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



tan/-^" + e j = tanh(-^ + ej; 



or, expanding and rejecting squares, &c. of e, 



tanh-7 —1 

 4 



2tanh^ 

 4 



which gives the roots correct to six places of decimals. 



For greater accuracy we may retain squares of e. We 

 then have 



__ j/2T-'-l-T 



T.+ l 



where m , 5tt 



l=tanh-7- , 

 4 



or (ii.) by observing that 



tan m = tanh m * 

 reduces to cos 2m cosh 2m = 1 (38) 



This is' the period equation for a free-free or clamped- 

 clamped bar when rotatory inertia is neglected, and the roots 

 of (38) have been found by Strehlkef. That the periods of 

 vibration of bars under these different terminal conditions 

 should coincide to this extent is obvious also from physical 

 reasons X . . 



The roots of tan n =tanh n are 



Using the values 



we find 



n = 



= 3-9266023 



%= 



= 7-0685827 



7i 2 = 



= 10-2101761 



n s = 



= 13-3517688 



rc 4 = 



= 16-4933614 



n 5 = 



= 19-6349541. 



7i 



-K' + r> 



72 



=H'-r> 



sy 



= (1 + 1\ 2 ), rejecting X 4 , : &c. 



* Greenhill, Mess, of Math. xvi. 1887. 



"j" Loc. cit. ante. 



+ Kayleigh, ' Theory of Sound,' vol. i. § 180. 



