171, 172] Equations of Mass Motion 151 



This is the hydrodynamical equation of continuity, expressing the 

 permanence of the molecules of the gas in other words, the conservation 

 of mass. 



172. The loss of each molecule of class A to the element means a loss of 

 momentum parallel to the axis of x equal to mu. Hence the total loss of 

 momentum parallel to the axis of as arising from this cause in time dt, is, 

 from expression (348), 



mdxdydzdt 1 1 1 ( u* ^- + uv =- + uw =- j [vf(u, v, w)] dudvdw (353). 



J J J \ ox oy oz i 



Let us write 



\\\u?f(u, v, w) dudvdw = it? (354), 



\uvf(u, v, w) dudvdw = uv (355), 



etc., 



so that u 2 , uv, etc. are the mean values of u z , uv, etc. at the point x, y, z, 

 Then expression (353) becomes 



(O \ O \ 

 ~- (vu 2 ) + ^- (vuv) + (vuw) J (356). 



With this must be compounded a gain of momentum caused by the 

 action of forces on the molecules. The gain of momentum parallel to the 

 axis of x to any single molecule is Xdt, where X is the component of force 

 parallel to the axis of as, including of course intermolecular forces and forces at 

 collision, acting upon the molecule in question. Combining the sum of these 

 gains with the loss given by expression (356) we find for the net increase of 

 momentum inside the element dxdydz in time dt, 



[, o r\ o \ ~i 

 2JT - mdxdydz (~- (vu*) + x- (vuv) + ~- (vuw) \\dt (357), 



where 5} denotes summation over all the molecules which were inside the 

 element dxdydz at the beginning of the interval dt. 



The total ^-momentum inside the element dxdydz at time t is, however, 

 mvu dxdydz. Hence the gain in time dt may be expressed as 



-j- (vu ) m dx dy dz dt, 



and equating this to expression (357), we obtain 



:^ g g g ~| 



T (vu ) + - (vu*) + ~- (vuv) + 5- (vuw) mdxdydz = ZX . . .(358). 

 dt das oy oz J 



These and the similar equations in y, z are the hydrodynamical equations of 

 motion of the gas, expressing the conservation of momentum, except in so 

 far as this momentum is changed by the action of external forces. 



