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The end and will fall during an equal time; the whole rise and we have discussed the various forms in which theory Tides and Hures fall will usually be less than in rivers; a very slight has been put for the purpose of explaining the grand Waves.

current will be directed towards the head of the bay facts of the Tides. SI

during the rise, and from it during the fall of the water. The variations of spring and neap tides, and the rela

Section II.-EQUILIBRIUM-Theory Of Tides. tion of the time of high water to the time of the moon's (14.) Before entering upon either of the theories Inade. passage over the meridian, follow the same general laws explaining the Tides, we must allude to their inade- quacy of all as in rivers. quacy, perhaps not to the explanation of the facts theories of

the Tidea, (9.) In long and narrow seas (for instance the English already observed, but certainly to the prediction of and the Channel) the tide in mid-channel follows the same laws

This inadequacy does not appear to arise cause of it. Dis as at a station near the mouth of a river, rising and fall- from any defect in the principles upon which the theory C++ ing in equal times, and running in a direction which may is based, (although perhaps our ignorance of the laws

be considered analogous to the direction up a river, for of friction among the particles of water, and between three hours before and three hours after high water; and water and the sides of the channels which contain it, in the opposite direction, for three hours before and three may be considered a failure of this kind,) but from the hours after low water. But near the sides of the channel, extreme difficulty of investigating mathematically the and especially near the mouths of bays or estuaries motions of fluids under all the various circumstances

branching from the channel, the change of tide follows a in which the waters of the sea and of rivers are found. en of very peculiar law. The water is never stationary, as in For the problem of the Tides, it is evident, is essen

river-tides, when changing from flow to ebb, but the tially one of the motion of fluids. Yet so difficult are direction of the current changes in 12 hours 20 minutes the investigations of motion that, till the time of Lathrough all the points of the compass. As a general place, no good attempt was made to determine, by rule

, supposing the observer's face turned in the direc- theory, the laws of the Tides, except on the supposition tion which is analogous to the direction up a river; that the water was at rest. Since that time theories of near the shores on his left hand the course of the tide- motion have been applied ; and it is hoped that in the current revolves in the same direction as the hands of present Treatise it will be found that something has a watch, and near the shores on the right hand it been added to the preceding investigations of motion, revolves in the opposite direction. Near the headlands possessing in some degree a practical character. Yet which separate different bays, there is usually, at certain the theory, even in this state, reaches very few cases.

times of the tide, a very rapid current, called a race. Indeed, throughout the whole of this subject, the Dis (10.) The elevations and depressions of tides in the selection of the proper theoretical ground of explana

open seas are much smaller than in contracted seas or tion is a matter of judgment. In some cases we may rivers, sometimes not exceeding one or two feet; the conceive that we are justified in using the Equilibriumstream of the tide is generally insensible.

theory ; in others the Wave-theory will apply, com(11.) In seas of small extent (as the Mediterranean) pletely or partially ; in a few cases, the results of ob**. the Tide is nearly insensible.

servation in one locality will be considered as we (12.) In some circumstances, phænomena which are fundamental set of experiments, upon which the expla

scarcely perceptible in ordinary localities become para- nation of the phænomena in other localities will be mount. Thus, in some positions near Behring's Straits, grounded without further reduction to theory; and as a the difference of morning and evening tides, which is last resource, in almost every case, we shall be driven carcely sensible in England, becomes so great that, in to the same arbitrary suppositions which Laplace certain parts of the lunation, there appears to be only introduced. Nevertheless, we conceive that our mate tide in the day. Other phænomena peculiar to thematical theory, pursued into some degree of detail, these localities, but less obvious to ordinary observa. will be far from useless. In the instances which it tian, will be noticed hereafter.

does not master completely, it will show that there are (13.) The phænomena which we have described must ample grounds for the arbitrary alterations of constants Pecessarily, for the most part, have been remarked by introduced by Laplace in his suppositions, to which we all nations dwelling on the borders of the ocean. Thus have more than once alluded. It will show that we Casar, in his account of the invasion of Britain, (De are precluded from further advance, partly by our Bello Gallico, lib. iv.) alludes to the nature of spring almost necessary ignorance of the forms of the bottom tides as perfectly well understood in connection with in deep seas, and partly by the imperfection of our

