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ON THE LENGTHS OF PENDULUMS, VIBRATING SECONDS IN difFERENT LATITUDES; AND THE Descent of HEAVY BODIES. [By John Baines, jun.]

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"PHILOSOPHERS," says General Roy," are not yet agreed in opi. nion with regard to the exact figure of the earth; some contending that it is an ellipsoid; others a spheroid; and some that it has no regular figure, that is, not such as would be generated by the revolution of a curve around its axis." But if the earth were in a state of fluidity at the commencement of its diurnal rotation, its figure, by the laws of gravity, must be that of an oblate spheroid. Accordingly the measurements of the French mathematicians in Lapland, in France, and at the equator, prove the earth's equatorial diameter is greater than its axis. Colonel Mudge, who conducted the most extensive geodesic operations that were ever carried on in this country, makes the earth's equatorial diameter 41,897,040 English feet, and the polar 41,616,084.* Now, (by art. 217, Simpson's Fluxions, Davis's edition,) the periodical time in which a body will revolve round the earth at its surface, when its velocity is just a counterpoise for the force of gravity there, is equal to the square root of the quotient of the earth's equatorial diameter in feet, divided by the space in feet which a heavy body falls from rest at the earth's surface, in the first second of time, multiplied by the circumference of a circle whose diameter is I. Hence, in the present example,

41897040÷16X3·141592—1614-00043.14159250708-53 seeds. But in circles having the same radii, the forces are inversely as the squares of the periodical times, (Ibid. art. 137,) and as the earth revolves on its axis in 23hrs. 56m. 48.-86164 seconds, we have the force of gra vity to the centrifugal force of a body at the equator arising from the earth's rotation, as

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1155:4, extremely near.

Put A then (by Simpson's Fluxions, art. 397), +

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will express the difference between the earth's equatorial diameter and its polar, the polar diameter being considered as unity. This, in numbers, is 004328854+000006413004335267; hence, the earth's axis is to its equatorial diameter as 1 : 1·004335267, or as 692: 695, extremely near. But the force of gravity at the equator is to that at either of the poles, inversely as the equatorial diameter is to the axis, that is, as 692: 695; consequently the diminution of gravity from the poles to the equator is

Although the earth's polar diameter is here mentioned, it is not for the purpose making use of it. It will be seen, by the following process, that the ratio of the earth's diameters is found independently of any consideration but that of gravity.

of

3

695

5ths of the gravity at the poles. Hence it appears that gravitation is different in different latitudes; and this difference arises from two causes, viz. the spheroidical figure of the earth, by which bodies on different parts of its surface are not equally acted upon by the attractive power, in consequence of their not being equally distant from its centre; and the centrifugal force arising from the rotation of the earth on its axis, by which the power of gravity is unequally diminished from the poles to the equator. However when the spheroid differs but little from a sphere, the diminution will be nearly as the square of the co-sine of the latitude. (Marrat's Mechanics, art. 449.) Therefore as 1 : 387525 (=the square of the co-sine of the latitude of London 51° 30′)::3: 1·162575, the diminution of gravitation from the poles to London; hence 693-837425 will express the force of gravity at London, when that at the poles is expressed by 695, and that at the equator by 692.

These things being premised, we will proceed to determine the lengths of pendulums vibrating seconds at the earth's surface in different latitudes. Now it has been found by experiment that a pendulum vibrating seconds in the latitude of London, is 39·125 inches long, and that a body falls from rest in the first second of time 16 feet. At Pello, lat. 66° 48' N. M. de Maupertius found that the length of the seconds' pendulum was 441-17 French lines, equal to 39-1824 English inches; and at Paris, latitude 48° 50′ 10′′ N. 440-57 lines, equal to 39-1291 English inches.

But because the length of the seconds' pendulum, in any latitude, is directly as the force of gravity there, (Marrat's Mechanics, art. 345,) we can, by calculation, find the length of a pendulum vibrating seconds in any latitude, from the experiments which have been made. Thus,

As 693-837425: 39-125:695: 39:19056 inches, the length of the seconds' pendulum at the poles; and

As 693-837425: 39125:692; 93-02139 inches, the length of the pendulum vibrating seconds at the equator.

Again, because the space descended from rest by a heavy body in the Arst second of time is directly as the length of the seconds' pendulum, we have

As 39.125 16:39·19056 16-11025 feet per second, at the poles. As 39-125 16:39-02139: 16-04074 feet per second at the equator. On these principles we have calculated the following Table of the lengths of the pendulum vibrating seconds, and the space fallen from rest by a heavy body in the first second of time on the earth's surface, for every third degree of latitude from the equator to the poles; the first in inches and latter in feet.

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Names of Places in or near the specified lat.
Goree; Sana, Arabia; Manilla, Philippines.
Pegu, India; Kingston; Arica, Peru, &c.
Surat; Mecca; Kesho, in Tonquin, &c.
Canton; Havanna; Santos, in the Brazils, &c.
Monfalout, Egypt; Agra, India; Kelveh, Persia.
Cairo; Shiras; Lassa; New Orleans, &c.
Damascus; Tripoli; Charlestown, &c.
Gibraltar; Balk, Tartary; Plymouth, N. Amer.
Lisbon; Badajos; Pekin; Washington, &c.
Bastia; Boston, N. America; Rome, &c.
Belgrade; Halifax, Nova Scotia ; Oczakow,&c.
Strasburg; Vienna; Presburg, &c.
Antwerp; Breslau; Dunkirk; London; Kiow.
York; Dantzic; Cavan, in Ireland, &c.
Riga; Aberdeen; Gotheborg, in Sweden, &c.
Christiana; Upsal; Petersburg, &c.
Drontheim; Wasa, Finland; Beresov, Siberia.
Tornea, Sweden; Touroukhansk, Siberia, &c.
Avievara, Lapland; Island of Vaigatch, &c.
North Cape, Lapland; S. part of Nova Zembla.
N.part of Nova Zembla; Lake Tamourskie, &c.j
Cape Ceverovostot chnoi, in Siberia.
North part of the island of Spitzbergen.
Never visited by man.

