404 VIII. NEWTONIAN POTENTIAL FUNCTION. 



where M is the mass, a, l>, c the coordinates of the center of mass, 

 A, B, C, D, E, Fj the moments and products of inertia of the body 

 at the origin. If we choose for origin the center of mass, and for 

 axes the principal axes of inertia at that point, we have 



a = b = c = D = E=F= 0, 



so that the second term of the development disappears, and the third 

 simplifies, so that we have 



* -* -2C)z* 



In all these developments, it is to be borne in mind that r is 

 greater than the greatest value of r r for any point Q in the body. 



If the body is a homogeneous sphere, all terms disappear except 

 the first. If the attracted point is at a considerable distance compared 

 with the dimensions of the attracting body, or if the body differs 

 but slightly from a sphere, the terms decrease very rapidly in 

 magnitude, so that the first is by far the most important. Thus 

 under these circumstances bodies attract as if they were concentrated 

 at their centers of mass, or were centrobaric ( 125). The correction 

 is in any case in which we are dealing with the actions of the planets, 

 given with sufficient accuracy by the second term in 133), from 

 which the moments causing precession were calculated in 96. In 

 161 we shall see how the terms depend upon the ellipticity of an 

 ellipsoid of revolution. 



149. Applications to Geodesy. Clairaut's Theorem. 



Although, as has been stated, the development 125) is not in general 

 convergent inside of a sphere with center at the origin which just 

 encloses the attracting body, on account of the divergence of the 

 series 105) when r r > r, still it may occur that the performance of 

 the integrations in 125) causes the latter series to converge even 

 within this sphere. At any rate for a body having the properties of 

 the earth, it has been shown by Clairaut 1 ), Stokes 2 ), and Helmert 3 ), 

 that the series 125) converges at all points on the surface of the 

 body, and also that for the earth the two terms in 133) represent 

 the attraction with quite sufficient approximation for applications to 

 the figure of the earth. In order to exhibit the surface harmonics 



1) Clairaut, Theorie de la Figure de la Terre, tiree des Principes de 

 I'Hydrostatique. Paris, 1743. 



2) Stokes, "On the Variation of Gravity at the Surface of the Earth.' 

 Trans. Cambridge Phil. Soc., Vol. VHI, 1849. 



3) Helmert, Geoddsie. 1884. 



