= by (4) 4, 0°.075; therefore a,=0°.825: and by proceeding in this way we obtain the following pairs of values by this formula. = $17=0.345 42 0.40 = = 3.419 a41=35.26 $41 = 3.600 38.86 24.314 That these results derived from the approximate formula (4) are thus far correct, we gather from the fact that the last pair, viz. a, 38°.86 and 4°.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. 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= of the longer station mass u2 and (by Art. 48) that Hence along the horizontal line at the attraction of slender prism = attraction of the prism at surface v 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 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), If B 30°, the coefficient = 1".1392 × 0.2580". 294. = = 65. We will take an example. Let the width of the lune B=30°, and let the 21 compartments from a, to a,, (see Table in Art. 60) be included. This will be a tract of country 5°.55 -0.75 -4°.8, or 334 miles in length, and the breadth at the mid-point will sin (5°. 55 +0.75) x the length of 30°= 0.055 × 30 × 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. 3o, 6o, and 9°; which nearly equal 208, 417, and 625 miles. The vertical thicknesses of these three divisions of the frustum are each = 3o 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. = Now since the greatest of u,u, take the arc for the chord without respect to all these quantities, и v = is less than 6o, we may U21 sensible error. Then, with the cosine of the angle of which is the cotangent. This enables us without difficulty d with the help of a Table of cosines and co-tangents to form the sum of the series in the last Article. The values of u are the first 21 values of a in the Table in Art. 60. From this Table we gather, that the Deflections caused at the station by the superficial mass one mile thick, when distributed uniformly through the depths 208, 417, 625 miles, are 3".385, 2".088, 1".478. The densities of the matter thus |