Analysis of the Mecanique Celeste of M. La Place. 953 



ner, how it is possible that the observed lengths of the se- 

 conds pendulum increase from the equator to the pole, nearly 

 in proportion to the square of the sine of the latitude, whilst 

 the variations of the observed degrees of the meridian differ 

 perceptibly from this law. For the same reason, the aber- 

 ration of the elliptic figure will be less sensible in the value 

 of the horizontal parallax of the moon, which is propor- 

 tioned to the terrestrial radius, than in the expression of the 

 length of the pendulum which is given by the differencia- 

 tion of the equation of equilibrium into which the radius of 

 the spheroid enters under a finite form. The preceding for- 

 mulas may also serve to verify the hypotheses necessarv for 

 representing the measured degrees of the meridian. The 

 author has made an application to that which Bouguer pro- 

 posed, namely, to suppose the increments of the degrees 

 from the equator to the pole proportional to the fourth 

 power oF the sine of the latitude j and he proves that this 

 law is inadmissible. 



The author applies these general results to the case in 

 which the 'Spheroid not being solicited by foreign im- 

 pulses is formed of elliptic layers, all of them having their 

 centres at the centre of gravity of (he fluid. We have seen 

 that this case is that of the earth supposed to be primitively 

 fluid ; and the author proves that it would still agree witli 

 it in the hypothesis of the figures of its layers being simi- 

 lar. He deduces from it, that then the radii diminish, and 

 the degrees increase from the equator to the pole propor- 

 tionally to the square of the sine of the latitude. He proved 

 also by the help of the same formulae, that on the most pro- 

 bable suppositions, suppositions which become necessary, 

 if the spheroid has been originally fluid, its oblateness must 

 be less than in the case of homogeneity. Finally, he esta- 

 blishes between the eliipticity of the earth, and the variation 

 of the pendulum from the equator to the pole, this remark- 

 able relation : Putting unity for the length of the pendulum 

 at the equator as much as the ellipticify of the earth sur^ 

 passes lluit which tvould take place in the ca<se nf homagC' 

 neity, in the same proportion is the total increase of the 

 pendulum from the equator to the pole exceeded Ln/ that 

 which would take place in the same case ; and reciprocally , 

 so that the sum of this increase of the eliipticity forms a 

 constant quantity. 



The author afterwards determines the attraction of sphe- 

 roids the surfaces of which are fluid and in equilibrium, an 

 hypothesis which takes place with respect tf) the earth, and 

 which it seems natural to extend to the other bodies of the 



svstcm 



