THE EQUILIBRIUM OF THE CIRCULAR-ARC BOW-GIRDER. 395 
‘ (3),- “al (1 — cos 8) cos (6 — 6,)d0 + a “@ — sin 8) sin (0 - 6,)d9, 
C 1 
where a is the total angle subtended by the beam. Integrating this expression 
and simplifying gives-— 
[2 —6, sin 2a -sin 20 sin 6, sin? a — sin? 6, 
| sin (a - 6,) — cos 6, | 5 tf ri tb 5 
= : ye Rr 
(3) = sin (a — 0;) — a cos (a — 6) + 0; — cos A 4 got) Se Sos a 
dO/ 6, ae 
Oy sin 0, sin? a — sin? 6, 
a 2 
a 
wr { T.. a—@ sin2a—sin26) sin @sin?a—sin? ‘| 
; =ONn= BSS eae a ate ae ae Ef) 
EI i sin (a — 0) -cos 0 \ ot 7; i 5 
1 
a 
. — T= = as 9 “ss . bel 
(a, sin (a6) ~a.c0s (a~ 6) +6 cos | * ceases — 
pipe 2 4 ; 
anne d 
CJ sin 6 sin? a — sin? 6 
es 9 
a 
Lat 
in 6, sin 2 
2 = 2/cox (ae (ae) simGy cosa = Cos 6, ) Sasa 
5 
wr 5} 
JBI Ei eee Bas 
ee a Sin za — sin 
— }(cos? a — cos? 6,) — cos 6, sin? a + oes AL 
2 1 
— — 
= | 2-2 cos (a—6,) — 2asin (a — 6,) +02 — 6,2+(a—6,) sin 6, 
sin 6, sin 2a 
ope + cos a — cos 6, — = + 4(cos? a — cos? 6,) 
a— @, sin 2a — sin 26, 
2 4 
At the free end 6,=0, and the deflection becomes— 
4 5 at 
wr sin 2a 
ai — cos a ~ (cos? a — 1) — sin?a + - od 
+ cos 6, sin? a — 
2EI 4 
a ae a — cosa — 2asina + a? + (cosa — 1) +sin2a—2 + sizer 
203 ae SE ier aa me pr yen | 
€.g., if a=, the deflection at the free end becomes 
_ al 3594 De 
ae le Froypal: 
Experimental Verification. — The experimental cantilever, 7+ = 10°10 ins., 
== Lex LO-* a 1°438 x 107°, a= 5, was loaded uniformly by a series of washers, 
giving w='0605 lb. per inch run. Under these conditions 
y. (calculated) =*502 in. ; y, (measured) = °510 in. 
