240 Prof. J. H. Jeans on Temperature-Radiation 



and potential energies which we have already seen to obtain 

 in the normal state, analysed into the energy o£ regular trains 

 of waves. 



We return to the discussion of this system in § 23. 



II. A Tube of Air. 



18. The next system to be considered will consist of the air 

 inside a tube of uniform cross-section, closed at both ends. 

 For simplicity, the molecules of air will be supposed to be 

 similar and infinitely small spheres : let them be N in 

 number. 



The normal state of this system is one with which the 

 Kinetic Theory of Gases has made us very familiar. As 

 regards position, the molecules are distributed absolutely at 

 random throughout the tube : as regards motion, each velocity- 

 component is distributed according to Maxwell's lav/. 



Let us consider the arrangement of positions first. Let 

 us imagine the tube divided into a great number n of cells, 

 each being of the same cross-section as the tube, and of 

 volume ay. 



An "arrangement by which the molecules are placed at 

 exactly equal distances apart, in some regular geometrical 

 order, is of course a possible arrangement, but is no more 

 typical of the normal state than would be a motion in which 

 each molecule had exactly the some velocity. So also an 

 arrangement in which each of the n cells into which the tube 

 is divided contained exactly the same number N/n of mole- 

 cules, is possible, but is not typical of the normal state. In the 

 normal state, the numbers of molecules in the different cells 

 will be distributed around the mean value N/n, according to 

 a law which can be determined. 



Consider a single arrangement in which the numbers of 

 molecules in the n cells taken in order are %, a g , . . . a n . In 

 the limit, when n is made infinite, a knowledge of the values 

 of a u a 2 , ... a n will be equivalent to a knowledge of the density 

 of the gas at every point of the tube. On expressing this, 

 by Fourier's theorem, as a series of circular functions, we 

 can represent the deviations from uniform density as due to 

 the superposition of trains of waves. 



Employing the conception of probability in the exact sense 

 in which I have defined it elsewhere *, the probability that an 



* Phil. Mag. [6] v. p. 597, or ' Dynamical Theory of Gases/ p. 53. 



