676 Mr. stokes, on THE VARIATION OF GRAVITY 



Now the differential coefficients of V„ with respect to r are Laplace's coefficients of the n"' 

 order as well as V„ itself; and since a series of Laplace's coefficients cannot be equal to zero unless 

 the Laplace's coefficients of the same order are separately equal to zero, we must have 



r~^"-nin+l)V„ = (6) 



ClT 



The integral of this equation is 



»^»=4+2»'-' 



where ¥„ and Z„ are arbitrary constants so far as r is concerned, but contain 9 and (p. Since these 

 functions are multiplied by different powers of r, V„ cannot be a Laplace's coefficient of the n"" 

 order unless the same be true of V„ and Z„. We have for the complete value of F 



Y y v. 



-^+-^ + -^+ ... + Z, + Z,ri- 



r r r' 



Now V vanishes when r = oo , which requires that Zg = 0, Z, = 0, &c. ; and therefore 



V= — +— + — + (7) 



r r r-* 



4. The preceding equation will not give the value of the potential throughout the surface of 

 a sphere which lies partly within the earth, because although V, as well as any arbitrary but finite 

 function of Q and (p, can be expanded in a series of Laplace's coefficients, the second member of 

 equation (3) is not equal to zero in the case of an internal particle, but to - inrp^'^, where p is the 

 density. Nevertheless we may employ equation (7) for values of r corresponding to spheres which 

 lie partly within the earth, provided that in speaking of an internal particle we slightly change the 

 signification of V, and interpret it to mean, not the actual potential, but what would be the poten- 

 tial if the protuberant matter were distributed within the least sphere which cuts the surface, in 

 such a manner as to leave the potential unchanged throughout the actual surface. The possibility 

 of such a distribution will be justified by the result, provided the series to which we are led prove 

 convergent. Indeed, it might easily be shewn that the potential at any internal point near the 

 surface differs from what would be given by (7) by a small quantity of the second order only ; 

 but its differential coefficient with respect to r, which gives the component of the attraction along 

 the radius vector, differs by a small quantity of the first order. We do not, however, want the 

 potential at any point of the interior, and in fact it cannot be found without making some hypo- 

 thesis as to the distribution of the matter within the earth. 



5. It remains now to satisfy equation (4). Let r = a (I + u) be the equation to the earth's 

 surface, where m is a small quantity of the first order, a function of 6 and (p. Let u be expanded 



in a series of Laplace's coefficients «„ + ti., + The term m„ will vanish provided we take for a the 



mean radius, or the radius of a sphere of equal volume. We may, therefore, take for the equation 

 to the surface 



r = a (l -1- Ui+ u.,+ ...) (8) 



If the surface were spherical, and the earth had no motion of rotation, V would be independent 

 of and (p, and the second member of equation (7) would be reduced to its first terra. Hence, 

 since the centrifugal force is a small quantity of the first order, as well as u, the succeeding terms 

 must be small quantities of the first order; so that in substituting in (7) the value of r given by 

 (8) it will be sufficient to put r = o in these terms. Since the second term in equation (4) is a 

 small quantity of the first order, it will be sufficient in that term likewise to put r = a. We thus 

 get from (4), (7), and (8), omitting the squares of small quantities, 



