AT THE SURFACE OF THE EARTH. 675 



about the axis of rotation, we know that the relative equilibrium of the earth itself, or at least 

 its crust, and the bodies on its surface, would not be affected by supposing the crust at rest, 

 provided that we introduce, in addition to the attraction, that fictitious force which we call the 

 centrifugal force. The vertical at any place is determined by the plumb-line, or by the surface 

 of standing fluid, and its determination is therefore strictly a question of relative equilibrium. 

 The intensity of gravity is determined by the pendulum ; but although the result is not 

 mathematically the same as if the earth were at rest and acted on by the centrifugal force, the 

 difference is altogether insensible. It is only in consequence of its influence on the direction 

 and magnitude of the force of gravity that the earth's actual motion need be considered at all in 

 this investigation : the mere question of attraction has nothing to do with motion ; and the results 

 arrived at will be equally true whether the earth be solid throughout or fluid towards the centre, 

 even though, on the latter supposition, the fluid portions should be in motion relatively to the 

 crust. 



We know, as a matter of observation, that the earth's surface is a surface of equilibrium, if 

 the elevation of islands and continents above the level of the sea be neglected. Consequently the 

 law of the variation of gravity along the surface is determinate, if the form of the surface be 

 given, the force / of Art. 1 being in this case the centrifugal force. The nearly spherical form 

 of the surface renders the determination of the variation easy. 



3. Let the earth be referred to polar co-ordinates, the origin being situated in the axis of 

 rotation, and coinciding with the centre of a sphere which nearly represents the external surface. 

 Let r be the radius vector of any point, Q the angle between the radius vector and the northern 

 direction of the axis, the angle which the plane passing through these two lines makes with a 

 plane fixed in the earth and passing through the axis. Then the equation (2) which V has to 

 satisfy at any external point becomes by a common transformation 



Let 0) be the angular velocity of the earth ; then U = — r-sin^O, and equation (l) becomes 



2 



V+—r^sm"e=c, (4) 



which has to be satisfied at the surface of the earth. 



For a given value of r, greater than the radius of the least sphere which can be described 

 about the origin as centre so as to lie wholly without the earth, V can be expanded in a series 

 of Laplace's coefficients 



V,+ V, +V,+ ...; 



and therefore in general, provided r be greater than the radius of the sphere above mentioned, 



V can be expanded in such a scries, but the general term r„ will be a function of )•, as well as of 



9 and 0. Substituting the above series in equation (3), and observing tliat from the nature of 



Laplace's coefficients 



I d f . ^dV„\ 1 d-V„ , ^,^ _^ 



sin0 - I + -. — r = - n(ti + i)V„, (5) 



sine dB\ dd I i^m' 6 d(f^' v ^ „, 



we get 



r d'.rV,, 1 



where all integral values of n from () to -x, are to l)e taken. 



