Since the node on the whole retrogrades pretty steadily, we may put ht on the second side of these equations, h being the mean regression. Hence & being the symbol of variation in i and , .. dλ = sin (nt + e' — N) di — i cos (nt + e' — N) EN This result differs but slightly from the measure obtained by geodesy; it is a little too small. But considering the minuteness of the quantity to be determined, the result is remarkable, and bears its testimony to the truth of the fluidarrangement of the earth's mass. Third Test. THE ELLIPTICITY OF THE SURFACE. 102. For the application of this and the next test the formulæ cannot be obtained without assuming a law of density in the distribution of the earth's mass. PROP. To find a law of density of the earth's mass. 103. In order to make the equation in Art. 80 for determining the ellipticity integrable, it is necessary to assume a law of density of the mass of the earth. Experiment has not yet determined what the law of compression in such a mass would be. integrable, we must assume the last term to be some multiple of (a) e, the other factor being a function of a. Suppose the negative sign being taken because the density decreases from the centre to the surface; q' is some function of a. The equation for the ellipticity becomes The only case in which this equation has been integrated is when q is constant. We must therefore assume it to be so*. The equation for the density then gives by differentiation and re-adjustment d2.pa da +q2. pa=0; .. pa= Q sin (qa + A). But p would be infinite at the centre unless A = 0. Hence * In order to explain how to differentiate a definite integral with respect to a quantity involved in the limits, let ƒƒ(x) = F (x) + const. ; In the case of the earth the constants Q and q must be found from the conditions, that the density at the surface is the density of granite; and that the mean density of the whole is twice that density. This last leads to the formula, 2 = mean density ÷ superficial density = mass ÷ (volume × superficial density) If D be the density of the surface or 2.75, then 104. If the law of density deduced in Art. 103 be used in equation (1) of Art. 75, then, neglecting the small term, or the increase of pressure varies as the increase in the square of the density. It was by assuming this law between pressure and density, that Laplace deduced the law of density which we have arrived at in another way, viz. by finding what condition is necessary to make the equation of ellipticities integrable (Mémoires de l'Institut, Tom. III. p. 496). PROP. To find an expression for the ellipticity of the strata, with the law of density deduced in the last Proposition but one, and to reduce it to numbers for the surface. 105. In order to integrate the equation of ellipticities in Art. 80, put $ (a) e d. p (a) € α 0 a'x'da'2 ; α α Multiply by a2 and differentiate; dx da + 2ax − 3ax+ qoa ["a'x'da' = 0. Divide by a and differentiate, and then divide by a; d2x The solution of this is x + Cq2 sin (qa + B) = 0, C and B being independent of a ; · ["a'a'da' = Cqa cos (qa + B) — C' sin (qa + B) ; In our case B=0, otherwise the ellipticity at the centre would be infinite, as is easily seen by expanding e in powers of a. Hence, if we substitute for (a), the ellipticity And the ratio of this to the ellipticity of the surface This gives the law of decrease in the ellipticity of the strata in passing down from the surface to the centre. By Art. 81, e being now the ellipticity of the surface, {' a'4 (a) (e — '}' m) = ['p'da, (a'e') da'= Q f* a sin qa' d a da' (a'e') da' a'3é (sin qa' - qa' cos qa') da'} by parts. = Q {a‘e sin qa + ["a"e' (sin qa' Substituting for e' from the ratio of ellipticities above, integrating and reducing, the integral in this expression (tan qa — ga) sin qa {69′a' — 15 – 9'a2 – 157a) 3 1 tan qa + qa 3 qa tan qa |