Some of the peculiarities of river mathematics. It will leave no doubt whatever that cirles, however, were not published in scientific works the first principles of our explanation are correct. till the beginning of the last century; and some of the Begging the reader to receive the first part of this paraproperties of the tides in the English and other chan- graph as an apology on the part of mathematicians for hels were not known till the end of that century. Upon applying to the motion of Tides a theory so evidently the whole, the statement above may be supposed to inadequate as the Equilibrium-theory, we shall now represent pretty well all that was known of the Tides proceed to give that theory, nearly in the terms of its about the year 1800: and it will serve to point out to proposers. the reader the leading facts, to whose explanation a

(15.) The popular explanation of the Equilibrium- Popular Theory of the Tides ought to be directed. In the pre- theory is very simple. If we conceive the earth to be explanasent century, the elaborate discussions of immense col- covered wholly or in a great degree with water, and tion of lections of accurate tide-observations by M. Laplace, consider that the attraction of the moon upon different

equili. Sir John W. Lubbock, and Professor Whewell, have particles (according to the law of gravitation) is in-theory. brought to light and reduced to law many irregulari- versely as the square of their distance, and is therefore ties which were before that time unknown. We prefer, greatest for those particles which are nearest to it; however, delaying the particular mention of these until then it will be obvious that the moon attracts the water


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Tides and

that side which is next to her, more than she attracts 90o distant from those vertically under the sun, is Tiles a Wares.

the great mass of the earth, and therefore tends to raise Trosor of gravity. Then he observes that, in the
the water from the earth on the side next to her; but points which are under the sun and opposite to the
she also attracts the great mass of the earth more than sun, the disturbing force of the sun tends to raise the
she attracts the water upon the side most distant from water, and is twice as great as the depressing force brium
her, and therefore tends to draw the earth from the water already found. He then considers that the same general Theor
on the side most distant from her ; which will produce effect will be produced if we put away the depressing Tices.
exactly the same effect as if a force tended to draw the force entirely, and augment the elevating force by the

Newt water away from the earth on that side. Thus the same quantity, and thus we may consider that the sole moon's action tends to raise the water on two opposite cause of the disturbance of the water is an elevating of thi sides of the earth; and similarly the sun's action tends force, at the point under the sun and the point opposite

vatio to raise the water on two opposite sides. The close to the sun, equal to Ty To'tiro of gravity; the elevatrelation, however, which the times of high water bearing force in other points being proportional to the to the times of the moon's passage, shows that the versed sine of double the sun’s altitude above the equil moon's influence in raising the tides must be much horizon of any point. In order to compute the effect briu greater than the sun's. If the sun and moon are of this force in raising the water, he compares this prod together, as seen from the earth, the elevations pro- force with the centrifugal force (oto of gravity) at the sun duced by these two bodies will coincide in place, and earth's equator, produced by the diurnal rotation of actic will therefore be added together. Thus Spring Tides the earth; it is therefore TTTTO,

of the centrifugal will be produced. In other relative positions of the force at the equator. Then, having found from his sun and moon, it may happen that the elevation pro- theory of the Figure of the Earth (supposed homoduced by the sun will occur at a place where the moon geneous) that the centrifugal force would raise the causes depression: the action of the sun there tends to Huids at the equator 85820 Paris feet, and supposing counteract that of the moon, and Neap Tides will be the proportion of the elevations produced by the tidal produced.

