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Density.

Method of

Figure of Paper will be found well worth the attention of any It was found at first, that a very small difference of Determina the Earth, practical person. The attraction of 960 prisms was temperature between the balls and the air produced tion of the calculated at each observatory; the attraction at the currents of air within the mahogany case which ma

Earth's North observatory was found to be to that at the South terially disturbed the experiment. The cautions subobservatory nearly in the proportion of 7:9. And the sequently used seem to have prevented any bad effect sum of the attractions was found (supposing the den- from this cause.

Disturbanc sity of the mountain equal to the mean density of the The method of observing was the following. The produced 1

large balls being in the midway position (their support- by curreni Earth) to be

of the Earth's attraction. But the ing rod at right angles to the deal rod) the position of "f air.

9933
sum of the disturbances in the direction of gravity as

the deal rod was read off from the scales n. The

observatio determined from the astronomical observations being large balls were then brought so as nearly to touch the = 11".6, the sum of the attractions was, in fact, = gra

case, sometimes in the positive position (in which their 1

attraction inade the rod move so as to increase the num

1 vity x tan 11".6 = x gravity =

X

ber on the scales) and sometimes in the negative posi17781

17804

tion. By their attraction on the small balls, the rod Earth's attraction, (allowing for centrifugal force.) was immediately put in motion, and vibrated backConsequently the density of the mountain was to the wards and forwards. The greatest extent of vibration was Earth's mean density as 9933 to 17804, or nearly as observed; the mean of two consecutive extreme points 5:9.

on one side was taken, and the intermediate extreme Mineralogi- But this supposed that the mountain was homo- point on the other side ; and the point midway between cal structure of the

geneous. The geological characters of its rocks were these was considered to be the point at which the rod mountain.

examined by Professor Playfair, (Phil. Trans. 1811,) would rest under the action of the balls and the torsion
he found that the upper part consisted of quartz whose of the wire. The time of passing the middle point was
mean Specific Gravity = 2.6398; and the lower part also ascertained, by observing the time of passing two
of mica slate and hornblend slate, whose mean Specific points near the middle, and then (when the middle
Gravity = 2.8326, and of limestone whose mean Spe- point was determined) calculating the time of passing
cific Gravity

= 2.7661. Two separate calculations it. This being done for vibrations separated by a con-
were made ; one on the supposition that the rocks siderable number, the time of vibration was accurately
were separated by vertical surfaces, and another sup- found. With the wire finally used, the change from
posing them separated by surfaces nearly horizontal. the position of rest without the action of the lead balls
The former gave for the Earth's mean density 4.559, to the middle position when they were applied, seldom
the latter 4.867, that of water being ). Dr. Hutton exceeded 3 divisions, (each division zgth of an inch,) and
(Phil. Trans. 1821) says that the number should be the time of vibration was about 7 minutes. With the
rather greater.

first wire the change was about 14 divisions, and the Cavendish's The next experiment was that of Mr. Cavendish on time nearly 15 minutes. apparatus the attraction of leaden balls. (Phil. Trans. 1798.) His The middle point is evidently the place where the for observe ing the

apparatus is represented in fig. 51, 1, x, are balls of lead attraction of the large balls is equal to the force of torattraction of about 2 inches in diameter, suspended to the ends of a sion of the wire. The time of vibration also depends leaden ba!!s light deal rod h, h. This is suspended by the wire lg, on both of these forces. For suppose that at the mid

forming, in fact, a balance of torsion. The piece to which dle point the distance of the small balls from the large
the top of the wire is attached, carries a wheel which is ones was A, and the space through which they had been
turned by the endless screw K F, so that the wire can be moved (to which the force of torsion is proportional) B;
twisted till the resting place of the balls is any required po- then putting Wand T to represent these forces respectively
sition. nn are small graduated scales carried by the rod.

W

=TB. The whole of this apparatus is enclosed in a mahogany at the distance 1, we shall have this equation

A?

. box, F E AB CDD C B A E F. At A and A are small And at the distance x beyond this, and further from the glass windows, and near these are scales serving as a

balls, the whole force tending to bring the balls to the vernier for measuring the motion of the scales n.

