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Let a, a, ag... $$,$, ... be the successive values of a and ¢ for the several compartments of a lune. These are connected by the following relations:

0, = +di, Ug = 0, +42, Suppose that for the first compartment ay = 0.75*, then

* This particular value is here used because the calculations following are taken from a Paper, by the author, in the Philosophical Transactions for 1855, upon Himmalayan Attraction, and three-fourths of a degree is about the distance of the nearest hills from the northern station of the Great Indian Arc of Meridian. Any other value for ay might have been taken. The results above deduced are perfectly general, and are applicable to any other similar problem. If compartments with elevations or depressions occur in the map, nearer to the station than three-fourths of a degree, or about 52 miles, their dimensions can easily be calculated backwards, i.e. towards the station, by the help of the formula

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The succeeding values, reckoning from a, and D, are here given, as they may be of use for reference.

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a-, = 00.28 0-9 : 09.028
a-10 = 0 .25 0-10 = 0.025
a-11 = 0.23 0-11 = 0.023
a-12 = 0.21 0-12 = 0.021
2-18 = - 0.19 0-13 = 0.019

: 0.17 0-14 = 0.017

= 0.15 0-15 = 0.015 Q-18 = 0.14 0-16 = 0.014 Q-17 = 0.13 0-17 = 0.013 &c.

&c.

0-14 a-15

by (4) $. = 0.075; therefore ag=0°.825: and by proceeding in this way we obtain the following pairs of values by this formula.

6.72 024 7.39 $25

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8.94 $272 =9 .83 $28

10.82 $20

Qy = 0°.75 0 = 0°.075 02

=0.83 , =0.083 ag = 0.91 = 0.091 a4

=1.00 de=0.100 ag

= 1.10 $ = 0.110 а ao - 1.21 = 0.121 Qq = 1.33

$ = 0.133 Ug = 1.46 de =0.146 ag = 1.61 de =).161 Q10 1.77 $10=0:177 du 1.95 u=0.195 0.42 = 2.14 $12=0.214 = 2.35

$18=0.235 Ch 14 = 2.59 $14 = 0.259 215 = 2.85 016 = 0.285 0 18 = 3.13 $18=0.313

-3.45 $1=0).345

= 3.79 018 = 0.379 Ang = 4.17 $19=0.417 220 = 4.59 $20 =0.459 Cha1 = 5.05 $21=0.505

0,9 = 50.55 022 = 0°.555 Cheg = 6 .11 029 = -0.611 ад

= 0.672

= 0.739 028 = 8 .13 $28 = 0.813 Qar

-0.894 = 0.983

= 1.089 - 11.91 $50=1.202 Ogi = 13.11 $1=1.326

= 1.462

=1.620 234 = 17.61 =1.800

= 1.992 28= 21.40 036 = 2.211 Qg7 = 23.61 082=2.456

= 2.732 Ogg = 28.79 089=3.054 = 31.84 0x0=3.419

= 3.600 = 4.314

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- 14.43 $32 - 15.99 фаз

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Q.35 = 19.41

35

ago

= 26.06 088

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a41

= 35.26 041 38.86 $42

That these results derived from the approximate formula (4) are thus far correct, we gather from the fact that the last pair, viz. Qgg = 38°.86 and =4o.314 sufficiently satisfy the test (3) when substituted." Beyond this pair, we cannot use (4), but must solve equation (1), or rather (3), by trial. This leads to the following pairs of values stretching to the antipodes.

689.94 фат 79.27 $ as

143 - 43°.17 018 = 4°.980 06 44 = 48.15 am = 5.783 53.93 $45 = 6.800

= 8.210

10°.330 dar

= 14.030 93.30 $9 = 23 .380 ago = 116.68 so imperfect.

а45

0.48

0 49

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60 .73 ФАв

61. The formulæ here deduced may be applied to find the effect on the plumb-line of any mountain-region, or hollow (as in the case of the ocean), so long as the angle subtended at the station by any part of it is such as to allow its square to be neglected

In the Philosophical Transactions for 1855 and 1858-9, the author has applied these principles to find the effect of the Himmalayas and the mountain-region beyond them on the plumb-line in India, and has found that the meridian deflection caused in the northern station of the Great Arc of Meridian (lat. 29° 30' 48", and long. 77° 42') is nearly 28", as far as the data regarding the contour of the mass can be ascertained ; and that the astronomical amplitudes between that and the next principal station (lat. 24° 7' 11"), and between that and the third (lat. 18° 3' 15'), are diminished by the quantities 15". 9 and 5".3. He has also shown that the meridian deflection between the first and third of these stations varies very nearly inversely as the distance from a point in the meridian in latitude 33° 30'.

