120 Cataclysmic Motion [OH. vi 



120. Multiplying equations (351) and (352) by b, c and subtracting, 



I"/* (X-6c)rfX M <T| 



''[Jo (& 2 +X)(c 2 +X)A + *7r^&~c~ f ~Fc ' 



The integral in this expression may be positive or negative, but the 

 integral plus 6/bc is found to be always positive. 



Since equation (353) has shewn that 6' is always positive, it follows 

 that the expression in square brackets in equation (354) is always positive, 

 so that this equation assumes the form 



7 2 



3- (b c) = (b c) x (a positive quantity) ......... (355). 



This shews that any initial inequality in b and c gives rise only to 

 oscillations about the value b c = 0. We may therefore suppose henceforth 

 that b = c throughout the motion. 



121. Putting b = c, equation (351) becomes identical with equation (352). 

 From equation (353) we obtain 



a Trpabc (0' 0) [c ^Trpabc (6' - 0)~\ /QK\ 



k ~ = ~ ~ (oov). 



a a 2 [c c 2 



Denoting each member of this equation by 77, equations (350) and (352) 

 assume the forms 



r ~ 

 Trpabc a 2 



1 ~ 



i 



-pabc c 2 



which are exactly identical with the statical equations (78) and (80) of 

 Chapter III except that //, has become replaced by /*, 77. 



Hence, exactly as in equation (85), it follows that 



(e) .............................. (359), 



Trp 

 where 



The values of the semi-axes of the spheroid are 



so that the value of 77 is found from equation (356) to be 



ad-cc 1 ./- 21 -5e 2 \~| 



* - 2u + c' = 3(1 - #) [** + e V + 3 (1 - #) (3 - f))\ ' ' < l > 



