Systems, and Planck's Theory of Radiation. 945 



fluid in this way., and the motion of the fluid becomes " steady- 

 motion"" in the usual hydrodynamical sense. The state of 

 the fluid is the same at all instants of time, so that we need 

 only discuss it at one single instant. 



The mass of fluid, considered at any single instant, maybe 

 compressed, distorted and dilated, in such a way as to become 

 of uniform density at every point*. After this distortion 

 a hydrodynamical steady motion taking place along the 

 distorted stream-lines ^ill represent all possible motions of 

 the dynamical system under discussion. Let us take new 

 orthogonal Cartesian co-ordinates in this new (distorted) 

 space, to be denoted by P l5 P 2 , . . . P„. 



4. The hydrodynamical condition for steady motion is 



s|g=o, (1) 



so that we have seen that corresponding to any system of 

 ]aws of motion of the dynamical system, at least one set of 

 co-ordinates can always be chosen such that equation (1) is 

 satisfied identically. And if there is one such set of co- 

 ordinates there must necessarily be an infinite number, for 

 a homogeneous fluid can be strained in an infinite number 

 of ways so as to remain homogeneous. 



For example, if the motion of the dynamical system is 

 governed bv Newtonian laws, one set of co-ordinates which 

 satisfy relation (1) is found in the Lagrangian co-ordinates 

 and momenta, while other sets are obtained by taking 

 Pi, P 2 , . . . P M to be any series of linear functions of the 

 Lagrangian co-ordinates and momenta such as determine a 

 set of orthogonal lines in the generalized space. 



5. The mass of fluid moving in the generalized space pro- 

 vides a basis for the introduction of the calculus of proba- 

 bilities. 



At this stage it may perhaps be permissible to draw 

 attention to a point which is often overlooked in the 

 application of this calculus to problems of statistical mechanics, 

 namely that any discussion of probabilities is meaningless 

 until the basis of calculation of the probability is clearly 

 stated. The question " What is the probability that the 

 entropy of a gas shall be W ? " is, unless a definite basis of 

 probability is stated, as meaningless as the question " What 

 is the probability that the temperature of a gas shall be T, or 

 that the gas shall be hydrogen ? " Also, for the application 



* This is obviously true for a 1, 2 or 3-dimensional space and a proof 

 by induction is easily constructed to extend to n- dimensions. 



