248.] THE PRINCIPLE OF VIRTUAL WORK. 



or, since a = (x-x')/l, /3 = (/-/)//, -f=(z-z')/l, 



Differentiating the relation 



we have 



hence the sum of the virtual works of the two tensions T 

 reduces to 



rs/; 



which is of course zero when the connecting rod is rigid. 



It thus appears that the internal reactions of a system of 

 points connected as above described, are eliminated from the 

 equation of virtual work by selecting the virtual displacements 

 so as to leave the lengths of the connecting rods or threads 

 unchanged. This can always be done when the rods and 

 threads are not elastic. When they are elastic, the equation of 

 virtual work will contain terms of the form T1. These terms 

 must then be determined from the known relation between the 

 tension and the length of an elastic rod or thread. 



248. It is somewhat difficult to prove the principle of virtual 

 'work for the most general case of any system of bodies although 

 this is the case in which it finds its most important application. 

 It is evident, however, that the principle will be true in this 

 general case provided that all the connections and reactions 

 between the different bodies constituting the system be ex- 

 pressed by means of forces and introduced into the equation 

 of virtual work. The difficulty lies in expressing the connec- 

 tions existing between the parts of the system by means of 

 forces. 



But most of these internal reactions can be shown to dis- 

 appear from the equation of virtual work, so that they need 

 not be taken into account. 



