172 On Rotating Elastic Solid Cylinders. 



accuracy of detail. That the results are exactly true is easily 

 shown as follows : — 



Consider a right cylinder of any length or form of cross 

 section, rotating round the axis formed by the centroids of the 

 sections, taking as before the centre of the cylinder for origin 

 and the axis for axis of z. We may write the internal equa- 

 tions in the form : — 



. . (115) 

 . . (116) 



dxx , dxy 

 dx dy 



dzx 



ooPpx-- 



= 0, 



dx dy 



dz 



n*py= 



= 0, 



dx dy 



dzz 

 dz 





= 0, 



. . (117) 



the notation being the same as before. Now multiply (115) 

 by Tjx and (116) by rjy ; then from the sum of these two 

 equations subtract (117) multiplied by z, and integrate 

 throughout the volume of the cylinder. Integrating the 

 terms involving the stresses by parts, we find the surface- 

 integrals vanish in virtue of the surface-conditions at the free 

 surface of an elastic solid, and thus obtain 



$${zz—v{mb + yy)}dxdydz + V^pS}{^+f)dxdydz = 0. 

 Now by the ordinary stress-strain relations, 



zz—r)(xx+yy)=E-^; 



thus 



dz 



I 



C %lx dy dz= -T^p*: 2 . 2A Z/E, 



where A is the area of the cross section, and k its radius of 

 gyration as in (108). 

 But 



J-dz=yi—y-.t=2y h 



J' 



where yi is the value of 7 at an end of the cylinder, and 

 \\yflx dy = Bl . A. 



Thus the formula (108) is exactly true for all forms of cross 

 section in all right cylinders long or short. 



