body relatively to the planet's surface is found by the process described as reducing the planet's centre to rest. The planet's centre will be at rest if every body in the system has impressed on it a velocity equal and opposite to that of the planet's centre; and this is accomplished by impressing on every body an acceleration equal and opposite to that of the planet's centre.

Let M, m be the masses of the planet and the satellite; r the radius vector of the satellite, measured from the planet's centre; p the radius vector, measured from same point, of the particle whose motion we wish to determine; and z the angle between r and p. The satellite moves in an elliptic orbit about the planet, and the acceleration relatively to the planet's centre of the satellite is (M-}-m)/r 2 towards the planet along the radius vector r. Now the centre of inertia of the planet and satellite remains fixed in space, and the centre of the planet describes an orbit round that centre of inertia similar to that described by the satellite round the planet but with linear dimensions reduced in the proportion of m to M-Fm. Hence the acceleration of the planet's centre is m/r 2 towards the centre of inertia of the two bodies. Thus, in order to reduce the planet's centre to rest, we apply to every particle of the system an acceleration m/r 2 parallel to r, and directed from satellite to planet.

Now take a set of rectangular axes fixed in the planet, and let M i r, Mgr, Mar be the co-ordinates of the satellite referred thereto; and let gyp, np, p be the co-ordinates of the particle P whose radius vector is p. Then the component accelerations for reducing the planet's centre to rest are - mM i /r 2, - mM 2 /r 2, - mM 3 /r 2; and since these are the differential coefficients with respect to pt, pn, of the function - mP - - Men-f-M3f), and since cos z=Mlle-Men-+-M31', it follows that the potential of the forces by which the planet's centre is to be reduced to rest is - m p cos Z.

r 1 The theory as presented in the Mecanique celeste is unnecessarily difficult, and was much criticized by Airy. Before the publication of the 9th and Toth editions of the Ency. Brit. it was necessary for the student to read a number of controversial papers published all over the world in order to get at the matter.

Now let us consider the other forces acting on the particle. The planet is spheroidal, and therefore does not attract equally in all directions; but in this investigation we may make abstraction of the ellipticity of the planet and of the ellipticity of the ocean due to the planetary rotation. This, which we set aside, is considered in the theories of gravity and of the figures of planets. Outside its body, then, the planet contributes forces of which the potential is M/p. Next the direct attraction of the satellite contributes forces of which the potential is the mass of the satellite divided by the distance between the point P and the satellite; this is ‘ 11r 2 -{- p 2 - 2rp cos z} To determine the forces from this potential we regard p and z as the variables for differentiation, and we may add to this potential any constant we please. As we are seeking to find the forces which urge P relatively to M, we add such a constant as will make the whole potential at the planet's centre zero, and thus we take as the potential of the forces due to the attraction of the satellite m tr 2 -{- p 2 - 2rp cos zi r It is obvious that in the case to be considered r is very large compared with p, and we may therefore expand this in powers of p/r. This expansion gives us m 3-P + p2 2 P + p 2 P 3 +. .. (, where P I = cos z, P2 = cos t z - 2 i P3 = cos 3 z - 2 cos z, &c. The reader familiar with spherical harmonic analysis of course recognizes the zonal harmonic functions; but the result for a few terms, which is all that is necessary, is easily obtainable by simple algebra.

Now, collecting together the various contributions to the potential, and noticing that '- 'r °' P1= mp cos z, and is therefore equal and opposite to the potential by which the planet's centre was reduced to rest, we have as the potential of the forces acting on a particle whose co-ordinates are p, pn, pf P - f - -(2 cos 2 z2)-}-' - 4 3 (I cos 3 z-2 cos z) -{- ... (I) The first term of (I) is the potential of gravity, and the terms of the series, of which two only are written, constitute the tide-generating potential. In all practical applications this series converges so rapidly that the first term is amply sufficient, and thus we shall generally denote V=2 p2 (cos t z - a) (2) as the tide-generating potential.2 At the surface of the earth p is equal to a the earth's radius.

§ 12. Form of Equilibrium. - Consider the shape assumed by an ocean of density a, on a planet of mass M, density S and radius a, when acted on by disturbing forces whose potential is a solid spherical harmonic of degree i, the planet not being in rotation.

If S i denotes a surface spherical harmonic of order 1, -such a potential is given at the point whose radius vector is p by V Si. (3) In the case considered in § II, 1=2 and S i becomes the second zonal harmonic cos 2 z - 3.

The theory of harmonic analysis tells us that the form of the ocean, when in equilibrium, must be given by the equation p= a -}-eiSi. (4) Our problem is to evaluate e i . We know that the external potential of a layer of matter, of depth e i S i and density o, has the value I / /I p i+l eiSi. Hence the whole potential externally to the planet and up to its surface is i al i + 1 2r 3 a> Si - i - 2 i - }- I (p/ ei Si. (5) The first and most important term is the potential of the planet, the second that of the disturbing force, and the third that of the departure from sphericity.

