FLUIDS AND THE DETERMINATION OF THE CRITERION. 133 
The Component Velocities in the Equations of Viscous Fluids. 
In no case, that Iam aware of, has any very strict definition of wu, v, w, as they 
occur in the equations of motion, been attempted. They are usually defined as the 
velocities of a particle at a point («, y, z) of the fluid, which may mean that they are 
the actual component velocities of the point in the matter passing at the instant, or 
that they are the mean velocities of all the matter in some space enclosing the point, 
or which passes the point in an interval of time. If the first view is taken, then the 
right hand member of the equation represents the rate of increase of kinetic energy, 
per unit of volume, in the matter at the point; and the integral of this expression 
over any finite space 8, moving with the fluid, represents the total rate of increase 
of kinetic energy, including heat-motion, within that space; hence the difference 
between the rate at which work is done on the surface of S, and the rate at which 
kinetic energy is increasing can, by the law of conservation of energy, only represent 
the rate at which that part of the heat which does not consist in kinetic energy of 
matter is being produced, whence it follows :— 
(a) That the adoption of the conclusion that the second term in equation (3) ex- 
presses the rate at which heat is being converted, defines u, v, w, as not representing 
the component velocities of points in the passing matter. 
Further, if it is understood that u, v, w, represent the mean velocities of the matter 
in some space, enclosing «, y, z, the point considered, or the mean velocities at a point 
taken over a certain interval of time, so that } (pu), = (pv), = (pw) may express the 
components of momentum, and zz (pv) — y= (pw), &e., &e., may express the com- 
ponents of moments of momentum, of the matter over which the mean is taken ; 
there still remains the question as to what spaces and what intervals of time ? 
(6) Hence the conelusion that the second term expresses the rate of conversion of heat, 
defines the spuces and intervals of time over which the mean component velocities must 
be taken, so that E may include all the energy of mean-motion, and exclude that of 
heat-motions. 
Equations Approximate only except in Three Particular Cases. 
8. According to the reasoning of the last article, if the second term on the right of 
equation (3) expresses the rate at which heat is being converted into energy of mean- 
motion, either pu, pv, pw express the mean components of momentum of the matter, 
taken at any instant over a space S, enclosing the point «, y, z, to which w, v, w 
refer, so that this point is the centre of gravity of the matter within S, and such 
that p represents the mean density of the matter within this space; or pu, pv, pw 
represent the mean components of momentum taken at «, y, z over an interval of time 7, 
such that p is the mean density over the time 7, and if ¢ marks the instant to which 
u, v, w refer, and t’ any other instant, =| (¢ — ¢’) p|, in which p is the actual density, 
taken over the interval r is zero. The equations, however, require, that so obtained, 
