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on line Piles and difference of their angular velocities, is entirely lost. from the former. Much confusion would be removed Tides and

Waves. crores

. It would be highly important to obtain solutions on by ascertaining these multipliers and applying them at

the supposition, that i differs from nor from 2 n by a once to the observations.
small quantity whose square may be neglected.

(599.) It will probably also be found that the time Correction (589.) Laplace's theory would be much more of high water requires a correction depending on the of time, si puderata valuable if it were extended so far as to include the whole vertical range, to make it comparable with that depending effects of friction. The methods of (315.), &c., would of the sea; and that the time of low water requires a

on range of

tide. probably apply with sufficient accuracy:

different correction. These should be ascertained, if 05

(590.) The additions to the theory, indicated in the possible, and applied. It would, perhaps, be best to it is brides three last articles, would be exceedingly valuable, even assume that such a correction is needed, and to deter

if quite independent. If they could be treated in mine its quantity from the observations themselves in
combination, their value would be very greatly such a manner that the epochs of highest tides and
increased.

mean lunitidal intervals shall synchronize. And in
(591.) In the theory of waves, the most important like manner, for the corrections to the heights, it might Correction

point by far is the theory of river-tides, in which the be best to determine the factors, so that the mean of height. of extent of vertical oscillation bears a sensible proportion height shall be uniform, and that the first proportion of

to the depth ; which we have partially treated in (192.), the Moon's mass to the Sun's, inferred from the semi-
&c. The following extensions would very much in- menstrual inequality of heights, shall be the same as
crease its value; it is apprehended that they would that given by the semimenstrual inequality of times.
introduce more of labour than of difficulty.

(600.) In the places where the diurnal tide, though Methods áte zde (592.) The investigations of (218.) and (260.) must sensible, is small, its effect in height at the time of high for diurnal ka variable be extended so as to include the terms depending on or low water may be considered independent of its tide. es la the vertical oscillation, at least to the third order; ob- effect on the time of the high or low water, and vice

servations showing that those terms are not only sen- rersâ. The best way of disengaging it, numerically,
sible but important. This investigation, if properly would probably be, to calculate a small approximate
conducted, will include the extension of the investiga- table of second differences of the heights, and, subtract-
tion of (309.).

ing from each observation of height the mean of the El triode (593.) The effect of friction must be introduced in preceding and following heights, to apply that comcombination with these investigations,

puted second difference. But where the diurnal tide jurion (594.) The investigation must, if possible, be effected is very large, the effect on height is not independent of lows and for the case where the vertical oscillation is very great; the effect on time. In this case we see no method so

as, for instance, where the low water leaves very small clear and easy as to calculate beforehand a few tables the

depth on the bottom. In this case it is hopeless to of the values of cos 0 + a cos (20+b) with different b very attempt a con verging series, and an independent and values of a and by the result will enable the experi

finite method must be tried. Much would be gained menter to judge how much the real epoch of high semiif this could be effected in the simplest case, as for diurnal tide differs from the time of highest water, and a rectangular channel, of uniform section, without what is the real epoch of diurnal tide. The same will friction.

apply to the times. (595.) In regard to observations, it is very desirable (601.) The whole of the inequalities should then be Methods sobo that simultaneous observations at short intervals should treated with reference to the theory of (451.), and so as for general er as be made on different points of some long tidal river, or to include the terms pointed out in (550.) and (554.).

inequalisome deep estuary, for examination of the change of For these it will not be sufficient to class together all the wave. These observations should be discussed as observations at which the declination was the same;

is mentioned in (479.). It is particularly desirable before this is done, the whole must be divided into two badeo that these observations should be made in very high categories, namely, those of declinations increasing and

spring tides and in very low neap tides, to discover of declinations diminishing, which are to be afterwards the laws of alteration of the various constants as subdivided by absolute declinations. In like manner, depending on the range of the tide.

as regards parallax, all the observations must be divided (596.) As a special locality, we may point out the into the two categories of parallax increasing and Dear various channels near the Isle of Wight as most parti- parallax diminishing; which are to be afterwards sub be of cularly requiring attention. It would be very useful divided by absolute Parallax. The whole of these

that simultaneous observations of a few tides should elements are to be taken for an epoch anterior by a be made at two or more points on Southampton water, quantity equal to the age of the tide. From the discustwo or more on the Solent, one or two on the eastern sion of these inequalities in time as well as in height, side, as at and beyond Portsmouth, and one or two on the mass of the Moon is to be inferred by the process the west side of Hurst Point.

