678 Mr. stokes, on THE VARIATION OF GRAVITY 



where a, /3, 7 are arbitrary constants ; and it is very easy to prove that the co-ordinates of the 

 centre of gravity of the volume are equal to aa, a(i, ay respectively, the line from which Q is 

 measured being taken for the axis of z, and the plane from which (p is measured for the plane of 

 xz. Hence the term u^ in (8) may be made to disappear by taking for origin the centre of gravity 

 of the volume. It is allowable to do this even should the centre of gravity fall a little out of the 

 axis of rotation, because the term involving the centrifugal force, being already a small quantity 

 of the first order, would not be affected by supposing the origin to be situated a little out of 

 the axis. 



Since the variation of gravity from one point of the surface to another is a small quantity of 

 the first order, its expression will remain the same whether the earth be referred to one origin or 

 another nearly coinciding with the centre, and therefore a knowledge of the variation will not 

 inform us what point has been taken for the origin to which the surface has been referred. 



7. Since the angle between the vertical at any point and the radius vector drawn from the 

 origin is a small quantity of the first order, and the angles 0, (p occur in the small terms only of 

 equations (8), (lO), and (12), these angles may be taken to refer to the direction of the vertical, 

 instead of the radius vector. 



ultimately equal to — . Comparing this with (10), we get F„ = E, and therefore, from the first 

 r 



8. If E be the mass of the earth, the potential of its attraction at a very great distance r is 



imately equal tc 



of equations (H), 



£=Ga=+f<..= a' = Go^(l+|m), (13) 



which determines the mass of the earth from the value of G determined by pendulum experiments. 



9. If we suppose that the surface of the earth may be represented with sufficient accuracy by 

 an oblate spheroid of small ellipticity, having its axis of figure coincident with the axis of rotation, 

 equation (8) becomes 



r = a{l +e(l-cos'^0)}, (14) 



where e is a constant which may be considered equal to the ellipticity. W^e have therefore in this 

 case M, = 0, u, = \- cos' 6, u,. = when w > 2 ; so that (12) becomes 



g=G{l - (|m -6)(i-cos-0)|, (15) 



which equation contains Clairaut's Theorem. It appears also from this equation that the value of 

 G which must be employed in (13) is equal to gravity at a place the square of the sine of whose 

 latitude is ^. 



10. Retaining the same supposition as to the form of the surface, we get from (10), on 

 replacing }'„ by E, and putting in the small term at the end uya^ = niGa' = mEa^, 



E K (i' 

 F= - + (e_im) -3 (i_cos'0) (U>) 



Consider now the effect of the earth's attraction on the moon. The attraction of any particle 

 of the earth on the moon, and therefore the resultant attraction of the whole earth, will be very 

 nearly the same as if the moon were collected at her centre. Let therefore r be the distance of the 

 centre of the moon from that of the earth, Q the moon's North Polar Distance, P the accelerating 

 force of the earth on the moon resolved along the radius vector, Q_ the force perpendicular to the 

 radius vector, which acts evidently in a plane passing through the earth's axis ; then 



dV dV 



^= -:r' ^ = -ir^' 



dr rdti 



