THE EOYAL AETILLEEY INSTITUTION. 
130 
and (3) with (4), 
cos O' _ R . cos 9 + W 
sin 9' “ R . sin 9 
while,, from (5), 
cos 9' C'B . sin 9 . cos y — AC . sin 0 
sin 9' C'B . sin 9 . sin y 
( 2 ') 
(30 
Combining, then, equation (20 with (S'), 
R . cos 9 + W‘ _ C'B . sin 9 . cos y — AC . sin 0 1 
R . sin 9 C'B .sin 9 . sin y 
and substituting in this for R . cos 9 its value in (2), for R . sin 9 its 
value in (1), and for sin 9 its value in (10, 
W- P . sin + W' __ cosjy __ AC sin_0 v / W^+ P 2 - 2P ClV. sin /3 > 
P . cos (3 sin y C'B sin y P . cos (3 
from which 
P 2 {C'B* . cos 2 08 - y) - AC 2 . sin 2 0} 
+ 2P {^(T 2 . sin 2 0 . sin 0 . W— C'B 2 . cos (/3 - y) sin y (7P+ TP')} 
= AC 2 . sin 2 0 . TP 2 - C"P 2 . sin 2 y (7P + 7P') 3 , 
and therefore 
2 7P(/P+ W') cos 08—y) sin y . sin /?} — AC' 2 . sin 2 0 . cos 2 /l. TP 2 
(7'P 2 . cos (j3 —"y) Y (W + ^0 — AC' 2 . sin 2 0 . sin /J. TF 
+ " CB 2 . cR^ff^yY^AC 2 ^, 2 ^ 
or 
P = 
fJLtl 7m , in C • v/C^{cos^W y .^T 
(7'P 2 . cos 2 (/2—y) —AC' 2 . si n 2 0 
(sin 2 y + cos 2 /2 — cos 2 (/3 — y) JPfV} — AC' 2 . sin 2 0 . cos 2 /2 . /P 2 
C'P 2 . cos (/3 - y) sin y (W + W) - AC 2 . sin 2 0 . sin /l . 7P" # 
"l y / iO O / / i \ < y >/ O V ■) “ * 
CB 2 . cos 2 (/^ — y) — ^ C' 2 . sin 
which, if the co-efficient of friction or tan 0 = becomes 
p= C^(l + ^)cos^-y)-^V 2 • - /ga ^+^)lcos^.^ + s in 8y .^ + 
(sin 2 y + cos 2 fi — cos 2 (j3 — y) WW 7 ] — y?0' 2 . ^. cos 2 /3 . /F 2 
C'^ 2 (1 + /x 2 ) cos (0 - y) sin y (7F + W ) — ^C' 2 . /j 2 . sin /? . W 
+ O'ii 2 (1 + y?) cos 2 03 - y) - AC'*. p* •-A^J 
19 
