302 
MR. S. DUNKBRLEY OR THE WHIRLING 
whence 
A + C = 0.(5), 
A - C = 0.(6), 
A'cosh + B' sinh wi? + C' cos wi/+ D'sin = 0 .... (7), 
A'cosh + B'sinh 7JiZ ~ C'cos 777^ — D'sin jn/= 0 .... (3). 
The elimination of A : B ; C ; D ; A': B' : C' : D' from these eight equations leads to 
2 sinh ml (a sin me sin me cos me cos 7)ie') 
— 2 sin ml (a sinh me sinh me + y8 cosh me cosh me) — 4 sin ml sinh ml 
+ ay8 {(cos me cos me' sinh me sinh me + sin me sin 7ne cosh me cosh me) 
— (sin me cos me cosh me sinh me + cos me sin me sinh me cosh m.c)} = 0 [A], 
where 
a = ^ = cu^VtuEI. 
Equation [A] is, of course, symmetrical with respect to c, e. 
If we imagine the pulley to be removed (hy putting W = 0 and I' = 0) the 
equation A reduces to 
sin ml sinli ml — 0, 
i.e., 
ml = 77 , 
a result already obtained in Case II., p. 290. 
As in Case VII., § 19, we cannot obtain a general solution to [A], which could be 
readily applied in any actual case. 
Second Method of Solution. 
21. The formulae obtained by considering the effect of the pulleys and the shaft 
combined have thus been shown, even in simple cases, to be absolutely useless for 
practical piu’poses. 
By the second method of solution the whirling speed of the pulley neglecting the 
shaft is first obtained. The general theory (Chapter II.) will have, therefore, to be 
slightly modified. 
Since 
w = 0 , 
