304 
MR. S. DUNKEELEY ON THE WHIRLING 
When X = c, that is, at the pulley, we have 
r/ 1 
whence 
and 
or 
and 
Let 
K - L = 0,21' ^ 
ax 
f?R _ ^ _ W g 
dx dx (j ^ 
(§ 7, equation 6), 
(§ 7, equation 5). 
cV'yjdx^ = — co^V. clyjdx, 
3 = _ W/^EI. (xhy, 
Ac + B = - ^{1 c2 + Be + C 
a = Wo,2/^EI, ^ = ro,2/EI, 
(3), 
• (4). 
so that yS = aid where h = .^/(p'l'/W), /' having the value assigned to it in § 7. 
The elimination of A : B : C : D from the four equations marked leads to 
whence 
+ iac3 — (/5c + 1 ) = 0 
0, 
2 _ 
^Yd 
6 — 
‘Id 
Id 
± 
V I ‘Idd V2d 
( 6 -w) +- 7 A 
[A] , 
[B] . 
24. Equation [A] may be put in the form 
, g 3 — «c" 4 
~ «c* - 12 ' ^ ’ 
If ac^ be < 3 or > 12, Id is negative, and therefore the equations do not hold. 
Hence, for whirling to be at all possible, ac^ must be > 3 and < 12 ; that is, 
o,2.Wc3/pEI must lie between 3 and 12. 
The speeds which these values give for any value of c may be termed the inferior 
and superior limits of the speed. 
The values of k corresponding to these limits are zero and infinity. In other words, 
if the shaft whirl at a speed which satisfies 
(£,2. Wc^/pfEI = 3 or 12, 
the effect of the inertia of the pulley is either zero or infinity. In the first case we 
