(29) 



Mass of Air confined by Walls at Constant Temperature. 3J9 



we have 



a __ 2® cos m x 

 a/3 + cos 2 /^ 

 For the pressure we have 



0+s/* = A 1 e-K(-cosm l +^^) + ..".". 



= 7j— cos m, . A x e- V + 



or in the particular case of (29), 



d + sa = 2® — 3^ 1 .- + (30) 



afj a/3 + COS 2 mi v ' 



If /3=0, we fall back upon a problem of the Fourier type. 

 By (22) in that case 



ma = ^7r(l, 3, 5, . . .) 



cos 2 ma = a?fi 2 /m 2 a 2 , 

 so that (30) becomes 



20 fe + ^v + --> • • • ^ 



or initially 



80/1 X , 1 . \ . _ 



^(p+ 35+52 + ...) »,«.e. 



The values of /* are given by 



/i= ^ (12 ' 32 ' 52 '--) ( 32 ) 



We will now pass on to the more important practical case 

 of a spherical envelope of radius a. The equation (8) for 

 (0—C) is identical with that which determines the vibrations 

 of air * in a spherical case, and the solution may be expanded 

 in Laplace's series. The typical term is 



d-(3={mr)-hJ nH (mr).Y n , . . . (33) 



Y n being the surface spherical harmonic of order n where 

 n=0, 1, 2, 3 . . ., and J the symbol of Bessel's functions. In 

 virtue of (6) we may as before equate — s/a — D, where D is 

 another constant, to the right-hand member of (33) . The two 

 conditions yet to be satisfied are that 0=0 when >*=a, and 

 that the mean value of s throughout the sphere shall vanish, 



* l Theory of Sound/ vol. II. cli. xvii. 

 Z2 



