344 
MR. S. DURKERLEY ON THE WHIRLING 
By making /g = co in tlie last equation, it becomes identical with that already 
investigated for the case of a single span overhanging on one side (Case XL, § 31, 
p. 313). 
51, In the case of three spans, one of the end ones of which is loaded, the three 
cases which at once suggest themselves for full investigation are :— 
(1) . The two unloaded spans zero. 
(2) . The two unloaded spans infinite. 
(3) . All the three spans equal. 
It has been shovAui that the first two cases have been already iiiA^estigated (Cases 
XII. and X.). It only remains to solve the third case when cdl the spans are equal. 
equation [A] on p, 341, becomes 
^Jdc-^e.p [71 + Gcj) 
+ a {7P + 2C| . 3ci + ICo) — k” {7l. Cj^ + cl + Gcj. + 4cA)] — 45Z^ = 0 [B], 
from wliich we immediately get 
jr, _ 1 IS/’V^Cj — ukcpcl {71- + 2q.3c, + 4:Co) 
3aqCo * a 3q'W(7^ + Gcj) — {71 .cl + cl + Gq. q* + 4:q'b 
so that for whirling to be at all possible (^ 24, p. 305.) 
a must be > 
15/'kjq 
7/" + 2cj (3cj + “ICo) 
< 
7/ ( q-'^ + el) + Cc; (q''^ + 4cq) 
71 + 6q 
and 
