THICKNESS OF THE EARTH'S CRUST. 109 111. The argument for the law of density hence deduced is, however, not very strong. For, as the strata are nearly spherical, almost any law might lead to right results by chosing the constants involved in the law rightly; especially as those results are the resultant effects of the whole mass, and not of the parts taken separately. 112. The Four Tests applied in the last twenty-four Articles give the following values of the ellipticity : § 4. The thickness of the Earth's Crust. 113. In the next Chapter we shall test the accuracy of the theoretical results we have now arrived at, considering the earth as a fluid mass, by an appeal to actual measurements. That it is not wholly fluid now, the existence of continents and ocean-beds attests. It is a question how far down this solidity prevails. It is supposed by some that the crust may be of comparatively small thickness, so thin as to be influenced in its position of elevation and depression by the fluid mass below, on which it is in fact imagined to float. Mr Hopkins has endeavoured to ascertain how far the interior of the Earth may at present be fluid, by calculating the value of the Precession upon the supposition of the mass being a spheroidal shell of heterogeneous matter, enclosing a heterogeneous fluid mass, consisting of strata increasing according to the law we have used. In three memoirs in the Philosophical Transactions of 1839, 1840, and 1842, he enters upon a complete investigation of this subject. We will give the evidence upon which he rests his conclusion that the crust is very thick. PROP. To trace the argument drawn from Precession to show that the crust is of considerable thickness. 114. Mr Hopkins has deduced the following formula (in which we have changed the notation to suit the present where P is the precession of the equinoxes of a homogeneous spheroid of ellipticity e, which by calculation 1 = 57" nearly if € = ; P' is the precession of the heterogeneous shell, the outer and inner ellipticities being e and e': this = 50"•1 by observation. 300 The success of the calculation depends upon a remarkable result at which he has arrived, that the precession caused by the disturbing forces in a homogeneous shell filled with homogeneous fluid, in which the ellipticities of the inner and outer surfaces are the same, is the same whatever the thickness of the shell. It is therefore the same for a spheroid solid to the centre. The formula above given is the relation of the amounts of precession in two shells, one heterogeneous and the other homogeneous; and, as the thickness is the quantity sought, neither of these amounts could be calculated, and therefore the relation expressed in the above formula would be of no avail. But in consequence of the property that the precession of the shell, when it and the fluid are homogeneous, is the same as that of the spheroid, this difficulty is overcome; and P can be calculated without knowing the thickness, and therefore P' will be known. We have shown (Art. 82) that the strata decrease in ellipticity in passing downwards: hence e"- e' is never negative, and the fraction on the right hand in the above formula is never negative, and is never so large as unity: let it ß. Hence = and therefore, because the ellipticity decreases in descending, the thickness must be greater than would correspond with an THICKNESS OF THE EARTH'S CRUST. 111 ellipticity of the inner surface of the shell equal to 7-8ths of that of the outer surface. If solidification took place solely from pressure, the surfaces of equal density would be surfaces of equal degrees of solidity. If we use the formula for finding e in Art. 105, and make qa = 150°, and the mean density = 2.4225 times the superficial 7 density (from Mr Airy's Harton calculation), then if é= ↓ e in the formula of Art. 105, we have, after reduction, a = 3 4 8 a, or the thickness equal to one fourth of the radius, or 1000 miles. If a smaller ratio of densities is used than 2.4225, the thickness is greater. (Mr Hopkins shows also that a ratio a little larger than 3 makes the thickness 1-5th of the radius: but this ratio is too large. The ratio generally used is about 2-2). But solidification depends upon temperature, as well as upon pressure. In his third memoir (Phil. Trans. 1842), Mr Hopkins shows that the isothermal surfaces increase in ellipticity in passing downwards. If temperature alone regulated the solidification, these surfaces would be the surfaces of equal solidity. But since both pressure and temperature have their effects, the ellipticities of the surfaces of equal solidity must lie between those of the isothermal and the equi-dense surfaces. Hence the surface of equal solidity at any depth will be more elliptic than the surface of equal density at that depth: and therefore the inner surface of the solid shell, of which the ellipticity is e, must be at a depth 7 8 corresponding to a stratum of equal density of smaller ellipticity thane, that is, at a greater depth than 1000 miles. 