Since the ocean must stand in a level surface, the expression (5) equated to a constant must be another form of (4). Hence, if we put p = a -}-e i S i in the first term of (5) and p=a in the second and third terms, (5) must be constant; this can only be the case if the The reader may refer to Thomson and Tait's Natural Philosophy (1883), pt. ii. §§ 798-821, for further considerations on this and analogous subjects, together with some interesting examples.

coefficient of i vanishes. Hence on effecting these substitutions and equating that coefficient to zero, we find 3ma 2 pro-a - o. are‘ +- 1 - r 3 + 21+ lei But by the definitions of S and a we have M=1-7r&a 3 =ga t , where g is gravity, and therefore 3ma2 2gr3 e i = 3c I - (21+ I)6 In the particular case considered in § II we therefore have p - a[1-4- Im3%/Sb3 (cos2z3)?

as the equation to the equilibrium tide under the potential V = 2r, p2 (cos 2 z - 3) .

If a were very small compared with the attraction of the water on itself would be very small compared with that of the planet on the water; hence we see in the general case that 1/ (I - 3c 2T+1)S/ is the factor by which the mutual gravitation of the ocean augments the deformation due to the external forces. This factor will occur frequently hereafter, and therefore for brevity we write Q b, - I - (21+I)S' and we may put (6) in the form _ 3ma2 e t 2gr3 bi Comparison with (5) then shows that V=gbi (id is the potential of the disturbing forces under which p =a-f-e i S i (I I) is a figure of equilibrium.

We are thus provided with a convenient method of specifying any disturbing force by means of the figure of equilibrium which it is competent to maintain. In considering the dynamical theory of the tides on an ocean-covered planet, we shall specify the disturbing forces in the manner expressed by (io) and (II). This way of specifying a disturbing force is equally exact whether or not we choose to include the effects of the mutual attraction of the ocean. If the augmentation due to mutual attraction of the water is not included, b i becomes equal to unity; there is no longer any necessity to use spherical harmonic analysis, and we see that if the equation to the surface of an ocean be p= a+S, where S is a function of latitude and longitude, it is in equilibrium under forces due to a potential whose value at the surface of the sphere (where p=a) is gS. In treating the theory of tidal observation we shall specify the tide-generating forces in this way, and then by means of " the principle of forced vibrations," referred to in § 7 as used by Laplace for discussing the actual oscillations of the sea, we shall pass to the actual tides at the port of observation.

In this equilibrium theory it is assumed that the figure of the ocean is at each instant one of equilibrium under the action of gravity and of the tide-generating forces. Lord Kelvin has, however, reasserted' a point which was known to Bernoulli, i but has since been overlooked, namely, that this law Theory. of rise and fall of water cannot, when portions of the globe are continents, be satisfied by a constant volume of water in the ocean. The necessary correction to the theory depends on the distribution of land and sea, but a numerical solution shows that it is practically of very small amount.

§ 13. Development of Tide -generating Potential in Terms of Hour - Angle and Declination. - We now proceed to develop the tidegenerating potential, and shall of course implicitly (§ 12) determine the equation to the equilibrium figure.

We have already seen that, if z be the moon's zenith distance at the point P on the earth's surface, whose co-ordinates referred to A, B, C, axes fixed in the earth, and at, an, a;', cos z =M1 - { - rtM2 where M I, M2, M3 are the moon's direction cosines referred to the same axes. Then, with this value of cos z, } cos 2 z-3 = 2 ?JMiM2+ 2 27J2M,2 2 M 22+2n1"M2M3+2EJM1M3 2 I Thomson and Tait, Nat. Phil. § 807. G. H. Darwin and H. H. Turner, Proc. Roy. Soc. (1886).

The axis of C is taken as the polar axis, and AB is the equatorial plane, so that the functions of, i, are functions of the latitude and longitude of the point P, at which we wish to find the potential. The functions of M1, M2 i M3 depend on the moon's position, and we shall have occasion to develop them in two different ways - first in terms of her hour-angle and declination, and secondly (§ 25) in terms of her longitude and the elements of the orbit.

Now let A be on the equator in the meridian of P, and B 90° east of A on the equator. Then, if M be the moon, the inclination of the plane MC to the plane CA is the moon's easterly local hour-angle. Let ho =Greenwich westward hour-angle; l =the west longitude of the place of observation; A = the latitude of the place; S = moon's declination: then we have M I =cos cos (ho - l), M2 = - cos sin(ho - l), M3=sin S, E=cos A, r t =o, i =sin A.

Also the radius vector of the place of observation on the earth's surface is a. Whence we find V - 3M(12 a cos t X cos 2 S cos 2 (ho - l) + sin 2X sin cos 6 cos (ho - 1) - sin e %) (a - sin2X) (13) The tide-generating forces are found by the rates of variation of V for latitude and longitude, and also for radius a, if we care to find the radial disturbing force.

The westward component of the tide-generating force at the earth's surface, where p =a, is dV/a cos Xdl, and the northward component is d VladX; the change of apparent level is the ratio of these to gravity g. On effecting the differentiations we find that the westward component is made up of two periodic terms, one going through its variations flour- twice and the other once a day. The southward cornponent has also two similar terms; but it has a third very small term, which does not oscillate about a zero value. This last term corresponds to forces which produce a constant heaping up of the water at the equator; or, in other words, the moon's attraction has the effect of causing a small permanent ellipticity of the earth's mean figure. This augmentation of ellipticity is of course very small, but it is necessary to mention it.

