222 Lord Rayleigh on General Theorems relating to 



And since the potential energy in either case is proportional 

 to the descent of the point Q, we see that the effect of the 

 constraint is to diminish this descent. 



The theorem under consideration may be placed in a clearer 

 light by the following interpretation of the function E. 



In forming the conditions of equilibrium, we are only con- 

 cerned with the forces which act upon the system when in 

 that position ; bat we may, if we choose, attribute any con- 

 sistent values to the forces for other positions. Suppose, then, 

 that the forces are constant, as if produced by weights. Then, in 

 any position, E denotes the work, positive or negative, which 

 must be done upon the system in order to bring it into the 

 configuration defined by V = 0. Thus, to return to the rod 

 with the weight suspended from Q, E represents the work 

 which must be done in order to bring the rod from the con- 

 figuration to which E refers into the horizontal position. And 

 this work is the difference between the work necessary to raise 

 the weight and that gained during the unbending of the rod. 

 Further, if the configuration in question is one of equilibrium 

 with or without the assistance of a constraint (such as the sup- 

 port at P), the work gained during the unbending is exactly 

 the half of that required to raise the weight ; so that E is the 

 same as the potential energy of the bending, or half the work 

 required to raise the weight. 



When the rod, unsupported at P, is bent by the weight at 

 Q, the point P drops. The energy of the bending is the same 

 as the total work required to restore the rod to a horizontal po- 

 sition. Now this restoration may be effected in two steps. We 

 may first, by a force applied at P, raise that point into its proper 

 position, a process requiring the expenditure of work. The 

 system will now be in the same condition as that in which it 

 would have been found if the point P had been originally sup- 

 ported; and therefore it requires less work to restore the confi- 

 guration V = when the system is under constraint than when 

 it is free. Accordingly the potential energy of deformation is 

 also less in the former case. 



We may now prove that any relaxation in the stiffness of a 

 system equilibrated by given forces is attended by an increase 

 in the potential energy of deformation. For if the original con- 

 figuration be maintained, E will be greater than before, in con- 

 sequence x of the diminution in the energy of a given deforma- 

 tion. A fortiori, therefore, will E be greater when the system 

 adjusts itself to equilibrium, when the value of E is as great 

 as possible. Conversely, any increase in V as a function of 

 the coordinates entails a diminution in the actual value of V 

 corresponding to equilibrium. Since a loss of freedom may be 



