﻿386 Mr. G. H. Livens on the Law of 



our disposal and may be chosen as simple as we like. The 

 simplest and most convenient thing to do is to make it 

 uniform throughout the whole of the space with which we 

 are concerned. 



This mass of fluid moving in the generalized space now 

 provides a basis for the introduction of the calculus of 

 probabilities ; but, as Jeans says, great care must be exer- 

 cised in settling the basis for the calculation of the prob- 

 ability. Of course for our analysis to be legitimate we are 

 not compelled to choose any one particular basis for the 

 calculation of the probabilities. We may select any basis 

 we please, and then the analysis will be legitimate if we 

 retain the same basis throughout the whole investigation. 



In the present instance we agree to state that the prob- 

 ability of the motion in any one type of coordinate, say p r , 

 being in any state A r is, on some definite scale, measured by 

 f r (a r ), where/,, is some, at present undetermined, function of 

 its argument a n which is the area, measured in definite units, 

 of the projection on the (p r g r ) plane of the volume occupied 

 by points representing systems in which the motion in the 

 p r coordinate has the characteristic A r : the relation between 

 the probabilities for different types of coordinate are deter- 

 mined as soon as we know the form of the function f r for 

 each of them. 



If we are dealing with a system, or part of a system, 

 comprising an entirely large number of one particular type 

 of coordinate for which the function / is the same (say m r 

 coordinates of type p r ), then we shall agree to say that the 

 probability of this system being in a state A } . is 



W r =¥ T (Y r ), 



¥ r being some definite function of Y r , which is the volume 

 occupied by the representative points for the system 

 characterized by the state A,.. It is, however, important to 

 notice that if the characterization of the state A r is general 

 and bears no reference to, or preference for any special 

 members of the coordinate system, the only really possible 

 functional forms for the functions f r and F P are such that 



j r (x) — ex*r, F r (.r) = C r x* r , 



where u r is a constant, the same for all the coordinates of 

 the specified type. This follows at once from the fact that 

 in these circumstances the coordinates share equally, one 

 with the other, the responsibilities implied in the specification 

 of A P , the probability that any coordinate is in the conditions 



