528 ON THE THEORY OF LUBBICATION [428
We denote, as before, the ratio of the extreme thicknesses (Aa/Aj) by k, and o still denotes b - a. For the total pressure we get from (7) and (8)
P -c- ( Sn-l (k'2+^l
l 2n-2
...... (23)
from which we may fall back on (15) by making n=l.
For example, if n — 2, so that the Curve of the second surface is part of a common parabola, P is a maximum at
P= 0-163 ^~, ........................... (24)*
when & = 2'3. The departure from (18) with k = 2'2 is but small. In order to estimate the curvature involved we may compare £ (/^ + A2) with the middle ordinate of the curve, viz.
I m (a + b)* = J {VAi + V(2'3 A^}2 = T58 Al5 which is but little less than
i (h, + ht) = I lh (1 + 2-3) = T65 Aj. It appears that curvature following the parabolic law is of small advantage.
I have also examined the case of n = oo . It is perhaps simpler and comes to the same to assume
h = eP* .................................. (25)
The integrals required in (7), (8) are easily evaluated. Thus
A2 2/3
/7/y> ^*™3pGt___ f\—3]
W/iA/ V """" (5
__= _^
making ff=o*'w - i j...........................^
In like manner
' gcfa _ ^ (1 + 2/gq) - 1 - 2g5
or ft f* t$ (~\ j.j-1 Q /•?/*/ \ ~l
Using these in (7), we get on reduction
6 or, since (3c = log k,
p =
, +
[* It may be proved that P has only one maximum when n=2, ft> 1. W. P. S.]ds unity but little, the right-hand member of (16) assumes an indeterminate form. When we evaluate it, we find