and the Partition of Energy in Continuous Media. 233 



energy of these sounds corresponds to that given in 

 formula (4). 



To represent the state of things imagined in § 3, suppose 

 that the walls behind the diaphragms are impervious to sound 

 and to all kinds of energy, and that the aperture in the wall 

 is stopped up with matter also impervious to energy. The 

 diaphragms must be supposed to be initially set into vibration, 

 and then left to themselves. The energy of the sounds they 

 emit will be reflected, absorbed, re-emitted, and so on. 

 Finally all sound will have become dissipated into heat. The 

 energy of the original sounds — the total store of energy in 

 the room — will figure as the heat of the air and of the 

 diaphragms. A person outside the room who uncovers the 

 aperture and listens will hear nothing at all, unless his ears 

 are sufficiently acute to hear the waves of air originating 

 in the random heat-motions of the molecules. For, as we 

 shall see, this random motion of the molecules can be re- 

 solved, by Fourier's theorem, into the motion of trains of 

 waves, and, in perfectly irregular heat motion, there is equi- 

 partition of energy between the different trains of waves, so 

 that the law of partition according to wave-lengths is given 

 by the formula 47rRTX -4 J\. This agrees with formula (3), 

 except for a numerical factor 2 \ which finds its origin in 

 the different energy-capacities of transverse and longitudinal 

 vibrations. 



It is found that the third question must be answered in the 

 affirmative. The reason for this answer will be found in 

 the concluding sections (§§ 22-30), and the reader who is 

 not interested in the abstract argument and analysis which 

 follow is advised to pass at once to these sections. 



The u Normal State" and Equipartition of Energy. 



8. We begin the mathematical discussion by proving the 

 law of equipartition in the form appropriate to the vibrational 

 energy of continuous media. It is important to exhibit the 

 proof in such a form as to make it clear that the law rests on 

 no assumptions of any kind, except the assumption that the 

 motion of the medium obeys the laivs of a conservative dynamical 

 system. 



Let L be a function of variables # l5 6. 2 , • . . 6 HJ 0^ 2 , . . . 



(W 

 and let these change in value so that -~ =0 l9 &c, while 



§Ldt is stationary in value. Let L consist solely of terms of 

 degrees two and zero in # l5 2 , . . . 6 n . 



