324 
MR. S. DUNKERLEr ON THE WHIRLING 
37. Values of 6^ in the equation co — 6^ ^^(^El/Wcg®), lohen the pulley lies between 
the free end and the centre of the spa.n, and — distance of pulley from free end. 
1 
1 
Values of ho = Cojl. 
Very 
small 
\ 
1 0 
1 1 X 
6 4 3 
1 
1 
2 
■ 
O 
II 
CO 
CL- 
Superior 
limit 
1-732 
1-737 
1-760 
1-851 
2-121 
3-873 
•2o 
1-677 
1-686 
1-716 
1-829 
2-115 
3-868 
•50 
1-500 
1-533 
1-598 
1-766 
2-098 
3-855 
•75 
I-I46 
1-300 
1-442 
1-693 
2-079 
3-838 
1-00 
0 
1-075 
1-309 
1-633 
2-063 
3-819 
1-25 
do. 
•938 
1-223 
1-591 
2-051 
3-800 
1-50 
do. 
•866 
1-167 
1-.562 
2-043 
3-785 
1-75 
do. 
•826 
1-1.36 
1-542 
2-036 
3-772 
2-00 
do. 
•801 
1-114 
1-529 
2 030 
3-761 
Inferior 
limit 
do. 
•728 
1-050 1-475 
2-012 
3-704 
) 
When the span is very long and the 23nlley is near the shoulder, so that cfl may 
be taken to be very small, a comparison of the results in §§ 36 and 25 shows that the 
effect of the free end is nil ; in other words, the speeds are the same as if the shaft 
merely overhung. If the pulley be near the free end of the span, so that cfl may 
be taken to be very small, a comparison of the results in §§ 37, 27, 32 shows that the 
effect of the shoulder is precisely the same as that of a free bearing. These results 
might, of course, have been anticipated, 
38. Comparing these results with those obtained in Case X., § 27 (that is, with 
the case of a pulley oji a shaft merely resting on a support at each end), we see that 
in the case where one end is fixed iii direction, the calculated speed for the pulley 
