THE EQUILIBRIUM OF THE CIRCULAR-ARC BOW-GIRDER. 397 
while, equating the torques at the weight, as obtained by working from both ends of 
the girder, 
T, cosa+R,r(1 -— cosa) —M,sina= —T, cos B- R,v(1-cos8)+M,sinB . : eG) 
The other two necessary relationships are obtained by expressing the fact that both 
slope and deflection at the weight are the same, whether the latter is considered as 
being at one extremity of the arc AW, or of the are BW. 
The slope at any point 0, between A and W is given by 
87 oi 
dy 2 x . = 
(2) “| My cos (8, ~ 0)00-+ 2 al Ty sin (6, - 8), 
and, on substituting for M, and T, from (4) and (5) and integrating, 
(3), - 
dé oy 
+503 
Similarly at any point between B and W, distant 0, from OB, 
(ia - 
The slope at the weight is obtained by writing 0, =a in the first, or 6,=8 in the second 
of these expressions, and is thus given by :— 
aat| Me {0, cos 0, + sin 6,} — (Ryr — T,)6, sin 6, | 
| (Te R,r)6, sin 6, + 2R,r(1 — cos 6,) - M,{sin 6, — 6, cos 6, | 
Faq] Mol cos 9, + sin 6,} — (Ryr - T,)6, sin 8 | 
+ 5t| (Bo- R,r)6,sin 6, + 2R,7(1 - cos6,) — M,{sin 6, - 6,cos 6} | 
v2 
. sraq| Me {acosa+sina} -(R,r—T,)o sin «| 
aN ae 
(aa)y > 
(LO) 
+ soy le — R,r)a sin a + 2R,7(1 - cosa) — M, {sina — acos a} | 
or by 
aM »{Bcos B+sin B} — (Ry T,)B sin B | 
(a) 2 ped 
according as the point W is considered as forming part of span AW or of span BW. 
On equating these two expressions, with the sign of the second changed since @ is 
measured in opposite directions in the two sections, a further relationship between the 
unknowns is obtained. 
+ sty) (To R,r)BsinB + 2R,r(1 -- cos B) - M,{sin B - pcos 8} | 
Deflections.—Assuming the supports to be at the same level, integrating oe to 
obtain the deflection, gives between A and W :— 
61 
i 
mal | M(4 cos #+sin 6} - (R,r — T,)d sin 6 |a6 
04 
+ ial | (te — R,r)@ sin 6+ 2R,r(1 - cos 6) — M, {sin 6 — 6 cos 6} |a8 