sketched in (555.); and the agreement of the different (597.) In regard to the reduction of long series of values of the mass will be the proof of agreement of tide-observations as applicable to particular ports, we theory and observation. shall only call the reader's attention to the following (602.) The best method of starting in these reducpoints.

tions cannot be the subject of general rule ; the age of (598.) It will probably be found, from the inquiries, the tide however should be determined as early as theoretical and experimental, to which we have alluded possible. When the lunar parallax correction is ascerabove, that the elevation of high water in rivers or tained, that part of it which applies uniformly in the

bays requires a certain multiplier to make its fluctua- same age of the Moon (depending on variation) should s-cb

. tion of range comparable proportionably with the fluc- be subtracted from all the observations, or rather from osao, tuation of range on the coast, and that the depression of the means of the groups, and then only can the semilow water in like manner requires a multiplier different menstrual inequality be found exactly. Each inequality,

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VOL. V.

sion.

Tides and when determined, should be subtracted from the obser- means for rendering the tide even coarsely sensible, in Tides and Waves. vations before investigating a new one.

places where its range is small, where it is partly Wares

. (603.) The same methods should be used for diur- masked by day-breezes and night-breezes, and where nal inequality, where it is conspicuous. But, as north the water is held in a state somewhat different from

Concler declinations and south declinations are not now to be that of the open sea by the rings of coral reef which confounded, it will be advantageous to express the surround so many of the islands.

places of the Sun and Moon by north-polar-distances (605.) In some smaller seas scrupulous attention Referens Points to instead of declinations.

should also be given to the distinction between the to the be noticed (604.) In regard to the tracing of cotidal lines, the actual time of high water and the time when the simple

plest for in the tide- principal defect, in marine localities which otherwise sine expressing the sea-tide reaches its maximum. It function observa

are well known, is in the Pacific Ocean generally. The is not unlikely that in this way some part of an appa- for dran tions in the Pacific

attention of those who are interested in defining these rent anomaly which Mr. Whewell has remarked to the ing cott Ocean. lines should be particularly directed to the devising of east of the Isle of Wight may be removed.

lines.

of tide.

INDEX TO THE ESSAY ON TIDES AND WAVES.

INTRODUCTION.

Article Actual forces of the Sun upon any particle of water ......

23 Section 1.ORDINARY PHENOMENA OF Tides. Disturbing forces of the Sun upon every particle.....

25 Article Condition of equilibrium when the density of the fluid is PHÆNomena of river tides ......

insignificant..

28
Semidiurnal tide...
Its time is related to the apparent position of the moon

Expansion of equations, supposing the elevation of water
2
small....

29 The interval between high water and moon's transit is

Expression for the elevation, and calculation for the Sun's
variable

3
effect...

..........30,31 Spring and neap tides

4

Elevation apparently, but not really double the depression 34 The duration of the fall is longer than the duration of the

Elevation produced by the Moon calculated..

35 rise

5

Calculation of forces supposing the density of water sen-
The water continues to run up the river after high water . 5

sible
High water occurs later for places higher up the river ....

....37, 39 6

Spheroidal form found to be possible, and ellipticity deter-
The progress of the tide is too rapid to be explained by the

mined...

40 transmission of the same body of water.....

Combination of solar aud lunar tides.
The duration of fall increases and that of rise diminishes in Tides of long period ..

45 ascending the river....

6
Diurnal tides.

16 The rapid rise sometimes produces a bore, sometimes

Semidiurnal tides

49 double or treble tides...

6 In contracted estuaries the tide is high. In ascending

Spring and neap tides

.51,52 Approximate expression for elevation of water

53 rivers the tide diminishes 7 Interval between moon's transit and high water

54 Bay-tides are simpler....

8
Channel-tides nearly similar to those of rivers, near the

General comparison of equilibrium-theory with observa-
tions ....

61 mouth.....

9 Rotation of the direction of the tide-currents

Its inadequacy, and its historical utility

64 9 Tides small in open seas and in small seas...

. 10, 11 Anomalous tides in certain localities....

12

Section III.-LAPLACE's THEORY OF Tides.
Spring and neap tides known to the ancients.

13
Suppositions limiting this theory.

65 A small vertical motion of the water implies large horizonSection II.-EQUILIBRIUM-THEORY OF TIDES.

tal motion; and all vertical forces except gravity may
be omitted ..

68
Inadequacy of all theories of the tides, and the cause of it. 14 Particles in a vertical line may be supposed to remain
Popular explanation of equilibrium-theory...
15 vertical.....

