468 Mr. H. W. Chapman on 



a case of steady motion; it is necessary to consider the stability 

 of this. 



If the root we start with be a, put /z. = a + V and we have 



.-. 2v=f( a )+yf%*\. 



But / / («)=0. 



Then, on solving this in the ordinary way, we see that the 

 motion is stable if f" [pi) is negative and unstable if f"[a) is 

 positive. 



9. The case in which the root considered is /j, = 1 requires 

 special attention. In this case the vanishing of /& does not 

 imply the vanishing of 6, for //, = — sin 06, and thus 

 sin = 0, so that p = whatever d may be. Consequently, 

 although the value of /u- turns back the body itself need not, 

 but may pass through the vertical position. 



We will therefore go back to the equation for 6 and make 

 6 small. 



It can be written 



sm 2 00 2 =f(6), where f(0) is obtained from (14). 

 /(#) is clearly an even function of #, so we have, neglecting 

 4 <9 2 and 6 { >, 



=-C o + C#-fC 4 0*,say. 

 Put 6 2 =u. 



Then we have 



du 



= 2dt. 



VC + C> + C 4 u- 



du 



2t + e = 



J vc +c 2 tt+c 4 u 2 



du 



This will give 

 if C 4 is positive, 





