Equilibrium and Initial and Steady Motions, 223 



regarded as an increase of stiffness, we see again how it is that 

 the introduction of a constraint diminishes V. 



The statical analogue of Thomson's theorem for initial motions 

 refers to systems in which given deformations are produced by 

 the necessary forces of corresponding types — for example, the 

 rod of our former illustration, of which the point P is displaced 

 through a given distance, as might be done by raising the sup- 

 port situated under it. The theorem is to the effect that the 

 potential energy V of a system so displaced and in equilibrium 

 is as small as it can be under the circumstances, and that the 

 energy of any other configuration exceeds this by the energy of 

 that configuration which is the difference of the two. 



To prove this^ suppose that the conditions are that -\\r v ^ 2 , 

 -^3, . . . . ty r are given^ while the forces of the remaining types 

 "4 r r+1 , ^+2, &c. vanish. The symbols yfr v &c, *P V &c. refer to 

 the actual equilibrium-configuration, and y^ l + Ayp t 19 i/r 2 -f A-^, 

 &c, ^, -f- A^j, ^P 2 -f A^F 2 , &c. to any other configuration subject 

 to the same displacement -conditions. For each suffix, therefore, 

 either Ayfr or ^ vanishes. Now for the potential energy of the 

 hypothetical deformation we have 



2(V + AV) = (¥ 1 +A¥ 1 )(^ 1 + A^ 1 )+.... 

 = 2V + %A^-f¥ 2 A<f 2 + .. .. 



+ A¥ 1 .o/r 1 + A^ 2 .^ 2 +.... 



+ A¥ 1B A^ 1 + A¥ 2 .A^r 8 + .... . . . (13) 



But by the reciprocal relation, 



% . A^r 1+ ¥ a . At 9 +. • .«A¥, .^+A¥ 2 . f 2 + . .. , 

 of which the former by hypothesis is zero. Thus 



2AV=A¥ 1 .Af 1 + A¥ 2 .A^ a + ...., . . . (14) 



as was to be proved. 



The effect of a relaxation in stiffness must clearly be to di- 

 minish V; for such a diminution would ensue if the configura- 

 tion remained unaltered, and therefore still more when the sys- 

 tem returns to equilibrium under the altered conditions. It 

 will be understood that in particular cases the diminution 

 spoken of may vanish. 



The connexion between the two statical theorems, dealing 

 respectively with systems subject to given displacements and 

 systems displaced by given forces, will be perhaps brought out 

 more clearly by another demonstration of the latter. We have 

 to show that the removal of a constraint is attended by an 

 increase in the potential energy of deformation. By a suit- 

 able choice of coordinates the conditions of constraint may be 



