326 Mr, Ivory oji the Variation of' Densitif afid Pressure 



integrcd f g d e from the centre to the surface. Substitute the 

 assumed value of e in the term on the left side ; then 



5 m <p X a' = 3/f d {a' e) + -^ a' + 3a' (F -f§ d e) ; 



and by taking the fluxions and dividing by 5 a* d a, 



"'"^IFIT = 3?^^ + ^ + 3{F-fsde). 



At the surface, F — fs d e = 0; and therefore, in that parti- 

 cular case, we obtain 



the symbols (p), {x), (-^^) denoting the particular values 



at the surface, or when « = 1. 



Differentiate the foregoing equation, then 



?« <p ; = 3 e d p; 



^ a da » 



and by substituting the value of e, 



\ a^ d a / a' do 



1 = X X — - X — i-. 



a aa j ^a'-da a da 



Again, let P denote the pressure at the distance a from the 

 centre ; then, the increment of the pressure upon the molecule 

 dyi ■=■ q d a, being proportional to the force urging d M to 

 the centre, we shall have 



d F = — g da X — ^^ , 



* rt- 



0j- d P 4 ""y f "" d a a d a 



If we now introduce a new symbol u, the equations that 

 }iave b6en investigated may be thus written, 



a3 dp 



X —i- =. — u 



f^ a^ da a da 



"^K^d'a-) , ,_^ !> (3) 



+ u X = 



J 



And these equations, which all depend upon the function u, 

 together with the formula (2), contain all the conditions of 

 the problem. 



Clairaut applied his equation to determine the progression 

 of ellipticities that would be the consequence of a proposed 

 law of density. He supposed the density proportional to 



some 



