104 Lord Rayleigh on the Law of 



it will introduce us to some of the points of difficulty. The 

 particles are supposed to move independently upon one 

 straight line, and the phase of any one of them is determined 

 by the coordinate x and the velocity u. At time t' the phase 

 of a particle will be denoted by (V, uf), and at time t the 

 phase of the same particle will be (x, w), where u will in 

 general differ from u', since we no longer suppose that V is 

 constant, but rather that it is variable in a known manner, 

 i. e. is a known function of x. The number of particles 

 which at time t lie within the limits of phase represented by 

 dx du is /(a?, u) dx du, and the question is whether this dis- 

 tribution is steady, and in particular whether it was the same 

 at time if. In order to rind the distribution at time t', we 

 regard x, u as known functions of x', u\ and transform the 

 multiple differential. The result of this transformation is 

 best seen by comparison with intermediate transformations in 

 which dx du and dx' du' are compared with dx dx' . We have 



dxdu — dx dx' x — - n (13) 



ax 



da/du'=dxdx'x^ (14) 



dx v 



In du/daJ of (13) x is to be kept constant, and in du'/dx of 

 (14) x' is to be kept constant. If we disregard algebraic 

 sign, both are by (8) equal to d 2 S/dxdct f , and are therefore 

 equal to one another. Hence we may write 



dxdu — dx'du'-, (15) 



and the transformation is expressed by 



f(x, u) dx du=/ 1 (x', u')dx ! du', . . . (16) 



where f x (x', u') is the result of substituting for x, u in f(x, u) 

 their values in terms of x', u'. The right-hand member of 

 (1 6) expresses the distribution at time t' corresponding to the 

 distribution at time t expressed by the left-hand member, as 

 determined by the law T s of motion between the two phases. 

 If the distribution is to be steady, fi(x', u') must be identical 

 with f(x', u 1 ) : in other words f(x, u) must be such a function 

 of (x, u) that it remains unchanged when (x, u) refers to 

 various phases of the motion of the same particle. Now, if 

 E denote the total energy, so that 



E=i^ + V P (17) 



then E remains constant during the motion ; and thus, if for 

 the moment we suppose / expressed in terms of E and x, we 



