THE DYNAMICAL THEORY OF GASES. 51 



Theory of a Medium composed of Moving Molecules. 



We shall suppose the position of every moving molecule referred to three 

 rectangular axes, and that the component velocities of any one of them, resolved 

 in the directions of x, y, z, are 



where u, v, w are the components of the mean velocity of all the molecules 

 which are at a given instant in a given element of volume, and 77, are 

 the components of the relative velocity of one of these molecules with respect 

 to the mean velocity. 



The quantities u, v, w may be treated as functions of x, y, z, and t, in 

 which case differentiation will be expressed by the symbol d. The quantities 

 rj, , being different for every molecule, must be regarded as functions of t 

 for each molecule. Their variation with respect to t will be indicated by the 

 symbol 8. 



The mean values of * and other functions of f, 77, for all the molecules 

 in the element of volume may, however, be treated as functions of x, y, z, 

 and t. 



If we consider an element of volume which always moves with the velocities 

 u, v, w, we shall find that it does not always consist of the same molecules, 

 because molecules are continually passing through its boundary. We cannot 

 therefore treat it as a mass moving with the velocity u, v, w, as is done in 

 hydrodynamics, but we must consider separately the motion of each molecule. 

 When we have occasion to consider the variation of the properties of this element 

 during its motion as a function of the time we shall use the symbol 9. 



We shall call the velocities u, v, w the velocities of translation of the medium, 

 and f, 77, the velocities of agitation of the molecules. 



Let the number of molecules in the element dx dy dz be JV dx dy dz, then 

 we may call N the number of molecules in unit of volume. If M is the mass 

 of each molecule, and p the density of the element, then 



p ....................................... (55). 



Transference of Quantities across a Plane Area, 



We must next consider the molecules which pass through a given plane 

 of unit area in unit of time, and determine the quantity of matter, of momentum, 



72 



