526 Dr. C. Chree on the Whirling and 
If b/a be very small, a first approximation is 
coth wa—cot ua=0, 
the least root of which (cf. (a) case 4) is 
poa=3'9266. 
This reduction to (a) case 4 only implies, what is physically 
obvious, that when the support O is close to an end it serves 
to fix the terminal direction of the shaft. If b/a in (4) is 
treated as small, and (0/a)? as negligible, we deduce 
fea 3°9266 (1 —b/3a), 22 ar 
whence 
w@? =237°7 (1—4b/3a) (HI /opat), . . (6) 
(6) As (4) is somewhat unmanageable except for special 
cases, I have tried several algebraic types of displacement, 
amongst them 
for AO, y=n{a*—4a*x—C(2*?—3a72) — Da}, 
for BO, y=nla4—4b a! —C/(a? — 3072’) es 
Here x is measured from A towards O, and 2’ from B 
towards O, and 
C= (8a? + ab—b")/2a, C! = (38? + ab —a*)/2b, 
De ab(a— b). 
Replacing in (7) by gop/24EI we should have the bending 
of the shaft under its own weight. 
From the kinetic method I find 

+o? = (3/5)(KI/ap) {3(@ + 6°) +5ab(@ 4+ 6°) — 5a°b(a+6)t + | Ta? +6°) 
65 12 7 127.7 I ; 113 8 173 7 6 8 5 75 S| 
ae a’ +C"b Dain Ps 70 o% +C'd Je oe )—,D(Ca —C ”)} . 8 
Putting 47=0 we have the critical angular velocity. The 
evaluation, though perfectly straightforward, is in general 
tedious. | 
For 6/a small, however, we easily find 
kK? + w= (4036/19) (1 —4b/3a) (HI /opat), . . . (9) 
and thence for the critical velocity 
w= 238'7(1—4b/3a)(El/opat), . . . (10) 
a result of course in close agreement with (6) (ef. also (8) 
and (4) of case 4, § 26). 
Again, if b=a we obtain from (8) for the period and 
