198 
PROFESSOR O. REYNOLDS ON THE THEORY OF LUBRICATION 
Multiplying equation (60) by sin 6 and integrating between 6 0 and 0 l , remember¬ 
ing that 6 0 =—6 1 
r o 
s j n 0d0 = 2A Q (sm 0 l — 9 1 cos 6 X ) 
J e 0 Re 
+ A / sm2g,sin <h _ 9ifAnik 
, Ao/ . n . , sin 36 cos 26 
-+-—1sm 6 cos 2(p — --- 
&c. 
&c. 
A 0 
1 - 2 «+i [sin (2n + 2)6 1 sin (2n + l)<j> 0 sin 2 n0 1 sin (2n + l)<f> 0 
2n + l\ 2n-\-2 2 n 
[ sin (271+1)^! cos 271(f) 
sin (2ti + 1)6 cos 271(f)] 
1 271-1 
2n + l 
• (61) 
Multiplying equation (60) by cos 6, and integrating from 6 0 to 6 1 
( e > v — Vrs n i n kin , sin 20-, . \ , A„/sin 36-, . _ . „ . n „ ,\ . 
I cos 6(16— —A 2 ( 6 1 cos (f> 0 -y- 1 cos (f > 0 ) + -y( — 4 sm 2%—^ sm 0 sm 2(f )) — &c. 
+ 
A 
2 V 6 
2 " +1 1 ' ^ + ysin (2n-\-2)6 cos (2w+l)</>—^y^sincos (2n-\- l)<^j 
2?i +1 2?i + 2 
+ “2^L2^+T sin ( 2w '+ 1 )^ sin —2^=l sin ^ 27i —l-^si 112 n^}. . (62) 
Multiplying equation (51) by cos 6 and integrating 
( e f cos 6 „ -p. /sin 26, sin<£ n , „ . . 
-J/ K = 2 B o sm 0 1 -B 1 (-J—^+^sm^ 
+ B 3 
— &c, 
+ B 
sin 3 6 l cos 20 o 
b sin cos 2^> 0 j 
\ sin (2?i -t-1 )6 l cos 2n<f> 0 
t sin (27i — l)6 1 cos 2 ii(f) 0 
[ 27^+1 
2n-l 
t> ("sin (211 + 2)^ sin (2n+l)4> 0 , sin 2nd x sin (2n + l)$ 0 ] 
2+2 + 2n j ' ' < 63 ) 
