348 
ilR. S. DUNKL:KLIi:y ON THE WHIRLING 
Considering tlie superior limit in each case, the increase of speed due to the 
additional span of length equal to the length of either of the two spans (the two 
unloaded spans being on the same side of the loaded span) is 1‘1 per cent, when near 
the inner bearing, 1'9 when one-third the span from the inner bearing (maximum 
advantage), 1’4 at the centre of the span, and zero at the outer bearing. Considering 
the inferior limits in each case, the increase of speed is 2‘9 j^er cent, when near the 
inner bearing, decreasing to i'3 at the centre of the span and '6 per cent, at the end 
bearing. 
IVe thus see that, in the present case, the effect of the second unloaded spam from 
the loaded one, in increasing the speed at ivhich the pyulley luill cause the shcft to 
ivhirl, can never he such as to cause the increase in the u'hirling speed to exceed 3 per 
cent, of that calculated on the assumption that the effect of that second unloaded span 
is altogether neglected. 
When the effect of the shaft is also taken into account, the increase in the whirling 
sjjeed due to the third span will be less than 3 per cent. (§ 62). 
Case XVI. 
55. Shaft supported ojs four beaeings, 4. 4) 4 feet apart respectively, 
AND LOADED WITH A PULLEY, WEIGHT W, AND MOMENT OF INERTIA I', ON THE 
MIDDLE SPAN OF LENGTH 4-THE PULLEY BEING DISTANT C^, C^ FEET FROM THE 
BEARINGS. 
Thus— 
Fig. 23. 
— 
1 
>t <-— 
I 
I 
K--C- 
Ji. 
-/ 
T-:—C- 
B 
c 
D 
We have, taking the origin at the bearing B (§ 21, equation 2), 
?/ = -- - x^'’ -|- Cx + D, from A to B, 
A' . r> 
y — -dil ~ xr -V Cx + D', from B to E , 
// 
y 
X? + ^ a;'" -h C'x + D", from E to C, 
A'" R'" 
y"' = ^ 7 - + C"x + D'", from C to D . 
6 
