680 Mr. stokes, on THE VARIATION OF GRAVITY 



Equations (22) shew that the co-ordinate axes are principal axes. Equations (23) give in 

 the first place 



fffa!"dm'=fffy"dm', 



which shews that the moments of inertia about the axes of x and y are equal to each other, as might 

 have been seen at once from (22), since the principal axes of ,t and y are any two rectangular axes 

 in the plane of the equator. The two remaining equations of the system (23) reduce themselves to 

 one, which is 



fffx'dm' - fff^'^dm' = |(e - ^m)Ea\ 



If we denote the principal moments of inertia by J, A, C, this equation becomes 



C -A = ^(e-^m)Ea\ (24) 



which reconciles the expression for the couple .ff" given by (18) with the expression usually given, 

 which involves moments of inertia, and which, like (18), is independent of any hypothesis as to the 

 distribution of the matter within the earth. 



It should be observed that in case the earth be not solid to the centre the quantities A, C must 

 be taken to mean what would be the moments of inertia if the several particles of whicli the earth 

 is composed were rigidly connected. 



12. In the preceding article the surface has been supposed spheroidal. In the general case of 

 an arbitrary form we should have to compare the expressions for F given by (10) and (ig). In the 

 first place it may be observed that the term i<, can always be got rid of by taking for origin the 

 centre of gravity of the volume. Equations (20) shew that in the general case, as well as in the 

 particular case considered in the last article, the centre of gravity of the mass coincides with the 

 centre of gravity of the volume. 



Now suppress the term ?«, in ii, and let u = u' + u", where u" = gW'C 3 ~ cos' 0). Then ?«' may 

 be expanded in a series of Laplace's coefficients ti'.^ + ii\ + ... ; and since Fq = E, equation (10) will 

 be reduced to 



V = E(-+-^u, + -^tc',...) (25) 



If the mass were collected at the centre of gravity, the second member of this equation would 

 be reduced to its first term, which requires that u',. = 0, it's = 0, &c. Hence (8) would be reduced 

 to r = a(l + u"), and therefore au" is the alteration of the surface due to the centrifugal force, and 

 au the alteration due to the difference between the actual attraction and the attraction of a sphere 

 composed of spherical strata. Consider at present only the term ?('„ of u'. From the general form 

 of Laplace's coefficients it follows that ati'., is the excess of the radius vector of an ellipsoid not much 

 differing from a sphere over that of a sphere having a radius equal to the mean radius of the ellipsoid. 

 If we take the principal axes of this ellipsoid for the axes of co-ordinates, we shall have 



u'2 = 6'(^ - sin' 9 cos^<p) + e"(^ - sin- 9s\a-(p) + e"\^ - cos' 6), 

 €, e", e'" being three arbitrary constants, and 9, (p denoting angles related to the new axes of x, y, z 

 in the same way that the angles before denoted by 0, tp were related to the old axes. Substituting 

 the preceding expression for ii^ in (25), and comparing the result with (19), we shall again obtain 

 equations (22). Consequently the principal axes of the mass passing through the centre of gravity 

 coincide with the principal axes of the ellipsoid. It will be found that the three equations which 

 replace (2.S) are equivalent to but two, which are 



A - yEa' = B - le"Ea = C - |e"'£o^ 

 where A, B, C denote the principal moments. 



The permanence of the earth's axis of rotation shews however that one of the principal axes of 

 the ellipsoid coincides, at least very nearly, with the axis of rotation; although, strictly speaking, tills 



