

466 Mr, H. W. Chapman on 



When p=p this must be satisfied identically, so that the 

 right-hand side must become + (A 2 )* ; thus the positive sign 

 is the proper one if p is positive, and the negative one if p 

 is negative. 



If ^ = 0, let us differentiate (15) with respect to t and 

 divide by pu ; this gives : — 



\aM a- cv I a La cr J 



4- 2^(\ 2 + C/r - An 2 + A 2 K) — -*-$- p 2 



(;; + ^){cf + (c-A),] 



\7S +A+c S +2C -^ +(C - AK (18) 



If we put/* = /A , /x vanishes and we get the initial value of 

 p. The sign o£ this is clearly the one to take in (17). 



Now the roots of the expression under the radical in the 

 numerator in (17) are clearly irrelevant ; this is at once seen 

 from its form in 6. 



Also the root of the denominator is irrelevant, being 

 clearly < — 1. 



From this we see that to every possible value of p there 

 correspond two and only two values of p, and these are equal 

 and opposite, and we can only pass from one to the other by 

 p passing through a root of the numerator of (17). 



There are three cases to be considered. A root, to be 

 relevant, must < 1 in absolute value and be placed so that p 

 is moving towards it. 



(a) The numerator has a real and relevant root, and the 

 first root it reaches is single. Then if this root is a, put 

 p — a + %, % being small. Then to a first approximation %= /3%^ 

 where ft is a constant. 



This oives %2 = i/3£, so that p reaches the root in a finite 

 time; the motion is then reversed and it goes back again till 

 it reaches another root. 



(/3) The first root reached is double. Then to a first 

 ap pr oxima tion 