force and the centrifugal force to be the same as the Newton's (16.) The theory of Newton is rather a collection of proportion of those forces, he obtains this result, that first theory hints for a theory than any thing else. In the Princi- the action of the sun would raise the water, in the parts of the mo- pia, lib. I. prop. 66, cor. 19, he has (by a remarkable immediately under and opposite to it, by 1 foot 114 tion of the

deduction from the Lunar Theory) considered the inches Paris measure, or a little more than 2 feet Eng.
motion of water in a canal passing round the earth in lish. Of the various steps of this process we shall here
or near to the earth's equator, and has arrived at the observe only that, though indirect, they are correct;
singular conclusion that the water would be lowest in and that the result (on the supposition of the earth's
that part which is most nearly under the body (the being homogeneous, and without rotation) represents
sun or moon) whose attraction causes the motion of correctly the elevation which the sun's action would
the water. This conclusion we shall find to be entirely produce.
supported by more complete investigations. In lib. (17.) In order to ascertain the effect which the N

III. prop. 24, he has modified this conclusion, and moon's action would produce, it is necessary to know ca His modi- seems to suppose that in free seas the high water ought the mass of the moon. For this there were in Newton's fied theory to follow the moon's transit over the meridian (con- time no direct means : and he was, therefore, obliged

th of motion.

ceiving, for the moment, the moon's attraction to be to refer to the phænomena of the Tides themselves, as for
the sole exciting cause of the Tides) in three hours, or observed in places where, from local causes, the rise of M
at least in less than six hours. To this he appears to the tide is very considerable. He quotes the observa-
have been led by erroneous reasoning of the same kind tions of Sturmy on the tides in the Severn, at the mouth
as that which, in lib. I. prop. 66, cor. 20, has intro- of the Avon, which give 45 feet for equinoctial spring
duced an incorrect inference as to the Solar Nutation tides, 25 feet for equinoctial neap tides : and those
of the Earth's axis. We shall find hereafter that the of Colepresse, on the tides at Plymouth, which give
introduction of friction into our theories of the motion 16 feet for the mean height (intermediate between
of water will lead to a conclusion somewhat similar. spring and neap) and 9 feet difference between springs
The only part in which he uses numerical calculation and neaps. Preferring the proportion deduced from
is in lib. III. prop. 36, and 37, the subjects of which the former, he considers the height of equinoctial
are, Invenire vim Solis ad Mare movendum,]n- spring tides to be to that of equinoctial neap tides as

venire vim Lunæ ad Mare movendum.The following 9:5. These tides (as will be seen hereafter) are in Newton's is his method of computation (the demonstration of the one case the effect of the moon augmented by the calculation different parts of which we defer till we treat of the effect of the sun ; and in the other case the effect of of the force

more complete theory of Bernoulli). First he refers the moon diminished by that of the sun. If no corof the Sun.

to the Lunar Theory for a calculation of the force rection were needed, we should infer at once that the which the sun exerts to draw the moon, when in quad- power of the moon is to that of the sun as 7:2. But ratures, towards the earth, and he finds it to be Newton remarks, that the greatest tides at Bristol do Totory part of gravity at the earth's surface. Then he not happen till 43 hours after syzygies,

bob aquarum remarks that the similar force upon the water at the reciprocos motus," meaning, probably, that the oscillaearth's surface, in the position distant 90° of terrestrial tions, like the oscillations of a pendulum, have a kind of are from the point to which the sun is vertical, is less inertia, which (on purely mechanical principles) prethan the force upon the moon, in the proportion in vents them from attaining their greatest magnitude till which the water's distance from the centre of the the force which causes them has past its greatest earth is less than the moon's distance from the centre magnitude. This we shall find, when we treat of of the earth, or in the proportion of 1 : 60.5 : and there- Waves, to be incorrect, except we take account of fore the force which depresses the water, at the points friction. Assuming this, however, Newton proceeds

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Tales and to correct for the position of the luminaries at the (21.) We have designedly used the word spherical Tides and Hi ares. instant of Bristol high tide: remarking that, as the for the form of the earth, because the investigation of Waves. suo is 18 ! degrees from the moon at spring tides, and the alteration produced in the form which, if undis

Tides the sel. Il. 90°4 1810 at neap tides, it is not the whole force of turbed, would be spheroidal, would prove rather

the sun which in one case increases and in the other troublesome, and would lead to no result which we spherical er of case diminishes the moon's effect, but the whole force shall not obtain without it. As the earth’s ellipticity earth as on

of the sun x cos 37°: and also that, as the moon's is small, (the difference between its major axis and its a spheroid. declination, 43 hours after an equinoctial syzygy, is

