The question then arises, Can the solid shell support the tangential thrust to which it would be thus exposed? By the application to this problem of an elegant theorem of Lagrange, I have proved that it cannot possibly do so, no matter what may be its thickness nor what its material, even were we to assume the latter not merely of the hardest and most resistant rocks we know anything of, but even were it of tempered cast-steel, the most resistant substance (unless possibly iridio-osmium exceed it) that we know anything about. Lagrange has shown that if P be the normal pressure upon any flexible plate curved in both directions, the radii of these principal curvatures being ρ' and ρ'', and T the tangential thrust at the point of application and due to the force P, then: P = T (1/ρ' + 1/ρ'') When the surface is spherical, or may be viewed as such, ρ' = ρ'' and P = 2T/ρ or, T = P × ρ/2 In the present case P is for a unit square (taken relatively small and so assumed as plane) of the shell, suppose a square mile, equal to the effect of gravity upon that unit, ρ being the earth's radius, and if we assume the unit square be also a unit in thickness, P is then the weight of a cubic mile of its material; and if we take (roughly) the earth's radius as 4,000 miles, the tangential pressure, T, is, on _each face_ of the cubic mile, equal to (4000/2) P, or equal to the pressure of a column of the same material of 2,000 times its weight. If the cubic mile that we have thus supposed cut out of the earth's crust at the surface were of the hardest known granite or porphyry, it would be exposed to a crushing tangential pressure equal to between 400 and 500 times what it could withstand, and so must crush, even though only left unsupported by the nucleus beneath, to the extent of 1/400 or 1/500 of its entire weight. And what is true here of a mile taken at the surface, is true (neglecting some minute corrections for difference in the co-efficient of gravity, etc.) if taken at any other depth within the thick crust.[F] The crust of our earth, then, as it now is, must crush, to follow down after the shrinking nucleus--if so be that the globe be still cooling, and constituted as it is; even to the limited extent to which we know anything of its nature--it must crush unequally, both regarded superficially and as to depth; generally the crushing lines being confined to the planes or places of greatest weakness; and the crushing will not be absolutely constant and uniform anywhere, or at any time, or at any of those places of weakness to which it will be principally confined, but will be more or less irregular, quasi-periodic, or paroxysmal: as is, indeed, the way in which all known material substances (more or less rigid) give way to a slow and constantly increasing, steady pressure.