AND VIBRATION OF SHAFTS. 289 
A-j-O = 0. . (l), 
B + D = 0.(2), 
A cosh me -f B sinh me — C cos me — D sin we = 0 . . . . (3), 
A sinh me + B cosh me C sin me — D cos me = 0 . . . (4), 
The elimination of A : B ; C : D, from these four equations, leads to either A = 0, 
B :i= 0, C = 0, D = 0, or to 
(cosh me + cos mc)^ — (sinh me + sin me) (sinh me — sin me) — 0, 
i,e., 
cosh me cos r/ic + 1 = 0.[A], 
The least value of me which satisfies this equation is 
wc= 1-87001.* 
Case II. 
10. Shaft, length I, merely resting on a bearing at each end. 
Thus— 
Fig. 4. 
f(r -/->, 
i I 
We have (§ 7, p. 280, equation 3) d^yldx'^ = whence 
y = A cosh mx -f- B sinh mx + C cos mx + B) sin mx. 
Taking the origin at the left-hand bearing, we have, when x = 0 or /, y = 0, 
d^yjdx^ = 0, whence 
A -j- C = 0.(1), 
A-C=0 .(2), 
A cosh ml B sinh ml + C cos ml sin ml = 0 . . . . (3), 
A cosh ml -f B sinh ml — C cos ml — sin rnZ = 0 . . . . (4). 
MDCCCXCIV.—A. 
* Poisson, ‘ Traite cle J^lecauique,’ vol. 2, § 528. 
2 R 
