Connected with the Earth's Field of Force. 151 



surface, whose coordinates are {x, y, z) and on Q, the 

 element symmetrically situated and of equal size, whose 

 coordinates are ( — x, y, z). If the intensity of pressure 

 at Q is greater than the intensity at P, then, so far as these 

 two elements are concerned, there will be a resultant 

 pressure tending to move the sphere towards the equator. 

 The intensity of pressure at any point of a homogeneous 

 liquid in equilibrium is (V — ^Vo)p, where V is the potential 

 of the field of force acting on the liquid, Vo is the value 

 of V for the free surface — here the geoid — and p is, as 

 before, the density of the liquid. The potential V may be 

 considered as consisting of two parts, (1) the part due 

 to the earth itself; this we shall consider as normal, i. e., 

 such that the geoid is a spheroid of revolution with an 

 ellipticity equal to the mean ellipticity of the earth; (2) 

 the part due to the attraction of the sphere; the direct 

 attraction of the sphere is obviously symmetrical about 

 the ^-axis ; there is also a small indirect effect due to the 

 slight heaping up of the liquid around the sphere owing to 

 the attraction of the latter ; this effect may also be taken 

 as symmetrical about the 0-axis.^^ The effect of that 

 part of "V due to the attraction of the sphere is therefore 

 the same at P and at Q and cancels out as far as the resul- 

 tant pressure is concerned. In calculating this resultant 

 pressure we may therefore use for V simply the normal 

 part of it, or that due to the earth alone. 



We shall assume that the normal part of the gravity 

 potential may be represented by a polynomial of the 

 second degree in x, y, and ^, or 



Y -Y =^ -9o z -{- a x' ^ b tf -\- G z' -\- 2 h X z (1) 



The absence of terms in x and y is explained by the fact 

 that the direction of the vertical at the origin coincides 

 with the ^-axis. The absence of terms in xy and yz is 

 explained by the fact that the a;-axis is in the meridian, 

 which is a plane of symmetry. The coefficient g^ is the 

 intensity of gravity at the origin; the other coefficients 



2» There is a very slight deficiency in symmetry in this indirect effect, 

 because the attraction of the sphere on the liquid in any given direction 

 draws the liquid from a region where gravity Js slightly different from 

 what it is in a region symmetrically situated with respect to the 0-axis. The 

 effect of the asymmetry is only a small part of the whole effect, which is 

 itself small, and so the effect of asymmetry may safely be neglected, even m 

 comparison with the minute quantities involved in the discussion. 



