WITH AN APPENDIX BY PEOFESSOR A. LODGE. 91 



The introduction of S,, from (10), (11) into (2) gives \f> H in terms of U supposed 

 known. 



When r is very great in comparison with the wave-length, we get from (4) 



v(r) = l " e ' (12) 



K\ / r +i V 1 ^/' 



so that 



^ = S^" e ~ (13). 



M 



In order to find the effect at a great distance of a source of sound localised on the 

 surface of the sphere at the point //,-=!, we have only to expand the complete value 

 of U in LEGENDRE'S functions. Thus 



U,, = i (2n + 1) P H ( M ) J *' UP H (/i) dp 



" JrfS . . (14), 



in which JJUc^S denotes the magnitude of the source, i.e., the integrated value of U 

 over the small area where it is sensible. The complete value of \\i may now be 

 written 



._ . , t 



"^- 



When n = 0, x-i ( c ) ~ C /l + !) X ( c ) ^ s t be replaced by c-^i ('') 

 If we compare (15) with the corresponding expression in " Theory of Sound," (3), 

 238, we get 



c" +1 (x-i ( c ) ~ ( + X ( c )l = " ^-"F. (/c) . . . (1(5). 



Another particular case of interest arises when the point of observation, as well as 

 the source, is on the sphere, so that, instead of r = oo , we have r = c. Equation (15) 

 is then replaced by 



4 



It may be remarked that, since i/> in (17) is infinite when ^ = + 1 and accordingly 

 P M = 1, the convergence at other points can only be attained in virtue of the factors 

 P fl . The difficulties in the way of a practical calculation from (17) may be expected 

 to be greater than in the case of (15). 



We will now proceed to the actual calculation for the case of c 10, or kc 10. 

 The first step is the formation of the values of the various functions x( 10 )> starting 

 from xo(10), ^ (10). For these we have from (5) 



N 2 



