77, 78] Equipartition of Energy 69 



where in the numerator we have as the value of /, 



and in the denominator we have as the value of /, 



Since c l does not involve vjj, we have 



i r + oo -i / 



J n = -7= - e-*V<fcc = 5 A / ^L 



vtok./ *. y IPS' 



/"., - I ^-s 2 r/r 



J d /-: 6 *** 



VA J - 



Substituting these values in the numerator and denominator of equation 

 (158) we find that the whole equation reduces to 



- ^ 



This equation shews that the mean energy of each momentoid is equal to vL^ 

 -JT , a quantity which is independent not only of the particular momentoid 

 chosen, but also of the type of molecule to which it belongs. 



For any type of molecule, three of the momentoids may always be taken 

 to correspond to the three components of the velocity of the centre of gravity. 

 Taking these to be the last three momentoids, we have 



E a = V + % [m a (U? + V* + W 2 ) + dV + C. 2 ?7 2 2 + . . . + Cn-rfn-s] -(160), 



and our result may be expressed in the form 



These equations express the law of Equipartition of Kinetic Energy; a law 

 of which the importance will appear later. 



Law of Distribution of Velocities and Density. 



78. From expression (156) it follows that if i} n -^ Vn-i and rj n are the 

 momentoids of translational energy, the number of molecules of type a for 

 which u, v, w lie within a small range dudvdw is 



N a dudvdw 



In the present case, 



f (t t 



/ V%1 %n VI 



