ξ and ξ þ dξ, η and η þ dη, ζ and ζ þ dζ,
ð4:106Þ
In addition, whose velocity components lie between the limits
u and u þ du, v and v þ dv, w and w þ dw,
ð4:107Þ
Moreover, let these molecules collide with other molecules under specific conditions; after a certain time their coordinates will lie between the limits
Ξ and Ξ þ dΞ, H and H þ dH, Z and Z þ dZ,
ð4:108Þ
Moreover, their velocity components will lie between the limits
U and U þ dU, V and V þ dV, W and W þ dW,
ð4:109Þ
Then at any time
dξ Á dη Á dζ Á du Á dv Á dw ¼ dΞ Á dH Á dZ Á dU Á dV Á dW:
ð4:110Þ
This is a general result. If at time zero the coordinates and velocity components of
arbitrary molecules (material points) lie between the limits (4.106) and (4.107) and
unspecified forces act between these molecules, [Because Boltzmann does not make
any assumption about the type of forces acting between the molecules therefore
shear forces between molecules are not excluded.] So that at time t the coordinates
and velocity components lie between the limits (4.108) and (4.109), then Eq. (4.110)
is still satisfied.
If, instead of the velocity components, one uses the kinetic energy x and the
velocity direction defined by the two angles α and β, to describe the action of the
forces, these variables would initially lie between the limits
ξ and ξ þ dξ, η and η þ dη, ζ and ζ þ dζ,
x and x þ dx, α and α þ dα, β and β þ dβ,
Then after the action of the forces lie between the limits
Ξ and Ξ þ dΞ, H and H þ dH, Z and Z þ dZ,
X and X þ dX, A and A þ dA, B and B þ dB,
And so
dξ Á dη Á dζ Á
ffiffi ffi
x
p Á dx Á φ α, β
ð Þdα Á dβ ¼ dΞ Á dH Á dZ Á
ffiffiffi ffi
X
p Á dX
Á φ A, B
ð
ÞdA Á dB
ð4:111Þ
158
4 Unified Mechanics Theory