canal to the equator, and disappearing for a sea situated at the pole: the econd part is a diurnal tide proportional to the sine of the latitude or of the inclination, being greatest when the luminary is furthest from the equinox, and vanishing when its declination vanishes."

He next proceeds "to inquire more particularly into the cause of the hitherto unintelligible fact, that the maximum of the spring tides in the most exposed situations, is at least half a day, if not a whole day, later than the maximum of the moving forces.

"Now it is easy to perceive that, since the resistance observing the lunar period is more considerable than that which affects the solar tide, the lunar tide will be more retarded or accelerated than the solar; retarded when the oscillation is direct, or when G2 B is [negative,] and accelerated when it is inverted, or when that quantity is [positive]; and that, in order to obtain the perfect coincidence of the respective high waters, the moon must be further from the meridian of the place than the sun; so that the greatest direct tides ought to happen a little before the syzygies, and the greatest inverted tides a little after; and from this consideration, as well as from some others, it seems probable that the primitive tides, which affect most of our harbours, are rather inverted than direct."

As a convenient epoch for dating the beginning of a series of tides, it is observed that the mean conjunction, at the beginning of 1824, happens exactly at mean noon of Jan. 1, in the time of the island of Guernsey or of Dorchester, and at 18m 49s Parisian mean time.

It is further observed respecting the effects of resistance, that this cause "tends greatly to diminish the variation in the magnitude of the tides, dependent on their near approach to the period of spontaneous oscillation, and the more as the resistance is the more considerable: and supposing, with Laplace, that in the port of Brest, or elsewhere, the comparative magnitude of the tides is altered from the proportion of 5 to 2, which is that of the forces, to the proportion of 3 to 1; the multipliers of the solar and lunar tides being to each other as 5 to 6,... we find that B `must be either .9380 or .6328, and the former value making the lunar tide only inverse, we must suppose the latter nearer the truth; and the magnitude of the tides will become 1.663 and 1.998, and . . A cannot be greater than .632. It seems probable, however, that the primitive tides must be in a somewhat greater ratio than this of 2 to 1, and 5 to 3, when compared with the oscillations of the spheroid of equilibrium; and if we suppose B= .9, and A still

1

= we should have [6.364] and [8.78] for their magnitude;" so that the actual elevations would be about 6 and 19 feet respectively.

"Now... the tangents of the angular measures of the displacement,

AG

GG

В'

give us 69° 50′ and 72° 40' for the angles themselves, when B = .6328; and

if B.9, these angles become 45° and 70° 24′ respectively; the difference in the latter case, 25° 24', corresponding to a motion of more than 24 hours of the moon in her orbit.

"It appears then that, for this simple reason only, if the supposed data were correct, the highest spring tides ought to be a day later than the conjunction and opposition of the luminaries; so that this consideration obviously requires to be combined with that of the effect of a resistance proportional to the square of the velocity, which has already been shown to afford a more general explanation of the same phenomenon."

It may easily be admitted that this theory may require much further illustration, and perhaps discussion, before it can be rendered very popular, or intelligible, in all its bearings; but in point of mathematical evidence, it may not be superfluous to insert here the reduction of the expression of the force acting on an oblique canal into the simple form which the author has adopted. without a demonstration, at the end of his paper.

Since the force ƒ = sin cos Alt. sin (Az. + Dev.) = sin cos Alt. (D′ sin Az. +D COS Az.); and sin Alt. = L sin Decl. + L' cos Decl. cos Hor. ; also cos Decl. sin Hor. L ; we have cos Alt. sin A. = cos Decl. sin cos Alt.

sin Az. =

Hor., and cos2 Alt. sin2 Az. = cos2 Decl. sin2 Hor.

=cos2 Alt. (1-cos2 Az.)

and cos2 Alt. cos2 Az. = cos2 Alt. cos' Decl. sin2 Hor.

