Mr. J. C. Maxwell on the Dynamical Theory of Gases. 193 



and t, in which case differentiation will be expressed by the 

 symbol d. The quantities £, r], £ being different for every mole- 

 cule, must be regarded as functions of / for each molecule. Their 

 variation with respect to t will be indicated by the symbol 8. 



The mean values of £ 2 and other functions of f, 7), f 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 con- 

 sist 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 hydrody- 

 namics, but we must consider separately the motion of each mo- 

 lecule. 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 d. 



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

 of the medium, and f, tj, £ the velocities of agitation of the mo- 

 lecules. 



Let the number of molecules in the element dx dy dz be 

 N 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 



MN = / 9 (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 quan- 

 tity of matter, of momentum, of heat, &c. which is transferred 

 from the negative to the positive side of this plane in unit of 

 time. 



We shall first divide the N molecules in unit of volume into 

 classes according to the value of f, 77, and f for each, and we 

 shall suppose that the number of molecules in unit of volume 

 whose velocity in the direction of x lies between f and % + d%, 

 rj and rj-t-drj, f and f+dfis dN ; dN will then be a function of 

 the component velocities, the sum of which being taken for all 

 the molecules will give N, the total number of molecules. The 

 most probable form of this function for a medium in its state of 

 equilibrium is 



d'N^-^e ^~d£d V d$ (56) 



In the present investigation we do not require to know the form 

 of this function. 



Now let us consider a plane of unit area perpendicular to x 

 Phil. Mag. S. 4 Vol. 35. No. 236. March 1868. 