62. The effect of the deficiency of matter in the Ocean south of Hindostan down to the south pole is also calculated, upon an assumed but not improbable law of the depth, and found to produce a meridian deflection northwards at the three stations specified of about 6", 9", 10".5 respectively; and 19". 7 at Cape Comorin.

63. It is possible that, the superabundant matter in mountain-regions having been heaved up from below, there may be a deficiency of matter below the mountains which would under certain circumstances have the tendency of counteracting their effect on the plumb-line. This Mr Airy has suggested in a Paper in the Philosophical Transactions of 1855, on the hypothesis that the deficiency is immediately below the mountains close to their mass. Upon the supposition that the mountains may have drawn their mass from the regions below through a considerable depth, by an extensive and small expansion of the matter in those lower regions, the author has calculated the modifying effect on the plumb

line in the Transactions for 1858-9. This has brought to light the fact, that a trifling deviation in the density from that required for fluid-equilibrium, if it prevail through extensive tracts, may have a sensible effect upon the plumb-line. The following Proposition, with which we shall close this Chapter, will show this. These questions, in themselves interesting as problems in Attraction, become still more so, as we shall see, in the determination of the Figure of the Earth.

PROP. To find the effect on the plumb-line of a slight but wide-spread deviation in density in the interior of the earth, either in excess or defect, from that required by the laws of fluid-equilibrium.

64. Suppose vertical lines drawn down through the four angles of any compartment to a depth d, and a surface uniting the four extremities drawn, so as to form the frustum of a pyramid of which the vertex is in the centre of the earth : draw also a vertical line of length d through the mid-point of the compartment. Suppose the height of the matter standing on the compartment to be uniform and equal to one mile. Let the several vertical prisms of which it consists be conceived to be distributed downwards uniformly through the depth d, the density of this lengthened prism will be less than that of the superficial rock in the ratio of 1:d. Let u and v be the distances of the extremities of this long prism from the station. Then the attraction of the short prism along the chord of the surface and (by Art. 48) that

u? of the longer = Hence along the horizontal line at the

attraction of slender prism station

attraction of the prism at surface Now in Art. 59, it has been shown that the attracting mass on any compartment may be considered concentrated in the mid-point. Much more may this be done with the horizontal layers of the frustum which are not of larger dimensions than the compartment, and are farther off from the station. Hence if u, and v, be the distances from the station of the extremities

mass

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mass

UV

u

v

of the vertical line d through the middle point of the mid-line of the compartment, the attraction of the mass on the compartment, and the deflection caused by it, must both be diminished in the ratio of u, to v, to find the effect of the same mass distributed through a depth d. Suppose the masses (one mile high) on n compartments of any lune are thus distributed; then by Art. 58, formula (2),

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If ß= 30°, the coefficient = 1".1392 x 0.258 = 0". 294.

65. We will take an example. Let the width of the lune B = 30°, and let the 21 compartments from an to Ag2 (see Table in Art. 60) be included. This will be a tract of country 59.55 – 0°.75 = 4°.8, or 334 miles in length, and the breadth at the mid-point will = sin 1 (5o. 55 + 0°. 75) x the length of 30°= 0.055 x 30 x 69.5 = 114 miles; and, by spherical trigonometry, the area is, in round numbers, 38,500 square miles. We will take three examples of depth which (for convenience of calculation) we will express in the length of degrees, viz. 3°, 6', and go; which nearly equal 208, 417, and 625 miles. The vertical thicknesses of these three divisions of the frustum are each = 3° = 208 miles. The widths, however, parallel to the horizon grow less in passing downwards. But owing to the convergency of the radii bounding the elementary prisms, the density increases in the distribution of the matter in exactly the same proportion that the area of the horizontal section diminishes. The amount of matter in the three divisions is therefore the same, and we may consider the volumes the same, and each equal to 38,500 X 208 = 8,008,000 cubic miles = 3-100,000th parts of the volume of the whole earth.

u

Now since the greatest of

u u

Un

is less than 6°, we may take the arc for the chord without sensible error. Then, with respect to all these quantities, the cosine of the angle of which z is the cotangent. This enables us without difficulty

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