202 
PROFESSOR O. REYNOLDS ON THE THEORY OF LUBRICATION 
As the load increases from zero, equation (80) shows that G moves away from O 
towards A. 
It also appears from equation (70) that y and cf) 1 diminish as <£ 0 diminishes, and that 
<f> x is positive as long as the equations hold. 
To proceed further it is necessary to retain all terms of the first order of small 
quantities. 
Retaining the first power of c only, equations (55) become 
A 0 — - 
-l-5c-y 
B 0 =l —3cy 
A x = 
l+3cy 
B 1 = 2c 
j 
. . (81) 
a 2 = 
l‘5c 
b 2 =o 
From (58) 
x(0 l J r 3c sin 9 X sin <£ 0 ) = — sin 9 X sin <£ 0 — l'5c9 x -{- l'5c- n ^— cos 2<p 0 . . (82) 
From (66) 
From (67) 
From (68) 
0= { — 2R 1 c(l , 5c-l-y)4*2K 3 (l —3cy)} sin 9 X 
+ 2K ] c(l , 5c-bx)# 1 cos 9 X 
+ {Kic(l + 3cx) —2K 2 c} —~sin <£ 0 
— {K 1 e(l + 3cx) + 2K 3 c}0 1 sin (f> 0 .(83) 
+|K 1 c 2 ^sin cos 2 <p 0 
- (K 1 c(l -h3c x )-2K 2 c}^ 1 - S -^ 1 ^ cos <f> 0 
l'5c 
sin 3 9 X sin 2 <f> 0 — sin 9 X sin 2<£ 0 ) . (84) 
M 
-jy 2 =— K 3 {2(1 — 3cx)^—4c sin 9 X sin <£ 0 ].(85) 
In the equations (82) to (85) terms have been retained as far as the second 
power of c, but these terms have very unequal values. As y and sin </> 0 diminish c 
increases, and the products of cy or c sin <£ 0 may be regarded as never becoming 
important and be omitted when multiplied by K x c or K 2 . 
Making such omissions and eliminating y between (82) and (83) 
2K„ 
• i K,c 
Sill <^) u = ^- 
o/, . 1 / sin 36,\ n /i ,, .sin 2& 
” 0 X sin G + c 11G sin 6 X — ) — 3(sm 6 X - 6 X cos 6 X )—^—^ 
, „ sin '2.6, \ . „ 
dj( 6 X ——-— ) — 2(sin 6 X — 6 X cos 0j) sin 6 y 
■ ■ (86) 
o 
