46 Mr. J. H. C. Searle on the Effect of 



.'. Putting n=no + r], 



where n is a root of cos n cosh n =l, . . (31) 

 and r) is small, 

 .'. cos (n + 7)) cosh (n + rj) — -~ A 2 (sin n cosh n + cos n sinh ?i ) 



— 1 — | A, 2 sin n sinh w = ; 



,\ remembering (31) 



n 

 rj(cos n sinh t^— sin ?i cosh n ) — ~ X 2 (sin n cosh w 



+ cos n sinh n ) — §A 2 sin w sinh w = 0. 



6 

 tan 7? + tanh n -j- — tan n tanh w 



_ _ WO X 2 ^0 . 



" * " 4 tan w — tanh wq 



By (31) this becomes 



wo- o ^ 2 / ; w 6 . n \ 

 "=-4- T( OOt 2 + » Cot 2> 



— 4«o P \ tan 2 - % tan 2 } 

 according as n lies in the 1st or 4th quadrant. 



Thus rv 1 k 2 ( n , a \ ,n ~] 



n = n Q \ 1— j-p-(« cot^-+b l« cot jj 



or rv 1 h? / w \ n ~| 



= "o[ 1_ - 4 72 ( w tan ^- -6jw tan^-J. 



The period ratio is 



i 1 k 2 f . n Q n ~] . n n \ 



p''Po = ±+2~p~} Uz cot T + 6 f 7? o cot 2 



or 1 /c 2 f , w ") h (" 



" + 2 I 2 " I ^° 2 J "° 2 J 



In the case of the fundamental, 



(32) 



n B tan "° = -4-04727596, as before. 

 %:-p = .1 + 24-7404148 — . 



In the case of a short thick bar this may become important. 



