520 Dr. C. Chree on the Whirling and 
we find for the frequency equation 

(?+o7)R+(P—e’)Q=S8, . . . (19) 
and so for the critical angular velocity 
o=S+(R—-QY). . 2... ae 
Obviously in general the evaluation of R and 8 is laborious. 
§ 22. (h) It will be found that none of the types (e), (f), 
- or (g) gives results which invariably accord well with ex- 
periment. It is clear that the most natural way of bending 
may be such that portions of the bar on opposite sides of the 
support at B have “centrifugal forces” acting on them in 
opposite directicns. This suggests the use of a type of dis- 
placement answering to an imaginary gravitational force 
oppositely directed on opposite sides of B. Such a type is 
for BC, y =n{a* — 4ca* + ber” +2A(3ca*°—xv*) —Ba}, t 21) 
Bel sy ew — ia — 41x? + 6a? + 2A! (3la”? —2*) — Bull, 
where ! = 2M /ap, Al 2 ap: 
a (8P? + 3apl*)/ap, ; 
Pl= —Mc—tap(? +c’). 
Putting for shortness 
R'=M213(2—c?) + 8(PP— Me')/op}?+ peopl +0") 
(22) 
i 
‘ 

= (P+ Mot) — = #0 (8PI+ Bopl)(IP +08) + = (El + M20) | 
2 23) 
aes 
(P +6) (SPP + Bap!) — 22 (BPP + 30pl") (PP + Me!) | 
= ser 1 (P40) + {Pl + Met + = (PP+M)}/op b, 
we find for the frequency equation 
Mtoe S 7 Ro aa 6) 4 
and for the critical angular velocity 
o= S/R’. ae 22) | (25) 
§ 23. Table III. compares the results obtainable from 
Dunkerley’s formula, corresponding to (5) with k=0, with 
those given by the simple formule (6) and (10) for the case 
when the mass of the shaft is neglected. 