If we consider the motion of a pendulum-bob under the influence of these forces during any one day, we see that in consequence of the semi-diurnal changes of level it twice describes an ellipse with major axis east and west, and the formula when developed shows that the ratio of axes is equal to the sine of the latitude, and the linear dimensions proportional to cos' 6. It describes once a day an ellipse whose north and south axis is proportional to sin cos 2X and whose east and west axis is proportional to sin 23 sin A. Obviously the latter is circular in latitude 30°. When the moon is on the equator, the maximum deflexion occurs when the moon's local hour-angle is 45°, and is then equal to 3 3 m a cos A. 2M r This angle is equal to o 0174" cos A. Attempts actually to measure the deflexion of the vertical have at length proved successful (see Seismometer).

III. - Dynamical Theory Of The Tides § 14. Recent Advances in the Dynamical Theory of the Tides. - The problem of the tidal oscillation of the sea is essentially dynamical. In two papers in the second volume of Liouville's Journal (1896) H. Poincare has considered the mathematical principles involved in the problem, where the ocean is interrupted by land as in actuality. He has not sought to obtain numerical results applicable to any given configuration of land and sea, but he has aimed rather at pointing out methods by which it may some day be possible to obtain such solutions. Even when the ocean is taken as covering the whole earth the problem presents formidable difficulties, and this is the only case in which it has been solved hitherto.2 Laplace gives the solution in bks. i. and iv. of the Mecanique celeste; but his work is unnecessarily complicated. In the 9th edition of the Ency. Brit. we gave Laplace's theory without these complications, but the theory is now accessible in H. Lamb's Hydrodynamics and other works of the kind. It is therefore not reproduced here.

In 1897 and 1898 S. S. Hough undertook an important revision of Laplace's theory and succeeded not only in introducing the effects of the mutual gravitation of the ocean, but Lord Kelvin's (Sir W. Thomson's) paper on the gravitational oscillations of rotating water, Mag. (October 1880), bears on this subject. It is the only attempt to obtain numerical results in respect to the effect of the earth's rotation on the oscillations of land-locked seas.

2 7 2 - 2 ? - M1 2 +M2 2 - 2 M3 2 (12) 3 3 (io) also in determining the nature and periods of the free oscillations of the sea.' A dynamical problem of this character cannot be regarded as fully solved unless we are able not only to discuss the " forced " oscillations of the system but also the " free." Hence we regard Mr Hough's work as the most important contribution to the dynamical theory of the tides since the time of Laplace. We shall accordingly present the theory briefly in the form due to Mr Hough.

The analysis is more complex than that of Laplace, where the mutual attraction of the ocean was neglected, but this was perhaps inevitable. Our first task is to form the equations of motion and continuity, which will be equally applicable to all forms of the theory.

§ 15. Equations of Motion. - Let r, 0, 4, be the radius vector, colatitude and east longitude of a point with reference to an origin, a polar axis and a zero-meridian rotating with a uniform angular velocity n from west to east. Then if R, H, be the radial, colatitudinal and longitudinal accelerations of the point, we have R dt2 - r (d8) 2 - r sine (dn) 2 41, - r dt (r' j) - r sin 0 cos 0 (d +n) H - r sin° dt [r2 sin20 (d +n) 1. If the point were at rest with reference to the rotating meridian we should have R = - n 2 r sin 0, E= - n 2 r sin 0 cos 0, H =o.

When these considerations are applied to the motion of an ocean relative to a rotating planet, it is clear that these accelerations, which still remain when the ocean is at rest, are annulled by the permanent oblateness of the ocean. As then they take no part in the oscillations of the ocean, and as we are not considering the figure of the planet, we may omit these terms from R and. This being so we must replace (+n)' as it occurs in R and E by (j4,) +2ndt Now suppose that the point whose accelerations are under consideration never moves far from its zero position, and that its displacements E, rt sin 0 in colatitude and longitude are very large compared with p its radial displacement. Suppose, further, that the velocities of the point are so small that their squares and products are negligible compared with n 2 r 2; then we have dr dp dt - dt a very small quantity; sin' 0d - dt (n sin 0), do di rc- - dt' Since the radial velocity always remains very small it is not necessary to concern ourselves further with the value of R, and we only require the two other components which have the approximate forms, V=' - 2n sin 0 cos 0 d - t' dt 2 H =sin 0a t 2 +2n cos B d t We have now to consider the forces by which an element of the ocean is urged in the direction of colatitude and longitude. These forces are those due to the external disturbing forces, to the pressure of the water, surrounding an element of the ocean, and to the attraction of the ocean itself.

If e denotes the equilibrium height of the tide, it is a function of colatitude and longitude, and may be expanded in a series of spherical surface harmonics e i. Thus we may write the equation to the equilibrium tide in the form.

r=a+e=a+Eei. Now it appears from (io) and (i i) that the value of the potential, at the surface of the sphere where p= a, under which this is a figure of equilibrium, is V = Egbiei.

