416 Prof. Stokes on the Communication of Vibration 



expressed in the form of* a definite integral. For n = we know 

 that 



IT 



f = f 2 -{E + Flog(rsin 2 £)j. cos (mrcos ?)d? . (18) 



is a third form of the integral of (15). It is not difficult to de- 

 duce from this the integral of (15) in a similar form for any in- 

 tegral value of n. Assuming 



and substituting in (15), we have 



+ mVJV# n dr=0. 



Assume 



a 2_ w 2 = 0j ...... (19) 



divide the equation by r a , differentiate with respect to r, and 

 then divide by r&. The result is 



dr* r ar r~ 



This equation will be of the same form as (15), provided 



« + /3=0, 



which reduces the coefficient of the last term but one to — (a+ I) 2 . 

 In order that this coefficient may be increased we must choose 

 the positive root of (19), namely n, which I will suppose positive. 

 Hence 



^ n=r *^,.-« x j r (30 ) 



gives 



*X* + 1 d Xn- ( w+1 >* y +m*y =0 

 ~^r + r dr r* Xn^ m Xn "> 



the same equation as that for the determination of yfr n+l . Hence, 

 expressing y^ n in terms of ty n from (20), writing n— 1 for n, and 

 replacing Xn-i by ty n , we have 



a formula of reduction which, when n is integral, serves to ex- 

 press \[/„ in terms of \f/ . We have 



+•-*$£)'+» < 21 > 



an equation which when applied to (18) gives the complete inte- 



