Figure of Paper will be found well worth the attention of any the Earth, practical person. The attraction of 960 prisms was calculated at each observatory; the attraction at the North observatory was found to be to that at the South observatory nearly in the proportion of 7:9. And the sum of the attractions was found (supposing the density of the mountain equal to the mean density of the 1 Earth) to be of the Earth's attraction. But the 9933 sum of the disturbances in the direction of gravity as determined from the astronomical observations being =11".6, the sum of the attractions was, in fact, = gra1 1 vity tan 11.6 = × gravity= X 17781 17804 Earth's attraction, (allowing for centrifugal force.) Consequently the density of the mountain was to the Earth's mean density as 9933 to 17804, or nearly as 5:9.

Mineralogical structure of the mountain.

for observe

But this supposed that the mountain was homogeneous. The geological characters of its rocks were examined by Professor Playfair, (Phil. Trans. 1811,) he found that the upper part consisted of quartz whose mean Specific Gravity = 2.6398; and the lower part of mica slate and hornblend slate, whose mean Specific Gravity 2.8326, and of limestone whose mean Specific Gravity = 2.7661. Two separate calculations were made; one on the supposition that the rocks were separated by vertical surfaces, and another supposing them separated by surfaces nearly horizontal. The former gave for the Earth's mean density 4.559, the latter 4.867, that of water being 1. Dr. Hutton (Phil. Trans. 1821) says that the number should be rather greater.

Cavendish's The next experiment was that of Mr. Cavendish on apparatus the attraction of leaden balls. (Phil. Trans. 1798.) His ing the apparatus is represented in fig. 51. x, x, are balls of lead attraction of about 2 inches in diameter, suspended to the ends of a leaden balls light deal rod h, h. This is suspended by the wire lg, forming, in fact, a balance of torsion. The piece to which the top of the wire is attached, carries a wheel which is turned by the endless screw K F, so that the wire can be twisted till the resting-place of the balls is any required position. nn are small graduated scales carried by the rod. The whole of this apparatus is enclosed in a mahogany box, FEA B C DDCBAEF. At A and A are small glass windows, and near these are scales serving as a vernier for measuring the motion of the scales n. These are illuminated by lamps L and L, and viewed by telescopes T and T from the outside of the room. The leaden balls W, W (each weighing 2,439,000 grains) are suspended by copper rods attached to the piece rr which is suspended to a beam. By means of a rope passing round the pulley M M, the balls W, W can be moved without entering the room. The support of the balance of torsion and its cases is independent of the walls.

The first wire by which the deal rod was supported was of copper silvered, of which one foot weighed 2.4 grains. After a few experiments with this, it was found that the attraction of the large balls made the rod touch the sides of the case; and a stiffer wire was then used.

&c. were made by Mr. Cavendish. This we have ascertained from an inspection of his papers, which we have had an opportunity of examining through the kind permission of his Grace the Duke of Devonshire.

The method of observing was the following. The produced large balls being in the midway position (their support- by current ing rod at right angles to the deal rod) the position of

the deal rod was read off from the scales n.

The

large balls were then brought so as nearly to touch the case, sometimes in the positive position (in which their attraction made the rod move so as to increase the num ber on the scales) and sometimes in the negative position. By their attraction on the small balls, the rod was immediately put in motion, and vibrated backwards and forwards. The greatest extent of vibration was observed; the mean of two consecutive extreme points on one side was taken, and the intermediate extreme point on the other side; and the point midway between these was considered to be the point at which the rod would rest under the action of the balls and the torsion of the wire. The time of passing the middle point was also ascertained, by observing the time of passing two points near the middle, and then (when the middle point was determined) calculating the time of passing it. This being done for vibrations separated by a considerable number, the time of vibration was accurately found. With the wire finally used, the change from the position of rest without the action of the lead balls to the middle position when they were applied, seldom exceeded 3 divisions, (each divisionth of an inch,) and the time of vibration was about 7 minutes. With the first wire the change was about 14 divisions, and the time nearly 15 minutes.

of air.

Method of observation

The middle point is evidently the place where the attraction of the large balls is equal to the force of torsion of the wire. The time of vibration also depends on both of these forces. For suppose that at the middle point the distance of the small balls from the large ones was A, and the space through which they had been moved (to which the force of torsion is proportional) B; then putting Wand T to represent these forces respectively

W

at the distance 1, we shall have this equation == A=TB. And at the distance r beyond this, and further from the balls, the whole force tending to bring the balls to the W middle point is T (B + ☛) · (A − x)2

+(T

- 2W) x nearly, =

(T-2W)

=TB

W

Method A calculat

x; and this is the force on which the time of vibration depends. Thus there are, in fact, two equations to be solved, from which the attraction of the balls and the torsion of the wire could be determined. Besides this, the attraction of the mahogany case was calculated. The attraction of the leaden balls being thus determined, and compared with the attraction of the Earth, the proportion between the Earth's mean density and the density of lead was found; and thus the Earth's mean density is obtained. The result of 29 experiments (as corrected by Dr. Hutton, Phil. Trans. 1821) is 5.31, that of water being 1. The smallest number given by one experiment is 4.86, and the largest 5.79.

