﻿390 Mr. L. R. Wilberforce on the Vibrations 



is seen to be transferred with almost perfect completeness 

 from ^'-vibrations to (^-vibrations and back again. 

 The condition for the vanishing of \ is of course 



A<ft 2 _B 



or, since to our order of approximation l — r(j>, 



A 



It is also of interest to produce, by means of a series of suit- 

 ably timed small impulses, the normal modes of vibration of 

 such a system and to demonstrate the permanence of each. 



If, however, the object in view is the determination of 

 elastic constants, it is convenient to arrange that (c— a) shall 

 be large compared with b. In this case, as we have seen, 

 pure x- vibrations and pure (^-vibrations are practically the 

 two normal modes, and the periodic times of the former and 

 the latter are given by 





Mlk 2 

 B 



If the mass m of the spring itself cannot be neglected, we 

 can allow for it, if small compared with M, by taking M + ^m 

 as the vibrating mass, and MP + J-mr 2 as its moment of 

 inertia*. 



Let us consider the case of a spring made of circular wire 

 of radius p. If we may assume the material to be homo- 

 geneous and isotropic, an assumption which is undoubtedly a 

 weak point of all methods of determining the elastic constants 

 of a material by experiments on wires, we have 



where E and n are respectively the Young's modulus and the 

 rigidity of the material. From the above we obtain 



E_2B Mff + ^mr* t£ 



n~ A X Mr 2 + J™r 2 'f 2 2 ' 



an equation involving only quantities easy of measurement, 



2 j, is 



n 

 * Lord Rayleigh's * Theory of Sound,' § 156. 



