134 Mr. R. H. M. Bosanquet on the Theory of Sound, 



of radius p at distance ?' is, as before, 



27rr^ + ir^pr + irp'^ ; 

 or putting p — a-\- hr, 



2 = TT { r'X2 + irk + ]^) + arijr + 2/.-) + a^ } . 

 The element diminishes in divergence as the surface increases ; 



and of that on 2 only -^ reaches Sq ; so that the total energy 



that reaches S^^ is measured by 



^ S So _ S^ 



S'-^ "" 2 "" t S2'^ • 



The surface 2 when i^—a has the value 



Suppose k to be a small fraction, such that the term which 

 involves it can be neglected for approximate purposes. We 

 have then to represent the law of variation of the function 



-^y2 ^Y ^ power of S, such that the two functions shall have 



the same value at r = and r = a; we neglect the difference 

 at great distances. 



Let 2 = ^jY, where r = a; then 2 = So when r = 0. 



When r=rt, 



2 = (3 + 7r)7ra^, approximately, 

 neglecting terms in k ; and 

 ^ = 7r(a-\-kry; 

 .'. 34-7r = (l + ^)^'^, approximately ; 

 .♦. •788 = 2.^M/.;, 

 where M = *434 (modulus of logarithms). 

 .*. ka;='d, nearly. 

 In the equation 



f °° ^S _ _ p rfS 



Putting 2=^, we get 



whence, since S = 7r(a + /tr)^, 



a 



