136 PROFESSOR O. REYNOLDS ON INCOMPRESSIBLE VISCOUS 
with the time, and is everywhere in the same direction, being subject to no variations 
in the direction of motion ; for suppose the direction of motion to be that of x, then 
since the periodic motion passes through a complete period within the distance 2a, 
= (pu’) will be zero within the space 
2a dy dz, 
however small dy dz may be, and since the only variations of the mean-motion are in 
directions y and 2, in which b and c may be taken zero, and du/dt is everywhere 
constant, the conditions are perfectly satisfied. 
The conditions are also satisfied if the mean-motion is that of uniform expansion or 
contraction, or is that of a rigid body. 
These three cases, in which it may be noticed that variations of mean-motion 
are everywhere uniform in the direction of motion, and subject to steady variations 
in respect of time, are the only cases in which the conditions (84), (8B), can be perfectly 
satisfied. 
The conditions will, however, be approximately satisfied, when the variations of 
u, v, w of the first order are approximately constant over the space S. 
In such case the right-hand members of equations (84), (8B), are neglected, and it 
appears that the closeness of the approximations will be measured by the relative 
magnitude of such terms as 
a u/dx, &e., t d?u/dt? as compared with du/dx, du/dt, &e. 
Since frequent reference must be made to these relative values, and, as in periodic 
motion, the relative values of such terms are measured by the period (in space or time) 
as compared with a, b, c and 7, which are, ina sense, the periods of w’, v’, w’, I shall 
use the term period in this sense, taking note of the fact that when the mean-motion 
is constant in the direction of motion, or varies uniformly in respect of time, it is not 
periodic, 7.e,, its periods are infinite. 
9. It is thus seen that the closeness of the approximation with which the motion of 
any system can be expressed as a varying mean-motion together with a relative- 
motion, which, when integrated over a space of which the dimensions are a, 6, ¢, has 
no momentum, increases as the magnitude of the periods of uw, v, w in comparison with 
the periods of w’, v’, w’, and is measured by the ratio of the relative orders of magni- 
tudes to which these periods belong. 
Heat-motions in Matter are Approximately Relative to the Mean-motions. 
The general experience that heat in no way affects the momentum of matter, shows 
that the heat-motions are relative to the mean-motions of matter taken over spaces of 
