108 Mr Bryan, On the beats in the [Nov. 24, 



quantities of the second order. This shows that in all cases of 

 practical interest the pitch is not perceptibly raised by the rotatory 

 motion, and the only noticeable effect is that of the beats already 

 described. On the other hand, if the cylinder is revolving very 

 rapidly, so that ay is comparable with II s , the vibrations will no 

 longer give the effect of beats at all*. 



The results of the last paragraph are very important because 

 they admit of extension to rotating shells in general, and afford us 

 an easy way of determining the nodal angular velocity and con- 

 sequent number of beats per revolution in other slowly-revolving 

 systems where the vibrations are not two-dimensional. It will be 

 seen that the nodal rotation depends exclusively on B'tB the 

 variation of the kinetic energy, and that the square of co may be 

 neglected throughout, since it only appears in the expression for 

 the frequency, and does not affect the latter to any appreciable 

 extent. All that is necessary, therefore, is to calculate #2£. We 

 proceed to apply this method to the vibrations which Lord Rayleigh 

 has investigated for a perfectly inextensible cylindrical shell of 

 length I closed by an inextensible disk at one end*f-. 



Taking the axis of the cylinder as axis of z, and the closed end 

 at z = 0, we have, if u denote the longitudinal displacement, 



B*® = dz\ {uBu + (v + 2anu) Bv + (w - 2gm>) Bw} aadd . . .(14), 

 Jo Jo 



omitting the terms — ao> 2 Bw — wafBw ; of which the first depends 

 on the undisturbed motion, while the second involves small quan- 

 tities of the second order. Neither of these omitted terms will 

 in any case affect the nodal rotation as they do not contain 

 differential coefficients with regard to the time. 

 Lord Rayleigh' s conditions of inextensibility are 



du dv du dv _ ,, „ x 



_ = 0, "+33 = 0, 35+« 5 = (15), 



and we suppose Bu, Bv, Bw to satisfy similar conditions. By means 

 of the second of these conditions we eliminate w, Bw and have, by 

 integrating by parts, 



B® =f l dzf*"iu&u + (v - ^ 2 -4&)S Sv\ aadB (16). 



From the first and third we see that u is a function of 6 and not of 

 z, and therefore, that 



_ z du 

 a dd' 



* By putting /3=0, yu=0 in (10) we may deduce the solution to the purely 

 kinetic problem of determining the oscillations of a rapidly revolving flexible 

 endless chain. 



t Proc. London Mathematical Soc. xni., page 5. Proceedings Royal Society, 

 Vol. XLV. 



