354 Lord Rayleigh on Vibrations and 



If we replace the reactions against acceleration by external 

 forces, we may obtain the solution of a statical problem. 



When a membrane of any shape is submitted to transverse 

 forces, all in one direction, the displacement is everywhere 

 in the direction of the forces. 



Similar conclusions may be formulated for the conduction 

 of heat in two dimensions, which depends upon the same 

 fundamental differential equation. Here the boundary is 

 maintained at a constant temperature taken as zero, and 

 " persistances " replace the periods of vibration. Any closing 

 in of the boundary reduces the principal persistance. In 

 this mode there can be no internal place of zero temperature. 

 In the steady state under positive sources of heat, however 

 distributed, the temperature is above zero everywhere. In 

 the application to the theory of heat, extension may evidently 

 be made to three dimensions. 



Arguments of a like nature may be used when we consider 

 a bar vibrating transversely in virtue of rigidity, instead of 

 a stretched membrane. In ' Theory of Sound,' § 184, it is 

 shown that whatever may be the constitution of the bar in 

 respect of stiffness and mass, a curtailment at either end is 

 associated with a rise of pitch, and this whether the end in 

 question be free, clamped, or merely " supported." 



In the statical problem of the deflexion of a bar by a 

 transverse force locally applied, the question may be raised 

 whether the linear deflexion must everywhere be in the 

 same direction as the force. It can be shown that the 

 answer is in the affirmative. The equation governing the 

 deflexion (w) is 



S( B S)= z > a 



where Tidx is the transverse force applied at dx, and B is a 

 coefficient of stiffness. In the case of a uniform bar B is 

 constant and w may be found by simple integration. It 

 suffices to suppose that Z is localised at one point, say at 



