FLUIDS AND THE DETERMINATION OF THE CRITERION. 157 
(3) The boundary conditions which with the continuity give 
7 
io — Oi da ay) (dy — dw! dz — 0 when y == a= en ee(ol) 
(4) The condition imposed by ‘symmetrical mean-motion 


i ( du | dw’ 
) de = 26f y?) GEA While alee a ODE 
ANY dz 
These conditions (1 to 4) must be satisfied if the effect on w is to be symmetrical 
however arbitrarily wu’, v’, w’ may be superimposed on the mean-motion which results 
from a singular solution. 
(5) If the mean-motion is to remain steady w’, v’, w’ must also satisfy the kine- 
matical conditions obtained by eliminating p from the equations of mean-mean-motion 
(38) and those obtained by eliminating p’ from the equations of relative-mean- 
motion (16). 
Conditions (1 to 4) determine an inferior Inmit to the Criterion. 
32. The determination of the kinematic conditions (5) is, however, practically 
impossible ; but if they are satisfied, w’, v’, w’ must satisfy the more general conditions 
imposed by the discriminating equation. From which it appears that when w’, v’, w’ 
are such as satisfy the conditions (1 to 4), however small their values relative to u 
may be, if they be such that the rate of conversion of energy of relative-mean-motion 
into heat is greater than the rate of transformation of energy of mean-mean-motion 
into relative-mean-motion, the energy of relative-mean-motion must be diminishing. 
Whence, when wu’, v’, w’ are taken such periodic functions of #, y, z, as under 
conditions (1 to 4) render the value of the transformation function relative to the 
value of the conversion function a maximum, if this ratio is less than unity, the 
maintenance of any relative-mean-motion is impossible. And whatever further 
restrictions might be imposed by the kinematical conditions, the existence of an 
inferior limit to the criterion is proved. 
Expressions for the Components of possible Relative-mean-motion. 
33. To satisfy the first three of the equations (50) the expressions for w’, v’, w’, must 
be continuous periodic functions of «, with a maximum periodic distance a, such as 
satisfy the conditions of continuity. 
Putting 
/ = 2n/a; and 7 for any number from 1 to ~, 
