328 
MR. S. DUNKERLEY ON THE WHIELTXG 
and 
< 
0 + i i I + cj) 
■i r + -J- / 
4^1 ' 3 ' 
If ocCj^^Cg^/S 1)6 GC|ual to tho first or SGCond of thoso expressions, the corresponding 
value of 0 ) gives the inferior or superior limit of the speed respectively. Moreover^ 
the 'period of whirl corresponding to the inferior limit of speed is identical ivith the 
natural p)eriod of vibration of the light shaft under the given conditions. 
The superior limit 
= inferior limit X a / ( i i i / 
V 4 G I 3 ‘ 
Let 
= Cj/Ai and Ig = c^jl, 
'■'O di + + 3 ^ (f + <^2) y. + C] (cg + 
^ 2 c^cf 
that is, a^ and are the ratios of the distance of the pulley from the middle bearing 
to the radius of gyration of the pulley and to either span respectively. Also, let 
%, be the corresponding ratios when the distance of the pulley is measured from 
the end bearing ; that is 
a .2 — c^jk and b.^ = cfl. 
Then the solution to ecjuation [B] may be put in either of the forms 
(4 4 - 3&i)- _ ^7) I ' I 1 I ^ 3 f ^ I 7i /1 I 
-^ - 36, ^ 1 + T + ^ + (1 _ hf “ ji _ 6^ + + T 1 - 
+ 
\/\i\ 
1 + i +1 + frAj 
4 
26i (4 + 3&j) 
”(i - hf 
a. 
1 
« -1 I 1 - 4 ,+ 
.[C], 
and 
a - hf 
+1 + /I 7, x3 “ ^1 r + ^3 + f (1 ~^ 3 ) 
(1 - hf 
i-K 
+ A/[3(i-62)(i H- + 1 + 
■h ^3 + f (1 ~ ^2) 
3 2&2 (7 353 ) 
+ % • 
(1 - h,f 
. [D]. 
As in Cases X., XL, and XII. (§§ 27, 32, 36, 37), by assuming certain values for «„ 
6 ,, or CTgj ^ 3 j corresponding values of acf or aef can be found, and so, for any 