= 1

cos Decl. sin' Hor. ; whence cos Alt. cos Az. = 1
Decl. sin2 Hor. ≤) + } (sin2 All. + cos2 Decl. sin2 Hor. ▲)2
finally,

(sin Alt + cos2

f = (L sin Decl. +L' cos Decl. cos Hor. ) (D' cos Decl. sin Hor. + D [1 − } ( sin2 Alt. + cos2 Decl. sin2 Hor. ▲) + } . ]); which may readily be more completely developed if required.

....

But for a lake obliquely situated at the equator, when L = 0, and L′ = 1, the expression becomes sin Alt. = cos Decl. cos Hor. L, and cos2 Alt. cos2 Az. 1 cos2 Decl. cos2 Hor. L cos2 Decl. sin Hor.

1- cos2

Decl. sin2 Decl., and cos Alt. cos Az. = sin Decl.; whence = cos Decl. cos Hor. (D' cos Decl. sin Hor. + D sin Decl.) = D' Cos2 Decl. sin cos Hor. + D sin cos Decl. cos Hor. L, which is the equation given at the end of the Article, agreeing with the equation of the form for a canal running east and west, in having for each luminary a semidiurnal tide which is greatest when the declination vanishes, and a diurnal tide increasing, on the contrary, as the sine of twice the declination increases. The two formulæ give the same result for a canal coinciding with a part of the equator, and they appear in other cases to represent the force for every part of the same oblique great circle, the deviation at the equator being equal to the latitude when it becomes perpendicular to the meridian.

LAPLACE, assisted by the indefatigable BOUVARD, has lately published a very valuable continuation of his Researches on the Tides, as a XIIIth Book of his

Mécanique Céleste, Febr, 1824. He has computed the results of about 6000 observations made at Brest since the year 1806, and has found them confirm in general those which he had obtained from the more ancient observations, There are also some new deductions, which may be made subservient to the further illustration of the principles laid down in the Supplement of the Ency. clopædia.

"I have considered," says Mr. Laplace, (P. 160,) " the tide of which the period is about a day. By comparing the differences of two high and two low waters, following each other, in a great number of solstitial syzygies, I have determined the magnitude of this tide and the time of its maximum, for the port of Brest. I have found its height very nearly one fifth of a metre, and one tenth of a day for the time that it precedes the time of the maximum of the semidiurnal tide. Though its magnitude is not one thirtieth of that of the semidiurnal tide, yet the generating forces of both these tides are nearly equal, which shows how differently their magnitude is affected by accidental or extraneous circumstances. This will appear the less surprising, when we consider that if the surface of the earth were regular and entirely covered by the sea, "the diurnal tide would disappear, provided that the depths were uniform throughout." In fact, the observed heights of the diurnal and semidiurnal tides are .2134, and 5.6m respectively, (P. 227); and the time that the diurnal tide precedes the maximum of the evening semidiurnal tide is .095a, (P.226). It is not quite clear that the words might not relate to the maximum resulting from the most perfect combination of the solar and lunar diurnal tides; but we may suppose, for the sake of the calculation, that the high water of the joint diurnal tide generally happens a little more than two hours earlier than that of the semidiurnal tide.

Now supposing, for the determination of the multiplier,

B

√[(GG−B)2+ AAGG] we assume the mean value of G, for the joint semidiurnal tide, about .98, and for the diurnal.49, B being about,9, and A=.1, the formula becomes=7.83, or if A=.2, 4.4 for the semidiurnal, and 1.327 or 1.234 respectively for the diurnal,

and Dr

D'

L

or must be such that D sin 2 Decl, x 1.327 may be to D' x 7.83

as .2134 to 5.6, or as 1 to 26.25; but sin 2 Decl. = sin 46° 55′.5 = .73045, and we have D x .9691 D' x 7.83 126.25 = D; D′ x 8.07 and D; D' 26.25 = 3.25 = cot 17° 6′, which must be the obliquity of the canal to 8.07

= 1;

the equator if A = .1, or if A = .2, 10° 30': either of which may possibly be near the truth, though the obliquity of the main channel of the Atlantic to the equator is probably greater. With respect to the times of high water, the