We may use this as specifying the external disturbing force due to the known attractions of the moon and sun, so that e i may be regarded as known.

But in our dynamical problem the ocean is not a figure of equilibrium, and we may denote the elevation of the surface at any moment of time by b. Then the equation to the surface may be written in the form r=a+b=a+Ebi, where 1, denotes a spherical harmonic just as e i did before.

1 Phil. Trans., 189 A, pp. 201-258 and 191 A, pp. 139-185.

The surface value of the potential of the forces which would maintain the ocean in equilibrium in the shape it has at any moment is Egb i b i. Hence it follows that in the actual case the forces due to fluid pressure and to the attraction of the ocean must be such as to balance the potential just determined. Therefore these forces are those due to a potential - Egb i b i. If we add to this the potential of the external forces, we have a potential which will include all the forces, the expression for which is - gEb i (b i - e i). If further we perform the operations d/ad9 and d/a sin 0d4 on this potential, we obtain the colatitudinal and longitudinal forces which are equal to the accelerations, and H.

It follows, then, from (i¢) that the equations of motion are dt2 - 2n sin e cos e at = - a E bi (F9 (bi - ei) 05) sin Bat2 +2n cos e B dt = - a sun Eb i d (b - e) It remains to find the equation of continuity. This may be deduced geometrically from the consideration that the volume of an element of the fluid remains constant; but a shorter way is to derive it from the equation of continuity as it occurs in ordinary hydrodynamical investigations. If t be a velocity potential, the equation of continuity for incompressible fluid is Sr-1 e (r 2 d-sin 0 50 SO) +5Bd (r si n Bd95r5c6) +S4 'd? (r r sine d5r SB) =0. The element referred to in this equation is defined by r, 0, 0, r+Sr, 0+50, 0-{-14,. The colatitudinal and longitudinal velocities are the same for all the elementary prism defined by 8, 4,, 0+50, 0+4, and the sea bottom. Then = dt' r s i n ed4 =sin 0- and, since the radial velocity is db/dt at the surface of the ocean, where r = a -}-y, and is zero at the sea bottom, where r = a, we have d? - r=a db Hence, integrating with respect to from TIT y dt r = a+y to r = a, and again with respect to t from time t to the time when b,, n all vanish, and treating y and Il as small compared with a, we have ba sin 0 - } - d e (y sin e)+ 4 (yr t sin 0) =0. (16) This is the equation of continuity, and, together with (15), it forms the system which must be integrated in the general problem of the tides. The difficulties in the way of a solution are so great that none has hitherto been found, except on the supposition that y, the depth of the ocean, is only a function of latitude. In this case (16) becomes i d 0 d0`'."sin 8) +yd? = o. (17) § 16. Adaptation to Forced Oscillations. - Since we may suppose that the free oscillations are annulled by friction, the solution required is that corresponding to forced oscillations. Now we have seen from 03) that e (which is proportional to V) has terms of three kinds, the first depending on twice the moon's (or sun's) hour-angle, the second on the hour-angle, and the third independent thereof. The coefficients of the first and second vary slowly, and the whole of the third varies slowly. Hence e has a semi-diurnal, a diurnal and a long-period term. We shall see later that these terms may be expanded in a series of approximately semi-diurnal, diurnal and slowly varying terms, each of which is a strictly harmonic function of the time.

Thus according to the usual method of treating oscillating systems, we may make the following assumptions as to the form of the solution e = Eei = Ee i cos(2nft+ sfi + a) b = bE i =Eh i cos(2nft+ + a) = Eb i x i cos(2nft +so +a) rt = Eb i y i sin(2nft+s0 +a) where e i, h i, x i, y i are functions of colatitude only, and e i, h i are the associated functions of colatitude corresponding to the harmonic of order i and rank s. For the semi-diurnal tides s = 2 and f is approximately unity; for the diurnal tides s = i and f is approximately 2; and for the tides of long period s=o and f is a small fraction. Substituting these values in 0E7) we have [ š th o)+ y y + d9(y b ixi s i n 9 s bi i h i a = o. Then if we write u i for h i - e i, and put m=n'a/g, from (18) in 05) leads at once to f 2 Eb;x i - f sin 0 cos 9Eb i y i = 4 m d B Ebiui, r . (20) f 2 sin 0Eb i y i +f cos 0Eb i x i = - 4m Ebiui. ¢m sin o (18) (14) (19) substitution (Eb i x i) (-cos 2 0)=-1 4m [d i . " -r" f sin B biu i? (Ebiyi) sin e 0(f2-cos" 0) _ c - 4m [ f B de zb i ui+ sin e`eiui Then substituting from (21) in (19) we have d y(sin d +f cos 0?b i u i) sin o f' -cos 2 0 [-° d+ sy Sln 0 biuiJ J sin 0 (f 2 -cos 2 0) This is almost the same as Laplace's equation for tidal oscillations in an ocean whose depth is only a function of latitude. If indeed we treat b i as unity (thereby neglecting the mutual attraction of the water) and replace Zu i and le i by u and e, we obtain Laplace's equation.

When u i is found from this equation, its value substituted in (21) will give x i and yi.

