Kinetic Tlieory of Gases. 603 



of this choice is that the fluid remains homogeneous throughout 

 the motion. This follows at once from Liouville's theorem*. 

 Not only does the density at every point of the fluid remain 

 constant, but the velocity at this point, being determined 

 solely by the coordinates of the point, will also remain 

 constant. We have therefore to discuss a case of hydro- 

 dynamical " steady motion. " There is no flow at infinity 

 across the boundary of the generalized space. For at infinity 

 one or more of the velocity-coordinates must be infinite, so 

 that E the total energy must be infinite ; whereas throughout 

 the motion of a point of the fluid E must remain constant. 



Comparison with Orthodox Theory. 



§ 9. Let us examine the relation between the procedure 

 now suggested and the usual procedure which rests upon 

 the calculus of probabilities. It may, in the first place, 

 be remarked that a problem of probability has only a 

 definite meaning when a certain amount of knowledge is 

 given and a certain amount withheld. For instance, sup- 

 pose we have an urn containing a number of masses of 

 different weights. If the weights are known, a question such 

 as " What is the probability that a weight selected at random 

 shall weigh less than an ounce ? " has a definite meaning and 

 a definite answer : if the weights are not given, the question 

 is meaningless and has no answer. So also in the Kinetic 

 Theory, it is meaningless to talk about the " probability " of 

 a system being in a specified state : the phrase only acquires 

 a meaning when it is understood that the system is selected 

 at random from a given definite series of systems in different 

 specified states. We shall take this given series of systems 

 to be the series represented by a homogeneous fluid filling 

 our generalized space. On this basis questions of probability 

 will have a definite answer. If we had selected a different 

 series of systems — represented, let us say, by a definite 

 heterogeneous fluid — the answer would be different : if we 

 neglect to specify our series o£ systems the problem is 

 meaningless, and there is no answer at all. 



§ 10. Consider, for instance, the question suggested 

 in § 2. "What is the probability, at a single definite 

 instant, that in the element of volume of which the coor- 

 dinates lie between ^ and x + dx &c, there shall be found 

 the centre of a molecule of which the velocities shall lie 

 between u and u + du &c. ?" 



* " On the Conditions necessary for Equipartition of Energy," Phil, 

 Mag. [6], iv. p. 585, equation (7); or J. W. Gibbs, ' Elementary Prin- 

 ciples of Statistical Mechanics/ Ch. I. 



