AND VIBRATION OF SHAFTS. 
325 
alone exceeds that in the case of a shaft free at both ends, in a certain ratio—that 
ratio depending on the position and size of the pulley. 
Considering the superior limits in each case, the increase of speed due to the 
shoulder is 100 per cent, at the shoulder end, decreasing to 91 at one-third the span 
from the shoulder end, 58 at the centre of the sjoan, and zero at the free end. 
Considering the inferior limits in each case, the increase of speed is 225 jDer cent, 
near the shoulder end, 51 at the centre of the span, and 19 per cent, near the free 
end. 
Case XIII. 
39. Shaft supported on three bearings, and 4 feet apart respectively 
AND LOADED WITH A PULLEY, WEIGHT W AND MOMENT OF INERTIA I' ON THE SPAN 
OF LENGTH 4> THE PULLEY BEING DISTANT FEET FROM THE MIDDLE BEARING AND 
C., FEET FROM THE END BEARING. 
Thus :— » 
Fig. 17. 
We have, taking the origin at the middle bearing B (§ 21, equation 2), 
y — ^ ^ + Cx + D, from A to B, 
y' — ^ O'x -J- D', from B to D, 
and 
It" = a:'^ + ^ a;‘^ + C'a; -f- D”, from D to C. 
j 6 2 
When a; = 0, 
y = 0, = 0, dyjdx — dy jd.x^ d~yldx?' — d^y jdx ^; 
whence we obtain 
D = 0. 
D'=0. 
C = C'. 
( 1 ), 
( 2 ), 
( 3 ) , 
( 4 ) . 
B = B' 
