210 riGURE OF A GRAYITATINC BODY. 



particle situated on the surface will be actuated by a force 

 preceisely equal and contrary to that which urges it in the 

 direction of the inclined surface. Hence, if the density of 

 tiie sea be supposed inconsiderable in comparifon with that 

 of the earth, the radius being 20,839,000 feet, the height of 

 a solar tide in equilibrium will be 2'0l66 feet, and that of 

 a lunar tide '8097. 

 Attraction of We must next inquire what will be the effect of the gra- 

 paris. vitation of the elevated parts, on any given supposition re- 



specting their density. Let us imagine the surface to be di- 

 vided by an infinite number of parallel and equidistant cir- 

 cles, beginning from any point at which a gravitating par- 

 ticle is situated, and let their circles be divided by a plane 

 bisecting the equatorial plane of the spheroid ; it is obvious 

 that if the elevations on the opposite sides of this plane be 

 equal in each circle, no lateral force will be produced; but 

 when they are unequal, the excess of the matter on one side 

 above the matter on the other will produce a disturbing 

 force. The elevation being every where as the square of the 

 distance from the equatorial plane, the difference, corre- 

 sponding to. any point of that semicircle in which the eleva- 

 tion is the greater, will be as the difference of the squares 

 of the distances of the corresponding points of the two semi- 

 circles, that is, as the product of the sum and the difference 

 of the distances : but the sum is twice the distance of the 

 centre of the circle from the equatorial plane, or twice the 

 pirie of the distance of the gravitating particle from the 

 plane, reduced in the ratio of the radius to the cosine of the 

 angular distance of the circle from its pole; and the differ- 

 ence is twice the actual sine of any arc of the circle, reduced 

 to a direction perpendicular to that of the plane, that is, re- 

 duced in the proportion of the radius to the cosine of the 

 angular distance of the given particle from the equatorial 

 plane. From these proportions it follows, that, in different 

 positions of the gravitating particle, the effective elevation 

 at each point of the surface, similarly situated with respect 

 to it, is as the product of the sine and cosine of its angular 

 distance from the equatorial plane, the other quantities con- 

 cerned remaining the same in all positions: the disturbing 

 attraction of ail the prominent parts varies therefore pre- 



ciseljr 



