176-179] Adiabatic Model 179 



From equation (445) it is at once seen that 6'a'b'c' = Oabc, so that the 

 point of bifurcation is determined by 



a' 5 b'c'J' AA - a*bcJ AA = 0. 

 Using relations (484), this may be put in the form 



% (^IAAA) + p C*44) + "2 (-*-AAC) (485). 



This equation, together with equations (475) (477) determine the values 

 of p, q, r and Ao> 2 at the point of bifurcation. 



178. In these equations a = b, so that p = q, and the system of four 

 equations reduces to the three equations 



2w (/) a-) abc ' 



in which d^ is given by equation (478) and d 3 is given by a similar equation, 

 or may be more readily derived from the relation 



2<t + d 3 = 0, 



f d\ 



which is necessarily satisfied since the potential I </> ( 173) is harmonic. 



The value of 8 1 for the configuration in question calculated from equation 

 (479) is found to be 0*00851, whence the solution of the equations is found 

 to be 



= - 0-016037, -, - 0-056337, CT . = - '04400 . . .(486). 



2 2 



179. It now appears that at the point of bifurcation the rotation 

 is given by 



- -04400 0>.-<7) ............ (487). 



This relation is exact only as far as first powers of (/o <*") ^o the same 

 degree of accuracy, the mean density p of the figure is given by 



so that equation (487) may also be put in the form 



-06827(^-0-) ............. ..... (488). 



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