182 Compressible and Non- Homogeneous Masses [on. vn 



From equation (450), the value of fl at a point on the #-axis is given by 



JF + 1 (7 - 1) (7 - 2) 



= F($), say. 

 The points on the #-axis at which 3ft/<k = are given by r- F ( Jf) = 0, or 



This condition can be satisfied either by making F' (jf) or djf/dx = 0. 

 The first condition cannot be satisfied except when (p <r)/p = 1 ; in this 

 case <r = and the equation merely reduces an equation which is auto- 

 matically satisfied when a = 0. Thus the true points at which 80/8*' 

 vanishes are given by d;(F/d# = 0. 



The condition that centrifugal force just balances gravity at the equator 

 is therefore that d$/dx = when Jff = 1 , and this is also the condition that 

 the surface jp = 1 shall have a double point on the axis of x. That the two 

 conditions must necessarily be identical is of course shewn by the analysis 

 of 151. 



The equation djf/dx = becomes 



Using the value of a?\o? provided by equation (495), we find as the 

 condition that djf/dx shall vanish on the boundary ( jf = 1), 



or, inserting numerical values, 



_ 2) _ 1-0509] 



po J 



"2)-'0510] + ...=0 ...... (496). 



po 



For a given value of 7, this equation determines a value of (p <r)/p Q 

 such that centrifugal force just balances gravity at the instant at which the 

 pseudo-spheroidal form is giving place to the pseudo-ellipsoidal. 



We may alternatively regard the equation as determining a critical value 

 of 7 when (p cr)/p is assigned. It is this latter use of the equation which 

 is of primary interest to us, the important case being (/3 &)/po =1- It 

 seems probable that the full series will be fairly rapidly convergent up to 