1 tartans about 22°, it is not the whole force of the moon that is minor axis being only about of either,) and as the

concerned, but the whole force of the moon x cos' 22°. whole elevation of the water, on the equilibrium

These corrections appear to us inconsistent with what theory, is but a few feet, the reader will have no chvas has gone before: for if the tides are increasing from difficulty in comprehending that the tidal elevation of the accumulated action of the sun and moon during a

the water on the spheroid, though without doubt long time, it seems clearly inaccurate to correct the theoretically different from that on a sphere, will results of observation for the places of those bodies at practically differ by a quantity which is quite insenthe very instant of observation. Then he observessible. In the same manner the reader will understand the tide that the moon is not, at syzygies, at her mean distance. that, supposing the water to be disturbed by the action produced All corrections applied, he finds that the force of the of the sun, and supposing the action of the moon to by each of moon is to that of the sun as 4:4815 to 1: and, there- be then introduced, the additional disturbance which it fore, as the sun's force would raise the water 1 foot, 11• will cause will be (as far as the senses can discover) ries the

ing lumina. inches, the moon's force would raise it 8 feet, 8 inches. the same as it would have caused if it had acted on same as if This, he remarks, is amply sufficient to account for all water not disturbed by the action of the sun.

And the other the motions of the tides.

thus the whole disturbance which the two luminaries (18.) The proportion of the moon's tidal force to the will produce upon the water surrounding a spheroidal sun's tidal force is used by Newton (as a different value nucleus will be found with sufficient accuracy by infound in nearly the same manner has been used by vestigating the disturbance which each of them, sepaLaplace) as the basis on which he calculates the moon's rately considered, would produce in the water surmass for application to other parts of the theory of rounding a spherical nucleus, and by adding those two gravitation. We shall see grounds hereafter for ques- disturbances together. tioning the propriety of this calculation.

(22.) Our first effort will now be directed to the (19.) Assuming that Newton intended here (as he estimation of the disturbing force of the sun upon Ama has done in several parts of Optics) only to exhibit, as

the water. We shall use the following notation :far as he was able, grounds for a numerical calcula

K, the mean density of the earth's spherical nucleus: tion relating to the subject of Tides, but not bearing R, its radius. directly upon any of its specific phænomena, we must

k, the density of the water : r, the radius of the allow that (in spite of the apparent inconsistency of his external spherical surface of the water when undiscorrections) it is a wonderful first attempt. That it turbed by the sun and moon. had no further meaning will be sufficiently evident, not (The density is supposed to be estimated by the only from the proposition already cited, lib. I., prop. acceleration which a cubical unit of matter acting by 66, cor. 19, but also from an examination of his 24th its attraction during a unit of time will produce in a proposition of the third book, and the first corollary of body whose distance is the unit of distance: the his 27th proposition. In these he has treated the gene- velocity and acceleration being referred to the same tal explanation of the Tides as a matter of Wave-theory units.) entirely, (though not without errors,) particularly in E, the whole mass of the earth and water. regard to the interference of semidiurnal tides, and in

9, the numerical expression, referred to the same explaining the small rise and fall at some islands in the units, for the acceleration which gravity at the earth's open sea by the oscillation of the whole mass of water surface causes in bodies falling freely. between the bounding continents. As a philosopher,

x, y, z, the rectangular coordinates of any point in we conceive Newton to have shown himself here the fluid, the centre of the spherical nucleus being the superior to his successors.

origin, and - being parallel to the line joining the (20.) In explaining the more complete equilibrium- centres of the sun and the earth. theory, we shall not confine ourselves to the methods of

D, the sun's distance: Dm, the sun's mean distance: Daniel Bernoulli, or any other writer, but shall present P, the sun's parallax : Pm, the sun's mean parallax : the theory in the form which appears most convenient, T, the periodic time of the earth’s revolution round

