338 
MR. S. DUNKERLEY ON THE WHIRLING 
in — c», then c., = / and the equation reduces to 
ri~ 
1 2 
+ n [I Cj® — PCi] — 1 = 0, 
the equation already obtained for the case of an overhanging sliaft working in a 
shoulder (Case IX., § 23, p. 305). 
From equation [A] we get 
aqCo -1 ac^c^ — 4 (q® + c^) ’ 
so that for whirling to be at all possible (see Case IV., § 24, p. 305), \olCiC .2 must be 
> V" and < 4 (cj^ + c^). 
If be equal to the first or second of these quantities, the corresponding 
value go) gives the inferior or superior limit of the speed respectively. Moreover, 
the period of whirl corresponding to the inferior limit of speed is identiccd with the 
naturcd period of vibration of the light shaft under the given conditions. 
Tlie 
superior limit = inferior limit X 2 ’ /yX 
1 ) 
Let 
a — cjh, h = cjl ; 
that is, a and h are the ratios of the distance of the pulley from the nearer 
shoulder to the radius of gyration of the pulley, and to the whole span respec¬ 
tively. 
Then the solution to equation [A], p. 337, may be expressed in the form 
As in Case X. (§ 27), by assuming certain values for a and h, the corresponding- 
values of aCj^can be found, and so for any particular value of Cj the value of w readily 
calculated. 
The following are the results obtained from equation [B]. The vertical columns 
give the values of 6 for different values of a, the value of h being fixed ; whilst the 
rows denote the values of 9 for different values of h, the value of a being kept the 
same. 
