342 
MR. S. DUNKERLRY ON THE WHIRLING 
whence, putting as before (§ 23, }). 305), 
a — Woi^/pEI, /3 =- w^r/EI, and therefore /8 = aP, where k = .^(^I'/W), 
A" - A' 
Ci^ + 
B" - vr , 
9 
+ (cr - C'") c*i + (D" - D'") = 0 . . (13), 
A" - A'" 
+ (B" - B'") c, + (D" - D'") = 0 . . . . (14), 
A" - A'" = _ a J L + iV + C"c, + D' 
■ ■ (15), 
(A" - A'") c, + (B" - B"') = -/3 V c,-+ B"o, + C" . . (16) 
The elimination of the lifteeii ratios 
A : B ; C : D : A' : B' ; C' : D' : A." ; B" : C" : D" : A'" : B'" : C" : D" 
from the sixteen equations marked leads to 
[(4 + 4) (4 + + ^hhl 
+ a 
'. + 4)44+ ;d + i-444l 
{(4 + 4) (4• C]® -f- . c4 + 4c/) + i44 i^i’ + 
4^ {(4 + 4) (34 + 44) + 44) - o. 
[A], 
a quadratic in o" which is not symmetrical with respect to c 
If in equation [A] we put /g == 0, it reduces to 
n ^2’ 
7,3,. 3,. 3 
A C/j Bq 
cc" 
(^1 + 4) 
+ “ [3 (44 + <^1 (/ji + ^Co) j 
-4M4 + l4) = o, 
(a ‘"1 {^1^ ■^^•2^) “ 1 “ 44(3 4 " ^‘i'" 2 )l] 
which is identical with the equation obtained independently, but not reproduced here, 
of tw'o spans, one of wdiich is loaded, the outer end of the shaft on the loaded span 
merely resting on a bearing, whilst the outer end of the unloaded span works in a 
shoulder. Thus— 