β

tangents =

a

AG
GG-B

become, if 4.2, at 72° 59′ and at s° 27' respec

tively; the former expressing the acceleration of the inverted semidiurnal tide, and the latter the retardation of the direct diurnal tide, by the effect of friction, the sum of the former and twice the latter is 89° 53′, or very nearly a right angle; so that the interval, thus computed, instead of one tenth of a day should be a little more than an eighth. It would, however, be necessary to compare the height of the water at different intervals before and after high water, in order to obtain the progressive magnitude of the diurnal tide with sufficient accuracy to allow us to place any reliance on the result of this computation.

With respect to the disappearance of the diurnal tide in an ocean of equable depth, no doubt the depth must be equable in order that it may disappear, but it must also be evanescent. In fact, it is not conceivable in what other manner the equability of depth can possibly produce such an effect; for there is no natural nor assignable relation between the period of revolution and that of diurnal tide; the effects are just the same as if the earth were at rest, and the attracting body moved round in a day, or in two days and it is impossible to admit the accuracy of any refined method of investigation, from which Mr. Laplace has obtained a result so clearly contradictory to the first principles of mechanics.

ii, An easy Method of comparing the Time indicated by any Number of CHRONOMETERS with the given Time at a certain Station. By the Rev. FEARON FALLOWS, M.A., F.R.S., Astronomer at the Cape of Good Hope.

LET a transit instrument, or even a sextant with an artificial horizon, be established in a conspicuous situation on shore, where a clock can always be regulated to true time: then provide a powerful Argand's lamp with a shutter, so as to be able to darken the lamp instantaneously; a few minutes before a certain hour in the evening, notice being previously given to the ships, let the lamp be lighted, and at the proper instant of time let it be darkened; this may be repeated several times at short known intervals. Then the errors of every chronometer on board of all the ships, from which the lamp can be seen, are immediately found. After a certain number of days, let the same be repeated, when the daily ship rates will be given, since they are only the differences of these errors divided by the number of days elapsed between the two sets of observations. It is evident that for greater truth these observations may be repeated at pleasure. No objection can be made from

the chronometer being generally below deck, as one person might have his eye upon it, and another immediately above him on the upper deck might give a stamp with his foot the instant the lamp is darkened.

The longitude of Cape Castle appears from eclipses of Jupiter's satellites to be about 18° 21′ E.

The height of Table Mountain above the sea was found,

3536

Entrance from the narrow passage on the top (5 obs.) 3430 F.
Highest western point (13 obs.)
Highest eastern point (11 obs.)

3545

iii. Easy APPROXIMATION to the difference of LATITUDE on a SPHEROID.

SUPPOSING the excess of the equatorial semidiameter to be known and equal to e, while the semiaxis is = 1, and having the linear dimensions of a portion of a perpendicular to the meridian, we may compute the difference of latitude and of longitude of its two extremities by considering the case of a sphere touching the surface of the spheroid in the given parallel of latitude, and having the same curvature with the perpendicular to the meridian at its origin, which must therefore be extremely near to the points that require to be compared with each other, so that they may be supposed to be in the surface of this sphere.

Now the local semidiameter will always be 1 + e cos3 L, L being the true latitude, whence, by taking the fluxion, we obtain for the tangent or the sine of the inclination of the surface, or the correction of the latitude, 2e sin cos L, consequently the sine of the corrected or geocentric latitude will be sin L 2e sin cos L cos L = sin L (12e cos2 L). Hence we find, by trigonometry, the normal to the equatorial plane (1 + e cos2 L) sin L (1 - 2e coso L); sin L = (1 + e cos2 L) (1 2e cos L) = 1 e cos L, e being a small fraction; and the normal to the axis, which is the radius of curvature of the perpendicular circle at its origin, = (1 + e cos2 L) cos (L- 2e sin cos L): cos L; but cos (L- 2e sin cos L) = cos L+ 2e sin cos L sin L = cos L (1 + 2e sin2 L) and this normal