130 Lord Rayleigh on the Passage of 



fi, v be integers, and (21) gives 



*^$ + &)> (28 > 



identical with (24) . 



If a> /3, the smallest value of k corresponds to /a=1, v = 0. 

 When v=0, we have k = fnr/a, and if the factor e i( ^ mz+pt) be 

 omitted, 



a= r &mkx, 6 = 0, c = coskx, . . (29) 



k 



P = 0, Q=- ? |-sin^, R = 0; . . . (30) 



a solution independent of the value of /3. There is no solu- 

 tion derivable from y^ = 0, v = 0, & = 0*. 



Circular Section. 



For the vibrations of the first class we have as the solution 

 of (2) by means of Bessel's functions, 



R = J n (kr)cosn0, (31) 



n being an integer, and the factor e i(mz+pt) being dropped 

 for the sake of brevity. In (31) an arbitrary multiplier and 

 an arbitrary addition to are of course admissible. The value 

 of k is limited to be one of those for which 



Jn(ArO=0 (32) 



at the boundary where r = r. 



The expressions for P, Q, a, b, c in (15), (16) involve only 

 dH/dx, dR/dj. For these we have 



^ = ^? cos 6- ~ sin 0= k J/(rtr) cos n0 cos 



dx dr rclo 



n 

 + - JJkr) sin n6 sin 



= i^cos(n-l)^J/+J;}+Pcos(ri + l)^{j^-^| 



= ikcos{n-l)0J n - l {kr)-±kcos {n+\)0 J n+l (kr), . (33) 

 according to known properties of these functions ; and in 



* For (18) would then become v 2 ^ = 0; and this, with the boundary 

 condition d^/dn = 0, would require that P and Q, as well as R, vanish 

 throughout. 