70 Newton's first theory of motion of the sea, and his modified Investigation of the equation of continuity..

72 theory...

16

Rectangular equations of motion of fluids.
Newton's calculation of the force of the Sun, and of the Changed to polar equations.....
elevation of water produced by it.....
16 Reduced by omitting several terms

78
Newton's calculation of the proportion of the Sun's force The forces divided into two parts: the pressures divided
to the Moon's...

17
into three parts .......

79 Newton's corrections apparently erroneous

17
Forces depending on the earth's attraction and on rotation

80
Merit of Newton's theory
19 Disturbing forces of the sun or moon

82 General problem of the equilibrium-theory

20 Equations for the pressure depending on the motion of the Tides the same on a spherical earth as on a spheroid... 21

water..

84 The tide produced by each of the attracting luminaries is Popular explanation of the terms depending on the earth's the same as if the other did not exist. 21 rotation

88

Article Distinction between forced wave and free wave.

278 Effect of the vertical force is insignificant ...

279 The sign and magnitude of the motion depend on the

relation of the period of the forced wave to that of a
free wave ......

280 Coefficient of force supposed variable

286

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Article Assumption of the form of solution, as regards longitude 89 Inferences from the assumption ....

91 Laplace's method of solution, as depending on the latitude 94 The equations reduced to one differential equation......

95 Equation corresponding to slow changes of force solved... 97 Equation corresponding to diurnal tide

100 No diurnal tide in height when the depth is uniform.. 102 Equation corresponding to semidiurnal tide......

107 Error in Laplace's process in solving this equation.. 111 Criticism on Laplace's process throughout.

116 Merit of Laplace's theory

117 Laplace's final assumption

121 The equilibrium of the sea is stable....

123 Precession is not affected by an oscillatory motion of the

125

Subsection 5.— Introduction of free Tide-Waves, forced Tide

Waves, and limits of Canals. Solution made more general by adding expressions for free waves .........

291 Force begins to act at a certain instant

294 Canal bounded at both ends

*96 Canal of small extent ......

300 Tides in a deep gulf....

303 Tides in a gulf, the elevation being supposed large

309 Canal between two tidal seas .

311

sea.....

SECTION IV.-THEORY OP WAVES IN CANALS.

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Subsection 7.— Theory of Waves in open Scas. Equations for three dimensions Solution expressing annular wares Solution expressing parallel waves Reflexion of waves from a straight boundary Form of wave in broad channels with shallow sides Revolving motion of the tide near the shore Cotidal lines in open seas do not represent the waves

347 351 353 355 359 363 367

Subsection 3.-Theory of Long Waves when the Vertical Oscilla

tion is large. Equations of continuity and pressure

194 Solution to first approximation

197 Solution to second approximation

198 Peculiarity in the form of the wave..

203 Rise occupies less time than fall..

206 Velocity of ebb-stream greater than that of flow-stream.. 209 Solution to third approximation

210 Investigation when there is a current-flow..

212 Investigation supposing the canal not rectangular

218 Limitations to the application of this investigation

222 Addition of solutions does not apply here..

223

Section V.-ACCOUNT OF EXPERIMENTS ON Waves. Weber's Wellenrinne, and methods of using it

374 Observed motion of individual particles

377 Russell's apparatus, and methods of using it

388 Abstract of Russell's observed velocities of waves, compared with theory

394 Experiments in triangular and trapezoidal channels .. 399 Experiments in channels of variable breadth and of variable depth

..400, 402 Peculiarities observed in canal-navigation

404 Theoretical explanation of them

.406, 411 Observations on sea-waves

414

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Subsection 4.-Theory of Waves under the Action of Forces. Investigation for solitary wave; theory of discontinuous functions

226 Forms of function which may be allowed here..

232 When the wave is very long, no maintaining force is neces

235 sary.

238 Depth of canal supposed variable : assumption of solution. Expression for force necessary to maintain the assumed

240 ware. Coefficients of horizontal and vertical motions. .243-247

248 Explanation of the breaking of waves ...

251 Investigation when the breadth is not uniform Cessation of flow occurs earlier than in uniform channel.. 256

260 Simpler investigation when the wave is long . Forces necessary to maintain waves increasing from wave

266

267 The force of the wind corresponds closely to this Forces necessary when the waves increase from time to time ...

270

271 The foree of the wind nearly similar to this Motion of water under the action of forces similar to tidal

274 forces ...

