194 
PROFESSOR 0. REYNOLDS ON THE THEORY OF LUBRICATION 
Putting —L for the effect of the external load and — M for the external moment 
per unit of length in the direction 2 , and assuming that there are no external 
horizontal forces, the conditions of equilibrium for the brass are 
( {'p sin 0—f cos 0}d0=O 
f [p cos 9-}-f sin 6}cl6=^ 
J R 
(44) 
(45) 
Substituting from equations (40) (41) (42) in equations (17) and (19), Section IV,, 
putting 
K^CE/nJ^ k _ 
/-Vo 
a 
(47) 
and remembering the boundary conditions, these equations become on integration 
dp _6ByLiU 0 c{ sin (6 — <f> 0 )— sin (0 X — <£ 0 )} 
dd a 2 {l + c sin (6 — $ 0 )} 3 
r _ 3/tU 0 c{ sill (6-<f> 0 )- sin (&-<ft 0 )} 
J «{1 +c sin (6 — 0)} 3 
_ /VJp _ 
{1 + c sin (6 — cf))} . 
(48) 
(49) 
26. The Method of Approximate Integration. 
The second numbers of equations (48) and (49) may be expanded so that 
lip ^ =A o+ A i sin (#-4) + A 2 cos 2(d-(/.) + , &c. 
+A 3 „ cos 2x[0 — <^ 0 )+A 2m+1 sin {(2x+l)(0—ifi)} 
——/^Bo+Bi sin ( 0 — <£o)+B 3 cos 2(0—<f>)+ &c. 
+ ] 0, cos 2x($— ^)+B*, +1 sin ,‘('2./'+ I)(/-'—<■/<); 
(50) 
(51) 
Putting 
X= 4) 
(52) 
