98-iooJ Stability 99 



= (267), 



On eliminating a, j3, 7 we have the relation 



L> l>' le" 



A/j_ j A/j 5 A/]^ 



& & ' & " 

 ^ ' I- " 



i^9t "'3 J rt '3 



which determines the condition for a point of bifurcation. 



On inspecting equations (264) in the light of relation (267), it becomes 

 clear that in general the solution for p, CJ, t is p = = t = oo , the ratio of these 

 quantities, from equations (266), being that of a : (3 : 7. With this solution 

 the third order terms ^ become identical with the previously found first 

 order terms P , and the attempt to extend the solution to the third order of 

 small quantities has failed entirely. 



It is, however, easy to find the condition that equations (264) shall have 

 a finite solution. For, assuming p, ff, t to be finite, and multiplying the three 

 equations (264) by the minors of &/', & 2 ", k." in the determinant (267) and 

 adding, we -obtain 



*,, *,', m, 



T, If' J& 



n/2, A 2 , iw\;> 



When, and only when, this relation is satisfied, there is a solution such 

 that p, (j, r are finite, and there is a genuine third order solution. 



After some transformation, this equation can be put in the simpler form 



=0 ..................... (268). 



#-** ............... (269) - 



The coefficients in brackets are known; the quantities 2eti, 2Bt 2 involve 

 p, q, r and so also So) 2 linearly (cf. equations (245) etc.), and the equation is 

 seen to be a linear equation for 8(o 2 . Carrying out the necessary computations, 

 the solution of the equation is found to be 



1^=0-0074231 .................. (270). 



This gives the value of 8&> a on the true linear series ; if we attempt to 

 carry the solution beyond terms of the second order with any other value of S&> 2 , 

 the solution simply lapses back to the first order solution already found. We 

 notice that o> 2 increases initially as we pass along the pear-shaped series. 



100. On inserting into equations (245) (248) the value of Sco 2 given by 

 equation (270), we obtain 



p= 3124954 ........................... (271), 



q = -0103164 ........................... (272), 



r = - 0-015236 ........................... (273), 



s = -0-256962 ........................... (274), 



72 



