594 Rotational Oscillation of Cylinder in a Viscous Liquid. 



of (15) which gives the forced oscillation, we can also write 

 (15) in the form : 



6 + 



L" \ d 2 a T , da , T ^ ; Di ,„ ,, 



pit* +L'af+M*-E^. . . (16) 



We see, therefore, that in consequence of the motion of 

 the liquid an apparent increase of the moment of inertia 

 arises. 



Putting 0+K=0> 



P 

 the particular solution of (16) becomes : 



E 



^(M-o'pty+L'y 



e Kpt-V) 



in which the phase-angle <j> is determined by the constants of 

 the differential equation. 



Resonance occurs for M — 0'p 2 =O 



or ' <y + L>-M = ..... (17) 



'PP 



Now L" is proportional to k and & = a / , so that we 



may conveniently write L" = N/)% N being a constant. 

 (17) is now replaced by 



flp'+NpS-MsO (18) 



This equation, which is bi-quadratic in ^/p 9 determines the 

 frequencies to which the system resounds. On closer ex- 

 amination there appears to be but one resonance-frequency. 

 Naturally we are only concerned with the "real roots p of 

 equation (18). There are found to be two such roots, one 

 for which y/p is positive, and another for which s/p is 

 negative. Now it follows from our calculation that we have 

 assumed \/p, which occurs in Jc, to be essentially positive. 

 For if we substitute a negative value for s/p in our equations, 

 we obtain a system of waves which moves from infinity 

 towards the cylinder. But the amplitude of this system is- 

 infinite at infinity, so that our first boundary-condition would 

 not be satisfied. 



Delft (Holland), 



March, 1919. 



