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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.
u U cot cos=
= cot cos=à
0.250 0.242 0.277 0.267 0.303 0.290 0.333 0.316 0.367 0.344 0.403 0.374 0.443 0.405 0.487 0.438 0.537 0.473 0.590 0.508 0.650 0.545 0.713 0.581 0.783 0.617 0.863 0.653 0.950 0.689 1.043 0.722 1.150 0.755 1.263 0.784 1.390 . 0.812 1.530 0.837 1.683 0.860
0.125 0.124 0.138 0.137 0.152 0.150 0.166 0.164 0.184 0.181 0.201 0.197 0.222 0.217 0.243 0.236 0.269 0.260 0.295 0.283 0.325 0.309 0.356 0.335 0.392 0.365 0.431 0.396 0.475 0.429 0.522 0.463 0.575 0.498 0.631 0.534 0.695 0.571 0.765 0.608 0.842 0.644
0.083 0.083 0.092 0.092 0.101 0.100 0.111 0.110 0.122 0.121 0.134 0.133 0.148 0.146 0.162 0.160 0.179 0.176 0.197 0.193 0.217 0.212 0.238 0.232 0.261 0.253 0.288 0.277 0.317 0.302 0.348 0.329 0.383 0.358 0.421 0.388 0.463 0.420 0.510 0.454 0.561 0.489
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 35.385, 2".088, 1".478. The densities of the matter thus 299th,
diffused in these three cases are about
adoth, of the density of the superficial rock. If we multiply the above deflections by 2, 4, 6, we have the deflections caused by matter, of 1-100th the density of superficial rock, distributed over the three depths, equal to 6".770, 85.352, 8".868. Retaining the first, subtracting the first from the second, and the second from the third, we have the three deflections caused by a mass of 208 miles vertical thickness (occupying 3-100,000 th parts of the volume of the whole earth and of density 1-100 th part of the density of the surface) the centre of which is at depths 104, 312, 521 miles : they are 6".770, 1".582, 0”.516. We may finally change the comparison between the density of this space, 3-100,000 ths of the earth’s volume, in its three situations, with the density of surface, to a comparison with the average density of the earth itself at the several depths at which the centre of the space lies. .
If D be the density of the surface, a the radius of the earth, the usually received law of density of the interior, determined from the Auid-theory, is
2aD Density at depth d sin
When d=100, 300, 500 miles, this gives the densities
1.14D, 1.43D, 1.71D. Multiplying the last angles by the ratios of these densities to D, we have finally the Deflections—caused by an excess or defect of matter, prevailing through a space equal to 3-100,000 th parts of the volume of the earth and 1-100th part of the earth's density at the centre of the space-equal 7".7, 2".3, and 0".9, the depths of the centre of the space being about 100, 300, 500 miles.
The form of the space in its three positions is shown in the diagram; viz. DG, FI, HK; and 0, P, Q are their centres. The particular form arises from the manner in which the mass is dissected, so as to make the calculation feasible. The result serves to show the kind of effect which slight but extensive variations from the density of fluid-equilibrium in the hidden regions below may have upon the plumb-line : and we shall find the use of this when we come to consider the Figure of the Earth : (see Art. 98).
66. Had the width DB been equal to the middle width at a, so as to make the boundaries BC, DE parallel, the effect
would have been very much greater. Moreover the defect or excess in density which we have taken, viz. 1-100 th, might have been chosen larger, and the deflections proportionably increased. For there are many kinds of rock, as granite, which differ so in density in the different specimens that the difference between the extremes is greater even than 1-10th of the mean.
And if this difference exists at the surface, it does not seem to be improper to suppose that great variations may exist also below, from the effect of the cooling down and solidifying of the crust, even much greater than 1-100 th.
FIGURE OF THE EARTH.
THE FIGURE OF THE EARTH CONSIDERED AS A FLUID MASS, AND THEREFORE CONSISTING OF STRATA
67. AFTER it was known that the earth is of a globular form, Newton was the first who demonstrated that it is not a perfect sphere. From theoretical considerations and also from the discovery that a pendulum moves slower at the equator than in higher latitudes, he arrived at the conclusion that its form is that of an oblate spheroid. This subject we propose to consider_fully in the present Chapter, on the hypothesis that the Earth was a fluid mass when it assumed its present general form. The calculation is one of great difficulty, and would indeed be impracticable did we not know that the figure differs but little from a sphere.
As a first approximation we shall inquire whether a homogeneous fluid mass revolving about a fixed axis can be made to maintain a spheroidal form according to the laws of
PROP. A homogeneous mass of Auid in the form of a spheroid revolves with a uniform velocity about an axis : required to determine whether the equilibrium of the surface left free is possible.
68. Let a and c be the semi-axes of the spheroid referred to three axes of rectangular co-ordinates, c being that about which it revolves: also let d = a' (1-6). The forces which act upon the particle (xyz) are the centrifugal force and the attraction of the spheroid parallel to the axes: these latter are given in Art. 12, and are