(19) 



Rotatory Inertia on the Vibrations of Bars. 41 



(25 + 1)1. Hence 



»* ?7 o[l -\^{ n o cot |° + 2 } w o cot^J 



If p = period when rotatory inertia is rejected, 

 p = „ . „ ,, „ retained, 



=i + 2 -". 



or 



l k- r ?? n ,^i * v? ft 



1+ 7^i fi tan^.-2J.Hotan-^ 



(20) 



n, 



When the order of the tone is moderate cot — ° becomes 

 closely = 1 and tan -° = — 1. 

 In this case, 



2- =1+ 5-^-.(« + 2) n , sensibly, . . (21) 



Lord Rayleigh *, by considering the addition to the kinetic 

 energy when rotatory inertia is retained and using the value 

 of the normal function and its derivatives (rotatory inertia 

 being neglected) integrated over the length of the rod, has 

 deduced a formula [(1) § 186] which becomes identical with 



Urn 



(20) when the ratio — is replaced by its equivalent in terms 

 of half- angles. 



It will be seen that the effect due to rotatory inertia 

 increases with the order of the tone and is smallest in the 

 fundamental. For this tone Strehlke f gives 



!? =1-875104, 



= 107° 26' 7"-96. 

 From this we find for the gravest tone, 



p: Po = 1 + 2-323889 £- (22) 



* Rayleigh, < Theory of Sound,' vol. i. § 186. 

 t Strehlke, Pogg. Ann. Bd. xxvii. p. 505 (1833). 