§ 17. Zonal Oscillations. - We might treat the general harmonic oscillations first, and proceed to the zonal oscillations by putting s=o. These waves are, however, comparatively simple, and it is well to begin with them. The zonal tides are those which Laplace describes as of the first species, and are now more usually called the tides of long period. As we shall only consider the case of an ocean of uniform depth, y the depth of the sea is constant. Then since in this case s=o, our equation (22), to be satisfied by u i or hi - ei, becomes sin od-biui +-4y a sin 0 h i =o.

f 2 - cos 2 0 This may be written d Ebiui+ 4 y a sin B 28f re_ sin od0+A =0, (23) where A is a constant.

Let us assume hi= C i P i, ei = EiPi where P i denotes the ith zonal harmonic of cos 8. The coefficients C i are unknown, but the E i are known because the system oscillates under the action of known forces.

If the term involving the integral in this equation were expressed in terms of differentials of harmonics, we should be able to equate to zero the coefficient of each dP i /de in the equation, and thus find the conditions for determining the C's.

The task then is to express f 2 - co s 2 0 ° P i sin Bdo in differentials sin o J z of zonal harmonics.

It is well known that P i satisfies the differential equation d e (sin 4Ç) +1(1+I)P i sin o=o. (24) Therefore f P i sin odor -i(i+I)sin o de , and 2 sin 0 Pi sin Bdo = -. 2 1 (f cos 0) ei (f )dPi I 2 dPi I de - i(i+1) sin ode Another well-known property of zonal harmonics is that sin O dP -= 221+ }- i) (Pi+i-Pi-1) If we differentiate (25) and use (24) we have 2) 21 + 1 ( d'de 1 - d ?8 1) +i(i+I)P i si n 0=0 . (26) Multiplying (25) by sin 0, and using (26) twice over, dP i(i+1) _ 1 dP 2_ dP i 1 dPi_dPi_2 sin g e x 21+ 22+3 (de do) + 21 - (de do) S f2 0 dPi_2 Therefore ore P sin Bdo - 5 f 2 - I 2 dPi I dPi +2 i(i +I)+(21 -I)(21 +3) "do + (21 +I)(21 +3) de This expression, when multiplied by 4ma/-y and by C i and summed, is the second term of our equation.

The first term is ?bi(C4-Ei) Vii. In order that the equation may be satisfied, the coefficient of each dP i /do must vanish identically. Accordingly we multiply the whole by y/4ma and equate to zero the coefficient in question, and obtain 4 a z-E:)? - (21 - ) y (21-3) t i(i+I) m (C (22-1)(22+3) C$ Ci+2 (2 +3) (22 (27) This equation (27) is applicable for all values of i from 1 to infinity, provided that we take Eo, C_1, E_ 1 as being zero.

We shall only consider in detail the case of greatest interest, namely that of the most important of the tides generated by the attraction of the sun and moon. We know that in this case the equilibrium tide is expressed by a zonal harmonic of the second order; and therefore all the Ei, excepting E2, are zero. Thus the equation (27) will not involve E i in any case excepting when 1=2.

If we write for brevity _ f 2 - 1 2 biy Li -i(i+1)+ (2 i- I) (21+3) 4ma' the equation (27) is C i+2 Ci - 2 - o. 28 (21+3)(22+5) LtiCi+( 21-3)(21-) () Save that when 1=2, the right-hand side is b 2 -yE 2 /4ma, a known quantity ex hypothesi. The equations naturally separate themselves into two groups in one of which all the suffixes are even and the other odd. Since our task is to evaluate all the C's in terms of E2, it is obvious that all the C's with odd suffixes must be zero, and we are left to consider only the cases where 1=2, 4, 6, &c.

We have said that Co must be regarded as being zero; if however we take Co = - 3b2yE2/4ma, so that Co is essentially a known quantity, the equation (28) has complete applicability for all even values of i from 2 upwards. The equations are It would seem at first sight as if these equations would suffice to determine all the C's in terms of C2, and that C2 would remain indeterminate; but we shall show that this is not the case. For very large values of i the general equation of condition (28) tends to assume the form C i+2 + Ci-2 + i ma 2 y Cc = o.

By writing successively i+2, 1+4, id-6 for i in this equation, and taking the differences, we obtain an equation from which we see that, unless Ci/Ci+2 tends to become infinitely small, the equations are satisfied by Ci = Ci+2 in the limit for very large values of 1.

Hence, if C i does not tend to zero, the later portion of the series for h tends to assume the form Ci(Pi+Pi+2+Pi+4...). All the P's are equal to unity at the pole; hence the hypothesis that Ci does not tend to zero leads to the conclusion that the tide is of infinite height at the pole. The expansion of the height of tide is essentially convergent, and therefore the hypothesis is negatived. Thus we are entitled to assume that C i tends to zero for large values of i.

Now writing for brevity ai /(21+I) (22+3)2(22+5), we may put (28) into the form Ci-2/Ci - Li - Ci/Ci+2 (21-3) (21 -I)- (21+I)(21+3) By successive applications of this formula we may write the righthand side in the form of a continued fraction.

