90 * LORD EAYLEIGH ON THE ACOUSTIC SHADOW OF A SPHERE, 



potential <// is supposed to be proportional throughout to e lhlt , but this time-factor is 

 usually omitted. The general differential equation satisfied by \jj is 



of which the solution in polar co-ordinates applicable to a divergent wave of the 

 n tu order in LAPLACE'S series may be written 



^ l = S, t r Xn (kr) ......... (2). 







For the present purpose we may suppose without loss of generality that k = 1. 

 The differential equation satisfied by x (r) is 



and of this the solution which corresponds to a divergent wave is 



Putting n = and n = 1, we have 



e~ ir (1 + ir)e~ ir 



Xo( r )-~> XiO') = L 



It is ea,sy to verify that (4) satisfies (3). For if x satisfies (3), ?' -1 x' satisfies the 

 corresponding equation for x+i- And r~ } e~ ir satisfies (3) when n = 0. 

 From (3) and (4) the following formulae of reduction may be verified : 



X '(>-)=-r Xn+l (r) ......... (6), 



r X '(r) + (2n + I) Xn (r) = Xa _ l ( r ) ....... (7), 



By means of the last, ^ 2 , ^ 3 , &c., may be built up in succession from XQ an d Xi- 

 From (2) 



d^/dr = S,,(nr"-\ n + r" x ' n ), 

 or, with use of (7), 



Thus, if U M be the w th component of the normal velocity at the surface of the 

 sphere (r = c), 



T7. = < f-ift,{ Xi . 1 (c)-(ii + l) x ,(c)J ...... (10). 



When n = 0, 



....... (11). 



