in the interior Paits of the Earth. 327 



some power of the distance from the centre. As the densities 

 decrease from the centre to the surface, the exponent of the 

 power must be negative ; and he found that the eUipticities 

 must be proportional to a certain positive power of the distance. 

 Therefore the density would be infinitely great, and the ellip- 

 ticity evanescent, at the centre ; circumstances which render 

 the assumption of Clairaut altogether inapplicable to the case of 



nature. If we put/s = ~ in the equations (3), we shall find, 



and this value renders the equation for x homogeneous^ and 

 conducts immediately to the solution of Clairaut. 



In the Memoirs of the Academy of Sciences for 1789, Le- 

 gendre proposed another law of densities which is equivalent 

 to making u equal to a constant positive number if, in the 

 equations (3). In this hypothesis the increment of the pres- 

 sure is proportional to the increment of the square of the den- 

 sity, which is the law of pressure assumed by Laplace, and 

 which he has shown* to agree both with the figure of the 

 earth as determined by observation, and with the mean density 

 of the matter of which it consists. 



If we make u = ?i% we obtain from the first of the equa- 

 tions (3), 



a^ -— ^ = — n^ fp a^ d a, 



da- -^ ^ 



which may be thus written, 



' ^ ^^rfT^ -(?«)= - «'/(? ^') « ^^« ; 



and, by taking the fluxions, 



dd (pa) o , , „ 



Now, by integrating this equation on the suj^position that the 

 central density is unit, we get 



whicli is the law of densities proposed by Lcgendre. Hence 

 we derive, 



y'f tr (hi = (sin an — a n cos a n), 



m = — (sin n — n cos n), 



sm n 



* Mican'ujuc Ci/csU; livrc ll"". cap. ii, § fi. 



Again 