We are upon the whole inclined to prefer this result to that of the observations on Schehallien. It cannot

pre

Figure of be denied that the greatest delicacy was necessary to the Earth, obtain any result, and that the determination is much less certain than the determination (such as it is) by Real pro- the Schehallien experiment. But we consider that the to quantity determined in the latter may be something rmer. very different from the attraction of the mountain. We have seen that at Arbury Hill, and some other places, there is evidence of disturbance to an amount nearly as great as the attraction of Schehallien, and without any known cause. May not such unknown causes have operated as well in the Schehallien experiment, and have increased or diminished the apparent effect of the mountain?

Considering the Earth's mean density as somewhat greater than 5, and the mean density of the rocks at the surface as 2.6, the proportion of the Earth's mean density to the superficial density is not very different from

that of 2: 1.

We have mentioned in our first Section the attempt made by the Baron de Zach to measure the attraction of a mountain near Marseilles. We have only to add that no calculation of the Earth's density was founded on these observations.

The attraction of a mountain might be found by observing the length of the seconds' pendulum on the top; if gravity should thus be found to be greater than gravity at the level of the sea in the same latitude diminished in the duplicate proportion of the distance from the Earth's centre, the excess would be attributable to the attraction of the mountain. Bouguer (as we have mentioned in Section 7.) observed the pendulum on one of the peaks of the Andes; but the circumstances were unfavourable, and we should have no confidence in the results. In the present century (see the Additions to the Milan Ephemeris) M. Carlini made similar observations in much more favourable circumstances at the hospice of Mont Cenis, at an elevation of 6375 feet Cenis, above the level of the sea. The pendulum we have described at the beginning of Section 7: twenty experi ments were made with it; they were reduced like the French measures, and corrected for elevation by the rule of inverse squares; the length at the level of the sea thus found was in mètres 0,993708. But the observations of Biot at Bordeaux, nearly at the level of the sea, corrected for the small difference of latitude, gave 0.993498. The difference is due to the attraction of the mountain mass. Representing this by a segment 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 the pendulum had been made exactly like the French pendulums, or if an invariable pendulum had been used.

erroneous

As it stands, there is one considerable source of error,* Conclusion. namely, the erroneous reduction for the effect of the air (mentioned in Section 7.) The barometer at Mont The result Cenis being several inches lower than at Bordeaux, this and doubterror would be serious. Besides, we could hardly trust ful if cor to a comparison between two places at so great a dis- rected. tance. On the whole, we do not think that any estimation of the Earth's density can be founded on this experi

is about 20,923,700 English feet.

2. The phenomena of precession and nutation give an ellipticity rather smaller; but as no result can be deduced from them except on an assumed law of density, this value cannot be put in opposition to the others.

3. As the results of the pendulum observations, the lunar inequalities, and the precessional phenomena, can only be used to determine the Earth's form by the intermediation of the principle of gravitation, the very near coincidence of the results is a strong argument in favour of the truth of that principle.

4. The same things make it highly probable that the Earth has once been in a fluid or semi-fluid state.

5. None of these results can be obtained without the admission of considerable anomalies, all of which, however, appear to be consistent with the principle of gravitation.

6. The mean density of the Earth considerably exceeds, and is probably double of the density of the superficial rocks.

7. The near agreement of the proportion between these as deduced from an assumed law with the proportion found by the experiments with leaden balls (where it is assumed in the calculation that the law of gravitation holds good at the distance of a few inches) makes it probable that the law is sensibly true to very small distances. G. B. AIRY.

Observatory, Cambridge, August 17, 1830.

dulum-balls being spherical,) the length at Bordeaux is 0.993553, and

*If we correct this, by doubling the usual reduction, (the pen

that at Mont Cenis, reduced for elevation, 0.993754; whence the mean density of the Earth = 4.59.

The author has discovered a small error in the Table of the observed lengths of the seconds' pendulum. The reduction applied to the length of the Paris pendulum for elevation above the sea is that due to the decimal pendulum. In consequence, the length at Paris, and all the lengths depending on it, ought to be increased by .00013; a quantity which does not sensibly affect any of the results. This correction applies to Nos. 18, 25, 31, 33, 40, 44, 47, 49; and half of it to Nos. 42, 45, 48. The multiplier 0.6 has been used (in reducing the foreign observations) in preference to 0.66, as it seems probable that the density at the Earth's surface is greater, and the mean density less, than Dr. Young

supposed. The multiplier 0.66 is adopted for the Postscrit English observations from the calculations of the respective observers; the effects of this inconsistency are not sensible.

