322 NEWTON S PEINCIPIA. 



formulae which we find in the second volume of Laplace s 

 &quot; Mecanique Celeste.&quot; We are enabled to determine in 

 some measure the form of the atmosphere of the heavenly 

 bodies. We have as yet neglected the effect of the centri 

 fugal force ; suppose the angular velocity to be co, the 

 co-latitude of the particle of atmosphere under considera 

 tion. Then including this force in our equation we have 



dP^-ILdx + rfx sin. d (x sin. 0), 



where the earth is considered to be a homogeneous sphere 

 whose particles attract according to the law of the inverse 

 square of the distance. Now along the free surface of 

 the atmosphere p is constant, and . . d p ; hence 



- ^ dx + co 2 x sin. d (x sin. 0) = 

 .-. const. = - + ~ . x 2 sin. 2 0. 



This therefore is the equation to the surface of the atmo 

 sphere. Let us compare the polar and equatorial diameters. 

 Call them 2 R and 2 R . When x = R we have 0, the 

 co-latitude, a right angle ; hence 



+ \ co 2 R 2 = const. 



1-X &quot;2 



when x = R , is nothing, hence 



= const. 



_ R -R _ ^_ R/ 3 



Now at the equator the centrifugal force is less than 

 gravity, that is 



