360 Proceedings of Boy al Society of Edinburgh, [july 15, 
of the two cases’ specially referred to by the examiners in their 
announcement, and prepares the way for the investigation of the 
less simple by a preliminary laying down, in §§ 27-29, and equations 
(7) to (12) below, of the fundamental equations of motion of a 
viscous fluid kept moving by gravity between two infinite plane 
boundaries inclined to the horizon at any angle I, and given with 
any motion deviating infinitely little from the determinate steady 
motion which would be the unique and essentially stable solution 
if the viscosity were sufficiently large. It seems probable, almost 
certain, indeed, that analysis similar to that of §§ 38 and 39 will 
demonstrate that the steady motion is stable for any viscosity, how- 
ever small ; and that the practical unsteadiness pointed out by 
Stokes forty years ago, and so admirably investigated experimentally 
five or six years ago by Osborne Keynolds, is to be explained by 
limits of stability becoming narrower and narrower the smaller is 
the viscosity. 
Let OX be chosen in one of the bounding planes, parallel to the 
direction of the rectilineal motion ; and 0 Y perpendicular to the 
two planes. Let the x y-, z- component velocities, and the pressure 
at ( x , y , 0 , t ) be denoted by U + u, v, and p respectively ; U denot- 
ing a function of (y, t ). Then, calling the density of the fluid 
unity, and the viscosity /a, we have, as the equations of motion,* 
du dv dw n . 
dx + dy + dz ' ' ’ 
s (U + «) + (U + «) s + t^(U + «) + w s = ^v*(U + «) — s + y8ml, 
dv ,,, . dv dv 
Tt + ^ + u ^ + v + 
c ^ 
^ 1 
II 
/xV 2 v 
dp T 
- -p - q cos I , 
dy J 
dw /TT . die die 
m + ^ +u ^ + % + 
dw 
w— = 
dz 
pV 2 W 
dp 
dz ’ 
pi U2 
where V 2 denotes the “Laplacian” — . + — + — — • 
uxr dy A dz 2. 
28. If we have u — 0, v — 0, iv = 0 ; j? = C — g cos I y + gx sin I ; 
these four equations are satisfied identically; except the first of (2), 
which becomes 
* Stokes’s Collected Payers, vol. i. p. 93. 