W

W

Method These are illuminated by lamps L and L, and viewed by middle point is T (B + o)

= TB telescopes T and T from the outside of the room. The

(A - x) leaden balls W, W (each weighing 2,439,000 grains)

2 W

2 W

x nearly, = (T are suspended by copper rods attached to the piece ror

As which is suspended to a beam. By means of a rope is the force on which the time of vibration depends. passing round the pulley M M, the balls W, W can be Thus there are, in fact, two equations to be solved, moved without entering the room. The support from which the attraction of the balls and the torsion of of the balance of torsion and its cases is independent of the wire could be determined. Besides this, the attracthe walls.

tion of the mahogany case was calculated. The attracThe first wire by which the deal rod was supported tion of the leaden balls being thus determined, and was of copper silvered, of which one foot weighed 2.4 compared with the attraction of the Earth, the proporgrains.

After a few experiments with this, it was tion between the Earth's mean density and the density found that the attraction of the large balls made the of lead was found; and thus the Earth's mean density rod touch the sides of the case; and a stiffer wire was is obtained. The result of 29 experiments (as corrected then used.

by Dr. Hutton, Phil. Trans. 1821) is 5.31, that of

water being 1. The smallest number given by one &c. were made by Mr. Cavendish. This we have ascertained from an inspection of his papers, which we have had an opportunity of experiment is 4.86, and the largest 5.79. examining through the kind permission of his Grace the Duke of We are upon the whole inclined to prefer this result Devonshire.

to that of the observations on Schehallien. It cannot

2

A* calculai

+(T

X; and this

erroneous

300'

fize

of be denied that the greatest delicacy was necessary to As it stands, there is one considerable source of error, * Conclusion. te karth, obtain any result, and that the determination is much pamely, the erroneous reduction for the effect of the air

less certain than the determination (such as it is) by (mentioned in Section 7.) The barometer at Mont The result Rezit prom the Schehallien experiment. But we consider that the Cenis being several inches lower than at Bordeaux, this spre

and doubt i quantity determined in the latter may be something error would be serious. Besides, we could hardly trust ful if corturmer. very different from the attraction of the mountain. We to a comparison between two places at so great a dis. rected.

have seen that at Arbury Hill, and some other places, tance. On the whole, we do not think that any estima-
there is evidence of disturbance to an amount nearly as tion of the Earth's density can be founded on this experi-
great as the attraction of Schehallien, and without any ment.
known cause. May not such unknown causes have
operated as well in the Schehallien experiment, and have

Section 12.-Conclusion.
increased or diminished the apparent effect of the
mountain ?

1. The measures of the Earth, the observations of Considering the Earth's mean density as somewhat pendulums, and the lunar inequalities, agree in showing greater than 5, and the mean density of the rocks at the that the Earth's form does not differ much from that of surface as 2.6, the proportion of the Earth's mean

an ellipsoid of revolution whose ellipticity is (we think density to the superficial density is not very different from certainly) greater than

1

and whose major semiaxis that of 2:1.

We have mentioned in our first Section the attempt is about 20,923,700 English feet.
made by the Baron de Zach to measure the attraction of 2. The phenomena of precession and nutation give
a mountain near Marseilles. We have only to add that an ellipticity rather smaller ; but as no result can be
no calculation of the Earth's density was founded on these deduced from them except on an assumed law of den-
observations.

sity, this value cannot be put in opposition to the others.
The attraction of a mountain might be found by ob- 3. As the results of the pendulum observations, the
serving the length of the seconds' pendulum on the lunar inequalities, and the precessional phenomena, can
top; if gravity should thus be found to be greater than only be used to determine the Earth's form by the inter-
gravity at the level of the sea in the same latitude dimi- mediation of the principle of gravitation, the very near
nished in the duplicate proportion of the distance from coincidence of the results is a strong argument in favour
the Earth's centre, the excess would be attributable to of the truth of that principle.
the attraction of the mountain. Bouguer (as we have 4. The same things make it highly probable that the
mentioned in Section 7.) observed the pendulum on one

