PROPERTIES OF MATTER IN THE GASEOUS STATE. 
805 
Section VIII.- —The Equations of Steady Motion. 
85. If Q is a quantity of such a nature that 2Q cannot change on account of any 
mutual action between the molecules within a unit of volume ; and further, if we 
assume that the molecules within a unit of volume at any instant are not subject to 
any influence other than those which they exert on one another, then whatever 
change may take place in an elementary volume must be on account of the excess of 
Q carried into the unit of volume over and above that which is carried out; and we 
have 
d%( Q)_ da je (Q) _ _ da !/ (Q) _ da : (Q) 
dt dx dy dz 
(55) 
( ~lt ~ * s ^ ie ra ^ e 2(Q) i s increasing at a point fixed in space. Hence if the 
condition of the gas is steady 
dX Q 
dt 
■ 0 
(56) 
Therefore if the condition of the gas is steady, we have 
rfg, *(Q) , rfg v(Q) , do---(Q) _ n 
dx dy dz ■ • • • r • • \ / 
86. If, therefore, we put Q = M, equation (57) gives us the condition of steady 
density. 
Whereas if we put successively Q = Mu, Q=Mr, Q = Mu;, we have from equation 
(57) the conditions of steady momentum in the directions of the axes. 
And if we put Q=M(w 2 fl- v 2 +wj we have the condition of steady pressure. 
The condition that the gas mag he subject to no distorsion or shear stress. 
87. In order that ou(Mr), ou(Mir), cr y (Mw), cr y (Mw), cr.(Mw), and cr,(Mr’) may 
respectively be zero for all positions of the axes, we must have 
cr. f (Mtt) = cr y (M( j = <T ; (Muj 
Therefore from the first of equations (46) and like equations 
(58) 
dpuV dpdV doc W 
S dx ' dx d o 
(59) 
These are the conditions that there shall be no tangential stress within the nas at a 
o o 
distance from a solid. 