The problem which we shall conceive to be presented the sun, or the length of a sidereal year : S, the sun's Real to us for solution is this: suppose the earth to be a

mass, estimated by the acceleration which it will probisa of spherical solid nucleus, either homogeneous, or conequili.

duce (in the same manner as for the density, above). sisting of a series of spherical concentric strata, (each

D', the moon's distance: D'm, the moon's mean disstratum having the same density and the same thick

tance : P', the moon's parallax : P'm, the moon's mean ness in its whole extent,) which nucleus is covered with parallax : 'T', the periodic time of the moon's revoluwater : and suppose the disturbing forces of the sun

tion round the earth: M, the moon's mass. and moon to act upon the water : to find the shape which the water will assume.

forces of (23.) The distance of the sun from the point whose co-ordinates are x, y, z, is {'+y*+(D—2)9}, and the the Sun


upon any attraction of the sun upon that point, according to the law of gravitation, is

in + y +(D-2)"


This force is in the particle of

the water.



Tides and direction of the line drawn from the point in question to the sun. Our expression for this force supposes it to be Tides and Waves.

Waves. estimated as an accelerating force ; the statical pressure which corresponds to it may be resolved into three pressures in the directions of x, y, z; and, by the principle that accelerations of a given particle are proportional to

Sect. II. the pressures which cause them, the accelerating forces which act in these directions may be deduced from the

Equiligiven accelerating force by the same laws of resolution as those for statical pressures. Thus we find for the brium resolved parts of the sun's accelerating force on the particle in question,

Theory o

In the direction of r

{r'+y+ (D-2)

In the direction of


{#*+y' + (D-2)}

In the direction of z

{ x + y +(D-2)}} Espan- (24.) Now the proportion of the earth's radius to the distance of the sun is extremely small; and the value of sions of the expres. X y

is necessarily smaller. It will be allowable, therefore, to expand these expressions approximately, sions. D'D' D'

retaining no higher powers of x, y, z, than the second. (Indeed these latter terms are wholly insensible for the sun; and we retain them only because, in the expressions which we shall infer by analogy for the forces of the moon, they may be considered sensible.) With this restriction, observing that


11 2:++2)-- 1 3: 12:9–31-3y2
1 +

{x?.+y* + (D-}D3 D D?

D: D

2 2D
we have

-Sr 3Srz
Sun's force in the direction of . =

D: D*

-Sy 3Syz
Sun's force in the direction of y

D D*

S 2S: S(67-3r-34)
Sun's force in the direction of 2 = +


2D Disturbing (25.) These expressions represent the whole force of the sun upon any particle. But it is evident that, to forces of

find the force which disturbs the form of the water in reference to the position of the earth, we must not use the

whole force of the sun upon any particle, but the excess of the sun's force on the particle above the sun's force particle. on the centre of gravity of the earth. In order to find the sun's force on the centre of gravity of the earth, we

must multiply each particle of the earth by the force which acts upon it; we must add together all these products, and we must divide the sum by the sum of all the particles of the earth. Now, using the 'expressions above, (which apply to the earth as well as to the water,) we may easily see that, if we multiply each particle

of the earth by the force and add all the products together, the sum will be 0, because for every particle
which has a certain positive value of x there will be another particle having an equal negative value of x, and
their products will, when added, destroy each other. The same remark applies to the terms depending on y, z,

yz. But it does not apply to the term or to that depending on x®, y', and zo.


(26.) Now for the term

we have only to remark that, upon multiplying it by each of the particles, adding

all the products, and dividing the sum by the sum of the particles, we again obtain For the other terms we

may proceed thus :--The sum of all the products of each particle by its value of zo, throughout the sphere, will be
the same as the sum of the products of each particle by its value of x' or y®, because, supposing the sphere at
one time divided by planes perpendicular to 2, and at another time by planes perpendicular to x or y, the sec-
tions for similar values of x, y, or z, will be similar. The sum, therefore, for 6zwill be equal to that for 3.x* +
3y', and, therefore, that for 6z? — 3r? — 3yo will be 0. The only remaining term, therefore, for the sun's force on

the centre of gravity of the earth, is in the direction of z.

