66 The Laiv of Distribution [CH. v 



The constants A a , A$ ... are to be determined from equations of the type 

 (140). The constant h depends on the total energy, and must be deter- 

 mined from equation (139). 



76. We have therefore proved that for a gas about which nothing is 

 known except the total energy, it is infinitely probable that the law of dis- 

 tribution of coordinates will be that expressed by equations (146). But in 

 the present case the function (S does not necessarily maintain the position 

 which it has been supposed up to now to hold, of being the only function of 

 the coordinates in the generalised space which remains constant throughout 

 the motion of the gas. In arriving at equations (146) it has in no way been 

 assumed that @ does hold this position, but if (5' does not hold this position 

 the importance of equations (146) is diminished, in that the information 

 then supplied is not precisely that which is of most importance. 



Let us suppose that there are other quantities %, ty ... functions of the 

 coordinates of the generalised space, which have the property, in common 

 with (, of remaining constant throughout every possible motion of the gas. 

 Then if we know the values of the quantities %, -v/r . . . as well as of (, the 

 information we require is the most probable law of distribution consistent 

 with these values of %-fy... and ($. This information is not supplied by 

 equations (146), for the law of distribution implied in these equations may 

 be inconsistent with these values of ^, ^r ..., and even if it is not, it is by no 

 means obvious that this law remains the most probable after the limitations 

 have been imposed on the values of ^, ^r 



If we know the values of %, ty . . . in practice, these values must be 

 deducible from the mass-properties of the gas, without reference to its 

 molecular properties. And consistent with the mass-equilibrium of the gas, 

 there are only six quantities other than the energy, which can be measured 

 from the mass-properties of the gas, these being the three components of 

 linear momentum and the three components of angular momentum. These 

 quantities are of course obtained by summation over all the molecules. They 

 remain constant, except in special cases, only when the gas has no boundary, 

 or when the boundary is either of a special shape, as in 38, or moves or 

 rotates with a given velocity. 



If some or all of these quantities remain constant throughout the motion, 

 the variation of ) will be subject not only to equations (141), (142), (143), 

 but also to some or all of the following equations 



(147), 



and two similar equations for v and w, 



(148), 



