The Hon. J. W. Strutt on Bessel's Functions. 331 



state them here. They may easily be proved from the ascend- 

 ing series. 



2»j_ I . . . . (8) 



whence may also be derived 



J«=-J„ — J„+i. ...... (9) 



The simpler forms approximated to, as z increases relatively to n, 

 should be noticed. 



One of the important applications of Bessel's functions is to 

 the investigation of aerial vibrations in cylindrical spaces, the 

 motion being perpendicular to the axis of the cylinder. The 

 general differential equation governing the small vibrations is 



where (ft is the velocity-potential (assumed to exist). If we 

 transform to polar coordinates in the plane xy, there results 



dt^ ~ \dr^ '^ rdr '^ r^ dd^ "*" dz^ J' ' ' ^ ^ 

 If now the motion be independent of z, and of such a period that 



277- 



-:^ = fc, we have (omitting the time factor) 



Suppose that we are considering the motion that can proceed 

 within a rigid cylinder of radius r=l. Whatever it may be, it 

 can be divided into simple vibrations of various periods, each 

 satisfying an equation like (12). Again, in (12) can by 

 Fourier's theorem be expanded in a series proceeding by sines 

 and cosines of multiples of 6. Considering the term containing 

 cos {nd + a)i substituting in (12) and dropping the factor con- 

 taining 6, we find 



which, on division by /c^, appears in the form (1). The general 

 integral of (13) may be written 



(^ = AJ„(/c7') +B (another function of r), 



but the function multiplied by B becomes infinite when r va- 

 nishes. In point of fact the function in question does not, pro- 



