674 Mr. stokes, on THE VARIATION OF GRAVITY 



preceding paragraph does not constitute the whole effect of the irregular distribution of land and 

 sea, since if the continents were cut off at the actual sea level, and the sea were replaced by rock 

 and clay, the surface so formed would no longer be a surface of equilibrium, in consequence of 

 the change produced in the attraction. In Arts 25 — 27, I have investigated an expression for the 

 reduction of observed gravity to what would be observed if the elevated solid portions of the 

 earth were to become fluid, and to run down, so as to form a level bottom for the sea, which in 

 that case would cover the whole earth. The expressions would be very laborious to work out 

 numerically, and besides, they require data, such as the depth of the sea in a great many places, 

 &c., which we do not at present possess ; but from a consideration of the general character of the 

 correction, and from the estimation given in Art. 21 of the magnitude which such corrections are 

 likely to attain, it appears probable that the observed anomalies in the variation of gravity are 

 mainly due to the irregular distribution of land and sea at the surface of the earth. 



1 . Conceive a mass whose particles attract each other according to the law of gravitation, and 

 are besides acted on by a given force /, which is such that if A", 1', Z be its components along 

 three rectangular axes, Xdx + Ydy + Zdz is the exact differential of a function U of the co- 

 ordinates. Call the surface of the mass S, and let V be the potential of the attraction, that is 

 to say, the function obtained by dividing the mass of each attracting particle by its distance from 

 the point of space considered, and taking the sum of all such quotients. Suppose 5' to be a 

 surface of equilibrium. The general equation to such surfaces is 



V+ U=c, (1) 



where c is an arbitrary constant; and since S in included among these surfaces, equation (l) 

 must be satisfied at all points of the surface S, when some one particular value is assigned to c. 

 For any point external to S, the potential V satisfies, as is well known, the partial differential 

 equation 



d'V d'V d'V 



d«r dy dz- 



and evidently V cannot become infinite at any such point, and must vanish at an infinite distance 

 from .S*. Now these conditions are sufficient for the complete determination of the value of V for 

 every point external to S, the quantities U and c being supposed known. The mathematical 

 problem is exactly the same as that of determining the permanent temperature in a homogeneous 

 solid, which extends infinitely around a closed space S, on the conditions, (1) that the temperature 

 at the surface S shall be equal to c — U, (2) that it shall vanish at an infinite distance. This 

 problem is evidently possible and determinate. The possibility has moreover been demonstrated 

 mathematically. 



If U alone be given, and not c, the general value of V will contain one arbitrary constant, 

 which may be determined if we know the value of V, or of one of its differential coefficients, at 

 one point situated either in the surface S or outside it. When V is known, the components 

 of the force of attraction will be obtained by inere differentiation. 



Nevertheless, although we know that the problem is always determinate, it is only for a very 

 limited number of forms of the surface .S" that the solution has hitherto been effected. The 

 most important of these forms is the sphere. When .S" has very nearly one of these forms the 

 problem may be solved by approximation. 



2. Let us pass now to the particular case of the earth. Although the earth is really 

 revolving about its axis, so that the bodies on its surface are really describing circular orbits 



