322 
MR. S. DUNKERLEY ON THE WHIRLING 
If be equal to the first or second of these quantities, the corresponding value 
of oj gives the inferior or superior limit of the speed respectively. The values of k, 
corresponding to these limiting values of oj, are zero and infinity, and if the shaft 
whirl at the speeds given by them it will do so in such a manner that the pulley 
still rotates in a plane perpendicular to the original alignment of the shaft. 
Moreover, the 2 )eriod of ivhirl corresponding to the inferior limit of speed is identiccd 
with the natural period of vibration of the light shaft under the given conditions. 
The 
superior limit = inferior limit X 
of 
+ip) + 
Let 
cq = c.//f, = c^d ; 
that is and by are the ratios of the distance of the pulley from the shoulder end of 
the shaft to the radius of gyration of the pulley and to the span respectively. Also 
let cq, be the corresponding ratios when the distance of the pulley is measured 
from the free end of the shaft; that is 
cq = cfh and b^ = cfl. 
Then the solution to equation [A], p. 820, may be expressed in either of the forms 
As in Cases X. and XL (§§ 27, 32), by assuming certain values for cq, b^, or a^, b^, 
the corresponding values of ac^ or ac.^ can be found, and so, for any particular value 
of c^ or C 3 , the value of w readily calculated. Two sets of results have thus been 
compiled. The first set (obtained from equation [B]) gives the values of aCy for 
different values of and b^, and is applicable when the pulley lies between the 
shoulder end and the centre of the span ; whilst the second set (obtained from 
equation [C]) gives values of ac.^ for different values of 0^3 and 63 , and is applicable 
when the pulley lies between the free end and the centre of the span. 
