220 Lord Rayleigh on General Theorems relating to 



corresponding thereto (viz. T) is greater than if the velocities 

 had remained unchanged; and this, again, is evidently greater 

 than the actual E (viz. T) of the original motion. The total 

 increase of energy is equal to the decrease which the alteration 

 of mass entails in the energy of the former motion, together with 

 the energy (under the new conditions of the system) of the dif- 

 ference of the old and new motions. If the change be small, 

 the latter part is of the second order. 



On the other hand, of course, any addition to the mass les- 

 sens the effect of the given impulses. 



A similar deduction may be made from Thomson's theorem, 

 which stands in remarkable contrast with that above demon- 

 strated. The theorem is, that if a system be set in motion with 

 prescribed velocities by means of applied forces of corresponding 

 types, the whole energy of the motion is less than that of any 

 other motion fulfilling the prescribed velocity-conditions. "And 

 the excess of the energy of any other such motion, above that of 

 the actual motion, is equal to the energy of the motion which 

 would be generated by the action alone of the impulse which, if 

 compounded with the impulse producing the actual motion, 

 would produce this other supposed motion." From this it fol- 

 lows readily that, with given velocity-conditions, the energy of 

 the initial motion of a system rises and falls according as the 

 inertia of the system is increased or diminished*. 



We now pass to the investigation of some statical theorems 

 which stand in near relation to the results we have just been 

 considering. The analogy is so close that the one set of theo- 

 rems may be derived from the other almost mechanically by the 

 substitution of "force" for "impulse," and "potential energy of 

 deformation " for "kinetic energy of motion." A similar mode 

 of demonstration might be used but it will be rather more con- 

 venient to employ generalized coordinates. 



Consider then a system slightly displaced by given forces 

 from a position of stable equilibrium, from which configuration 

 the coordinates are reckoned. The potential energy of the dis- 

 placement V is a quadratic function of the coordinates ^r v ^/r 2 &c. 



V = i a u ty\ + i « 22 ^2 + • * • + a i& if 2 + "as^fa + • • • • (7) 

 If, then, 



E'=%t\ + ^' 2 +-..--V', ... (8) 



where ' ) ¥ v W^, &c. are the forces, E' will be an absolute maxi- 

 mum for the position actually assumed by the system. In 

 equation (8), V is to be understood merely as an abbreviation 



* See a paper by the author on Resonance, Phil. Trans. 1871, p. 94. 



