hersey: vibrations of elastic systems 441 



Nothing that has been said prevents $ in (1), and therefore 

 c in (5), from being a multivalued function, corresponding to the 

 existence of more than one mode of vibration. From this it 

 might be imagined that the condition for compensation (5) 

 would not apply to more than one mode of vibration at a time; 

 so that, if an alloy were so chosen as to compensate the fundamen- 

 tal, the pitch of the harmonics would alter with temperature. 

 Closer examination shows that, for any two frequencies which 



stand in a constant ratio - - = k, where k is independent of FOIS- 

 TS 



son's ratio, the values of c will be identical, and compensation 

 will be possible simultaneously, if possible at all. 



The relation of frequency to stiffness. From equation (3) of the 

 previous paper, S = LE <t> (a) = Lh\I/(<f). Comparing with (1) 

 above, 



V- / M (6) 



n 



in which the form of / will depend on the generalized shape. 



In passing it may be noted that the corresponding expressions 

 in terms of torsional stiffness are sometimes preferable. It 

 can be shown that the torsional stiffness or rigidity 



S t = UE 4n (a) = L-V fc Or) (7) 



so that 



n = jJ St M<r) (8) 



L 1 m 



From (7), also, 



S t = (C t +l)a-C tl 3 + 3y 



in which 



e, = (i + „)^u + „)£,o g (A) v (9) 



I 



y/ a<r \ & t / 



\ 



The use of (7), for example, as a recipe for model experiments, 

 would render unnecessary the St. Venant theory of the torsion of 



