[ 252 ] 



XXXV. Analysis of the Mccanique Celeste of M.hApL/iC^. 

 By M. BiOT. 



[Continued from p. 209] 



J. HE author also considers the general case in which the 

 spheroid, supposed to be always fluid at its surface, may 

 contain a solid nucleus of anv civen figure not differ- 

 ing much from the sphere. The radius drawn from the 

 centre of gravity of the spheroid to the surface, and the law 

 of gravity at this surface, have certain general properties, 

 which the author has found out, and which are the more 

 important as being independent of all hypothesis. The 

 first consists in this, that in the stale of equiHbrium, the 

 fluid part of the spheroid must always be arranged in such 

 a way that the centre of gravity of the external surface may 

 coincide with that of the spheroid. The permanent slate 

 of equilibrium in which the celestial bodies are, also bring'J 

 to lisiht some properties of their radii ; for this state requires 

 that these bodies should turn, if not exactly, at least very 

 nearly so, round one of their three principal axes. Hence 

 result certain conditions which their radii must satisfy, and 

 these are explained by the author in a very simple manner. 

 He afterwards obtains, by the differenciatiou of the gene- 

 ral equation of the equilibrium of spheroids, the law of 

 gravity at their surfaces ; and he deduces from it the length 

 of the seconds pendulum, which is proportional to this gra- 

 vity. Finally, the developed expression of the radius of the 

 spheroid gives him the osculating radius, and consequently 

 the degree of the meridian. These formulie possess the valu- 

 able advantage of being absolutely independent of the interior 

 constitution of the spheroid ; i. e. of the figure and density 

 of its layers. They depend solely on the expression of its 

 radius, with which they are connected by very simple rela- 

 tions. On comparing these relations with each other, we find 

 that the parts of the radius which enter under a finite form 

 into the expression of gravity and length of the pendulum, 

 undergo two successive differenciations, in order to pass into 

 the expression of the degree of the meridian, and would 

 consequently undergo three in the variation of two conse- 

 cutive degrees ; and as the diflerential of a quantity raised 

 to any given power is always multiplied by the index of this 

 power, it results, that terms scarcely sensible in themselves 

 in the expression of the length of the pendulum might, if 

 elevated to high powers, become considerable in the varia- 

 tion of the degrees j which explains in a very simple miLn- 



