526 ON THE THEORY OF LUBRICATION [428
x.P {htfdx -,,Pa?da! so that TTf= ~~hr~-tą\ "Is"......................W
3/jiU ja 'i Ja h
By (7), (8), (9) x is determined.
As regards the total friction (F), we have by (6)
Comparing (7) and (10), we see that the ratio of the total friction to the total load is independent of p and of U. And, since the right-hand members of (7) and (10) are dimensionless, the ratio is also independent of the linear scale. But if the scale of h only be altered, F/P varies as h.
We may now consider particular cases, of which the simplest and the most important is when the second surface also is flat, 'but inclined at a very small angle to the first surface. We take
h = mss, .................................(11)
and we write for convenience
b-a = c, 7/2/7*! = b/a = k, .......:.............(12)
so that
m=(k-l)hl/c............................(13)
We find in terms of c, k, and 7^
H== 2Mi
P r* ( 9 (If 1 M 1 1 j fJ ^/C X )l
\p,U (k-X i ~\2 /, 2 1 *°%e 'c //, "L'i "1 ' .................. - JL) Hi [ K -J- 1 J 7c2 - 1 - 27c log k
F Ji! 2 -l)log7c-2(&-l)2' .....................
P~~ c i J(Jfc + l)loe:A;-6(A;-l) ...................
U being positive, the sign of P is that of
log/. a^-1).
If k >I, that is when A2 >hi, this quantity is positive. For its derivative is positive, as is also the initial value when k exceeds unity but slightly. In order that a load may be sustained, the layer must be thicker where the liquid enters.
In the above formulse we have taken as data the length of the bearing c and the minimum distance 7/a between the surfaces. So far k, giving the maximum distance, is open. It may be determined by various considerations. Reynolds examines for what value P, as expressed in (15), is a maximum, (fifl