J3 THE DYNAMICAL THEORY OF 



of heat, Ac. 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 (, 17, and for each, and we shall suppose that the 

 number of molecules in unit of volume whose velocity in the direction of x lies 

 between and (+<l(, i) and y + drj, and C + d is 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 



(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 moving with 

 a velocity of which the part resolved parallel to x is '. The velocity of the 

 plane relative to the molecules we have been considering is u' (u + g), and since 

 there are dN of these molecules in unit of volume it will overtake 



such molecules in unit of time, and the number of such molecules passing 

 from the negative to the positive side of the plane, will be 



Now let Q be any property belonging to the molecule, such as its mass, 

 momentum, vis viva, &c., which it carries with it across the plane, Q being 

 supposed a function of or of , 77, and , or to vary in any way from one 

 molecule to another, provided it be the same for the selected molecules whose 

 number is dN, then the quantity of Q transferred across the plane in the 

 positive direction in unit of time is 



or u-u'QdN+^QdN ........................... (57). 



If we put QN for fQdN, and ~QN for tfQdN, then we may call Q the 

 mean value of Q, and fQ the mean value of Q, for all the particles in the 

 element of volume, and we may write the expression for the quantity of Q 

 which crosses the plane in unit of time 



(u-u')QN+JQN .............................. (58). 



