Vibrations in Solid and Hollcw Cylinders. 347 



In a hollow circular section the radius of gyration round 

 the perpendicular to the plane through the centre is given by 



K 2 =(a 2 + b*)/2, 



so that (43) is in agreement with (3) and (22). 



The fact that (43) is merely a special case of (3) or (22) 

 may seem to indicate that our separate treatment of the 

 hollow cylinder, or tube, is quite unnecessary. I can only 

 say that having regard to the methods by which (3) and (22) 

 were arrived at — more especially to the fact that in establishing 

 (6) I was dealing with solid cylinders — I had long felt the 

 desirability of an independent investigation. 



§ 14. The complete determination of the constants A, A 7 , 

 C, C^, and of the several displacements, strains, and stresses 

 to the degree of accuracy assumed in (43), though not a very 

 arduous labour, would require more time than seems warranted 

 by the physical interest of the problem. I thus confine my 

 further remarks to the form of the longitudinal displacement 

 w. Substituting their approximate values for tbe J's and 

 Y's from (39) in (32), we find 



piDJfJb cos M sin (pz — e) = 



- A(l - i/zV) ~ A'{ (1 - i|*V) log fir + JpV \ 



Now, considering only their principal terms, it is easily 

 seen that A'/ A and C/C are both of the order (p 2 ab) 2 . Thus, 

 to the present degree of approximation, we may leave the A' 

 and G / terms in iv out of account. Also confining ourselves 

 to principal terms, we easily find 



-A = Cp 2 /fJLCC 2 



2(m—n) ~ m + n 

 Hence, employing the two last of equations (40), we deduce 

 i{ Afi 2 - (Cp 2 /y.a 2 )\ 2 \ H- { - A + Cp 2 /f*« 2 } 



We thus have from (44), to the degree of approximation 

 reached in (43), 



iv = iv cos kt sin (pz — e) (1 — i^pV) , . . (45) 



where iv is a constant which depends on the amplitude of the 

 vibration. 



The expression (45) for iv is exactly the same as I found 



