298 
MR. S. RUNKERLET OX THE WHIRLING 
It will be noticed that in some of these experiments the observed speed is greater, 
and in others less, than the calculated speed. 
With the exception of Expei’iment 39, the experiments amply verify the theory. 
From Experiment 39 it would appear that when the ratio of the spans is one-half, the 
calculated speed is less than the observed speed. It, therefore, errs on the right side. 
It appears from Experiment 40, that the same formula as for a — \ holds, to a 
sufficient degree of approximation, until the ratio of the spans is ecjual to f. 
Case VI. 
17. Shaft, length /, fixed in direction at each end. 
Thus— 
Fig. 9. 
Taking the origin at A, we have (§ 8, p. 287, equation 7) 
y = A cosh + B sinh mx + C cos mx + D sin mx 
when X = 0, or /, 
y = 0, 
dyjdx = 0 ; 
whence 
A-bC — 0. (1), - 
B + D = 0.(2), 
A cosh ml + B sinh ml -f- C cos ml -V V> sin ml = 0 . . . . (3), 
A sinh ml + B cosh ml — C sin ml + D cos ml = 0 . . . . (4). 
The elimination of A : B ; C : D from these equations leads to 
cosh onl cos ml — 1 = 0. 
The least value of ml which satisfies this equation is 
ml = 4'74503.^ 
* Poisson, ‘ Traile de Mecaniqiie,’ vol. 2, § 528. 
