1004 T1i£ Problem of the Whispering Gallery. 



Another example might be taken from tlio vibrations of 

 air within a spherical cavity. In the usual notation for 

 polar coordinates (r, 6, <f>) we have as a possible velocity- 

 potential yjr = (kr) -i J n+ % (kr) sin" 6 cos {hat — ncfy), and the 

 discussion proceeds as before. 



So far as I have seen, the ultimate form of J n (z) when n 

 i> very great unci z a moderate multiple of n has not been 

 considered. Though unrelated to the main subject of this 

 note, I may perhaps briefly indicate it. 



The form of (2) suggests the application of the method 

 employed by Kelvin in dealing with the problem of water 

 waves due to a limited initial disturbance. Reference may 

 also he made to a recent paper of my own*. 



When n and : arc greal the only important part of the 

 range of integration in (2) is the neighbourhood of the place 

 or places, where c sin r«> — nco i> stationary with respect to ca 

 These are to be found where 



«j = rt/z, (0) 



from which we may infer that when ~ is decidedly less than 



w, the total value of th Pal is .-mall, as we have already 



to be the case. When c>??, a), is real, and according 



i would admit <>f an infinite series of values. Only one, 



iver, «'t' tie-'' into consideration, since the actual 



range of integration is from to rr. We suppose that z is 



so much greater than >/ that <-/>, has a sensible value. 



The application of Kelvin's method gives at once 



/ { - V" -::-i na,,-^,-M _ _ 

 V \ws/ \/{ sm M >} 



We may test this by applying it to the familiar case where 

 - is so much greater than // as to make w^ — ^tt. We find 



J» W=^/(^) • cos I*- W- i*}> • • (in 



the well known form. 

 As an example of (10), 



J„(2«)= v /(^ 7 r J ).cos {(y^M n-{ir) 



(12) 



Although in (2) n is limited to be integral, it is not difficult 

 to recognize that results such as (3), (5), (12), applicable to 

 large values of n, are free from this restriction. 



* Phil. Mag. xviii. p. 1, imnieliately preceding Nicholson's paper 

 quoted. 



