Rotatory Inertia on the Vibrations of Bars. 51 



Put n = n Q + r], 



where n is a root o£ tan n =tanh n . 

 (sin n + rj cos >i )(cosh n + rj sinh >i ) 



— (cos n — 7] sin n ) (sinh ?i Q + 77 cosh ft ) 



= 2 X 2 [cos w sinh n + sin n cosh ?i — 2>i cos n cosh ft ] . 



But sin n Q cosh n = cos n sinh ?? , 



2 2 

 2?7 sin % sinh n = -j- -jj [2 sin % cosh vi — 2/i cos ?i cosh ra ] . 



2 2 



V 



r 7T C n o cot n Q — 1] cot n . 



4 / 



n==n [l— - — {yi cot yiq— l}n cot >i ] . 



1 k 2 

 .'. ^:_p =l.+ p-p{n cotn -l}n cotn Q} . . (44) 



which for higher harmonics becomes 



1 K 2 



For the fundamental, 



p .^=1 + 5-7562650^-. 



VIII. 1. Pivoted-pivoted Bar. 

 f = d cos 7^+ d' sin 7^ + d" cosh y 2 % + d' / ' sinh 7 2 f , 



^=of 



hen f =0, 



d + d"=*0 ; 



and y i 2 d = y 2 2 d' / , 

 d=d /f =0. 



... / =1^ sin 7!^ + ^" sinh 7 2 f. . . . (45) 



Also / = 0) / 



^y Jwhen ?=-=/'• 



0=^sin7 1 Z / + rf"'sinh7/ J 



and 0= — d'y 2 sin 7^' + y 2 2 d n/ sinh 7 2 Z', 



d'"=o, 



and sin7 x ^ = 0. 



E 2 



