330 
Mil. S. DUNKERLEY ON THE WHIRLING 
42. Values of 0^ in the equation w = \/(i/EI/Wco®) when the pulley lies between 
the free end and the centre of the span, and = distance of pulley from free end. 
Values of = c^jl. 
Very 
small 
1 
1 0 
1 
6 
1 
4 
1 
3 
1 
2 
c\ 
O 
II 
e 
CO 
<1> 
Supei’ior 
limit 
1-732 
1-735 
1-747 
1-795 
1-936 
2-908 
•25 
1-677 
1-682 
1-700 
1-764 
1-920 
2-907 
•50 
1'500 
1-524 
1-567 
1-676 
1-880 
2-905 
•75 
1-146 
1-273 
1-385 
1-578 
1-837 
2-902 
1-00 
0 
1-022 
1-225 
1-485 
1-801 
2-900 
1-25 
do. 
•872 
1-123 
1-428 
1-775 
2-898 
1-50 
do. 
•796 
1-063 
1-386 
1-757 
2-896 
1-75 
do. 
•755 
1-027 
1-365 
1-745 
2-895 
2-00 
do. 
•731 
1-004 
1-346 
1-738 
2-894 
Inferior 
limit 
do. 
•661 
•932 
1-285 
1-671 
2-890 
By a comparison of these two sets of results it will be noticed that the same pulley 
placed at equal distances from the middle bearing and the end beariug of the shaft 
whirls at different speeds, those near the middle bearing being higher than those near 
the end bearing. Moreover, if the span he very long and the pulley he near the heading, 
so that cjl may he tahen to he very smcdl, it will be seen that, whilst the superior 
limits in the two cases are the same, the ratio which the inferior limit bears to the 
superior limit is less when the pulley is near the end bearing than when it is near the 
middle bearing. Also the superior limits when the pulley is near either bearing are 
the same as those obtained in Case X., § 27, Case XL, § 32, and also in Case XII., § 37, 
provided the pulley is near the free end of the span. The superior limit in any of 