Let Then we have or Thus C 2 = 3.5K2Co; C4=3.5.7.9K2K4Co; C6 = 3. 5.7. 9 . I I.13K 2 K 4 K 6 Co, &c. If we assume that any of the higher C's, such as C14 or C16, is of negligible smallness, all the continued fractions K2, K4, Ks, &c., may be computed; and thus we find all the C's in terms of Co, which is equal to-3b 2 yE 2 /4ma. The height of the tide is therefore given by lt = Zh i cos(2nft+ a) = - 4m 2 Y E 2{3.5 K P +3.5. 7.9K2 K 4 P 4 + ... } cos (2nft + a). It is however more instructive to express h as a multiple of the (21) (22) dPi, (25) ai-2 ai +2 Li - L i+2 - Li+4 - Ci -2 /Ci ai -2 (21-3)(211) - Ki' Ci/C12 = (21 - (21 - I) (22+ I)Ki.

- I - 4maE(us+ei) =o.

sin 0 (21 - I)(22+I) de Co - 0 5.7 -L4C4+ II.13 - .

ai equilibrium tide e, which is as we know equal to E P cos (2nft+a). Whence we find b = 4 2a P2 { 3.5K P +3.5.7.9K K P +3.5 ...13K K K P ... } m The number f is a fraction such that its reciprocal is twice the number of sidereal days in the period of the tide. The greatest value of f is that appertaining to the lunar fortnightly tide (Mf in notation of harmonic analysis), and in this case f is in round numbers 1/28, or more exactly f 2 = .00133. The ratio of the density a of sea-water to S the mean density of the earth is 18093; which value gives us b2 = I - 5 = 89144.

The quantity m is the ratio of equatorial centrifugal force to gravity, and is equal to 1/289. Finally, /a is the depth of the ocean expressed as a fraction of the earth's radius.

With these numerical values Mr Hough has applied the solution to determine the lunar fortnightly tide for oceans of various depths. Of his results we give two: - First, when =7260 ft. = 1210 fathoms, which makes y/4ma =1/40, he finds b =p { 2669P - 1678P4+. 0485P6 - o081P + 0009Pia - 0001P ... }. If the equilibrium theory were true we should have e b= P {P2}; thus we see how widely the dynamical solution differs from the equilibrium value.

Secondly, when y =58080 ft. =9680 fathoms, and y/4ma =1/5, he finds b=P { 7208P - 0973P4+ o048P6 - 0001P8... }.

From this we see that the equilibrium solution presents some sort of approximation to the dynamical one; and it is clear that the equilibrium solution would be fairly accurate for oceans which are still quite shallow when expressed as fractions of the earth's radius, although far deeper than the actual sea.

The tides of long period were not investigated by Laplace in this manner, for he was of opinion that a very small amount of friction would suffice to make the ocean assume its form of equilibrium. In the arguments which he adduced in support of this view the friction contemplated was such that the integral effect was proportional to the velocity of the water relatively to the bottom. It is probable that proportionality to the square of the velocity would have been nearer the truth, but the distinction is unimportant.

The most rapid of the oscillations of this class is the lunar fortnightly tide, and the water of the ocean moves northward for a week and then southward for a week. In oscillating systems, where the resistances are proportional to the velocities, it is usual to specify the resistance by a " modulus of decay," namely the time in which a velocity is reduced by friction to e1 - or 1/2.78 of its initial value. Now in order that the result contemplated by Laplace may be true, the friction must be such that the modulus of decay is short compared with the semi-period of oscillation. It seems certain that the friction of the ocean bed would not reduce a slow ocean current to one-third of its primitive value in a day or two. Hence we cannot accept Laplace's discussion as satisfactory, and the investigation which has just been given becomes necessary. (See § 34).

§ 18. Tesseral Oscillations. - The oscillations which we now have to consider are those in which the form of surface is expressible by the tesseral harmonics. The results will be applicable to the diurnal and semi-diurnal tides Laplace's second and third species.

If we write a=s/f the equation (22) becomes d (sin cos o) Eb u (0- cos 8d +s 2 cosec o) Ebiui do s2 - a' cos' 0 52-a2 COS' 0 +- sin 0 2h = 0. (29) If we write D for the operation sin d, the middle term may be arranged in the form a cot 0 (D +a cos o) (Eb i u i) Ebiui s 2 -02 cos' 0 sin 0' Therefore on multiplying by sin 0 the equation becomes (D -a cos o) [(D± cos ') (biu)] - (' E b i u i ) + m sin BEh = o. (30) We now introduce two auxiliary functions, such that Eb h - e -Ebiui =(D -0- cos 0)T+ (s 2 - a' cos' 0)4>. (31) It is easy to prove that (D+a cos 0)(D -a cos 8) =D 2 -s'+a sin' o + (s'-a' cos' 0), (D-a cos 0)(D+a cos 0) = D 2 -s 2 -a sin' 0+ (s 2 - a' cos' o). Also (D +a cos 8) (s 2 -a 2 cos' 0) I = (s 2 - a 2 cos t 0) (D +a cos B) +2a 2 sin 2 8 cos 0 (33) Now perform D+a cos 0 on (31), and use the first of (32) and (33), and we have (D +a cos 0)(Eb i u i) =(D2-s2-1-0- sin e 0+s 2 - a 2 cos28) ?.

