354 
MR. S. DURKERLEY OR THE WHIRLIRG 
pulley. Considering the superior limits in each case, we see that the increase of 
speed due to two additional spams (each equal in length to the first span), one on 
each side, is, as regards the superior limits, 10 per cent, near the bearing, and 34'4 
per cent, at the centre of the span ; and, as regards the inferior limits, it is 417 per 
cent, near the bearing, and 34*4 per cent, at the centre of the span. 
Again, comparing the case under discussion with Case XIV., § 48, in which the two 
end spans are zero (that is, the shaft works in a shoulder at each end), we see that 
the increase of speed, due to the two shoulders, is, as regards the superior limits, 100 
per cent, near the bearing, and 48"3 per cent, at the centre of the span; and, as 
regards the inferior limits, 135 per cent, near the bearing, and 48'3 per cent, at the 
centre of the span. 
Comparing the present case with the results obtained in Case XIII., §§ 41, 42 (that 
is, with the case of two equal spans, one of which is loaded), we see that the increase 
of speed due to the extra span (the loaded span being in the middle) is, as regards the 
superior limit, zero when the pulley is near the inner bearing, increasing to 13"6 per 
cent, at the centre, 21'6 (maximum advantage) at one-third the length of the span 
from the free end, and decreasing to 10 per cent, when near the free end ; and, as 
regards the inferior limits, the increase of speed is 5 per cent, when near the middle 
bearing, 14’3 at the centre of the span, and 30*5 per cent, at the outer bearing. 
Finally, comparing the results in the present case with those obtained in Case XV., 
§§ 52, 53 (that is, with the case of three equal spans, one of the end ones being 
loaded), it will be noticed that the results in the latter case are the higher wdieu the 
pulley is near the bearing, as regards the superior limits, but less as regards the 
inferior limits. By further referring to what was proved in § 54, p. 346, ive may 
infer that if, in the 'present case, an additional equal sqoctn he added on each side 
[making, in all,f ve spans, the middle one being loaded), the effect of those additional 
spans, in increasing the speed at which the pulley will cause the she ft to tvhirl, will 
never he such as to cause the increase in the whirling speed to exceed one or two per 
cent, of that calculated on the assumption that the effects of the two additional sqmns 
are cdtogether neglected. 
When the effect of the shaft is also taken into account, the Increase in the whirling 
speed, due to the two additional spans, will be still further reduced (§ 62). 
This result, and that obtained in § 54, are extremely imjDortant, as they practically 
limit any problem to the case of three spans. In other words, in the case of a 
continuous shaft, supported on bearings, placed at equal distances apart, and loaded 
with a pulley on one of the spans, the whirling speed, due to that pulley, obtained by 
considering the loaded span, and the span or sjjans immediately adjacent to it on 
either side, is sufficiently accurate for practical purposes. 
