392 PROFESSOR A. H. GIBSON ON 
Where the beam is of uniform section, the only case here considered in detail, 
this becomes 
d are 6) are 6 
(2),- = nl Me cos (6, - 6)ds-+ = fe sin (0, — 6)ds ; 
or, since, if 7 is the radius of the are, 
dy 1 = 
ds 1+ dg’ 
d onal 
NG a ee 
or “lc cos (0, — 0)d6 + G ae sin (6, — 6)d6. 
; ds=rdo ; 
§ 2. Cracutar-ARc CANTILEVER WITH Loap W art FREE Enp. 
Let a (fig. 1) be the angle subtended by the are. 
O 
ecu ix 
LX io 
Fic. 1. 
Then, 
M?=W x CQ=Wrsin (a - 6), 
ABN ee Bor Wat —cos(a—6)}; 
(3). a [= (a— y cos (6, — 6)d6 + pate — cos (a — 6)} sin (8, — 0)d6. 
On integrating and simplifying, this becomes— 
d Wr W) 
ee = oH] | sin (a — 6,) +sind,sin a + + Je 21 ~ 008 6) +6, 8in (a — 6,) — sin 6, sina | ae hy 
As 6, is any angle between o and a, on writing 6, = 9 in this expression and integrating 
ietseee the limits 6, and 0, we get the coaeten at 0. 
1 
N 3 
Yo) = aor sin (a — 6) + sin sin a} d6 + 303 | (20 — cos 6) + Osin (a - 6) —sin 6sin a}d0 
0 
Wr? 
= eo cos (a — 6,) = cosasin 6, | fh ee — sin 6,) + 0, cos (a — 6) | 
20S + sin (a ~ 6,) + sin a (cos 6, — 2) 
(2) 
