128 HELMH0LT2 ON THE CONSERVATION OF FORCE. 



laws of which may be readily deduced from our principle ; the 

 conservation of the centre of gravity, and the manifold elastic 

 vibrations which continue without fresh excitement^until, through 

 the friction of the interior parts and the yielding up of motion to 

 exterior bodies, they are destroyed. In fluid bodies, liquid (evi- 

 dently also elastic, but endowed with a high modulus of elasticity 

 and a position of equilibrium of the particles) as well as gaseous, 

 (with low modulus of elasticity and without position of equili- 

 brium) motions are in general propagated by undulations. To 

 these belong the waves on the surfaces of liquids, the motion of 

 sound, and probably also those of hght and radiant heat. 



The vis viva of a single particle Am in a medium which is 

 traversed by a train of waves, is evidently to be determined from 

 the velocity which it possesses at its position of equilibrium. 

 The general equation of waves determines, as is known, the 

 velocity u, when a^ is the intensity, X the length of the wave, ot 

 the velocity of propagation, x the abscissa, and / the time, as 

 follows : — 



{- 



For the position of equilibrium u h = a, hence the vis viva of 



the particle Am during the undulatory motion -Ama^ is pro- 



portional to the intensity. If the waves expand spherically 

 from a centre, masses continually increasing in bulk are set in 

 motion, and hence the intensity must diminish, if the vis viva is 

 to remain the same. Now as the masses embraced by the waves 

 increase as the square of the distance, the law follows as a con- 

 sequence, that the intensities diminish in the reciprocal ratio. 



The laws of reflexion, refraction and polarization of light at 

 the limit of two media of different wave- velocity, are known 

 to have been deduced by Fresnel from the assumption that the 

 motion of the limiting particles in both media is the same, and 

 from the conservation of vis viva. By the interference of two 

 trains of waves we have no destruction of vis viva, but merely 

 another distribution. Two trains of waves of the intensities a^ 

 and b^ which do not interfere, give to all points on which they 

 strike the intensity a^-f 6^ ; if they interfere, the maxima possess 

 the intensity (a-\-b)% that is, 2ab more, and the minima {a— -by, 

 just as much less than a^ + 6*. 



u = a. co%\ —-[x~at) • 



