Rotatory Inertia on the Vibrations of Bars. 49 



.\ substituting in (36) 



(l + fX 2 ) S in«(l + ^)cosh«(l->-) 



= cos n ( 1 4- -j- J sinh n ( 1 — j- ) . 



•"• (1 + 1^ 2 ) ( sni n + ~r cos n ) ( cosn ft -jr sinh w) 



nX 2 . N , . . nX 2 , N 

 = (cos n -sm nj(smn n — cosn n). 



.'. sin n cosh n — cos n sinh n 



= — — 2;i cos n cosh n — 6 sin rc cosh rc . 



Put n = n + r], 



where n is a root of tan ;? = tanh n , .... (39) 



and 7j is small. 



(sin n + r) cos 7i )(cosh n -f- r\ sinh w ) — (cos n — 97 sin n ) (sinh w -f 7; cosh n ) 



X 2 

 = — — ['2w cos ?? cosh h + 6 sin ?2 cosh n ~\ 



(sin 7? cosh n — cos « sinh n ) + t;(cos ?? cosh n + sin n sinh n 



+ sin ?i sinh n — cos n cosh ?i ) 



X 2 

 — — — [2« cos n cosh n + 6 sin n cosh n ~] . 



.*. 77 = — j- [wo c °t 7i o c °th h + 3 coth n ~] 



X 2 



■ = — j- [;?o co ^ ?z o + 3 J cot ?2 , 



which for higher harmonics becomes 



^ 



X 2 • 

 ?;= — — (>? + 3). 



Hence 1 /c 2 . . ' 



ra==w {l — - -p- (n Gotn -}-6)n eotn o { f , 



^ : p = .1 + 2 ^T ("0 cot »o + 3) »(reot ra ; . . (40) 

 for the gravest tone this becomes 



2>:p = 1 + 13-6155760^-^ 

 PA*7. 1%. S. 6. Vol. 14. No. 79. Jt% 1907. E 