} (s 2 -a 2 cos t 8)(D-{-a cos 0)4)+2a 2 sin 2 0 cos 0 1. (34) The functions 4, and are as yet indeterminate, and we may impose another condition on them. Let that condition be (D 2 - s 2 +0- sin' 0) _ -202 sin e 0 cos 0 c b. (35) Then (34) may be written (D+a cos 0) (niui) 1 +(D+a cos 8)43. s 2 -a 2 cos' 0 Substituting from this in (30), and using the second of (32), the function disappears and the equation reduces to (D 2 - $ 2 - a sin' 8)(1)+ 4 y a sin' e 2;14=0. (36) c os 8 Since by (35) -a 2 cos' 0 - 2 sin' (D2-s' +a sin' 0) 4,, (31) may be written Eb u = [D - a cos 8+1 s n 6 sin' 8)] 4 Y+s 2 4). (37) The equations (35), (36) and (37) define and 4), and furnish the equation which must be satisfied.

If we denote cos 0 by the zonal harmonics are defined by _ i Py 2` I 2 ?(dµ) (µ2 1)i ' The following are three well-known properties of zonal harmonics: [(I _,.,2) dP2]-i(i-} I)Pi =o, (38) (i+I)P - (21+I)µP +iP _ =0, (39) dPi+i dPi_i dµ - µ = (21-i- d (40) If Pi s? are the two tesseral harmonics of order i and rank s, it is also known that Pi = (1 - / 22) z8 a BPi µ Let us now assume h i = C:P, 'e =E:P:, = EaP R, '43= E1331P:.

These must now be substituted in our three equations (35), (36), (37), and the result must be expressed by series of the Pi functions. It is clear then that we have to transform into Ps functions the following functions of PR, namely in' (e D 2 - s 2 cr sin' 8)PL, cos OP:, s [D_ cos o+2 cos ° 0 (D 2 -s 2 +cr sin 2 0)]P:. If we differentiate (38) s times, and express the result by of the operator D, we find ( D 2 - s 2) p.4 +i (i + I) P 4 sin' 8 =0. Again, differentiating (39) s times and using (40), we find (i-s+ I)P:+1- (21+I) cos 0 P i+(i + s) P ti 1=o. Lastly, differentiating (41) once and using (38), (40) and 2(i-s+I) s (1+1)(2+S) DPi = i+ Pi+1 21+I P i - 1 By means of (42), and (44) we have sin' 0(D'-s2ta sin 8)Pi=[-2(2+ I) Ea1F, P:= s s cos 8 Pt121+I P-+ 21+I Pi+it [D-a + 12 s 2 +a sin'8)]F - (i-s+1)[a - I)1P8+1 s in 60°(D2 2(21+I) (i+s)[a+(2+I)(i+2)1 s P i-1 Therefore the equations (35), (3 6), (37) give i+s i s+I [a{_i(i+I) +a}P:+2a2s 21 + IP8-1+ 21+I P ti+1 ] =0, M[0:1 -i(i+I) -a}P;+ y 732a C;Pi] =o, a [c:_ E)P+a (i -s 2(2[Z+I(i-1)1p: +1 +(2+s){a"i"(2+I) (2-}'2)1Pa - ist`s2P:] =o. 2(22+I) (41) (32) means (43) Since these equations must be true identically, the coefficients of P. in each of them must vanish. Therefore ai{a-2(2+I)}+2r 2 2 } ?i+ 1 2 +31+131 _ 122- S I -f i{a+2(2+I)I+ ? a - C a =o, (Ci -E;)-1-ai - 1(1-s)[a+(1-I)(i-2)] bi 2(21 - I) 'a ' If we eliminate the a's and g's from the third equation (45), by means of the first two, we find

8 L'C3 C8 - yb E° (46) 4ma +771+2 i t2 - 'i, ' a s2 (22 - S2) [a+ (2 -  I)(2-2)] L -  1)1(422 - I)[a - (1 - I)i}[a+2(2+I)] [(i+I)2-s2][a+(i+2)(i+3)I 'Ybi + 14(11 - 0 2 - '11 a - (i + I) (i+2)][a+i(i+1)]_ 4ma - (2-s)(i- s- I) - (210(213)[ ° -- (i _ -(i+s+i)(2+s+2)  ni+2 (21+3) (21 +5)[0- (i+ I) (i+2)1 In the case of the luni-solar semi-diurnal tide (called in the notation of harmonic analysis) we have i = 2, s =2, a =2. Hence it would appear that these formulae_ for Ly and E fail by becoming indeterminate, but i and s are rigorously integers, whereas a depends on the " speed " of the tide; accordingly in the case referred to we must regard terms involving (i-s) as vanishing in the limit when approaches to equality with i (11). For this particular case then we find and to=o. 5 The equation (46) for the successive C's is available for all values of i provided that C_ 1, E_ 1 , Co, E 0 are regarded as being zero.

As in the case of the zonal oscillations, the equations with odd suffixes separate themselves from those with even suffixes, so that the two series may be treated independently of one another. Indeed, as we shall see immediately, the series with odd suffixes are satisfied by putting all the C's with odd suffixes zero for the case of such oscillations as may be generated by the attractions of the moon or sun.

