AND VIBRATION OF SHAFTS. 
301 
Case VIII. 
20, Shaft length /, merely resting on a support at each end, and loaded 
WITH A PULLEY, WEIGHT W AND MOMENT OF INERTIA V AT DISTANCES C, C FROM THE 
SUPPORTS. 
Thus— 
Fig. II. 
Taking the origin at A, we have (§ 8, p. 287, equation 7) from A to B, 
y — A. cosh inx + B sinh mx + C cos mx + 1^ sin mx, 
and from B to C, 
ij = A' cosh mx + B' sinh mx + C' cos mx + D' sin mx, 
where dashed letters refer to the right, and undashed letters to the left of the pulley. 
At the pulley, when x — c, we have (§ 8, p. 287, equation 13) 
(A — A') sinh ?nc + (B — B') cosh me + (C — C') sin m.c — (D — D') cos me 
(A cosh me + B sinh me + C cos me + D sin me) 
( 1 ). 
Again, from equation 14, p. 287, we have 
(A — A') cosh me + (B — B') sinh me — (C — O') cos me — (D — D') sin me 
=-(A sinh me + B cosh me — C sin me V> cos me) . . . . (2), 
when 
whence 
x — e, y — y\ dyjdx = dy'jdx, 
(A — A') cosh me + (B — B') sinh me + (C — C') cos me + (h) — D') sin me = 0 (3). 
(A — A') sinh me + (B — B') cosh me — (C — C') sin 7ne + (D — D') cos me = 0 (4). 
Again, when x = 0, or I, 
7 / = 0 , 
dy'yjdx'^ = 0 , 
