190 
PROFESSOR O. REMOLDS OX THE THEORY OF LUBRICATION 
Therefore 
and 
2 W(Lrf) 
12 /a dh 
K 3 dt 
Ej —0, &c. 
O x = — 1 + 
n 
W) 
p —n= 
12 fi a 2 e 2 \x 2 z 2 
IT o ! + c 2 ta 2 "'"c* — 1 
(23) 
(2‘0 
From equations (19) 
A 
,24/t a 2 c' 
o o 
>70 
^ h 3 a? + c} X dt 
f *= z F 
24yU, rf 
A 3 a 2 + c 2 dt 
(25) 
supposing surfaces horizontal and the upper surface supported solely by the pressure 
of the fluid. The conditions of equilibrium in this case are obvious by symmetry. 
The centre of gravity of the load must be vertically over the centre of the ellipse. 
Since by symmetry 
voI) 
(26) 
And 
Therefore integrating 
’CKO 
pxclxdz — 0 
rM^pzdxd^o 
•Vo 
c [ a \/( 1 --) 
1 c f x dxdz =0 
[ 
rr v '^) f dxdz 
• 0- 0 
w=rfv- 
p — ft dxdij 
o J o 
3 firr a 3 c 3 dJi 
t — 
h 3 a 2 + c 2 <7 
0)177 a 3 c 3 / 1 1 
(a 2 + c 2 )W\V 7q 2 
(27) 
(28) 
(29) 
t being the time occupied in falling from /q to /q. 
24. Plane Surfaces of unlimited Length and parallel in the Direction of z. 
The lower surface unlimited in the direction x and moving with a velocity — U. 
The upper surface fixed and extending from x = 0 to x=a. This case corresponds 
with Case 4, Section TIL 
