and 



y2T + ja^c^ T - i(ri Va(c P| - c va ) p = Q (||51) 



K a Pa c a *a 



Equation II-50 can be written in terms of P alone by taking the Laplacian and substituting equations 11-51 and 

 II-50 for V 2 T and T, respectively, into the resulting equation. The result of this procedure is 



V 4 P + [ m2Y ' + ia) P°a C Pa 1 V 2 P + i( ° 3p °a C Pa p = Q (11-52) 



L C a K a J K a C a 



Equation 11-52 is reduced by considering it as a quadratic equation in V 2 . Then 



(V 2 + k la 2 ) (V 2 + k 2a 2 ) P = , (11-53) 



where 



2( kl .„ ) 2 = Sfflfc. + **»*» ± J^PoaCpa f , 

 1a 2al C a 2 ^ K a L 



C0 2 K a 2 Y a 2 _ 2iC0K a ( Ya ~ 2) 1 > 

 Po a 2 C a 4 C pa 2 Po a C a 2 C pa J 



(11-54) 



The second and third terms under the radical are much smaller than 1 for the ranges of parameters selected for 

 this study, so that the radical can be approximated as 



(1 - & * 1 -■§- . (l, - 55) 



Thus, 



k 2 _ ioPo a c Pa <d 2 (Ya - 1) _ ico 3 K aYa 2 

 la kT- + —c? 4p 0a c a ^c pa ' C'- 56 ) 



and 



i, 2 J^_ io) 3 K a Y a 2 



k2a " c a 2 + 4p 0a c a <c pa - (H-57) 



An examination of the relative magnitudes of the terms in equations II-56 and II-57 shows that 



k 2 ~ ic °Poa c p a (II-58) 



1a * ~K 



or 



and 



k 1a «(1 +|) (^^-f (H-59) 



a c a ' (II-60) 



12 



