NOTES. 235 



Of this the part due to the flow of matter across the yz-face nearest the 

 origin is 



(it + a) - \ 2&amp;lt;? (u * a)dx dydz, 



and that due to the opposite face is 



u + a) + J J^ 2Q (u + a) dx\ dydz. 



Calculating in the same way the parts due to the remaining pairs of 

 faces, we find for the total variation in the amount of Q in the element, 

 due to flow of matter across its walls, the expression 



~ 



In this let us first put Q = m, the mass of a molecule. Then 

 equating (3) and (4), and taking account of the relations (1) and (2) we 



find 



dp d . pu d . pv d . pw _ ^ 



dt dx dy dz 



the equation of continuity. 



Next let us put Q = m (u + a), the ^-momentum of a molecule. 

 Change in the amount of momentum contained in the space dxdydz is 

 produced not only by the flow of matter across the boundary but also 

 by the action of force from without. The external impressed forces 

 X, 7, Z increase the momentum parallel to x at the rate 



^mXdxdydz ........................... (5); 



and the effect of the internal forces in the same direction is 



(6), 

 dy dz J 



where we have introduced the general specification of the state of stress 

 at (x, y, z) from Art. 175. Equating then (3) to the sum of (4), (5), 

 and (6), and making use of (1) and (2) as before, we find 



d . pu d . pn 2 d . puv d . puw 

 dt dx dy dz 



~ 



