414 Prof. Stokes on the Communication of Vibration 



using circular functions directly, I will employ the imaginary 

 exponential e' mat , i denoting as before V — 1, and will put ac- 

 cordingly V = e imat \J, Ubeing a given function of 0,and(/> = -\/re" na '. 

 For a given value of r, ty may by a known theorem be developed 

 in a series of sines and cosines of and its multiples, and there- 

 fore for general values of r can be so developed, the coefficients 

 being functions of r. If yjr n be the coefficient of cos n6 or sin n6, 

 we find 



dr* + r dr r ^n + rn^ n -K). . . (lb) 



If we suppose the normal velocity of the surface of the cylinder 

 to vary in a given manner from one generating line to another, 

 so that U is a given function of 0, we may expand U in a series 

 of the form 



U = U o +U 1 cos0 + U 2 cos20 + ... 

 + U' 1 sin0 + U 2 'sin20 + ... 

 On applying now the equation of condition which has to be sa- 

 tisfied at the surface of the cylinder, we see that a term XJ n cos nO 

 or U' B sin n0 of the nth order in the expression for U will intro- 

 duce a function ^Jr n of the same order in the general expression 

 for (f>. Now the only case of interest relating to an infinite cy- 

 linder is that of a vibrating string, in which the cylinder moves 

 as a whole. The vibration may be regarded as compounded of 

 the vibrations in any two rectangular planes passing through 

 the axis, the phases of the component vibrations, it may be, being 

 different. These component vibrations may be treated separately, 

 and thus it will suffice to suppose the vibration confined to one 

 plane, which we may take to be that from which is measured. 

 We shall accordingly have 



U = U lC os0, 

 Uj being a given constant ; and the only function ^ n which will 

 appear in the general expression for <b will be that of the order 1. 

 Besides this we shall have to investigate, for the sake of compa- 

 rison, an ideal vibration in which the cylinder alternately con- 

 tracts and expands in all directions alike, and for which accord- 

 ingly U is a constant U . Hence the equation (15) need only 

 be considered for the values and 1 of n. 



For general values of n the equation (15) is easily integrated 

 in the form of infinite series according to ascending powers of r. 

 The result is 



• _a „ ! T 1 ^ 2 >' 2 m 4 ;- 4 I"". 



Yn ~ T 1 2 (2 + 2») + 2 . 4(2 + 2r^(4 + 2n) '"J { 



■„ _ f mV 2 m 4 t A "1 



+ r 1 "~ 2(2 -2») + 2 . 4(2-2/i) (4-2n) ""jJ 



(16) 



