340 
Mil. S. DUNKERLEY ON THE WHIRLING 
work in a shoulder the calculated sjDeed for the pulley alone exceeds that in the case 
of a shaft free at one end and working in a shoulder at the other, 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 two 
shoulders is zero at the shoulder end, increasing to 27 per cent, at the centre of the 
span, and 100 per cent. a.t the free end. Considering the inferior limits in each case, 
the increase of speed due to the two shoulders is 2 per cent, near the shoulder end, 
increasing to 32 at the centre of the span, and 180 per cent, near the free end of the 
shaft. 
Again, comparing the results obtained in the present case with those obtained in 
Case X., § 27 (that is, with the case of a shaft merely resting on a bearing at each 
end) wm see that, considering the superior limits in each case, the increase of speed 
due to the two shoulders is 100 per cent., whatever be the position of the pulley; 
whilst considering the inferior limits the increase of speed near the bearing is 233 per 
cent., decreasing to 100 at the centre. 
Case XV. 
.50. Shaft supforted on four bearings, /j^, L, and feet apart respectively, 
AND LOADED WITH A PULLEY, WEIGHT W, AND iMOMENT OF INERTIA I', ON THE 
OUTER SPAN OP LENGTH -THE PULLEY BEING DISTANT C] FROM THE INNER, 
AND C 3 FEET FROM THE OUTER BEARING. 
Thus- 
Fig. 19. 
We liave, taking the origin at ihe bearing, C (§ 21, equation 2). 
y — ^ + up + Cx + V, from A to B, 
- ^ + C'x + D', from B to C, 
and 
x^ + ^ x~ "b C'x + D", from C to E, 
/// p/// 
r A'’ + ; x" -f- C'x + D'", from E to D 