In the determination of the length of the seconds' pendulum at Königsberg, a correction is included which is not applied to any of the other observations. The reason is that the effect of this correction is to increase the length of the seconds' pendulum; and it appears that the observations on which Bessel principally relied, give a smaller length than those made in the usual way. The difference depends probably on the difference of the apparatus employed.

We propose, in this article, to enter at some length into the mathematical theories, and the experimental observations, applying to the two subjects of Tides and Waves of water. But we do not intend to treat them with the same extension. We shall give the various theories of Tides in detail sufficient to enable the reader to understand the present state of the science which regards them; and we shall advert to the principal observations which throw light either on the ordinary phænomena of tides, or on the extraordinary deviations that occur in peculiar circumstances. In thus treating the Tides, it will be necessary for us to enter largely into the theory of Waves. We shall take advantage of this circumstance for the introduction of several propositions, not applying to the theory of Tides, but elucidating some of the ordinary observatons upon small Waves. But these investigations will be limited to that class which is most closely connected with tides, namely, that in which similar waves follow each other in a continuous series, or in which the same mathematical process may be used as when similar waves follow each other. In this class will be included nearly all the phænomena of waves produced by natural causes, and therefore possessing general interest. But it will not include the waves of discontinuous nature produced by the sudden action of arbitrary causes, which have been the subject of several remarkable mathematical memoirs, but which possess no erest for the general reader.

The general plan of this Essay will be as follows:We shall describe cursorily the ordinary phæno

IV. We shall give an extended Theory of Waves on water, applying principally to the motion of water in canals of small breadth, but with some indications of the process to be followed for the investigation of the motion of Waves in extended surfaces of water.

V. The results of a few Experiments on Waves will be given, in comparison with the preceding theory. VI. We shall investigate the mathematical expressions for the Disturbing Forces of the Sun and Moon which produce the Tides, and shall use them in combination with the theory of Waves to predict some of

the laws of Tides.

VII. We shall advert to the methods which have Tides and been used, or which may advantageously be used, for Waves. Observation of Tides, and for the Reduction of the Observations.

VIII. We shall give the results of extensive observations of the Tides, as well with regard to the change of the phænomena of tides at different times in the same place, as with respect to the relation which the time and height of tide at one place bear to the time and height at other places, and shall compare these with the results of the preceding theories, as far as possible.

And as Conclusion, we shall point out what we consider to be the present Desiderata in the Theory and Observations of Tides.

Tide.

tion of the

(2.) The first and most important change is, that the Semidisurface of the water rises and falls regularly twice in urnal every day. A short series of observations will show however that this statement is not quite correct; the tides of each succeeding day are somewhat later than those of the preceding day: the average retardation. from day to day being about 40 minutes. In a short Its time is time he will find that the times of occurrence of high related to water bear a very close relation to the time of the the appaMoon's appearance in certain positions; and that the rent posilanguage of the persons who are most accustomed to moon. observe the tides conveys at once this relation. Thus, at Ipswich, high water occurs when the moon is south nearly at London Bridge high water occurs when the moon is nearly south-west: at Bristol, it takes place when the moon is E.S.E. These are rude statements, but they are sufficiently accurate for many purposes; and they show at once the close connection between the time of high water and the time of the moon's passage over the meridian. In fact, so completely is this recognized, that, in order to give the time of high water upon any day, it is usually thought sufficient to

*We commence with this case, because, judging from the notions of sea-faring persons upon many points connected with the Tides, which are correct as regards rivers, but incorrect as regards the sea, (some of which will hereafter be indicated,) it is the case from which ideas of tidal movements have usually

been taken with the greatest facility. 2 K*

The interval be

water and moon's transit is variable.

The rules, as we have mentioned them, indicate the time of only one high water in the day but the reader must understand that there will always be another high water in the same day, preceding or following that which we have mentioned by 12 hours 20 minutes nearly. On those days, however, in which high water occurs within 20 minutes of noon, there is no other high water on the same civil day.

(3.) On closer examination it will be found that the interval between the time of the moon's passage over tween high the meridian and the time of high water varies sensibly with the moon's age. At new moon, full moon, first quarter, and third quarter, (or rather on the day following each of these phases,) the interval between the time of the moon's passage and the time of high water is nearly the same: but from new moon to first quarter, and from full moon to third quarter, the high water occurs earlier than would be inferred by using that same interval; and from first quarter to full moon, and from third quarter to new moon, it occurs later than the same interval would give it.

