XI. Essay towards a First Approximation to a Map of Cotidal Lines. By the Rev. W. WHEWELL, M.A. F.R.S. Fellow of Trinity College, Cambridge.

Read May 2, 1833.

Introduction.

EVER since the time of NEWTON, his explanation of the general phenomena of the tides by means of the action of the moon and the sun has been assented to by all philosophers who have given their attention to the subject. But even up to the present day this general explanation has not been pursued into its results in detail, so as to show its bearing on the special phenomena of particular places,-to connect the actual tides of all the different parts of the world,—and to account for their varieties and seeming anomalies. With regard to this alone, of all the consequences of the law of universal gravitation, the task of bringing the developed theory into comparison with multiplied and extensive observations is still incomplete; we might almost say, is still to be begun.

DANIEL BERNOULLI, in his Prize Dissertation of 1740, deduced from the Newtonian theory certain methods for the construction of tide tables, which agree with the methods still commonly used. More recently LAPLACE turned his attention to this subject; and by treating the tides as a problem of the oscillations rather than of the equilibrium of fluids, undoubtedly introduced the correct view of the real operation of the forces; but it does not appear that in this way he has obtained any consequences to which NEWTON's mode of considering the subject did not lead with equal certainty and greater simplicity; moreover by confounding, in the course of his calculations, the quantities which he designates by λ and ', the epochs of the solar and lunar tide (Méc. Cél. vol. ii. p. 232. 291.), he has thrown an obscurity on the most important differences of the tides of different places, as Mr. LUBBOCK has pointed out.

LAPLACE also compared with the theory observations made at Brest from the year 1711 to 1715; and showed that the laws which, according to the theory, ought to regulate the times and heights of the tides, may, in reality,

be traced in the averages of this series of observations. In pursuance of his advice also, a new series of observations was undertaken at the same port, with the intention that it should be continued, at least, during one period of the motion of the nodes of the moon's orbit. The new observations were begun in 1806, and have since been carried on without interruption. Of the observations thus made, LAPLACE subjected to a mathematical discussion those for sixteen years, beginning with 1807; and M. BOUVARD, who performed the requisite calculations, employed nearly 6000 tide-observations. In our own country also, Mr. LUBBOCK has given the results of the examination of about 13,000 tide-observations, made at the London Docks, from 1808 to 1826, in a Memoir recently published in these Transactions. These results are very important, in consequence of their consistency with theory and with each other; the calculations by which they were obtained were performed by Mr. DESSIOU; and the task which he has thus executed, is, perhaps, in the amount of labour, and in the judicious and systematic mode of its application, not inferior to any of the most remarkable discussions of large masses of astronomical or meteorological observations by other modern calculators.

But in the meantime no one appears to have attempted to trace the nature of the connexion among the tides of different parts of the world. We are, perhaps, not even yet able to answer decisively the inquiry which BACON suggests to the philosophers of his time, whether the high water extends across the Atlantic so as to affect contemporaneously the shores of America and Africa, or whether it is high on one side of this ocean, when it is low on the other; at any rate such observations have not been extended and generalized.

It will easily be understood that we may draw a line through all the adjacent parts of the ocean which have high water at the same time; for instance, at 1 o'clock on a given day. We might draw another line through all the places which have high water at 2 o'clock on the same day, and so on. Such lines may be called cotidal lines; and they will be the principal subject of the present essay.

It might perhaps be supposed at first that we have now considerable materials for drawing such cotidal lines upon our maps. The time of the tide has been observed and recorded over a large portion of the earth's surface, by residents or by voyagers, during the last two centuries; and we have in many works

tables of the establishment of a long list of places. There are, however, in these statements, certain errors and imperfections, which prevent our being able as yet to determine the course of the cotidal lines with accuracy, or even to obtain with certainty a first approximation to these lines. But before we explain the defects of our observations, it will be proper to say a few words on the general properties of the cotidal lines.

The cotidal line for any hour may be considered as representing the summit or ridge of the tide-wave at that time; in which expression we mean, by the tide-wave, that protuberance of water upon the surface of the ocean which moves along the seas, and by its motion brings high-water and low-water to any place, at the time when the elevated and the depressed parts of the watery surface reach that place. The cotidal lines for successive hours represent the successive positions of the summit of this wave; and if we suppose a spectator, detached from the earth, to perceive the summit of the wave, he will see it travelling round the earth in the open ocean once in twenty-four hours, accompanied by another at twelve hours distance from it; and both sending branches into the narrower seas; and the manner and velocity of all these motions will be assigned by means of a map of cotidal lines.

I now proceed to endeavour to determine, first, from the laws of the motion of water, what the form of such lines may be expected to be; second, from the tide observations which we possess, what their form appears to be in reality.

Sect. I. On Cotidal Lines as determined by the laws of fluids.

1. Tides on a globe covered with water.-If we suppose the whole surface of the terrestrial globe to be uniformly covered with water, it is easy to see what must be the nature of the form and motion of the cotidal lines. The tides would be, in their main circumstances, entirely governed by the moon. High water at every place, in the same latitude, would follow the transit of the moon at the same interval of time*. The points at which it was high water at a given moment would therefore be situated in a meridian, at a certain distance from the meridian in which the moon was (or at least in some curve symmetrical with regard to the equator). There would be one such curve having reference to the moon, and another, having reference to the point * We here consider the moon as moving in the equinoctial.

immediately opposite to the moon; and these curves would each revolve round the earth, from east to west, in something more than twenty-four hours. If we suppose one cotidal line to be drawn through all points at which it is high water at 1 o'clock on a given day, a second cotidal line through all places where it is high water at 2 o'clock on the same day, and so on, there will be twentyfour such similar lines on the whole surface of the globe, cutting the equator at equal intervals, like so many meridians. And since the circumference of the earth is about 25,000 miles, it is obvious that any one of these cotidal lines would travel with a velocity of above one thousand miles an hour at the equator, and with a velocity of about six hundred miles an hour in our latitude. This is the velocity with which the summit of the tide-wave would travel on this supposition.

2. Derivative tides.-If on such a globe as we have been considering, a continent were interposed, occupying a great extent of latitude, it is clear that. the motion of the cotidal lines must become quite different from what it was in the uninterrupted ocean. On the western side of such a continent the tidewave could no longer proceed as if the continent were not there; for the supply of water and of pressure brought by the tide-wave advancing from the east, on which its further motion westwards altogether depends, is entirely intercepted. The tide on the western side of the continent must be produced by the water and the pressure which comes from the north, south, and west, and will be governed by laws different from those which regulate the primary or uninterrupted tide. And the same may be said of the tides produced in any seas of which the extent is much intercepted by land.

In order to see the general character of such cases, let us take the case of a tide which is entirely derived from the primary tide, and is not affected at all by the direct action of the sun and moon. Suppose the surface of the southern hemisphere to be entirely occupied by water, and the northern hemisphere to be principally land. Let a considerable inland sea run northward from the equator towards the pole. The tide-wave of the southern ocean, as it passes the entrance of this sea, will send off a derivative undulation, which will advance northwards up the sea, being impelled entirely by the mechanical action by which undulations are propagated in fluids. If we suppose the depth and other circumstances which would affect the motion of this derivative wave to