ami IHstriltution of Sound. 30)^ 



Since every element dd' contributes equally, it suffices to 

 take 0=0. so that the inteo^ral to be evaluated i> 



(■ 



sin ('2kc<mW) ,^ ,-.^. 



2/cc sin iH ' 



or, if i(9=</,, 2/v = .r. 



,ri- .in(..sin<^) 



Jo •^■^»^<^ 

 The integral (26) may be expressed by means of Bessel's 

 function J^, for 



2 r^'^ 



J.. (a:) = - ) cos (.i- sin </>) d^, 

 so that 



Jo ■^Jo ^™'^ 



Thus, if eonstact factor? be disregarded, we get 



1 l" 



(■21) 



in which x='2kc. Since (27) reduces to unity when c, or 

 j:, vanishes, it represents the ratio in which the work done is 

 diminished when a source, originally concentrated at the centre, 

 is distributed over a circular arc of radius c. 



The case of a source uniformly distributed over a circular 

 disk of radius c is investigated in my book on the ^ Theory 

 of Sound ^ *. According to what is there proved, the factor,. 

 analogous to (27j, expressing the ratio in which the work 

 done is diminished when a source originally concentrated at 

 the centre is expanded over the disk, has the form 



i={-^'} » 



where, as usual, 



Another case of interest is when the distribution takes 

 place over the surface of a spliere of radius c. In (25) 

 we have merely to introduce the factor sin 6, equal to 

 2 sin \d cos \6, so that we get instead of (26) 



. I ^ sin ix sin <f>) .7,4.^ n 



41 1^ ^^COS (j>d<t)= -(1 — COr^A-). 



Jo '^* ^ 



* MacMillan, 1st edition, 1878 ; 2nd edition, 1896, § 302. 



