1890.] vibrations of a revolving cylinder or bell. 109 



This enables us to eliminate v, 8v, and therefore by integrating by 

 parts we find 



m=s [o dg S7{ U "»("" r^M)) Buaad0 - 



We have not yet assumed the density a to be independent of z. 

 Making this assumption and integrating with respect to z we have 



m = fi M -*SS( a -^- 4 » §)} "»•••(«). 



and this shows that the differential equation for u may be written 

 in the form 



®-£„i 



Z 2 <f / d 2 il . du\ -r, ( d 2 \ , , 

 a 2 W{ U -oW-* CO Td) =F W)( u) > 



the right-hand side containing no differential coefficients with 

 regard to the time. Taking u proportional to cos (s<£ + pt) we have 



P 2 + i -2 « 2 (/ + sY - 4ms P ) = F(- s 2 ). 

 This may be put in the form 



\p — SH?!*— ; as). 



where -S7 s is a function of s. Comparing this form with (10), the 

 corresponding form for the two dimensional oscillations, it is easy 

 to see that, in the present case, the nodal rate of revolution will be 



Vs 2 



CO, 



S 2 + W72+1 



3a 2 

 ZV 



and the number of beats per revolution will be 



3a 2 



s 2 + S-l 



2s 



2 3a 2 , n 



S+ W 2 + 1 



But we may generalise still further. In any surface of revolu- 

 tion, one of Lord Rayleigh's conditions of inextensibility is 



w+ l=° < 19 >- 



If we form #2D, using this and the other two conditions, it is evident 

 that we shall arrive at an equation for p of the form 



\p 2 + (1 + s 2 )p 2 — 4<a)sp = a function of s (20), 



VOL. VII. PT. III. 10 



