AND VIBRATION OF SHAFTS. 
327 
the equation already obtained for the case of a shaft resting freely on two supports at 
the ends, and loaded with a pulley distant c., from the bearings (Case X., 
§ 26, p. 308). 
If Zj = 0, the equation reduces to 
7 . 2 . 4 . 3 
0 ^2 1 
I 3b 
3 
1 3 
- v = 0 . 
the equation already obtained for the case of a shaft resting freely on a support at one 
end and working in a shoulder at the other (Case XII., § 35, p. 320). 
If = Cg = 00 the equation reduces to 
(^1 + 3 (4 + Cl) — Jr ^Ci + y ) j — 1 = 0, 
the equation already obtained for the case of a shaft, span l-^, and overhanging^a 
distance c^, the pulley being at the extremity (Case XI., § 31, p. 313). 
40. In the case of two spans, one of which is loaded, it is, of course, useless to 
completely solve the many cases which might occur. The three cases which at once 
suggest themselves for full investigation are— 
(1.) Unloaded span zero. 
(2.) Unloaded span infinite. 
(3.) Unloaded span equal to loaded span. 
It has been shown that the first two cases have been already investigated (Cases 
XII. and X.). It only remains to solve the third case volien the two spans are equal. 
If 
equation [A] becomes 
a 
7 . 2 . 3 . 3 
2 ''' b H 
T + 
Cl ( C, 3 
C 
+ T 
+ A (<>1* +« 
;’)}1-27= = 
0. [B], 
from which we immediately get 
Td = 
1 2 i c, V + Cl. C 3 + j c,)_ 
aCjCj ^ cicicl (4 <1 + ^ 0 (^1 • ^2* + I+ 3" C 
so that for whirling to be at all possible we must have (see Case IX. § 24, p. 305), 
2 CiCjZ'^ 
3.3 
A c.v 
a 
> n 
P + Cl (Cj + 5 C'l) 
