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. 79) that the strata decrease in ellipticity in passing downwards : hence é' – € 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 E 7 7 g €; and therefore, because the ellipticity decreases in descending, the thickness must be greater than would correspond with an 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 & in Art. 78, and make qa= 150°, and the mean density = 2.4225 times the superficial density (the second of the values in Art. 80), then if ε = 3 the formula of Art. 78, we have, after reduction, a= 4 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 7 e in a, or any depth will be more elliptic than the surface of equal density at that depth : and therefore the inner surface of the 7 solid shell, of which the ellipticity is €, must be at a depth 8 corresponding to a stratum of equal density of smaller ellip 7 ticity than that is, at a greater depth than 1000 miles. 8 In the above reasoning B 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. 85. Professors Hennessy and Haughton have both written upon this subject: see Phil. Trans. 1851, and Transactions of the Royal Irish Academy, 1852. The first makes the thickness be 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.) PROP. To show what influence the present aspect of the surface of the Earth has upon the argument for the thickness of the crust. 86. The following considerations are sufficient to show that the crust of the Earth must at the present date be very thick. The above diagram represents a vertical meridian section of Hindostan, the Ocean, and the crust of the Earth, through 0, or Cape Comorin. AbcD is the average form of the mountain mass: AB= 140, BC= 230, Bb= Cc=2.5 miles : mn=t, Ar=ť the thickness of the crust below m (any point on the table-land) and A: arc AM=a, area of AbmM=K; G its centre of gravity, Gg vertical; rg=k, perpendicular to Gg; Mm=h; ħ and ke the middle points of mn and Ar; he perpendicular to mn; re=y, to he. The mass Mr is held in equilibrium by its weight, the downward pressure of the overlying mass MA, the upward pressure of the fluid below, and the force of adhesion at the joints mn and Ar. Since the crust has, by hypothesis, been formed by the solidification of the fluid, its density at any point will be very much the same as the fluid was at that point. We will at present assume it to be the same. Hence the weight of Mr = the upward pressure of the fluid, and the weight of MbA tends to break the crust, and is sustained by the adhesion at the joints. Let C be the length of rock, of a unit section, the weight of which equals the average force of adhesion on a unit of surface. C= 1-5th mile, may be considered to be the greatest limit of C (see Phil. Trans. 1855, p. 102). If the point m sink, the joints Ar and mn will open at A and n, and an opening will take place at some other point on the left. The equation of moments of the forces acting on Mr taken about r is K. k=C.t.y+C.ť.ft =C.t{r- ft+h-(r - t') cos a} + {C.t"; :. 2K.k={t"? + 2te' cos a – + +2 (r vers a + h) t} C... (1). Take the case of m being at c, then k=rg = 255 miles : a= 5° 19', cosa = 0.9957, r vers a = 17.2 miles: take K=C6 (omitting ABb), this = 230 x 2.5 = 575 square miles; h=2.5; :t'? + 2t't - † + 39.4t =1466250 = (1212) nearly. If t be very small, ť = 1212 nearly; this is a condition which no law of cooling could bring about. Also ť cannot be small, otherwise t would be negative. If t = t', then each is greater than 800 miles. Formula (1) may be applied to find the least thickness of the crust beneath S, any point on the Ocean south of Cape Comorin, to prevent its bed Os being broken up by the lava from below. 'Make 0 the centre of moments: Op=ť, sq=t, OS=d, Ss=-h. Suppose that the depth of the Ocean increases uniformly with the distance from 0, and is 3 miles at 25° distance from 0, i.e. between Madagascar and Australia ; then h=14 sin la. Also K= area OSs; and K. k, the moment about 0 of the several elementary portions, = 18.7r* sino ja by integration. Hence formula (1) becomes t2 + 2tt' cos a –ť + 2 (r vers Q – 14 sin ja) t=187m2 sin fa. If a is so taken, that the coefficient of t may be neglected, .. t2 + 2t't cos a –ť = 1877* sin ja. As before, neither t nor t can be small. If t= t'; then each = r sin 10 V 93.5 sin ta = 1000 miles, when a = only 10'. In this case the coefficient of t= 118, which may be neglected. Therefore, as before, the thickness must be very great to prevent the crust being broken through. It has been assumed, that the density of the crust is everywhere the same as the fluid from which it was formed by solidification. Suppose, however, that it is more dense, then the tendency of the crust to break, in the first case, will be greater than we have made it, though the tendency in the second will be less. The reverse will be the case if the crust is lighter than the fluid from which it is formed. So that in any case a fracture must take place somewhere, either beneath the mountains or beneath the ocean, unless the thickness is very great. As both the mountains and the oceanbed retain their positions of elevation and depression, we have no alternative to choose but that the thickness of the crust is very great. 87. The result of the whole proves that the crust must be very thick: and, as Mr Hopkins's calculation appears to be free from objection, and in fact to be the only one on which any reliance can be placed, we may conclude that the thickness is at least 1000 miles. The present form of the surface in mountains, table-lands, continents, and oceans has been, no doubt, acquired from a process of expansion and contraction which the crust has undergone during the ages since it was first consolidated. Geology teaches us that these elevations and depressions of vast regions are at this present day going on. We may, therefore, fairly conclude-especially with this evidence that the crust is so thick-that the present varieties of the Earth's contour have grown from this cause, and have not arisen in any way from the operation of hydrostatic principles. This does not in any way contravene the hypothesis that the Earth was once a fluid mass, and has received its general figure from that condition. The fact that its mean form, as measured by geodetic operations, coincides with the fluid-form calculated upon an assumed, but (à priori) very probable, law of density, is an unanswerable argument in favour of the hypothesis of original fluidity. And the coincidence of the calculated amount of Precession, on this law of density, with its observed amount, is a very strong evidence that that law of density is the law of nature. |