36 The Law of Distribution [CH. in 



with a continuous fluid, we shall suppose the different points in this fluid 

 to move, as directed by the dynamical equations of the gas, along the paths 

 in our generalised space as stream-lines, and we shall then examine the 

 motion of this fluid. 



It is obvious that the initial distribution of density of this fluid may be 

 chosen quite arbitrarily. We therefore choose that the initial distribution 

 shall be homogeneous. The advantage of this choice is that the fluid remains 

 homogeneous throughout its subsequent motion. This result follows from a 

 general theorem which will be proved later ( 72), but we now proceed to 

 give a separate proof for the special case at present under consideration. 



37. It has been seen that throughout the motion which takes place 

 between two collisions, all the velocity coordinates u a , v a , w a> M&...etc., 

 remain constant for any single path. 



Consider a series of systems starting simultaneously with the same values 

 of these velocity coordinates, but having positional coordinates lying between 

 x a and x a 4 dx a , y a and y a + dy a , z a and z a + dz a> 

 x b and x b + dxi>, ...etc (56). 



Let these systems move for a time dt, and let it be supposed that no 

 collision occurs during this interval, then it is clear that at the end of the 

 interval the various positional coordinates will have values lying between 



a a + u a dt and x a + u a dt + dx a ... etc., etc., 

 while the velocity coordinates of course remain unaltered. 



Hence the element of generalised space occupied by these systems remains 

 unaltered in shape, size and orientation, but has in the course of the time dt 

 moved parallel to itself a distance u a dt parallel to the axis of % u , v a dt parallel 

 to the axis of y a , etc. It follows that the density of the fluid with which the 

 element of generalised space may be supposed to have been filled, has 

 remained constant through this rectilinear motion. 



Just as rectilinear motion leaves the velocity coordinates unchanged 

 while altering the positional coordinates, so a collision leaves the positional 

 coordinates unchanged while altering the velocity coordinates. There are two 

 types of collisions to be discussed collisions between molecules and the 

 boundary, and collisions between pairs of molecules. 



As a specimen of the former, consider a collision between molecule A and 

 the boundary: this leaves all the coordinates unchanged except u a , v a ,w a . 

 Considering a series of systems in which before collision all the coordinates 

 except u a , v a , w a have the same values for each member of the system, whilst 

 i<a^a, MO, lie between u a and u a + du a> v a and v a + dv a , tv a and w a + dw n , 

 we see that after collision all the coordinates will remain unaltered except 

 u a , v a , w a , while these will lie within a new set of limits. Now in fig. 2 