Spring and Neap Tides.

The dura

the dura

rise.

the river after high

(4.) If the observer examines the height of the water, he will find that the height at high water and the depression at low water are not always the same. On the days following new moon and full moon, high water is higher and low water lower than at any other time: these are called Spring Tides. On the days following the first and third quarters, high water is lower and low water higher than at any other time: these are called Neap Tides. The whole variation of height at spring tides is nearly double that at neap tides. There are other variations of height depending on other circumstances; but they require, for the most part, very numerous observations to establish the fact of their existence, and to give a measure of their amount. In many places, however, the tide which occurs at one certain part of the day (the afternoon for instance) is, during one half of the year, sensibly higher than the other tide which occurs upon the same day, and, during the other half of the year, sensibly lower.

(5.) Upon examining the circumstances of a single tion of the tide, the following facts will attract notice. The interval fall is from high water to low water is greater than that from longer than low water to high water: the difference between these tion of the intervals is sensibly greater at spring tides than at neap tides. The current in the river runs upwards for The water some time after high water, and after changing its dicontinues rection, continues to run downwards for some time to run up after low water, when it again changes its direction and runs upwards. This phænomenon is often so much misrepresented in the language of nautical men, that the mistake deserves particular notice. From the habit of observing tides in places where the current ceases at high water and at low water, sailors conceive that high water may always be inferred from the cessation of the current; and therefore it is not unusual for persons on the banks of the Thames to say that "it is high water in the centre of the channel long after it is high water at the shore." The observer

water.

Ordinar

who is not convinced of the absurdity of supposing Tides ar the water in the middle of the channel to stand at one Waves time considerably higher and at another time considerably lower than at the shore, will satisfy himself Sect, I most easily as to the general fact by stationing himself Phan at one of the central piers of a bridge, (as London mena o Bridge,) when he will see that the water continues to Tides. run upwards even after its surface has dropped nearly two feet.

curs

toor

(6.) Now suppose that the observer examines the state of the tide in different parts of the same river. Commencing with the mouth of the river, (for instance Margate or Sheerness on the Thames, or Swansea or Cardiff on the Severn,) he will find that there is very little difference, or perhaps none which is appreciable, between the interval from high water to low water, and that from low water to high water. He will also find that the current runs up the channel for a long time (sometimes approaching to three hours) after high water, High and runs down the channel for as long a time after low wate water. In going up the river, he will find that the time for p of high water occurs later and later, but yet that the high velocity with which high water travels up the river is so the 1 great as entirely to banish the idea of explaining the Tide by supposing the same mass of water to have been moved all the way up the river. For instance, if at Margate the The high water occurs on a certain day at twelve o'clock, it gress will occur at Sheerness at 24 minutes past one, at Graves- tide end at 15 minutes past two, and at London Bridge at a few minutes before three; having thus described in less plai than three hours a course of about 70 miles. He will the also find that the interval from low water to high mis water diminishes as he goes up the river: thus, on the the lower parts of the Severn, the rise and fall occupy little wal. more than six hours each; but at Newnham on the Th Severn the whole rise of the water is effected in an tion hour and a half, the descent occupying nearly eleven inc hours. In cases like the last-mentioned, the first rise an of the tide is sudden, and if the banks of the river are nis shoaly, the water spreads over the flat sands with a ase roaring surf, which travels rapidly up the river, pre- the senting the phænomenon called a bore or boar, (some- Th times bour's-head,) in French barre or mascaret. In ris other cases, however, when the difference of durations tin of rise and fall is considerable, there are in each high bo water two, or sometimes three distinct rises and falls tin of the water. The phænomena of bore and double ble tide are always much more conspicuous in spring tre tides than in neap tides.

to b

bod

rist

du

tua

tide

(7.) If the estuary or mouth of the river contracts In very much, the elevation and depression of the water will tra become very great. Thus at the entrance of the Bristol Channel the whole rise at spring tides is about 18 feet, hig at Swansea about 30 feet, and at Chepstow about 50 feet. Similar high tides occur at St. Malo and other parts of the great bay formed on the northern coast of France by the projection of land towards Cherbourg, and tides still higher in the head of the Bay of Fundy (Baie Française) on the Eastern coast of North America. In But when the tide has fairly entered a river, its range ing of elevation and depression generally diminishes. Thus the at Newnham, on the Severn, the range is reduced to about 18 feet, and it is still less at Gloucester.

din

(8.) Quitting now the phænomena of river-tides; if Bay observations are made in a bay communicating with are the open sea, the results will be found to be much more simple. The water will rise during 6 hours 10 minutes

I