Problem of the Whispering Gallery. 1003 



of (2) evidently corresponds to small values of <w. If z=n 

 absolutely we may write ultimately 



IT 72 " 1 ("° 



«!« (") = - I cos n((o— sin ay)d w— — \ cos n(co— sin <wWa) 



TTJO 7Tj 



= - 1 cos— v- rfw=- - cos a° da 



■""Jo b w\n/ Jo 



= r(i).2-f3-i7r- l 7z-i, . (3) 



one of Nicholson's results. 



In like manner when n— z, though not zero, is relatively 

 small, (1) may be made to depend upon Airy's integral. 

 Thus 



.J, (*)=-( cos{(n-5)» + Ja» 8 }rfw. . . (4) 

 '•"Jo 



In the second of the papers above cited Nicholson tabulates 



zh J n (z) against 2*1123 (n — z)/z*. It thence appears that 



2*4955 i , , 01 , i /K . 



zi = n+ 9 » J = ?i + 1-1814 ?i 3 . ...(§) 



The maximum (about 0*67) occurs when 



2 = w + "51w*, (6) 



and the function sinks to insignificance (0"01) when 



2 = n-l'5?i J (7) 



Thus in the membrane problem the practical range is only 

 about 2*7 n$. 

 In like manner 



, 1*0845 i , -io<o 2 /o\ 



so that in the aerial problem the practical range given by 

 (7) and (8) is about 2'lrfi. 



To take an example in the latter case, let ?i = 1000, repre- 

 senting approximately the radius of the reflecting circle. 

 The vibrations expressed by (1) are practically limited to an 

 annulus of width 20, or one fiftieth part only of the radius. 

 With greater values of n the concentration in the imme- 

 diate neighbourhood of the circumference is still further 

 increased. 



It will be admitted that this example fully illustrates the 

 observed phenomena, and that the clinging of vibrations to 

 the immediate neighbourhood of a concave reflecting wall 

 may become exceedingly pronounced. 



3U2 