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Ditto.

The Poles.

Deductions.

1. The length of the seconds' pendulum is 16917, or nearly one-sixth of an inch greater at the poles than at the equator.

2. A heavy body falls 16951 of a foot, or 2 inches more in the first second of time at the poles than it does at the equator.

3. Pendulums which vibrate seconds at the earth's surface will lose time, when removed to places considerably above it; aud this is evidently owing to the diminution of the force of gravity. Thus, a pendulum which measures true time at the earth's surface, will lose one minute per day, when removed to the summit of a mountain whose perpendicular height is 14,600 feet. Hence, an allowance ought to be made when the ratio of the earth's diameters is calculated from observations made on the pendulum in different latitudes, if the elevation of the places of observation above the level of the sea be not the same.

4. If the earth revolved on its axis in 1h. 24 m. 30s. and '32 thirds, bodies on its surface would weigh nothing, but be as liable to fall off as to stop on. If it revolved in more time than the above, they would stop on but if it revolved in less, they would fly off.

5. If the earth's rotation on its axis were stopped, the weight of bodies

1 287.76

4

th, or

1151

at the equator would be increased part of the whole; but at the poles, their weights would not be altered. Moreover, as the weight of a body at any place is always proportional to the force of gravity there, it follows, that a body which weighs one pound or 16 ounces at the poles, would only weigh 15.93095 ounces at the equator, while the earth is revolving on its axis, and 15·98631 ounces, if its rotation should be stopped.

QUESTION 4. By W. Godward.

IT is required to find that number whose 6th power being taken from its 5th, shall have the greatest remainder possible?

QUESTION 5. By Aaron Arch, York.

To construct the plane triangle there are given the base, the vertical angle, and the ratio of a line from the vertex intersecting the base in a given angle, to the difference between the segments of the base made by the intersecting line.

QUESTION 5. By the same.

Three men agree to drink a quart of ale out of the same tankard: it is required to determine the divisions of the vessel made by the surfaces of the liquor at the time each man ceased to drink, supposing the height 71⁄2 inches.

Original Poetry.

THE following verses were written by a learned friend of mine, a Physician, now no more, whose practice was very extensive some years ago in this wapentake. His hours of leisure were devoted much to the Roman muses, and I have by me other productions of his, of the same kind, some of which have been printed. As these verses, I believe, have never yet appeared in public, they are at your service, and I should be happy to see a translation of them from some of your poetical correspondents. Wapentake of Strafford and Tickhill, Feb. 1818.

ROCHE-ABBEY, 1777.

QUANTIS illecebris ornata hæc vallis amona,
Quas natura dedit!-monachis vecordibus olim
Sacra fuit vanæque superstitionis alumnis.

Ecce sibi quales isti retinere solebant
Blanditias! en, quas Pietas construxerat ædes
Devia, quam subitæ jam devenêre ruinæ !
Fallor? -an augurio tandem meliore Patronus

PAGANUS.

Nobilis atque bonus sociales vertit in usus.
Dumque sibi solumque suis, secludere spernens
Tantas delicias, in publica commoda cedit,
Arte polit mirâ, quam, naturæ æmula, doctis
Carminibus cecinit dulcissima Musa Masoni.
Hinc ubi erant tantum Cypici Arcadiique recessus
Et per prata latex, blando cum murmure repens,
Elysios, en, nunc campos rivumque tuemur,
Cui vel Callirhoë vel dulcis cedat Ilyssus !

Rupibus est facilis clivi descensus ab altis,
Undique, quæ placidave mundo sejungere vallem
Et contra vulgi curas munire videntur.

"Sed" revocare gradum, vanumque revisere mundum, "Hic labor, hoc opus est."-Hic ducere leniter ævum Quod superest, quàm dulce foret! Quàm grata senectus, In valle huic simili, cui det fortuna quietem,

VERSES TO LAURA.

DEAR Laura smiles, but not for me,-
Why beauty's charm so oft renew?
I love that smile-'tis true as thee,

And hope believes it more than true!
I love thee!-O that aught should sever

Hearts link'd by nature's truest will;
Reason-resolve-may whisper never,
The heart remains a truant still.
Thou wert the first, the only one,

Dear to my heart, fair to my eye,
Thy kindness drew a splendid zone
O'er youth's unclouded morning sky.
Those past endearments yet awhile
Survive the promises they gave;
Lovely, though hopeless, still they smile
Like roses blown on beauty's grave.
O might those smiles so oft, so sweetly
Turn'd on me, be ever mine;
How fain the heart, how indiscreetly,
Would it answer-they are thine.
How dear to lose the pensive mind
A moment 'midst ideal joys!
But, O! how sad to turn, and find

A baseless charm, and fled the prize.
The eye might once delight to trace
(Where yet, e'en yet 'tis bliss to rove,)
Each line of softness in thy face,

Where goodness was the throne of love;
Where, dawning in the maiden charm,

The modest, youthful bride is seen,

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