528 Dr. C. Chree on the Whirling and 
Lagrange’s equations lead to 
sc SE a 4 20+ 3¢) 
[ M(H? + 0%) BEL) Goo? + Bb $30 } 
x fe 2 «?) )~3EI 4 Tal + ab 3q + 
af 2 «Seb aa ae 
98D") Goa ay =0. . (18) 


Omitting 4?, we obtain an equation for the critical angular 
velocity which—allowing for a misprint—agrees with Dun- 
kerley’s equation A (J. c. p. 326). It is worth noticing that 
the terms independent of / or w in (18) reduce to 
36(HI)*a’(a + 6) + 4c? (a—c)*(40-+ 3¢) t. 
The quadratic (18) splits into two factors, representing pure 
transverse and oscillational vibrations, when 
c?/(a—c)?=(46+4+ 6e)/(46+30), 
=1 when b/c is very big (¢. (c) case 2), 
=2 when 6/c is very small (¢7. (c) case 4). 
If in (18) we altogether neglect I’ we find for the critical 
velocity 
wo” = 12H Ia?(a+b) +[Me(a—e)*$4a(b+e)—e}]. (19) 
(e) If we assume I’ small to begin with, we may replace 
the displacements in (d) by the simpler type 
in OU, y=n|[ —2abce(3ac— 2a? —c*) a + 2a?(a +b) (3cx? — x*) 
—c(dac+ 2ab —c*) (8axv?—2*)], 
in CA, y=n| — 2c°a? (a+) + 2ac(@b + 3a7¢ + 2a7b) & 
—c(3ac+ 2ab— &) (8a2? —2°*) |, 
in OB, y/=n[2abe(3ac— 2a? —c?) ve’ 
— (ac/b)(3ac — 2a? —c’) (3ba? — w*)1. 
. (20) 
In the above 2 is measured from O to A, and 2’ from O 
to B. | 
Replacing » by (gM/EI)/12a?(a+b), we should have the 
bending of the shaft due to the weight of M at C. 
