329, 330] Maxwells Equations of Transfer 277 



and hence the total amount of Q lost by motion into and out of the 

 element is 



dxdydzdt III (u ^- + v^- +w^-j (vf)\ Qdudvdw ...... (644). 



JJJ l_v a* oy a*/ v 'J 



If we write 



r r r 



1 1 1 uQf(u, v, w) dudvdw = uQ, etc., 



expression (644) can be put in the form 



dxdydzdt ^- (vuQ) + ^~ (vvQ) + ~- (vw( 



Another cause of change in 2Q is supplied by the action of external 

 forces on the molecules. For any single molecule, we have 



dt du dt dv dt dw dt m\ du dv dwj ' 



where X, Y, Z are the components of external force, so that in time dt the 

 total value of SQ experiences an increase 



dxdydzdt - \X flPl + F f ?S\ + Z fpY] . . .(646). 



m [_ \ouj \ovj \dwjj 



Lastly, SQ may be changed by collisions between molecules. If Q is any 

 one of the quantities which have previously been denoted by ^, %2"-%5> 

 namely the mass, energy, and the three components of momentum of a 

 molecule, there is no such change, but if Q is any other function of the 

 velocities such changes will occur. In general let us denote the increase 

 in 2Q which is caused in the element dxdydz by collisions in time dt by 



dxdydzdt&Q (647). 



Expressions (645), (646) and (647) now contain between them, the effect 

 of all possible changes in 2,Q. The value of SQ is however given by 

 expression (643), so that the change in Q in time dt will be 



-r.(vQ) dxdydzdt. 



Comparing the two different values which have been obtained for this 

 change, we have 



^ (,;)= -ri^^-f i 



dt [dx dy 



+ "-\X(^} + Yr^}+Z^)\+&Q (648). 



771 



