HEMI-SPHERICAL SHELL. 59 having the same equator, i. e. of a meniscus in the other hemisphere taken together with the meniscus we have been considering, being the distance from the nearest pole. Hence if we take the difference of these we have the attraction of a thin hemi-spheroidal meniscus on a point on its own surface: the formula becomes, attending to the directions of the attraction, Horizontal attraction = h = (0·1446 sin & — 0·6958 sin 24 + 0·0244 sin 3p) - g... (2). $ = 180° – 0, so that 0 and 4 in (1) and (2) are each measured from the pole of the attracting meniscus, and in each case the attraction is reckoned positive towards the pole of the attracting meniscus. Ex. 4. A somewhat simpler example for the reader to work out is this, To find the tangential attraction of a hemispherical shell of small uniform thickness upon any point in the surface of the whole sphere. If the calculation be first made for points, as in the last example*, 91°, 120°, 135°, 150°, 180° from the pole of the shell, the results will be and the following formula will approximately embrace other points in the hemisphere opposite to the hemi-spherical shell: Horizontal attraction h = (2·2054 sin ✪ + 1·9842 sin 20 + 0·7606 sin 30) — g ... (3). = As the tangential attraction of a whole spherical shell on any point is zero, it follows that the tangential attraction of * The first point is here taken 91° and not 90° (that is, 1° or about 70 miles from the edge of the shell) because otherwise the square of the ratio of the height of the mass on the nearest compartments to the distance from the point attracted could not be neglected. See end of Art. 60. a hemi-spherical shell on any point on its own surface will equal the above with its sign changed: or if o be the angle from the pole of the shell it will be h (2·2054 sin † — 1·9842 sin 24 + 0·7606 sin 34) 9 and being reckoned in each case from the pole of the shell. In each case the attraction is reckoned positive towards the pole of the attracting meniscus. Ex. 5. Suppose we take a hemi-spheroidal meniscus of thickness h at its edge, and no thickness at the pole. The attraction of this will be found by subtracting the results of Ex. 3 from those of Ex. 4: they give h (2·0608 sin @ + 1·8884 sin 20 +0-7362 sin 30)g... (5), a h and (2·0608 sin - 1·2884 sin 24+0·7362 sin 34) — g 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 plumbline in the Philosophical 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. This has been recently verified by the observations and calculations of Professor Schmeizer, who has shown. that within a distance of 19 miles the plumb-line varies by 16" near Moscow without any apparent cause, and that it varies in such a way as to indicate a deficiency of matter EFFECT OF EXCESS OR DEFECT IN MASS BELOW. 61 below. See Monthly Notices of Ast. Soc. Ap. 1862. The following Proposition, with which we shall close this Chapter, will show that this is possible. 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 a four-sided space drawn upon the surface of the earth, bounded on two sides by great circles passing through the station where the plumb-line is and making an angle B with each other, the other two sides being parts of circles of which the station is the centre; let & be the angular distance of these two circles measured along the surface, and the distance of the middle of o from the station. We shall take 2° 52′ 40′′ (= 200 miles), and B = 30°, and shall find how small may be that a mass of small uniform height covering the space should attract the station as if it were collected into the middle point of p. The area of the space = = a2ß {cos (0 — 14) — cos (0 + 14)} = 2a2ß sin 10 sin 0, = and the chord between the mid-point and the station being 2a sin 10, the attraction of the mass collected at the midpoint and resolved along the tangent = 2pha3ß sin 1 sin cos2 10 cos 10 = phẞ sin 10 sin 10 But by Art. 58 the attraction of the mass Put ẞ= 30o = π÷ 6, 4 = 2° 52′ 40′′ = 0·016π ; .. 4 sin3 10 = 3 sin2 0 (0·01 +0·012) = 0·066 sin2 0 ; .. sin 08 sin 10 nearly; .. 0=24 = 400 miles. = = Hence the centre of the space may be as near as 400 miles to the station, and yet the whole mass be supposed to be collected into its centre. The area a2ßp sin 0=2a2ßp2 = (ap)2 very nearly = (200) miles, or the space is equal to a square of 200 miles each way.. 65. Now suppose the height of the matter on this space to be 1 mile, and suppose every small vertical prism of it to be distributed uniformly downwards into a slender prism to a depth d. Thus the whole superficial mass 1 mile thick will be distributed through a depth d, and form an attenuated mass the density of which is one dth part of that of the superficial rock. As the mass at the surface may be collected into its middle point, much more may that in any horizontal section of this attenuated mass, because the section is further from the station than the space at the surface. Hence the whole attenuated mass will attract the station as if it were collected uniformly into one vertical prism drawn down from the central point of the surface to the depth d. Let u and v be the distances of the extremities of this prism from the This will also be approximately the horizontal attraction for all distances not exceeding 30° from the station. Hence deflection of the plumb-line EFFECT OF EXCESS OR DEFECT IN MASS BELOW. 63 Ex. We may give any values to u and v so long as u is not less than 400 miles. We shall take u= 400, 600, 800, 1000 miles successively. The calculation will be facilitated by using a table of tangents and secants, observing that u÷d is the tangent of the angle of which v÷d is the secant. Hence the following Table: Depth Distance of the mid-point of the space from the in miles. This Table enables us, with the formula above, to tabulate the deflections as follows: The densities of the masses distributed through the depths 100, 200, 300, 900, 1000 miles are severally inversely proportional to those numbers. Hence by multiplying the lines of numbers in this table successively by 1, 2, 3, 9, 10 we shall have the deflections of masses having the same volumes as |