FLUIDS AND THE DETERMINATION OF THE CRITERION, 127 
been assumed to be so taken that P is the centre of gravity of the system within S. 
The relative positions of P and S being so defined, the shape and size of the space 8 
requires to be further defined, so that wu, &., may vary continuously with the position 
of P, which is a condition that can always be satisfied if the size and shape of S may 
vary continuously with the position of P. 
Having thus defined the relation of P to S and the shape and size of the latter, 
expressions may be obtained for the conditions of distribution of u, for which = (ME€) 
taken over § will be zero, 7.e., for which the condition of mean-momentum shall be 
satisfied. 
Taking §,, 7, &c., as relating to a point P, and 8, wu, &c., as relating to P, another 
point of which the component distances from P, are a, y, 2, P, is the C.G. of S,, and 
by however much or little S may overlap S,, 8 has its centre of gravity at a, y, z, 
and is so chosen that u, &c., may be continuous functions of «, y, z w may, 
therefore, differ from u, even if P is within §,. Let w be taken for every molecule of 
the system §,. Then according to assumption (2), ¥ (Mw) over 8, must represent the 
component of momentum of the system within §,, that is, in order to satisfy the 
condition of mean momentum, the mean-value of the variable quantity wu over the 
system S, must be equal to u, the mean-component velocity of the system §), and 
this is a condition which in consequence the geometrical definition already mentioned 
can only be satisfied under certain distributions of uw. For since w is a continuous 
function of x, y, z, M (uw — u,) may be expressed as a function of the derivatives of wu at P, 
multiplied by corresponding powers and products of «, y, z, and again by M ; and by 
equating the integral of this function over the space 8, to zero, a definite expression 
is obtained, in terms of the limits imposed on #, y, z, by the already defined space 8, 
for the geometrical condition as to the distribution of w under which the condition of 
mean momentum can be satisfied. 
From this definite expression it appears, as has been obvious all through the 
argument, that the condition is satished if wv is constant. It also appears that there 
are certain other well-defined systems of distribution for which the condition is 
strictly satisfied, and that for all other distributions of ~ the condition of mean- 
momentum can only be approximately satisfied to a degree for which definite 
expressions appear. 
Having obtained the expression for the condition of distribution of u, so as to 
satisfy the condition of mean momentum, by means of the expression for M (wu — w’), 
&c., expressions are obtained for the conditions as to the distribution of €, &c., in 
order that the integrals over the space 8, of the products M (w€), &c. may be zero when 
={[M (uv — u,)| = 0, and the conditions of mean energy satistied as well as those of 
mean-momentum. It then appears that in some particular cases of distribution of u, 
under which the condition of mean momentum is strictly satisfied, certain conditions 
as to the distribution of €, &c., must be satisfied in order that the energies of mean- 
