64 POTENTIAL ENERGY OF A DISPLACEMENT. 



+ terms of the second and higher orders, 



where H y H l9 ...,H n are constants, in so far as different 

 displacements are concerned. 



In the first place, it is manifest that H= o ; because 

 if no displacement be made, no energy is consumed. In 

 the second place, H lf ---- , H n must also be each zero, 

 because the position O is one of equilibrium ; and there- 

 fore, by the principle of virtual velocities, the work done 

 by small twists about the screws of reference must be zero, 

 as far as the first power of small quantities is concerned. 

 Finally, neglecting all terms above the second order, on 

 account of their minuteness, we see that the function V, 

 which expresses the potential energy of a small displacement 

 from a position of equilibrium, is a homogeneous function 

 of the second degree of the n co-ordinates, by which the dis- 

 placement is defined: 



69. The Wrench evoked by Displacement. When the 

 body has been displaced to P, the forces no longer equi- 

 librate, for a certain wrench has been evoked. We now 

 propose to determine, by the aid of the function V, the 

 co-ordinates of this wrench, or, more strictly, the co- 

 ordinates of the equivalent reduced wrench ( 66) upon a 

 screw of the complex, by which the freedom of the body 

 is defined. 



If, in making the displacement, work has been done by 

 the agent which moved the body, then the equilibrium 

 of the body was stable when in the position O, so far as 

 this displacement was concerned. Let the displacement 

 screw be 0, and let a reduced wrench be evoked on a 

 screw rj of the complex, while the intensities of the com- 

 ponents on the screws of reference are i^", . . . . , ij n ". 

 Suppose the body be displaced from P to an excessively 

 close position P, the co-ordinates of P, with respect to 



