AND ITS APPLICATION TO MR. B. TOWER’S EXPERIMENTS. 
189 
Equations 17 and 19 are the general equations of equilibrium for the lubricant 
between continuous surfaces at a distance h, where h is any continuous function of 
x and z, and g is constant. 
22. For the further integration of these equations it is necessary to know the exact 
manner in which x and z enter into h, as well as the function which determines the 
limit of lubricated surfaces. 
These integrations have been effected either completely or approximately for certain 
cases, which include the chief case of practical lubrication. 
Complete integration has been obtained for the case of two parallel circular or ellip¬ 
tical surfaces approaching without tangential motion. This case is interesting from 
the experiment, treated approximately by Stefan, * of one surface-plate floating on 
another in virtue of the separating film of air. It is introduced here, however, as 
being the most complete as well as the simplest case in which to consider the 
important effect of normal motion in the action of lubricants. It corresponds with 
Case 2, section III. 
Complete integration is also obtained for two plane surfaces 
between the limits at which 2^=11 (the pressure of the atmosphere) 
5C=0, x=a, 
the surfaces being unlimited in the direction of 2 . This corresponds with Case 4, 
section III. 
For the most important case, that of cylindrical surfaces, approximate integration 
has been effected for the case of complete lubrication with the surfaces unlimited in 
the direction of 2 . Case 9, section III. 
Section Y.— Cases in which the Equations are Completely Integrated. 
23. Two Parallel Plane Surfaces approaching each other, the Surfaces having 
Elliptical Boundaries. 
Here h is constant over the surfaces, and when 
a> + <* K P ". 
. . . (20) 
U 0 , U 1 are zero. 
Equation (17) becomes 
/ d d \ 12/u, dh 
W+d*P h s dt ' ' 
. . . (21) 
The solution of which is 
P = ^)\ 
f+? + C,}+E 1 VLin? + &c. . . . 
. . . (23) 
* Wien. Sitz. Ber., A r ol. 69 (1874), p. 713. 
