126 PROFESSOR O. REYNOLDS ON INCOMPRESSIBLE VISCOUS 
possible simultaneous distribution of certain quantities im space, and in no wise 
depends on the physical significance of these quantities. Yet the physical significance 
of these quantities, as defined in the equations, becomes so clearly exposed as to 
indicate that further study of the equations would elucidate the properties of matter 
and mechanical principles involved, and so be the means of explaining what has 
hitherto been obscure in the connection between thermodynamics and the principles 
of mechanics. 
The geometrical basis of the method of analysis used in the kinetic theory of gases 
has hitherto consisted :— 
(1) Of the geometrical principle that the motion of any point of a mechanical 
system may, at any instant, be abstracted into the mean motion of the whole system 
at that instant, and the motion of the point relative to the mean-motion ; and 
(2) Of the assumption that the component, in any particular direction, of the 
velocity of a molecule may be abstracted into a mean-component-velocity (say w) 
which is the mean-component velocity of all the molecules in the immediate 
neighbourhood, and a relative velocity (say €), which is the difference between u 
and the component-velocity of the molecule ;* wv and € being’so related that, M being 
the mass of the molecule, the integrals of (ME), and (Mw€), &c., over all the molecules 
in the immediate neighbourhood are zero, and =[M (wu + €)?] = }{M(u? + €*)].t 
The geometrical principle (1) has only been used to distinguish between the energy 
of the mean-motion of the molecule and the energy of its internal motions taken 
relatively to its mean motion; and so to eliminate the internal motions from all 
further geometrical considerations which rest on the assumption (2). 
That this assumption (2) is purely geometrical, becomes at once obvious, when it is 
noticed that the argument relates solely to the distributicn in space of certain 
quantities at a particular instant of time. And it appears that the questions as to 
whether the assumed distinctions are possible under any distributions, and, if so, 
under what distribution, are proper subjects for geometrical solution. 
On putting aside the apparent obviousness of the assumption (2), and considering 
definitely what it implies, the necessity for further definition at once appears. 
The mean component-velocity (w) of all the molecules in the immediate neighbour- 
hood of a point, say P, can only be the mean component-velocity of all the molecules in 
some space (S) enclosing P. w is then the mean-component velocity of the mechanical 
system enclosed in §, and, for this system, is the mean velocity at every point within 
S, and multiplied by the entire mass within § is the whole component momentum 
of the system. But,according to the assumption (2), 7 with its derivatives are to be 
continuous functions of the position of P, which functions may vary from point to 
point even within S$; so that wv is not taken to represent the mean component-velocity 
of the system within S, but the mean-velocity at the point P. Although there seems 
to have been no specific statement to that effect, it is presumable that the space S has 
* “Dynamical Theory of Gases,” ‘Phil. Trans.,’ 1866, pp. 67. + ‘Phil. Trans.,’ 1866, p. 71. 