(27.) Subtracting this term, therefore, from the force in the same direction upon the particle under considera-
tion, we have the following expressions for the sun's disturbing force,

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+ Ds


Tides and


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(28.) We shall now proceed to investigate the form which the water covering the solid nucleus will receive from The den

the action of these forces in addition to the attraction of the nucleus and the mutual attraction of the particles of sity of the La ty of water. And first we may remark that, if the attraction of the particles of the water is insensible, (or if the den- Huid, sup

posed in. sity of the water is insensible in comparison with that of the nucleus,) the problem is very simple. Referring to

significant. our Treatise on the FIGURE OF The Earth, section 2, article 7., we find that the condition for the possibility of Mathemaequilibrium of the water is that Xdx+Ydy+Zdz shall be a complete differential, or, in more correct language, tical condi

DU dU dU

tion which that it shall be possible for us to find some function U, such that =X,

=Z; X, Y, Z, being the determines dx

dz dy

the form of whole forces in the directions of x, y, z. In article 9. of the same Treatise it is shown that, when the forces are the fluid produced by attraction to any number of particles, this condition is always satisfied; and, therefore, it is satis- when in fied here (which will also be easily seen on substituting the expressions which we shall immediately exhibit). equili

. In article 8. of the same Treatise, it is proved that the form of the external surface will be determined by making Xdr+Ydy + Zdz=0, or U=C. To apply this now, we must add, to the expressions above, the resolved parts of the attraction of the nucleus. That attraction is the same as if all the matter of the nucleus were collected at

47 • R’K
its centre; and it is, therefore,

The resolved parts in the directions of x, y, z, are respectively
3 *** + y + 1
47 ROK..
47 R’Ky

4* R’Kz
3 '(x* + y +z')1'

3°(r* +y +z)!'

3 °(2x + y + z')} Hence the whole forces acting on any particle of the water are


4. R'KI Sr 3Sr:

3 (r + y +=')! Do D

47 Ꭱ°Ky Sy 3Syz
dy 3 (x* +yo+za) DS Do
47 R'Kz 2Sz S(6z — 3.ro - 3y)

3 (r+y+z)l' D'

2D* From these we easily find

47 ᎡᏦ S(2z? — x*—y) S(2-3-3r 2-3y :)

2D4 and the equation to the external surface of the water will therefore be

Equation 47 R'K S(2z'— -—y?) S(2z8 — 3.réz- 3y®z)

face of the 3° (x +yo +28)* 2D8

2D* We may remark, that the very same equation would have been obtained if we had considered only the distarbing force which acts in the direction of a tangent to the Earth's surface. For, the equation which we have zed for the external surface amounts to this, that the whole force is perpendicular to the external surface. Therefore the inclination of the surface of the water to the surface of the sphere will depend entirely on the proportion of the tangential force to the force directed towards the centre of the sphere. The only tangential force is the tangential disturbing force, which must therefore be retained; but the force directed to the centre of the sphere consists of the attraction of the sphere and the minute disturbing force ; and it is indifferent, for the inclination of which we have spoken, whether we retain that minute portion or not. If we retain it, we consider all the forces; if we omit it, we use no disturbing force but that which is tangential. We shall see hereafter that a similar rule is true when we consider the forces producing the motion of the sea.

(29.) Since the difference of the form from a spherical form will be exceedingly small, we may for Expansion (+*+y+z), which is the distance of any point at the surface from the sphere's centre, put r+9 (then q is the supposing elevation of the water above the height which it would have had if undisturbed by the attraction of the sun): differ little and in substituting this expression in the first term on the right hand side of the equation we may neglect the from a square of q; and in substituting it in the factors of the other terms, which are exceedingly small, we may omit sphere q entirely. In these small terms, therefore,

2'--y=2z-(-2)=3z* -p,

2 (22 – 3x – 3y)=x{22-3(7-2)}=2 (5z* - 3r*); and in the larger term

1 1


s+9 Substituting, the equation becomes



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