350 
ME. S. DUNKERLEY ON THE WHIRLING 
whence 
i(A'-A")ci^+i(B'-B")c,2 + (C'-C'')c, + (D'-D") = 0 . (13), 
1 (A'- A") + (B'- B") Cl + (C'- C") = 0. (U), 
A'-A"=-a(iAV + iB'ci3+C'ci+D') . (15), 
(A' - A") Cl + (B' - B") - - y8 (1 A' ci^ + B Ci + C') . (16), 
where, as in § 23, p. 305, 
« = Wc^VpEI, /3 = co-r/EI, and ^ = «P, where k = v/(pI7W). 
The elimination of the fifteen ratios 
A : B : C : D : A' : B' : C' ; D' : A" : B" ; C" : D" : A'" : B'" : C'" ; D'" 
from the sixteen equations marked leads to 
. 3^- kW^ {12 {If.2 + 4 ^ 1 ) + 9 C1C3 + 1644} 
+ “ [ i {34 (ciCg + If I ’E 4%) + ^ 1 % (4 "h 4) “ 1 " ‘^4441 
— I F {9C1C3 (Ci^ + C33) + 34 % (C3® + 4 c/) + 34 ci (Ci^ + 403^) + 444 (Ci^ + 03^)]] 
— 4^ [ 34 ® + 4 (44 + 44 + 44)} — o.[A], 
a quadratic in w", symmetrical with respect to (Z3, Ci) and (/g, Cg). 
If in equation [A] we put 4 = 0, it reduces to 
. i FCi V (i C3 + 14) 
+ «• [3 C] V {C34 + 4(4 + i C3)] - ciP {4C3 (42 - 
— 4 ^ (4 + 14 ) = ^5 
3C1C2) + i 4 (ci 3 + 4C33)}] 
which is identical with the ecjuation obtained independently, but not reproduced 
here, of two spans, one of which is loaded, the outer end of the shaft on the 
loaded span workiug in a shoulder, whilst that of the unloaded span merely rests 011 
the support. Thus 
Fig. 24. 
