516 Dr. C. Chree on the Whirling and 
calculated from (3) and (8) on his hypothesis (1) § 1, and with 
those which are given by the formule (15) and (20) singly. 
TasLE [].—Critical revolutions per minute. 

















Pulley I. Pulley ins 
3) 6 (GNU) se colina aeell 3) & (8 . 
Bi. | Observed. | (°) © ee | 5) | (20) |, Observed. Dankotley (15) | (20) 
2} 921 807 | 901] 899 || 769 745 | 747] 769 
1/3] 952 933 | 951 | 940 || 808 793°) 
16 | 1044 1036 | 1113 | 1050 || 942 953. / ee 
1/8 | 1101 1069 | 1179 | 1079 || 1007 1018 | |S eee 
1/11} 1128 1089 | 1241 | 1100 |) 1046 1055 | ... | 1074 
182 1150 1117 | 1362 | 1121 || 1130 1115.) ae 

Formula (15) is derived from a type of displacement which 
treats the shaft as massless, so its failure to agree well with 
experiment when the load is near one of the ends was to be 
anticipated. The simple formula (14) would give results in 
close agreement with those calculated by Dunkerley. 
Case 3 (Dunkerley’s Cases ITI. and XI). 
§ 19. Shaft supported at end A and at a second point B, 
A B c 
ee, 
with BC overhanging (AB=/, BC=c). Taking B as origin 
of coordinates wv, y for BC, and a’, y' for BA, we have: 
At e=0, y=0 with continuity in dy/dx and d?y/da’, 
5 c=, dy/da?=d*y/dz’*=0 (when there is no load), 
i dyn. 
(a) Unloaded shaft: Huler-Bernoulli solution. The dis- 
placements are cumbrous to record. The equation deter- 
mining p (cf. Dunkerley, /.c. equation (A) p. 291) is 
(cosh wl sin wl —sinh pl cos wl)(cosh pe sin we—sinh pe cos pc) 
—2 sinh pl sin pl(1 + cosh pe cose) =0. . 2 
Except for special values of ¢/l this is somewhat intractable. 
On his p. 292 Dunkerley specifies 3°08 as the limiting value 
to which pl approaches when c¢// is indefinitely reduced. 