Earth has once been in a fluid or semi-fluid state. of the peaks of the Andes; but the circumstances were 5. None of these results can be obtained without the unfavourable, and we should have no confidence in the admission of considerable anomalies, all of which, lowresults. In the present century (see the Additions to ever, appear to be consistent with the principle of gravithe Milan Ephemeris) M. Carlini made similar obser- tation. vations in much more favourable circumstances at the 6. The mean density of the Earth considerably ex

hospice of Mont Cenis, at an elevation of 6375 feet ceeds, and is probably double of the density of the superde: Ceais, above the level of the sea. The pendulum we have ficial rocks.

described at the beginning of Section 7 : twenty experi- 7. The near agreement of the proportion between these
ments were made with it; they were reduced like the as deduced from an assumed law with the proportion
French measures, and corrected for elevation by the found by the experiments with leaden balls (where it is
rule of inverse squares; the length at the level of the assumed in the calculation that the law of gravitation
sea thus found was in mètres 0,993708. But the obser- holds good at the distance of a few inches) makes it pro-
vations of Biot at Bordeaux, nearly at the level of the bable that the law is sensibly true to very small distances.
sea, corrected for the small difference of latitude, gave
0.993498. The difference is due to the attraction of

G. B. AIRY. the mountain mass. Representing this by a segment

Observatory, Cambridge,

August 17, 1830. of a sphere, 1 geographic mile in height and 11 in diameter at the base, of Specific Gravity 2.66, the mean density of the Earth is calculated to be 4.39. Perhaps this experiment would have been more satisfactory if dulum-balls being spherical,) the length at Bordeaux is 0.993553, and

* If we correct this, by doubling the usual reduction, (the pen. the pendulum had been made exactly like the French

that at Mont Cenis, reduced for elevation, 0.993754; whence the pendulums, or if an invariable pendulum had been used. mean density of the Earth = 4.59.

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POSTSCRIPT.

Figure of The author has discovered a small error in the Table supposed. The multiplier 0.66 is adopted for the postseris the Earth. of the observed lengths of the seconds' pendulum. The English observations from the calculations of the re

reduction applied to the length of the Paris pendulum spective observers; the effects of this inconsistency are
for elevation above the sea is that due to the decimal not sensible.
pendulum. In consequence, the length at Paris, and all In the determination of the length of the seconds'
the lengths depending on it, ought to be increased by pendulum at Königsberg, a correction is included which
.00013; a quantity which does not sensibly affect any is not applied to any of the other observations. The
of the results. This correction applies to Nos. 18, 25, reason is that the effect of this correction is to increase
31, 33, 40, 44, 47, 49; and half of it to Nos. 42, 45, 48. the length of the seconds' pendulum ; and it appears

The multiplier 0.6 has been used in reducing the that the observations on which Bessel principally relied, foreign observations) in preference to 0.66, as it seems give a smaller length than those made in the usual way. probable that the density at the Earth's surface is The difference depends probably on the difference of the greater, and the mean density less, than Dr. Young apparatus employed.

1

TIDES AND WAVES.

INTRODUCTION.

We propose, in this article, to enter at some length into VII. We shall advert to the methods which have Tides and

Waves. To the mathematical theories, and the experimental obser- been used, or which may advantageously be used, for

rations, applying to the two subjects of Tides and Observation of Tides, and for the Reduction of the Wares of water. But we do not intend to treat them Observations. with the same extension. We shall give the various VIII. We shall give the results of extensive obsertheories of Tides in detail sufficient to enable the reader vations of the Tides, as well with regard to the change to understand the present state of the science which of the phænomena of tides at different times in the same regards them ; and we shall advert to the principal place, as with respect to the relation which the time observations which throw light either on the ordinary and height of tide at one place bear to the time and phænomena of tides, or on the extraordinary devia- height at other places, and shall compare these with the tions that occur in peculiar circumstances. In thus results of the preceding theories, as far as possible. treating the Tides, it will be necessary for us to enter And as Conclusion, we shall point out what we conlargely into the theory of Waves. We shall take ad- sider to be the present Desiderata in the Theory and rantage of this circumstance for the introduction of Observations of Tides. sereral propositions, not applying to the theory of Tides, but elucidating some of the ordinary observabons upon small Waves. But these investigations will Section I.-ORDINARY PHÆNOMENA OP Tides. be limited to that class which is most closely connected with tides, namely, that in which similar waves follow (1.) If we suppose an observer stationed on the Phænoeach other in a continuous series, or in which the same bank of a river, * at no great distance from the sea, mena of mathematical process may be used as when similar (for instance, on the bank of the Thames any where River

