Functions of Ugh order to the Whispering Gallery. 103 



vibrations. The velocity-potential <j> is expressed by means 

 of polar coordinates r, 6, and will be assumed to be propor- 

 tional to cos nd, attention being concentrated upon the case 

 where n is a large integer. The problem is to determine the 

 motion at a distance due to the normal vibration of a cylin- 

 drical surface at r=a } and it turns upon the character of the 

 function of r which represents a disturbance propagated 

 outwards. If Dn(kr) denote this function, we have 



<£= e ikYi cos nO . J) n (Jcr) ,, . ~ . . (9) 

 and D n (z) satisfies Bessel's equation 



D„" + iD 1 /+(l-J)D,=0. . . . (10) 

 It may be expressed in the form 



J Jnn T 



!)„=-". -— ", (11) 



sm nir 



which, however, requires a special evaluation when n is 

 an integer. Using Schlafli's formula 



1 C* 

 J n (z) = -I cos (z sin 6-n 6) d6 



-! mn, [f\-^«nh^ (j2) 



17 Jo 

 n being positive or negative, and z positive, we find 



».(«)- ,?J« M +—) B e de 



1 C v i C* 

 1 sin(ssin0-w0)rf0 \ cos (z sin 0—n0)dd, (13) 



77 'Jo ^Jo 



the imaginary part being — J„(~) simply. This holds good 

 for any integral value of n. The present problem requires the 

 examination of the form assumed by D n when n is very great 

 and the ratio zjn decidedly greater, or decidedly less, than 

 unity. 



In the former case we set n . = .; sin a, and the important 

 part of D n arises from the two integrals last written. It 



