AND VIBRATION OF SHAFTS. 
331 
these cases is the inferior limit obtained in Case IX., § 25, and also in Case XTI., § 36, 
provided the pulley lie near the shoulder end of the span. 
43. Comparing the results in §§ 41, 42 with those obtained in Case X., § 27 (that is, 
with the case of a pulley on a shaft merely resting on a support at each end), we see 
that in the case of two equal spans the calculated speed for the pulley alone exceeds 
that in the case of a single span (equal in length to either of the two equal spans) 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 extra 
span is 10 per cent, when near the middle bearing, 24 (maximum advantage) when 
one third the span from the middle bearing, 19 at the centre of the span, and zero at 
the end bearing. 
Considering the inferior limits in each case, the increase of speed is 35 per cent, when 
near the middle bearing, decreasing to 18 at the centre of the span and 8 per cent, 
near the end bearing. * 
44. Experimental Results. The same remarks apply here as in § 28, p. 310. 
The following are the mean results of experiments made with different spans and 
with different positions of pulleys I. and II. (p. 285) on those spans. The shaft without 
the pulley has been investigated in §§ 15, 16, whilst the calculated speeds for the 
pulleys alone have been calculated from equation [A], § 39, p. 325, or, in the case of 
equal spans, from equation [B], § 39, p. 326. 
