166 Mr. J. Buchanan on a Laiv of Distribution of 



or a maximum -minimum condition. It is of course assumed 

 that the fluid has a '"coarse-grained" structure. 



In the proof given below, the important assumption is also 

 made that the molecular energy of the fluid is wholly due to 

 the linear motion of the molecules ; or, rather, that if any 

 portion of this energy exists in the forms of molecular rotation 

 and vibration, this portion remains constant in value, and 

 may therefore be left out of account in what follows. 



Proof. 



By the term " element of volume at a point " is to be 

 understood a very small volume enclosing the point. This 

 elementaiy volume is supposed to be extremely small compared 

 with the whole volume occupied by the fluid, but very large 

 compared with the average space occupied by a molecule of 

 the fluid. 



The molar velocity of such an element of volume of the 

 fluid may be defined in precisely such terms as those in which 

 Clerk-Maxwell defines the velocity of a gasf : — If we determine 

 the motion of the centre of gravity of all the molecules within 

 the element of volume, then the molar velocity of the ele- 

 mentary volume of fluid may be defined as the velocity of the 

 centre of gravity of all the molecules within that region. 



In an element of volume at a point P let there be N mole- 

 cules. Let us take three rectangular axes of reference, and 

 let us denote the components of the linear velocity of the nth 

 molecule by u n , v n , iv„, and its mass by m n . 



A summation extended throughout the elementary volume 

 we will denote by %, whilst the sign of integration will indicate 

 a summation extended throughout the whole mass of fluid. 



Then we can write 



E = ij2(«* + t£+«>>„ (1) 



l>2,i» n =? l m n v n , > (2) 



r'2ni n = %i» n v n ; ) 



where E denotes the molecular kinetic energy of the whole 

 mass of fluid, and a, h, c express the components of the molar 

 velocity of the element of volume at the point P. 



The fluid being supposed to have been set in motion initially 



* The method of proof is taken from Meyer's Einetische Theorie der 

 Gase, see § 118 et seq. 

 t 'Theory of Heat,' p. 311. 



