Equilibrium and Initial and Steady Motions. 221 



for the right-hand member of (7), and the displacements yfr { &c. 



are any whatever. 



dV 

 In the position of equilibrium, since then "^j = -tt- &c, 



E=v=i(^t. + ••••); (9) 



and thus 



B-w-*(*,ti+---)'+-y-c*i:# / i+"-) 



Now, by a reciprocal property readily proved*, 



¥ 1 ^ 1 +...=¥* 1 ^ 1 +..., . . . (10) 

 and also 



V'=£nf ,+ ..., (11) 



where 'f 1 &c. is the set of forces necessary to maintain the con- 

 figuration i^i &c- Thus by (10) and (11), 



E-E'=i(^-n)(t.-f .)+—, • • (12) 



a positive quantity representing the potential energy of the 

 deformation (^j— ^0 &c. Thus E' attains its greatest value 

 E in the case of the actual configuration, and the excess of this 

 value E over any other is the potential energy of the displace- 

 ment which must be compounded with either to produce the 

 other. So far the displacement represented by yjr\ &c. is any 

 whatever ; but if we confine ourselves to displacements due to the 

 given forces and differing from the actual displacements only by 

 reason of the introduction of constraints limiting the freedom of 

 the system, then E/=V'; and the theorem as to the maximum 

 value of E' may be stated with the substitution of V for E\ 

 Thus the introduction of a constraint has the effect of dimi- 

 nishing the potential energy of deformation of a system acted 

 on by given forces ; and the amount of the diminution is the 

 potential energy of the difference of the deformations, f 



For an example take the case of a horizontal rod clamped 

 at one end and free at the other, from which a weight may 

 be suspended at the point Q. If a constraint is applied hold- 

 ing a point P of the rod in its place (e. g. by a support situ- 

 ated under it), the potential energy of the bending due to the 

 weight at Q is less than it would be without the constraint 

 by the potential energy of the difference of the deformations. 



* By substituting ¥.= €£. & c ., ¥'i= 4?-'- &c - 

 dy\r 1 dy\r'i 



t Compare Maxwell's "Theory of Heat,' p. 131. 