277 Motions of particles at the surface

Section V1.--EXPRESSIONS FOR Solar And LUNAR TIDES, con

SIDERED AS WAVES IN CANALS. Forces arising from the earth's rotation are not required .. 419 Expression for the disturbing force of the Sun or Moon.. 426 Expression for the force, as depending on the time..... 428 Canal supposed to be a small circle and disturbing body in the equator

4.29 Canal supposed to be a great circle, and the disturbing body to have any declination

435 Disturbance of mean level computed

437 Diurnal tide

439 Semidiurnal tide

441 Relation of the sign of the expressions to the depth of the

444 Effect of inequalities in the distance and angular velocity of the disturbing body

446 Further modifications produced by friction

452 Simultaneous action of two bodies..

454 Mass of the Moon, as first inferred, will probably be too

great; the error depending on the depth of the canal. 455 In consequence of friction, the tide will depend on an anterior position of the bodies..

459

sea.

to wave

575

Article Subsection 3.-On the Laws of the Tides in varying positions of The relation of the time of a river-tide to the time of a sea

the Sun and Moon. tide not always the same ..... 463

Article
Value of the theory of waves as applied to tides

466 Dependence of mean level on the Moon's declination 533
Semimenstrual inequalities of semidiurnal tide

535 SECTION VII.-METHODS USED FOR MAKING AND REDUCING

Proportion of effects of Sun and Moon.

.537—540
TIDE-OBSERVATIONS.
Retard of the tide and age of the tide

.541-546
General law of the semimenstrual inequalities..

547 Imperfection of ordinary observations 468 Corrections for lunar declination ...

550 Objections to Mr. Whewell's proposed method 469 Term omitted by all investigators of observations

551 Cautions for accurate observations.... 471 Laplace's deductions of the mass of the Moon.

535 Self-registering tide-gauge.... 473 Corrections for lunar parallax . .

556 Methods of reducing the observations of a single tide.... 479 Solar corrections

560 Laplace's methods of reducing observations for different Diurnal tide

561 positions of the Sun and Moon 480 Instances of large diurnal tides..

569 For the age of the tide...... 490 Effect of barometrical pressure on the tide.

572 For the effect of the Moon's motion in right ascension.... 481 Effect of wind on the tide...

573
For changes of declination and parallax, diurnal tide, &c.. 482
For proportion of Sun's effect to Moon's effect .484, 486
Lubbock's methods

489 Subsection 4.-On the Progress of the Tide over the Ocean.
Whewell's methods of treating Lubbock's results alge-
braically ...

492 The cotidal lines of the Northern Atlantic are probably Use of graphical constructions by Lubbock and Whewell. 493

accurate ..

575 Methods of tracing cotidal lines ...

498

Those of other seas are very doubtful Confusion produced by the ignorance of mariners. 502 Rapid changes of depth, &c., may be treated as for free

576

Where cotidal lines are crowded, the tides are large. 576 Section VIII.-COMPARISON OF THEORY WITH OBSERVATIONS. Distortion of the cotidal lines by the shoals round islands. 577

Inference of depth from velocity of tide

578 Subsection 1.- On the individual Tides in Rivers and Estuaries. These remarks do not apply to large seas ..

579

579 Instances of the change in the magnitude of the tide

The Atlantic tides cannot be treated as derivative........

505 Intervals from high water to slack water

507
Failure of theory as regards the transmission of the tide-

580
Elevation of the mean level in ascending a current river .. 508
Change in the form of the wave

509

The cotidal lines may not represent the ridges of the tide-
waves.

581 Double and triple tides

511 The Bore

514
The interference of wave-ridges may explain the cotidal

392 Mathematical expression for tide at Deptford

515

lines west of America, .. Several tides in succession on the Amazons

516
Laws which, notwithstanding the failure of several parts of

585 Expression for tide at Southampton....

517

the theory, apply in every place Anomalous tides on the south coast of England.

519 Subsection 2.- On the individual Tides in small Seas.

CONCLUSION.-ON THE PRESENT DESIDERATA IN REGARD TO TIDES. In the Mediterranean and Adriatic 521 Extensions of Laplace's theory.

587 In the English Channel...

522
Extensions of the theory of river-tides...

592
In the Irish Channel .....
524 Simultaneous observations of river and bay-tides...

595 In the German Sea....

.525, 527
Observations near the Isle of Wight .....

