436 Lord Rayleigh on the Fundamental 



If p be the longitudinal density, we have by the Principle of 

 Virtual Velocities as the variational equation of motion, 



SV+jyySy=0; (2) 



where 8y refers to a hypothetical variation of y, which is subject 

 only to the condition of not violating the connexion of the system. 

 In the formation of (2) we have., as is usual, neglected the 

 reaction of the elements of the bar against the acceleration of 

 rotation. 



In order to deduce the ordinary form of the differential equa- 

 tion and the terminal conditions, we require to transform the 

 expression SV. By the usual method we find 





f 



-m}-»{2*}*»C&**' 



and therefore, as in the Calculus of Variations, 



B &+*'-? < 3 > 



The integrated terms give us the conditions to be satisfied at the 

 extremities, namely 



££-* 3*f < 4 > 



There are three cases to be considered. At a clamped end 



d/u 



8y and 8-j- vanish, and the equations (4) are satisfied without 



dii 

 any further supposition. At a free end By and 8-^ are arbi- 



trary, and hence 



^-°> s*-° ( 5 ) 



are the conditions to be satisfied in such a case. The third case 

 occurs when the end is constrained to be a node, but the direc- 

 tion of the rod is left free. Since 8 -^ is arbitrary, the condi- 

 tions are 



y=°> 3=° < 6 ) 



The general equation (3) and the terminal conditions are the 

 same as would be found without the use of the potential energy 

 by the ordinary method, 



