AT THE SURFACE OF THE EARTH. 679 



whence we get from (l6) 



^=^' +^ (« " a'») 75- (i- cos°0), Q = 2(e-lm)^-sin6lcos0 (17) 



The moving forces arising from the attraction of the earth on the moon will be obtained by 

 multiplying by M, where J/ denotes the mass of the moon ; and these are equal and opposite to 

 the moving forces arising from the attraction of the moon on the earth. The component J\IQ of 

 the whole moving force is equivalent to an equal and parallel force acting at the centre of the earth 

 and a couple. The accelerating forces acting on the earth will be obtained by dividing by E ; and 

 since we only want to determine the relative motions of the moon and earth, we may conceive equal 

 and opposite accelerating forces applied both to the earth and to the moon, which comes to the 

 same thing as replacing E by E + M in (17). If A' be the moment of the couple arising from the 

 attraction of the moon, which tends to turn the earth about an equatoreal axis, A' = AfQr, whence 



. . MEa- . ^ 

 A = 2(e - im) — sm6 cos 9 (18) 



The same formula will of course apply, mutatis mutandis, to the attraction of the sun. 



11. The spheroidal form of the earth's surface, and the circumstance of its being a surface of 

 equilibrium, will afford us some information respecting the distribution of matter in the interior. 

 Denoting by ,r', y', z' the co-ordinates of an internal particle whose density is p, and by a;, y, z 

 tho.se of the external point of space to which V refers, we have 



V 



'Iff 



p dx dy dz' 



|(.^•-,^•')^ + (y-2/T + (^-^')■1*' 



the integrals extending throughout the interior of the earth. Writing dm for p' dx dy dz, 



putting X, n, V for the direction-cosines of the radius vector drawn to the point {x, y, z), so that 



■r = Xr, y = fir, z = vr, and expanding the radical according to inverse powers of r, we get 



1 A 1 ^ ' 



''= -fffdm' y'S.-Jifx dm' +-^^1{3X' - 1) ffjx"- dm' + — l.X/x JJfw' y dm' + ...(19) 



S denoting the sum of the three expressions necessary to form a symmetrical function. Comparing 

 this expression for V with that given by (10), which in the present case reduces itself to (iG), we 

 get Y„ = fjjdm' = E, as before remarked, and 



fffx'd?n' = Q, fffy' dm' = 0, jj'jz'dm = 0, (20) 



12(3V-- 1) fffx'-dm' + S^Xiu fffx'y'dm' = (e-^ra) Ea-(^- cos=0); (21) 



together with other equations, not written down, obtained by equating to zero the coefficients of 



i,i&c.in(l9). 



Equations (20) shew that the centre of gravity of the mass coincides with the centre of gravity 

 of the volume. In treating equation (21), it is to be remarked tliat X, (u, f are not independent, but 

 connected by the equation X" + //" -t- i/° = 1. If now we insert X^ + n' + v' as a coefficient in each 

 term of (21) which does not contain X, iuL,or v, the equation will become iiomogeneous with respect to 

 X,(i, I', and will therefore only involve the two independent ratios which exist between these three 

 quantities, and consequently we shall have to equate to zero the coefficients of corresponding powers 

 of X, /J, V. 15y the transformation just mentioned, equation (21) becomes, since cos ti = v, 



S (X'^ - J ^/ - ^v") ///V'' dm' + :s-:i\,xfffx'y'dm' = (e- ^m) £0' (^X* + 1^' - f"') J 

 and we get 



fffvy'dni' = 0, fffy'z'dm' = 0, fffz'x'dm' = 0, (22) 



Sf{a:'"-dm' - }Jffy'"-dm' - yffz'^-dm' = fffy'^dm' - yffz'^dn,' - IJffx'dm'] 



= -ijjfz"dm' + iffjx"dm' + Uffy'dm' = }^{c - ^m) Ea\ \ 



4 s 2 



