144-147] CONSERVATIVE SYSTEMS. 131 



It is clear that this is the case also whenever the force between any 

 two particles is a one-valued function of their distance and does not depend 

 upon any other variable. 



There is a more general case possible, viz., when the force between two 

 particles is a one- valued function of their distance and of the other distances 

 between particles of the system ; provided such functions satisfy certain 

 conditions. 



147. Potential Energy of Stretched String. In the case of gravitation 

 the work done by internal forces in any displacement can be calculated 

 directly, even for the case of a continuous body, but in other cases, where the 

 internal forces themselves are only assigned as regards their resultants across 

 small elements of area, the work done by the forces constituting these 

 resultants is calculated indirectly by the application of the Principle of 

 Virtual Work. (Article 157.) 



We shall carry out the calculation for an extensible string. 



Suppose an element of the string of natural length As , which will 

 presently be taken infinitesimal, is in equilibrium under the tensions at its 

 ends and any bodily forces, and suppose e is the extension of the element, and 

 X the modulus of elasticity. Then, with sufficient approximation, the length 

 is (1-f e) As , and the tension at either end is Xe. The bodily force acting on 

 the element is of the order Xf.As , viz., of the order of the difference of 

 tensions at the ends. 



Now let the ends be slightly displaced so that the length is increased, and 

 let the extension become e + Ae, so that the length is (1+e + Ae) As . The 

 work done by the tensions at the ends is Xe . Ae . As . The work done by the 

 bodily forces is of the same order as the product of this quantity and As . 



Let A W be the work done by the internal forces during the displacement, 

 then, with sufficient approximation, the equation of Virtual Work is 

 ATF + Xe. Ae. As = 0. 



It thus appears that the work done by the internal forces within the 

 element ds Q of the string in an indefinitely small displacement de is dW\ 

 where 



dW=-d9 Q .\c.d*. 



Hence the work done by the internal forces in the string, when it is 

 stretched from its natural length until the extension at any element becomes 

 e, is W, where 



W 



=- (ds [ e 

 J Jo 



the integral being taken along the string. 



When the string is uniformly extended we find 



92 



