L. On the Variation of Gravity at the Surface of the Earth. By G. G. Stokes, M.A., 

 Fellow of Pembrohe College, Camhridge. 



[Read April 23, 1849.] 



On adopting the hypothesis of the earth's original fluidity, it has been shewn that the surface 

 ought to be perpendicular to the direction of gravity, that it ought to be of the form of an oblate 

 spheroid of small ellipticity, having its axis of figure coincident with the axis of rotation, and that 

 gravity ought to vary along the surface according to a simple law, leading to the numerical relation 

 between the ellipticity and the ratio between polar and equatoreal gravity which is known bv the 

 name of Clairaut's Theorem. Without assuming the earth's original fluidity, but merely supposing 

 that it consists of nearly spherical strata of equal density, and observing that its surface may be 

 regarded as covered by a fluid, inasmuch as all observations relating to the earth's figure are reduced 

 to the level of the sea, Laplace has established a connexion between the form of the surface and 

 the variation of gravity, which in the particular case of an oblate spheroid agrees with tlie connexion 

 which is found on the hypothesis of original fluidity. The object of the first portion of this paper 

 is to establish this general connexion without making any hypothesis whatsoever respecting the 

 distribution of matter in the interior of the earth, but merely assuming the theory of universal 

 gravitation. It appears that if the form of the surface be given, gravity is determined throughout 

 the whole surface, except so far as regards one arbitrary constant which is contained in its com- 

 plete expression, and which may be determined by the value of gravity at one place. Moreover 

 the attraction of the earth at all external points of space is determined at the same time ; so that 

 the earth's attraction on the moon, including that part of it which is due to the earth's oblateness, 

 and the moments of the forces of the sun and moon tending to turn the earth about an equatoreal 

 axis, are found quite independently of the distribution of matter within the earth. 



The near coincidence between the numerical values of the earth's ellipticity deduced independ- 

 ently from measures of arcs, from the lunar inequalities which depend on the earth's oblateness, 

 and, by means of Clairaut's Theorem, from pendulum experiments, is sometimes regarded as a 

 confirmation of the hypothesis of original fluidity. It appears, however, that the form of the surface 

 (which is supposed to be a surface of equilibrium,) suflices to determine both the variation of gravity 

 and the attraction of the earth on an external particle*, and therefore the coincidence in question, 

 being a result of the law of gravitation, is no confirmation of the hypothesis of original fluidity. 

 The evidence in favour of this hypothesis which is derived from the figure and attraction of the 

 earth consists in the perpendicularity of the surface to the direction of gravity, and in the circum- 

 stance that the surface is so nearly represented by an oblate spheroid having for its axis the axis 

 of rotation. A certain degree of additional evidence is afforded by the near agreement between 



i 



' It has been remarked by Professor O'Brien, {Mathematical 

 Tracts, p. 56) that if we have given the form of the earth's sur- 

 face and the variation of gravity, we have data for determining 

 the attraction of the earth on an external particle, the earth being 

 supposed to consist of nearly spherical strata of equal density ; so 

 that the motion of the moon furnishes no additional confirmation 

 of the hypothesis of original fluidity. 



If we have given the component of the attraction of any mass, 

 liowever irregular as to its form and interior constitution, in a di- 

 rection perpendicular to the surface, throughout the whole of the 

 surface, we have data for determining the attraction at every ex- 

 ternal point, as well as the components of the attraction at the 

 surface in two directions perpendicular to the normal. The corre- 

 sponding proposition in Fluid ^lotion is self-evident. 



