54 Lord Rayleigh on BesseVs Fumtlons as applied 



n, regarded as susceptible of continuous variation.- It will he 

 shown that each root increases continually with n. 



Let us contemplate the transverse vibrations of a membrane 

 iixed along the radii # = and 6 = /3 and also along the 

 circular arc r=l. A typical simple vibration is expressed 

 by* 



w = Jn (*<J>r) . sin n$ . cos (z<£>t), . . . (1) 



where z$ is a finite root of J„ (~) = 0, and n — 7r//3. Of: 

 these finite roots the lowest z^ gives the principal vibration, 

 i. e. the one without internal circular nodes. For the vibra- 

 tion corresponding to z^ the number of internal nodal 

 circles is s— 1. 



As prescribed, the vibration (1) has no internal nodal 

 diameter. It might be generalized by taking n — fir/ ft, where 

 v is an integer ; but for our purpose nothing would be gained, 

 since /3 is at disposal and a suitable reduction of /3 comes to 

 the same as the introduction of v. 



In tracing the effect of a diminishing /3 it may suffice to 

 commence at /3 = tt, or n = l. The frequencies of vibration 

 are then proportional to the roots of the function J 1# The 

 reduction of j3 is supposed to be effected by increasing 

 without limit the potential energy of the displacement (to) 

 at every point of the small sector to be cut off. We may 

 imagine suitable springs to be introduced whose stiffness is 

 gradually increased, and that without limit. During this 

 process every frequency originally finite must increase f, 

 finally by an amount proportional to dfi ; and, as we know, 

 no zero root can become finite. Thus before and after the 

 change the finite roots correspond each to each, and every 

 member of the latter series exceeds the corresponding member 

 of the former. 



As ft continues to diminish this process goes on until 

 when /3 reaches ^7r, n again becomes integral and equal to 2. 

 We infer that every finite root of J 2 exceeds the correspond- 

 ing finite root of J x . In like manner every finite root of J 3 

 exceeds the corresponding root of J 2 , and so on. 



I was led to consider this question by a remark of Gray 

 and Mathews J — " It seems probable that between every pair 

 of successive real roots of J n there is exactly one real root 

 of J n +i. It does not appear that this has been strictly 

 proved ; there must in any case be an odd number of roots 



* ' Theory of Sound/ §§ 205, 207. 



t I.e. §§88, 92 a. 



t Bessel's Functions. 1895, p, 50. 