Tides. waves follow each other. In this class will be included below London Bridge, on the Humber below the nearly all the phænomena of waves produced by mouth of the Trent, or on the Severn below the Pasnatural causes, and therefore possessing general interest. sages,) he will remark the following changes in the But it will not include the waves of discontinuous state of the water. nature produced by the sudden action of arbitrary (2.) The first and most important change is, that the Semidicauses, which have been the subject of several remark- surface of the water rises and falls regularly twice in urnal able mathematical memoirs, but which possess no

Tide. every day. A short series of observations will show interest for the general reader.

however that this statement is not quite correct; the Pet The general plan of this Essay will be as follows:- tides of each succeeding day are somewhat later than Vice I. We shall describe cursorily the ordinary phæno- those of the preceding day : the average retardation mena of Tides.

from day to day being about 40 minutes. In a short Its time is II. We shall explain the Equilibrium-Theory of Tides, time he will find that the times of occurrence of high related to insluding the first tidal theory given by Newton, and water bear a very close relation to the time of the the appathe more detailed theory of his successors, especially Moon's appearance in certain positions; and that the rent posi; Daniel Bernoulli.

language of the persons who are most accustomed to III. We shall give a sketch of Laplace's investiga- observe the tides conveys at once this relation. Thus, tions, (founded essentially on the theory of the motion at Ipswich, high water occurs when the moon is south of water,) in the general form in which he first attempted nearly': at London Bridge high water occurs when the the theory, as well as with the arbitrary limitations moon is nearly south-west : at Bristol, it takes place which he found it necessary to use for practical appli- when the moon is E.S.E. These are rude statements, cation.

but they are sufficiently accurate for many purposes ; IV. We shall give an extended Theory of Waves on and they show at once the close connection between water, applying principally to the motion of water in the time of high water and the time of the moon's canals of small breadth, but with some indications of passage over the meridian. In fact, so completely is

process to be followed for the investigation of the this recognized, that, in order to give the time of high motion of Waves in extended surfaces of water.

water upon any day, it is usually thought sufficient to V. The results of a few Experiments on Waves will be given, in comparison with the preceding theory.

VI. We shall investigate the mathematical expres- * We commence with this case, because, judging from the sions for the Disturbing Forces of the Sun and Moon

notions of sea-faring persons upon many points connected with

the Tides, which are correct as regards rivers, but incorrect as which produce the Tides, and shall use them in com

regards the sea, (some of which will hereafter be indicated,) it bination with the theory of Waves to predict some of is the case from which ideas of tidal movements have usually the laws of Tides.

moon.

the

been taken with the greatest facility. 241*

2 **

VOL. V.

Ware

mendo

water and

curs

for p

toor to b.

Neap

Tides and state the time of high water on the days of new moon

who is not convinced of the absurdity of supposing Tides as Waves.

and full moon, when the moon passes the meridian at the water in the middle of the channel to stand at one twelve o'clock nearly. This time is called the Esta- time considerably higher and at another time consiblishment of the port. Then to find (roughly) the derably lower than at the shore, will satisfy himself Sect, i time of high water on any other day, it is only neces- most easily as to the general fact by stationing himself Prdinar sary to add the Establishment to the time of the moon's at one of the central piers of a bridge, (as London passage.

Bridge,) when he will see that the water continues to Tides, The rules, as we have mentioned them, indicate the run upwards even after its surface has dropped nearly time of only one high water in the day : but the reader two feet. must understand that there will always be another high (6.) Now suppose that the observer examines the water in the same day, preceding or following that which state of the tide in different parts of the same river. we have mentioned by 12 hours 20 minutes nearly. Commencing with the mouth of the river, (for instance On those days, however, in which high water occurs Margate or Sheerness on the Thames, or Swansea or within 20 minutes of noon, there is no other high Cardiff on the Severn,) he will find that there is very water on the same civil day.

little difference, or perhaps none which is appreciable, The inter- (3.) On closer examination it will be found that the between the interval from high water to low water, and val be- interval between the time of the moon's passage over that from low water to high water. He will also find tween high the meridian and the time of high water varies sensibly that the current runs up the channel for a long time moon's with the moon's age. At new moon, full moon, first (sometimes approaching to three hours) after high water, High transit is quarter, and third quarter, (or rather on the day fol- and runs down the channel for as long a time after low wate variable.

