Manchester Memoirs, Vol. Hi. (1908), No. 8. 



5 



Hence if w' is the angular velocity at the end of this 



cycle, 



Vh 



. fl; 



2r \/r + /&-/ 



i.e., the mean angular acceleration is directed towards the 

 position of relative equilibrium, and is proportional to the 

 angular displacement. The mean motion is therefore a 

 simple oscillation 



Vh 



6 = asm 



h^^k 



,/+e 



(0 



The actual motion is evidently of the nature shewn in 

 the diagram {Fig. 2), in which the time is plotted hori- 



Fig. 2. 



zontally and the angular displacement vertically : the two 

 boundaries are sine curves and the successive vertices are 

 equidistant in time. 



The preceding synthetic investigation brings out the 

 essential characteristics of the motion. The impulses are 

 constant in magnitude, and the effect of any impulse in 

 changing the angular velocity is proportional to the 

 angular displacement ; secondly, the angular displacement 

 at B algebraically exceeds half the sum of the two at A 

 on either side, by an amount proportional to the displace- 



