(25) 



Functions of high order to the Whispering Gallery. 105 



We may now, following Stokes, compare the actual motion 

 ■at a distance with that which would ensue were lateral 

 motion prevented, as by the insertion of a large number of 

 thin plane walls radiating outwards along the lines = 

 constant, the normal velocity at r=a being the same in both 

 cases. In the altered problem we have merely in (23) to 

 replace D^, D n ' by D , D '. When z is great enough, D n (r) 

 has the value given in (16), independently of the particular 

 value of n. Accordingly the ratio of velocity-potentials at 

 n distance in the two cases is represented by the symbolic 

 fraction 



D '(fa) 



»„'(**)' 



in which 



D '(*a) = -v(^)V^+*»> (26) 



W^e have now to introduce the value of D n \ka). When n is 

 very great, and ka/n decidedly less than unity, t is negative 

 in (20), and e t is negligible in comparison with e~ f . The 

 modulus of (25) is therefore 



/ n/te_ _ \* t g -»P-* ft» 



\sinh/3cosh/3/ e ' 01 sinh*/3 " ' ' ^ J 



For example, if n=2kd, so that the linear period along the 

 •circumference of the vibrating cylinder (2ira/n) is half the 

 wave-length, 



cosh £=2, /3=1-317, sinh/3=i"7321, tanh£=-8660, 



iind the numerical value of (27) is 



e-- 4510 »+V(l-732), 



When n is great, the vibration at a distance is extraordinarily 

 small in comparison with what it would have been were 

 lateral motion prevented. As another example, let n = 1*1 ka. 

 Then (27) = 6-' 027 ^ A /('4587). Here n would need to be 

 about 17 times larger for the same sort of effect. 



The extension of Stokes' analysis to large values of u 

 only emphasizes his conclusion as to the insignificance of the 

 effect propagated to a distance when the vibrating segments 

 are decidedly smaller than the wave-length. 



We now proceed to the case of the whispering gallery 

 supposed to act by "total reflexion." From the results 

 already given, we may infer that when the refractive index 