596 Captain Hewett's observations in the German Sea.. 528

Methods for reducing long series of tide-observations..... 595 Races

529

Observations of tides in the Pacific ocean, for cotidal lines. 604 Rapid currents

530

Improvement of cotidal lines by attention to the laws of Correspondence of mean levels....

531 individual tides .....

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CHAPTER I.

establish the fit proportions which the several parts of Part I.

an edifice should bear to each other, in order to fulfil The Origin of Architecture.

the above conditions, and to render the whole capable

of producing an effect which might be pleasing to the les Beractice If we pass over the time when there existed no imple- eye. Now the Art of Architecture is a collection of

illing ment capable of excavating a rock, or felling a tree, and rules for constructing buildings conformably to these
began with when the human savage, who drew his subsistence from principles.

the Earth or waters, retired for the night to the shelter As this Art must have had its origin in some very The proto.
of some natural cave or hollow tree, we shall come to simple and rude construction, and as its early progress type of Art
a period when the practice of building began in the must have been made by insensible steps, neither the
World, and this was, probably, as early as the formation primitive form of a building nor the successive improve-
of the first Societies. As soon as a number of indivi- ments it underwent were likely to excite such notice at
duals or families had united themselves together for the the time they were made, as to cause the memory of
purposes of defence, or of more effectually obtaining them to be preserved. It is therefore hardly to be ex-
the means of support

, habitations, larger and more pected that any Tradition, still less that any Historical
commodious than those afforded by Nature, would be document should remain, concerning either the one or
required.

the other, of a date anterior to the establishment of a The means resorted to by different Tribes of people to considerable number of general principles of construcprocure the necessary protection from the inclemency tion. Afterwards, these becoming objects of study for of the weather, may be reasonably supposed to vary persons who were called upon to exercise Architecture according to the mode by which each people obtained as a profession, would necessarily lead to an inquiry its subsistence, to the materials for building which hap- whether they were founded upon any original inodel, pened to be found in the places occupied, and, perhaps, and what that model might have been. to some peculiarities of character with which each people Vitruvius, who wrote on the Art, in the time of Au- Probable might be endowed.

gustus, or of one of his immediate successors, has, in form of the A pastoral nation, of which each family must change the Ist Chapter of his IId Book, indulged his own este its place of abode as often as its flocks consume the fancy, or recorded that of some more ancient Philoso

productions of the Earth about the station it occupies, pher, respecting the origin of the Arts and Sciences, and will have little inducement to erect permanent buildings; particularly that of the Art of building. Having given and, in a region nearly level, tents, or at most, light a fabulous account of the origin of fire, he goes on to timber-buts, which can either be removed at pleasure, state that, by the comfort men derived from it, they or abandoned without loss, will, probably, be the only were induced to form themselves into Societies, and dwellings it finds necessary to construct.

erect buildings to shelter themselves from the severity Nearly the same thing may be said of a people inha- of the climate. Such buildings, he thinks, would rebiting a mountainous country, or the sea-coast, and seinble the huts constructed by men who occupy the seeking its support from hunting or fishing ; for the lowest grade in Society; and, to give an idea of that families or communities which pursue those modes of primitive style, he describes the mode of building pracliving, being thinly scattered over an extensive terri- tised by the Colchians, a people who probably were tory, will, in most cases, find it convenient to fix their then in the same rude state as are now the inhabitants

abodes in caves formed by Nature or Art in the rocks. of the Islands in the South Seas. wa cip. But when men applied themselves to the cultivation He says, it consisted in fixing trees upright in the homes of the ground, in a region where an ungrateful soil ground, side by side, so as to include the space to Siste compelled them to the constant performance of a re- be inhabited; the distance between the rows of trees

gular succession of labours in order to acquire the equalling the length of the trees intended for the cobe an Art, means of subsistence, being necessarily attached to one vering. The roof was laid over the tops of the upright

spot, perhaps for life, it is conceivable that their dwell- trees, and above these, other upright trees were placed,
ings would be of a permanent nature ; and, consequently, in a manner siunilar to those below. Thus the building
it is among such a people that, through successive im- was raised to the height required. A roof was formed
provements, the practice of building may be expected to at the top, he says, by raising beams across from the
rise to the dignity of an Art. This would take place when, four angles, so as to unite in a point; the sides and
in process of time, men had learned to give to their build- roof were filled up with boughs, and the interstices
ings such internal arrangements as were suited to the were stopped by chips and clay.
purpose for which they were intended; to make the It is, however, by no means necessary to suppose
exterior forms characteristic of those purposes; and to that the square or rectangular form which Vitruvius

237

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YOL. V.

21

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