lowing each of these phases,) the interval between the water. In going up the river, he will find that the time time of the moon's passage and the time of high water of high water occurs later and later, but yet that the

high is nearly the same : but from new moon to first quar- velocity with which high water travels up the river is so the i ter, and from full moon to third quarter, the high great as entirely to banish the idea of explaining the Tide water occurs earlier than would be inferred by using by supposing the same mass of water to have been moved that same interval; and from first quarter to full all the way up the river. For instance, if at Margate the The moon, and from third quarter to new moon, it occurs high water occurs on a certain day at twelve o'clock, it grest later than the same interval would give it.

will occur at Sheerness at 24 minutes past one, at Graves. tide Spring and (4.) If the observer examines the height of the end at 15 minutes past two, and at London Bridge at a water, he will find that the height at high water and few minutes before three ; having thus described in less

plai Tides.

the depression at low water are not always the same. than three hours a course of about 70 miles. He will the On the days following new moon and full moon, high also find that the interval from low water to high mis water is higher and low water lower than at any other water diminishes as he goes up the river :- thus, on the the time: these are called Spring Tides. On the days fol- lower parts of the Severn, the rise and fall occupy little lowing the first and third quarters, high water is more than six hours each ; but at Newnham on the

Th lower and low water higher than at any other time: Severn the whole rise of the water is effected in an tio these are called Neap Tides. The whole variation of hour and a half, the descent occupying nearly eleven inc height at spring tides is nearly double that at neap hours. In cases like the last-mentioned, the first rise ani tides. There are other variations of height depending of the tide is sudden, and if the banks of the river are on other circumstances; but they require, for the most shoaly, the water spreads over the fat sands with a asc part, very numerous observations to establish the fact roaring surf, which travels rapidly up the river, pre- the of their existence, and to give a measure of their senting the phænomenon called a bore or boar, (some- Tb amount. In many places, however, the tide which times bours-head,) in French barre or mascaret. occurs at one certain part of the day (the afternoon for other cases, however, when the difference of durations tin instance) is, during one half of the year, sensibly of rise and fall is considerable, there are in each high higher than the other tide which occurs upon the same water two, or sometimes three distinct rises and falls day, and, during the other half of the year, sensibly of the water. The phænomena of bore and double ble lower.

tide are always much more conspicuous in spring tre The dura- (5.) Upon examining the circumstances of a single tides than in neap tides. tion of the tide, the following facts will attract notice. The interval (7.) If the estuary or mouth of the river contracts In fall is

from high water to low water is greater than that from very much, the elevation and depression of the water will tra longer than low water to high water : the difference between these become very great. Thus at the entrance of the Bristol the duration of the intervals is sensibly greater at spring tides than at Channel the whole rise at spring tides is about 18 feet, lug rise. neap tides. The current in the river runs upwards for at Swansea about 30 feet, and at Chepstow about 50 The water some time after high water, and after changing its di- feet. Similar high tides occur at St. Malo and other continues rection, continues to run downwards for some time parts of the great bay formed on the northern coast of to run up after low water, when it again changes its direction France by the projection of land towards Cherbourg, the river

and runs upwards. This phænomenon is often so and tides still higher in the head of the Bay of Fundy after high

much misrepresented in the language of nautical men, (Baie Française) on the Eastern coast of North America. In
that the mistake deserves particular notice. From the But when the tide has fairly entered a river, its range inz
habit of observing tides in places where the current of elevation and depression generally diminishes.
ceases at high water and at low water, sailors conceive at Newnham, on the Severn, the range is reduced to
that high water may always be inferred from the ces- about 18 feet, and it is still less at Gloucester.
sation of the current; and therefore it is not unusual (8.) Quitting now the phænomena of river-tides ; if Ba
for persons on the banks of the Thames to say that observations are made in a bay communicating with are
“ it is high water in the centre of the channel long the open sea, the results will be found to be much more
after it is high water at the shore.” The observer simple. The water will rise during 6 hours 10 minutes

wat

nis

In ris

du bo tin

tua tida

water.

Thus the

din

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