z 



(17) 



414 SCIENCE PROGRESS 



Following the procedure set forth in dealing with the 

 general case, we first put 



F 1 (*,*') = -F 9 (s,s') =0 (15) 



One solution of (15) is 



z = z' = o (16) 



It is not, however, this point of equilibrium in which we 

 are interested, but the one given by the other solution of 

 (15), namely 



kk'p p' - gq' r 

 ~ k{k'p'-q) - Cl ) 



kk'pp^_-ql r \ 

 Accordingly we introduce as new variables 



Xi — 1 Z C- 1 



x-i — z' — Ci 

 and obtain 



—J = [q — k'C^)x x + k'(p — Ci)x2 — k'x x Xi. 



—j- 2 = k{p' — C^)Xi + W — kCi)x2 -~ kx x x<i 



or, in the notation of our general discussion, 



dxi 

 lit 

 dxi 



(18) 



(19) 



— CL\\X\ ~f~ di^Xi> -p Q*\\2X\X% 



at 



dt 



0>2\X\ ~I~ Q22X2 ~\~ &2i2X\X-2 . 



(20) 



Assuming now, for the sake of illustration, that the con- 

 stants in (19) are capable of assuming all kinds of values, 

 we may distinguish the following cases : 



1. Case in which the roots \, X 2 , of (11) are real (and dis- 

 tinct). 



The determinantal equation (11) here takes the form 



&1\ A. d\2 



G-21 d<n — A, 



=0 (21) 



In the immediate neighbourhood of the origin the product 

 terms in (20) are negligible. 



