370 Lord Rayleigh on Forced Harmonic 



resulting kinetic energy T is proportional to the square of 

 either, being equal to i^ij^i^. Now Thomson's theorem 

 asserts that the introduction of a constraint can only increase 

 the value of T when yjr x is given. Hence, whether ^ be 

 given or not, the constraint can only increase the ratio of J T 

 to y\r\, or of J ^ clt to ^r lm This form of the statement virtually 

 includes both Bertrand's and Thomson's theorems, which are 

 thus seen to be merely different aspects of the same truth. If 

 the velocity be given, the impulse is a minimum in the absence 

 of constraint. If the impulse be given, the velocity is under 

 the same circumstances a maximum. Calling the ratio of 

 \ x £ r 1 dt to tyi the moment of inertia of the system when sub- 

 jected to forces of the type in question, we may say that this 

 moment can only be increased by the introduction of a con- 

 straint forcing the motion to follow a different law from that 

 natural to it. 



In close analogy to this theorem there are two others, rela- 

 ting to equilibrium and to steady motion resisted by viscous 

 forces, of at least equal importance*. They may be thus 

 stated. 



Conceive a system to be displaced from stable equilibrium 

 by a force of specified type. If the corresponding displace- 

 ment be given in magnitude, the force is a minimum — or if 

 the magnitude of the force be given, the displacement is a 

 maximum, — when there is no constraint. Or we may say that 

 the stiffness of the system, with respect to the kind of force in 

 question, is increased by constraint. Examples, in illustra- 

 tion of the general proposition, are given in the papers already 

 cited. 



The third theorem depends upon the properties of the dissi- 

 pation-function, and its most interesting application is to the 

 conduction of heat and electricity. To take the latter case, if 

 an electromotive force be applied to any system of conductors, 

 the "resistance " to steady currents can only be increased by 

 the imposition of a constraint, such for example as the rupture 

 of a contact. 



Hitherto we have supposed the forces to be either instanta- 

 neous or steady ; and the three theorems depend upon the 

 functions T, F, and Y, expressing respectively kinetic energy, 

 dissipation, and potential energy, only one of them being 

 supposed to come into consideration at a time. We have now 



* Phil. Mag. Dec. 1874, "A Statical Theorem " ; March 1875, " General 

 Theorems relating to Equilibrium and to Initial and Steady Motion." 

 See also ' Theory of Sound,' ch, iv. 



