AND VIBRATION OF SHAFTS. 
351 
If we further put /g = 0, the equation reduces to that already obtained for the case 
of a shaft working in a shoulder at each end (Case XIV,, § 47). 
If /g z= oo2', instead of 0, we obtain the equation for the case of a shaft working in 
a sleeve at one end and merely resting on a bearing at the other (Case XII., § 35). 
If in equation [A] we put /g = coy it reduces to that already obtained for two 
spans, one of which is loaded (Case XIII., § 39). If in addition l^ — ca we obtain 
Case X., § 26. 
56. In the case of three spans, the middle one of which is loaded, the three cases 
which at once suggest themselves for full investigation are— 
(1.) the two unloaded spans zero, 
(2.) the two unloaded spans infinite, 
(3.) all three spans equal. 
It has been shown that the first two cases have been already investigated (Cases 
XIV. and X.). It only remains to solve the third case when all the spans are equal., 
If l^ = l^ =: l^ = I, equation [A] reduces to 
a3 . -gig- JcW^ {2SP + OCiCg) 
+ « {71^ + 5c,a,) - I F [(9c,C 3 + 7P) (c,^ + c.-^) + 9A,c., (c,^ + c/)}] 
-15/!^=0 .. [B], 
from which we get 
= 
12/ 15/dc,^3 “ k + ScjCg) 
aqco (28/“ + Qqco) — 4 {(9c, C 2 + 7/“)(c,-^ + c.I) -f- 9/c,c., (c," + 63 '")} ’ 
so that for whirling to be at all possible we must have (see Case IX,, § 24, p, 305), 
and 
, o o 15/Vco 
4 V > 
^ (9c,Cg -f 7P) (c,=^ + Cg^) + O/qCg (c,^ -f c^~) 
^ ‘ 28/“ -f 9 c,C2 
If ac,^C2®/3 be equal to the first or second of these expressions, the corresponding 
value of (0 gives the inferior or superior limit of the speed respectively. Moreover, 
the period of ivhirl corresqoondinc) to the inferior limit of speed is identical ivith 
natural period of vibration of the light shaft under the given conditions. 
The 
superior limit 
. „ . T -i. //(9 CiC2 -f 7l~){f -t- ch) -f9/c,Co(c,“ -1- ch) 7/“ + 5c,Cc 
= 2 X mfenor l.m.t X l f ^ X “15^^ 
