THE EQUILIBRIUM OF THE CIRCULAR-ARC BOW-GIRDER. 393 
At the free end 0,=a, and we have— 
Wr? : Wr? : : ; 
= —— — — _ \ 5 A 3 
aw le cos a sin «| + ar | 30 4 sin a+ sin a cos a| é ; ‘ (3) 
Asa check on the validity of the reasoning leading to these results the deflection 
at the weight may be calculated by equating the resilience of the beam to the work 
5 
ee 
on 
< 
Hire, 2: 
done during deflection. Taking, for convenience, the origin at the free end (fig. 2), 
M,=Wrsin 6; T,= Wr(1 — cos@) ; 
and, if / be the length of the beam, the resilience is given by— 
1 1 - s i 
1 2 gly T 2d es W272 eg W2r3 15 2 ee - 
EI ! Me tat | ods= ar sin? 6d6 + CT ( cos 6 + cos? 6)d6 
Be wl: —cosasina , 3a—4sina+sinacos Z| 
= ’ 
4 ile as CJ 
and, since this = = x deflection at weight, 
. Yw= 
Wr a—cosasina , 3a—4 sina +sin a cos a 
2 
EI CJ 
which is identical with equation (8). 
E.y., a===90", 
_Wrf 2 ,157-4]_wgl 7854 , 3562 
1S = | sart CI ] ie ae Fey | 
3m 
If a= 27 = 135° 
ie 
WH) Gees one eo “1-4981 18716 
We irae Saar = 2 | oa af 1742 
NGS onl RE — We a CI | 
Experimental Vervfication.—Further verification was obtained by the results of 
a series of experiments on such a cantilever of round wire ‘1624 inch diameter. This was 
