AND VIBRATION OF SHAFTS. 321 
When a? = Z, 
y' = 0, dry' jclx^ = 0, 
whence 
1A'Z3 + 1BW2 +C7+D'= 0.(5), 
A7 + B' = 0.(6). 
When x= c-^ (at C), we have 
dX^jdx — dldjdx — — Wlg.co^y (§7, equation (5)), 
and 
L — R — wR' dy/dx (§ 7, equation (G)); 
whence we obtain, putting as before (§ 23, p. 305) 
a = WwV^EI, /3 = coRyEI, and /3 = al\ where F = 
(A-A') = -afiAcy + iBcy + Cc, + D).(7), 
(A-A')c,+(B -B')= -^(lAcy + Bcj + C) . . . (8). 
The elimination of the seven ratios 
A : B : C : D : A' : B' : C' : D' 
from the eight equations marked leads to 
a quadratic in co", which is not, of course, symmetrical with respect to c^, c.^. 
If I = 00 , then Cg = I, and the equation reduces to 
Fc-y + a (;| — FCj) — 1 = 0, 
the equation already obtained for the case of an overhanging shaft working in a 
shoulder. (Case IX., § 23, p. 304.) 
From equation [A] we get 
^3 _ y . - «Cik-y (Iq + Fa) 
«q - (cy + ’ 
SO that for whirling to be at all possible (see Case IX., § 24, p. 305) we must have 
> l\l{lc^ + Ic^) and <12 + :icy). 
2 T 
MDCCCXCIV.—A. 