7 In the above reasoning ẞ has been neglected. If its value be used, it strengthens the argument for a greater thickness than 1000 miles. We may, therefore, safely conclude that 1000 miles is the least thickness of the solid crust. In the calculation it has been assumed that the transition from the solid shell to the fluid nucleus is abrupt. This will hardly be the case. The above result will therefore apply to the effective surface, lying near the really solid shell. But in consequence of the tendency, as shown above, of every cause being to prove that the crust is really thicker than 1000 miles, we may safely take this to be its least limit*. 115. Professors Hennessy, Haughton and W. Thomson have written upon this subject: see Phil. Trans. 1851, Transactions of the Royal Irish Academy, 1852, and Phil. Trans. 1862. The first makes the thickness lie between 18 and 600 miles. But in his calculation he assumes that the shell is so rigid as to resist, without change of form, the internal pressure which arises from the inner surface ceasing to be one of fluid equilibrium: an assumption which cannot be considered admissible. Moreover he supposes that in cooling the outer shell will contract less than the fluid nucleus; which can hardly be true. Mr Haughton's investigation is simply a problem of densities, and determines nothing whatever regarding the ratio of the solid to the fluid parts of the Earth. (See Philosophical Magazine, Sept. 1860.) Mr Thomson, in his paper on the "Rigidity of the Earth," confirms Mr Hopkins' result, as will appear from the investigation with which this Chapter closes. 116. In the last edition we gave an argument in favour of a great thickness, arising from the tendency of the weight of the enormous mass of the Himmalaya mountains to break down the crust if it is thin; and of the fluid mass below extensive ocean-beds to burst them up under the same circumstances. Subsequent calculations, however, on another subject have shown that the crust below the mountains must be of less density, and that below ocean-beds of greater density than the average, so as to produce a very considerable compensation. If this is the case the downward pressure in the first instance *It will be observed that the ellipticity of the surface of the earth is in these pages always denoted by the symbol e, the mean radius of the surface being a. Owing, however, to the want of variety of Greek type this same symbol has occasionally been used to denote the ellipticity of an internal stratum of nean radius a. No confusion will arise from this, as the meaning of the symbol in each case is sufficiently obvious. RIGIDITY OF THE EARTH. 113 and the upward pressure in the second might be too feeble to produce fracture even if the crust be thin. (See Arts. 132-135.) 117. We will conclude this Chapter by giving some account of Professor W. Thomson's remarkable investigation regarding the "Rigidity of the Earth," in the first place making a calculation which will be of use. He accepts the result of Mr Hopkins' calculation, and indeed thinks that it might have been pressed further. His own investigations, an abstract of which we now give, point out that the earth's mass must possess such a degree of rigidity as to be altogether inconsistent with a crust of moderate thickness. PROP. A mass in the form of a prolate spheroid of small ellipticity consists partly of a nucleus of spherical shells concentric with the spheroid, the remaining portion of the spheroid being homogeneous: to find the form of the level surface, or surface at every point of which the resultant attraction of the whole mass acts in the normal, passing through the equator of the spheroid. 118. Let a and c = a (1+e) be the semi-axes, suppose a sphere of radius a to be inscribed touching the spheroid in its equator: p the density of the outer part of the spheroid, N.p the mean density of the mass within the inscribed sphere. We may conceive the mass made up of a homogeneous prolate spheroid of density p, and a concentric sphere of radius a and mean density (N-1) p. Then by Art. 14, the attractions of the homogeneous spheroid parallel to the axis on a point xyz 4 3 3 4 πρ 4 3 3 πρ If X, Y, Z are the total attractions parallel to the axes at the point xyz, then, that their resultant at that point may act in the normal, we must have P. A. Xd. + Ydy + Zdz = 0, 8 |