35** Anahjsis of the Micnn'iqne Celeste ofM. La VlacS^ 



system of the world. He afterwards gives an extremely 

 simple expression for the law of gravity at the surface of 

 honiooeiieous spheroids in equilii^jrioni, whatever may be 

 the index of the power to which the attraction is propor- 

 tional : for this purpose he makes use of the equation 

 wh,ich takes place at the surtace of spheroids differmg very 

 li-ttic from the sphere; and he deduces that in general, if 

 the spheroid be a homogeneous fluid and endowed with a 

 rotatorv motion, the gravity varies from the equator to the 

 pple proportionally to the square of the sineot the latitude; 

 and what is singularly remarkable, this viiriation vanishes 

 when the attraction is proportional to the cube of the di- 

 stance; so that in this case the gravity ai ihe surface of ho- 

 mogeneous spheroids is every where the same, whatever 

 may be their rotatory motion. 



In the preceding inquiries the author has supposed the 

 effect of the centrifugil, ^>rce and of the foreign attract 

 tions to be very small, with respect to the attraction of 

 the spheroid, which has admitted of his neglecting the 

 square and the other powers of thes^e forces, as well as 

 quantities of the same order : but he shows that it is easy 

 to extend the same analysis to a case in which it may be 

 necessary to preserve them. At last he arrives at this im- 

 portant conclusion,- — that the equilibrium is rigorously pos- 

 sible, although it is only by successive approximations that 

 •we can assicn the figure which satisfies it. Such is the re- 

 sult of M. La Place's labours on the attractions of sphe- 

 roids. The uniform and direct manner in which this the- 

 ory, so abstract and difricult, is derived bv simple differcnci- 

 ations Irom a single iuudamer.fal equation, is doubtless oneof 

 the most remarkable tilings that analysis has ever etiected. 



In order to compare the preceding results with observa* 

 tions, it is necessary to know the curve of the terrestrial 

 meridians, and that which we trace by a course of geodesic 

 operations. If by the axis of rotation of the earth, and by 

 the zenith of a place on its surface, we imagine a plane 

 proloneed to the heavens, this plane will there trace the 

 circuinference of a great circle, which will be the meridian 

 of the place ; and all the poinis on the surface of the earth, 

 which will have their zenith in this circumference, will 

 be under the same celestial meridian. These points are 

 therefore such that the normals, drawn through them to 

 the surface of the earth, are all parallel to one and the 

 same plane. According to this condition the author deter- 

 mines the curve which they fi)rm on the surface. This 

 €urve> which js the terrestrial meridian, is plane if tlie 



spheroid 



