294 MR. S. DUNKERLEY ON THE WHIRLING 
and when x — 
y z= 0, d^yjdx^ = 0. 
Hence, 
A + C = 0 .... .(1), 
B+D = 0.(2), 
A cosh ml + B sinh ml + C cos ml -f D sin ml = 0 . . . . (3), 
A cosh ml + B sinh ml — C cos ml — D sin m? = 0 . . , . (4). 
The elimination of A : B : C : D from these equations leads to 
coth ml = cot ml. 
To solve this equation, draw the curves of coth ml and cot ml. The points of 
intersection of y = coth ml with y = coi ml will give values of ml which satisfy the 
equation coth ml = cot ml. 
Fig. 7. 
From the diagram, it will be seen that the. first value of ml is less than tt + i’’’ 
by a small quantity. It may be shown that, to a sufficient degree of approximation, 
