187-188] PROPERTIES OF BESSEL's FUNCTIONS. 311 



this requires that I I w 1 w 2 rdSdi' = (vii), 



Jo jo 



where w ly w 2 are any two terms of the expansion (v). If w l , w 2 involve 

 cosines or sines of different multiples of 0, this is verified at once by integra- 

 tions with respect to 6 ; but if we take 



W-L oc J 8 (Jc-^r] cos s#, w 2 oc J a (k 2 r) cos 50, 

 where k lt k z are any two distinct roots of (6), we get 



'Jc^r) rdr = ( viii). 



o 



The general results of which (iv) and (viii) are particular cases, are 



--jJ^(Jca} (ix), 



and 



/.(V)*(V)*-rrr 



j o KI - 



(x). 



In the case of ^ = k 2 the latter expression becomes indeterminate ; the 

 evaluation in the usual manner gives 



a 1 ' * ? t 



o *" 2^ 2 



For the analytical proofs of these formulae we must refer to the treatises cited 

 on p. 305. 



The small oscillations of an annular sheet of water bounded by 

 concentric circles are easily treated, theoretically, with the help of 

 Bessel's Functions ' of the second kind.' The only case of any 

 special interest, however, is when the two radii are nearly equal ; 

 we then have practically a re-entrant canal, and the solution 

 follows more simply from Art. 178. 



The analysis can also be applied to the case of a circular 

 sector of any angle*, or to a sheet of water bounded by two 

 concentric circular arcs and two radii. 



188. As an example of forced vibrations, let us suppose that 

 the disturbing forces are such that the equilibrium elevation 

 would be 



(16). 

 * See Lord Rayleigh, Theory of Sound, Art. 339. 