For the semi-diurnal tides i=2, s=2, and f is approximately equal to unity. Hence the equilibrium tide is such that all the Ei, excepting E2, are zero.

For the diurnal tides i=2, s= i, and f is approximately equal to I. Hence all the Ei, excepting El, are zero. Since in neither case is there any E with an odd suffix, we need only consider those with even suffixes.

In both cases the first equation among the C's is -L 2 8 C +na C 71 ' 2 E 2 = 2 or r) It follows that if we write tZC C = -4 ma E2 (s=2 or I), the equation of condition amongst the C's would be of general applicability for all even values of i from 2 upwards.

The symbols E g o, 77 3 2 do not occur in any of the equations, and therefore we may arbitrarily define them as denoting unity, although the general formulae for t and n would give them other values. Accordingly we shall take o C 0 = CO= -4maE2 (S = 2 Or i). With this definition the equation O(S=2 or I) is applicable for i =2, 4, 6, &c.

It may be proved as in the case of the tides of long period that we may regard C2/C: +2 as tending to zero. Then our equation may be written in the form =Li ?iC Gi+2' and by successive applications the right-hand side may be expressed in the form of a continued fraction. Let us write r Li- L:+2 - Hence our equation may be written i C Ca H n: TJ 1l C3 It follows that H a H:H3 H3H:H 3 Ca 2 ' a 2 6 a s s ' 1 74.77 8 4, 7 g s ' Then since we have defined n2 and 4maE2, all the C's are expressed in terms of known quantities. Hence the height of tide b is given by b=Zh i cos(2nft+S4h+a) -4 b2 8 3 8 H2H4 H2H,2Hs ma E2 cos(2nft+ s? a) [HP +__ 1 _ :+ But the equilibrium tide c is given by c=E2P2 cos (2nft +4+ a).

Hence we may write our result in the following form, which shows the relationship between the true dynamical tide and the equilibrium tide. - _ 7b2 3 3 1 4 11 13 H 8 H 8 H s b 4ma ?12 P 2-i- 77 4 From a formula equivalent to this Mr Hough finds for the lunar semi-diurnal tide (s=2), for a sea of 1210 fathoms (y l 4 ma 40) b = 3 Io 39 6P9 + '5799 8P 1 - ' 19273 P s + 03054 1... .

This formula shows us that at the equator the tide is " inverted," and has 2.4187 times as great a range as the equilibrium tide.

For this same ocean he finds that the solar semi-diurnal tide is " direct " at the equator, and has a range 7.9548 as great as the equilibrium tide.

Now the lunar equilibrium tide is 2.2 times as great as the solar equilibrium tide, and since 2.2 X2.4187 only 5.3, it follows that in such an ocean the solar tides would have a range half as great again as the lunar. Further, since the lunar tides are " inverted " and the solar " direct," spring tide would occur at quarter moon and neap tide at full and change.

We give one more example from amongst those computed by Mr Hough. In an ocean of 9680 fathoms (y/4ma =1/5), he finds p 2 I. 7646P1- 06057P1:+.001447P6...

At the equator the tides are " direct " and have a range of 1.9225 as great as the equilibrium tide. In this case the tides approximate in type to those of the equilibrium theory, although at the equator, at least, they have nearly twice the range.

We do not give any numerical results for the diurnal tides, for reasons which will appear from the following section.

§ 19. Diurnal Tide approximately evanescent.

The equilibrium diurnal tide is given by e=E2 P2 cos(2nft+4+a), where f is approximately z and the associated function for i =2, P2 =3 sin B cos 0. Now the height of tide is given by b= MC1Pi cos(2nft +a), and the problem is to evaluate the constants Ci. If possible suppose that b is also expressed by a single term like that which represents e, so that b = 3C2 sin 0 cos 0 cos(2nft+cb+a).

Then the differential equation (22) to be satisfied becomes d sin +fu c cos O - c f9 d0+ sing y(Cz-E2) sin 9 d6 f2-cos29 sin 0(f2 -cos2 B) +4ma C2u=o, where u is written for brevity in place of sin 0 cos 0. Now when f is rigorously equal to 2, it may be proved by actual differentiation that the expression inside the brackets { } vanishes identically, and the equation reduces to C2 =o.

We thus find that in this case the differential equation is satisfied by zero oscillation of water-level. In other words we reach Laplace's remarkable conclusion that there is no diurnal rise and fall of the tides. There are, it is true, diurnal tidal currents, but they are so arranged that the water level remains unchanged.

In reality f is not rigorously 2 (except for the tide called K ) and there will be a small diurnal tide. The lunar diurnal tide called 0 has been evaluated for various depths of ocean by Mr Hough and is found always to be small.

§ 20. Free Oscillations of the Ocean

Mr Hough discusses the various types of free oscillations of the ocean. They are very complex, and consist of westward waves and eastward waves of very various periods. He finds, as was to be expected, that if, for an ocean of given depth, a free wave very nearly coincides in period Whence - O'.

(45) + s (2 +S+I)[a+(i+2)(i+3)] 2(21+3)