790 
PROFESSOR 0. REYNOLDS ON CERTAIN DIMENSIONAL 
It also appears that within the limits of the necessary approximation, terms 
multiplied by IT 3 , V 3 , W 3 , or differentials of the second order, may be neglected; so 
that cr(Q) is expressed in terms of p, a, U, V, W, and their differential coefficients of 
the first order multiplied by some one or other of the several values of s. 
U, V, W, are then at once found by putting Q=M, so that cr r (M), oy(M), and 
cr-(M), are respectively u, v, and iv, which form the left sides of three equations (48) 
in which U, V, and W respectively appear on the right side. 
It is difficult to give an intelligible sketch of so complicated a series of operations, 
but what has been stated above may serve to indicate the general scheme of this 
section. 
Mean component velocities of the molecules which pass through an element. 
77. It has been already pointed out that when the condition of the gas varies, the 
mean component velocities of all the molecules which in a unit of time pass through an 
element are not, to the same degree of approximation as they would be if the gas were 
uniform, the doubles of the mean component velocities of the molecules in the element at 
the same instant. 
To express this, suppose that the condition of the gas varies only in the direction of 
x, so that the mean momentum in any direction perpendicular to x carried across all 
surfaces is zero. 
Then taking a rectangular element, so that its edges are parallel to the axes, and its 
edges parallel to x are indefinitely short compared with its edges perpendicular to x, 
the only momentum carried through the element will be by molecules entering and 
leaving the faces perpendicular to x ; and since the condition of the element remains 
unchanged the aggregate momentum of the molecules which enter must be equal to 
the aggregate momentum of the molecules which leave. 
u + 
The aggregate momentum which enters at the face on the left is <x r (Mw), or as it 
Vj + 
may be written 2 (Mir), while the aggregate momentum which enters on the right is 
— <x f (M«) or —2 (Mir). 
Therefore the whole momentum hi the direction of x carried through the element in 
a unit of time is 
(tI(Mm)-o-I(Mw) or 2(Mw 2 )-2(Mw 3 ) 
And since the aggregate mass of the molecules which pass through the element in 
the same time is 
— <x,(M) or 2 (Mm) — 2(Mff) 
the mean component velocity of all the molecules which pass through the element in 
a unit of time is 
U+ 'll— u+ u — 
< x,,.(M»)-cr,(M»,) 2 (Mu 2 )-2(Mu 2 ) 
aC(M) - <7i (M) 2 (Mu)—2 (Mk) 
