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The differential equation for tfie forced oscillation in complex 

 notation is as follows : 



^ d^a da 



6 -- + L— + Ma = Ee'i>^ (15) 



dt^^ dt ^ ' 



Here in onr case L is a complex quantity L=^ L'-\- iL", where 



L' = (4.T ft 7^^ + 1/2 jr [I k R') 



L" = 1/2 jr ft kR \ 



If we onlj concern ourselves with the particular solution of (15) 

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

 form : 



L"\ d*t( da 



^+-]-~ + L'~ + Ma = ±Je'pt (16) 



p J dr at 



We see therefwe that in consequence of tiie motion of the licpiid 

 an apparent increase of tiie moment of inertia arises. 

 Putting 



L" 



P 

 the particular solution of (16) becomes : 



E 



e'(/^'— ?) 



in which the phase-angle (f is determined by the constants of the 

 dilferential equation. 



Resonance occurs for M — <9' p'' zzz 

 or 



6p' -\- L"p — M -=0 ...... (17) 



Now L" is proportional to k and k=:.\X — , so that we may 



y ft 



conveniently write L" =.' Np^, N being a constant. 

 (17) is now replaced by 



dp^^Npn — M=0 (18) 



This equation which is bi-quadratic in V p determines the frequencies 

 to which the system resounds. On closer examination 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 

 of such, one for which V^ p is positive, and another for which \/ p 

 is negative. Now it follows from our calculation that we have 

 assumed |//>, which occurs in k to be essentially positive. For if we 

 substitute a negative value for \^p in our equations, we obtain a 

 system of waves which moves from infinity towards the cylinder. 



