1883.] "Apparently Neutral" Equilibrium. 413 



This equation is of the form 



— -1-7 = c9 — mv 3 , 

 9 dt> 



where c is very small compared with m, which may be put equal 

 to \a\ and in fact we are investigating the character of oscillations 

 under a restoring force proportional to the cube of the distance 

 disturbed by a small force proportional to the distance. 



8. We have 



g \dtj - 

 where /3 is the amplitude of the swing ; and therefore 



kcIO 



t = 



9 h jA + ce 2 -^md i 

 d9 



k . - r —d(f> 



where cos (6 = Jp . 0, k 2 = — - — , 



P + 2 



A = |/»/3 4 - c/3 2 , 

 q — p = -r , which is small, 



qp 



A 

 m 

 2A' 



so that q +p = sj 2 ^ (l + 4/ ^) , approximately. 



Also —7= is a small quantity which we shall call e. 

 mp 



Hence t- ^ \ T^W 



g h - jA{p + q)JJl- tf sin 2 </> 



c \ f —d<f> 



= k . (mgFfi (1 + £«) J F (I , h) - F (</>, ft) J ; 



where cos</> = i/p . = | (1 - he), tf = \{l+e). 



29—2 



