AND VIBRATION OP SHAFTS. 
345 
If be equal to the first or second of these expressions, the corresponding 
value of (xi gives the inferior and superior limit of speed respectively. Moreover, 
the period of whirl corresponding to the inferior limit of speed is identiccd with the 
natural penod of vibration of the light shaft under the given conditions. 
The 
superior limit = inferior limit X /y / f 
i) + Gq , 7/' + 2q (3/ + e,) 
Let 
7/ + 6q 
= c^jh and = c^jl; 
X 
tliat is, and are the ratios of the distance of the pulley from the inner bearing 
to the radius of gyration of the pulley and one of the spans respectively. Also, 
let ctg, b .2 be the corresponding ratios when the distance of the pulley is measured 
from the end bearing ; that is, 
— c^jh and = c^//. 
Then the solution to equation [BJ may be put in either of the forms 
2(7 + 65,).«.d =7(1 + 
5,3 
(i - If 
+ 4 + 
5d 
(1 - hi 
,3/ «1 
7 + 85, - 25d 
1-L 
+ A/h(i+n-rT«') + c'^.('i+ 
(1 - h)- 
(1 - Lfi/ 
— o, 
7 + 85, + 25,- 
1-5, 
, 605, (7 + 65,) 
+ «i 
(1 - 5,)3 
2 (13 - 65^). uf 
3 
= 71 + 
(1 - h)' 
+ 6(1 1 + 4 
(1 - hf 
— 0.1 
[C.] 
1 - 5„ 
+ 2(3 + 
+ a/[7(i+++,') +Ml-'>»)('+ 4 
(1 - fe)*/ 
(1 - 53 ) 
1 - 5., 
+ 2(3+y 
605^ (13 - 65a) 
+ C(.: 
(1 - hf 
[D.] 
As in Cases X.-XIV. (§§ 27, 32, 36, 37, 41, 42), by assuming certain values for 
or a. 2 , h^, the corresponding values of ac,®, or ac^ can be found, and so, for any 
particular value of or Cg, the value of co is readily calculated. Two sets of results 
have thus been compiled. The first set (obtained from equation [C]) gives the 
values of ac,^ for different values of a, and 6,, and is applicable when the pulley lies 
MDCCCXCIV.—A. 2 Y 
