512 Dr. C. Chree on the Whirling and 
which is certainly in close agreement with the result (18) 
obtained by Dunkerley’s hypothesis. 
(g) If instead of (13) we assume the type (5), but regard 
I’ as small, we obtain in place of (20) _ 
lye fergie ain MP: 51 
~1246EI | 32El 9 BI” Ce 

Case 2 (Dunkerley’s Cases II., VIII., and X.). 
Shaft supported at both ends. 
§ 12. At each end we have 
ya y/dr7 =), 
(a) Unloaded shaft: Euler-Bernoulli solution. 
The displacement is of the type 
ySosin pe... 2). 
where « is a constant, and the equation for p is 
plaom,ot. OL er 
where / is the total length of the shaft. 
From this we have 
eo? = (Kl /opl*)=— 97:41 (EL /apl*).. |.) a 
(6) Instead of the EKuler-Bernoulli method for the unloaded 
shalt, we may assume 
y= na(l 2+ 2). |. 
If » were replaced by gop/24EI this would represent the 
bending of the shaft under its own weight. 
Assuming 7 «& cos kt, we have 
Sy 
oa 
T=4(9? + 09") a, spl’, | 
630° a 
3 
V =49°(24/5) HIP? | 
whence we have for the frequency equation 
note 2 +-0?=(3024/31)(Elepl'),. . : . (6) 
and for the critical velocity 
w?=97-55E1/apl!, oo 4) 
which presents an exceedingly close agreement with (3). 
§ 13. (c) Mass (M, I’) on massless shaft at a point C 
(AC=a, BC=b, atb=l). 
a A C ze 
Assuming Rayleigh type formula, and employing Lagrange’s 
