FLUIDS AND THE DETERMINATION OF THE ORITERION. 149 
— 18uC, (27/a)? «a3 
the integral of the first term over the same space cannot be greater than 
pC,e°C 2a? 
Then, by the discriminating equation, if the mean-energy of relative-mean-mction is 
to be maintained, 
pC,* is greater than 700 p/a?, 
or 
par ae ae y 5) 
= (iy) = 700. Site Won ee ek OO) 
is a condition under which relative-mean-motion cannot be maintained in a fluid of 
which the mean-mean-motion is constant in the direction of mean-mean-motion, and 
subject to a uniform variation at right angles to the direction of mean-mean-motion. 
Tt is not the actual limit, to obtain which it would be necessary to determine the actual 
forms of the periodic function for w’, v’, w’, which would satisfy the equations of 
motion (15), (16), as well as the equation of continuity (13), and to do this the 
functions would be of the form 
/ 9 
> E cos {o(nt + aaa ; 
\ t 
where r has the values 1, 2, 3, &e. It may be shown, however, that the retention of 
the terms in the periodic series in which 7 is greater than unity would increase the 
numerical value of the limit. 
24. It thus appears that the existence of the condition (26) within which no 
relative-mean-motion, completely periodic in the distance a, can be maintained, is a 
proof of the existence, for the same variation of mean-mean-motion, of an actual 
limit of which the numerical value is between 700 and infinity. 
In viscous fluids, experience shows that the further kinematical conditions imposed 
by the equations of motion do not prevent such relative-mean-motion. Hence for 
such fluids equation (26) proves the actual limit, which discriminates between the 
possibility and impossibility of relative-mean-motion completely periodic in a space a, 
is greater than 700. 
Putting equation (26) in the form 
»/ (du/dy)* = 700 p/pa’, 
it at once appears that this condition does not furnish a criterion as to the possibility 
of the maintenance of relative-mean-motion, irrespective of its periods, for a certain 
condition of variation of mean-mean-motion. For by taking a? large enough, such 
relative-mean-motion would be rendered possible whatever might be the variation of 
the mean-mean-motion. 
