71-73] Extension to Complex Molecules 63 



If the fluid is to represent the motion of the system, the equations of 

 motion of the fluid must be equations (128) and (129). From these we get, 

 by differentiating with respect to p s> q s and adding, 



dps , dq* _ ft 



o *\ "j 



dp* dq s 



whence equation (130) takes the form 



In other words, p remains constant throughout the whole motion of any 

 element, so that if the fluid is initially homogeneous it will always remain so, 

 This is the general theorem of which the result proved in 37 was a particular 

 case. We now see that the result there proved is true for a generalised space 

 representing any conservative dynamical system, provided the coordinates in 

 this space are the generalised coordinates of this system in the sense required 

 for the Hamiltonian equations (128) and (129). 



Application to a Gas. 



73. We now suppose our system, as before, to be the gas contained 

 in a vessel. We shall suppose this gas to consist of 



N a molecules of type a ; 



Nft molecules of type y8, and so on. 



A molecule of type a will be specified, let us suppose, by n a positional 

 coordinates, and the corresponding n a momenta 2w a coordinates in all. 

 The whole system will therefore be specified by 



ZN a n a = N a n a + N ft n ft + ... 

 positional coordinates and an equal number of momenta. 



Let E a , Ei) denote the energies of molecules A, B, ... when free from 

 intermolecular forces, but when in position in the permanent field of force. 

 Then E a , for instance, is a function of the 2n a coordinates of position and 

 momentum which belong to molecule A, and if q s , p s are a pair of correspond- 

 ing coordinates of position and momentum belonging to this molecule, 

 we have 



The total energy of the gas is 



= E a + E b + ...+& ........................ (133), 



where <1> is the total potential energy arising from the intermolecular forces. 

 This quantity <t> is a function solely of the positional coordinates of the 

 various molecules: it does not depend on velocities. Hence in the right- 

 hand member of equation (133) none of the terms will depend on q s except 



