179-182] Adiabatic Model 181 



It is readily seen that the appropriate form to assume for Q is one con- 

 sisting of terms of degrees 6, 4 and 2 in , TJ, f. 



A general discussion of the solution of this equation will be found else- 

 where*. For our immediate purpose we may consider the particular solution 

 at the point of bifurcation at which the pseudo-spheroidal form becomes 

 unstable, thus extending the solution already obtained in 174 to the 

 second order of small quantities. 



181. In this solution x and y enter symmetrically. Let us write cr 2 for 

 f 2 + ?7 2 , and assume for Q the value 



(493). 



On carrying out the necessary calculations and solving equation (492) 

 we find 



R= 0-4155 -T7799 (7 -2) + 0-4908 ( 7 -2)(2 7 -3) 



3S= 01894 -1-4585 (7 -2) + 0-5000 (7 -2) (27- 3) 



3T= 0-0506 -0-4124 (7 -2) + 01698 ( 7 -2)(2 7 -3) 



U= 0-00346 -0-0375 ( 7 - 2) + 0-01922 (7- 2) (2 7 - 3) 



r= 0-08755 - 010962 (7- 2) 



s = -0-01727 +0-04871(7-2) 

 t = - 0-007862 + 0-02511 (7 - 2) 



u = - 0-00550 -0-02651(7-2) 



v= 0-00778 +0-03195(7-2) 



= _ (&Z5Y [0-01292 + 0-05495 (7 - 2)] 



\ o / 



This completes the determination of the equation of the boundary as far 

 as the second order of small quantities. 



182. The lengths of the intercept on the #-axis are determined by the 

 equation jf = 1, where 



............ (494), 



and the solution of this equation is found to be 



*- L , *P] ^O-^VP , r , ^u (I 2p\(2L 2p\l 

 2 J V Po ) L 6 <* 4 F U 4 aV I a 4 + Wj + 



.................. (495). 



* Bakerian Lecture 1917. Phil. Tram. R. S. (not yet published). 



