408 PROFESSOR A. H. GIBSON ON 
Since, from symmetry, M,=M,; T,=,; it follows that :— 
M, cos ¢— T, sin d= wr? a - 2G — 2 — sin 2¢) ; 
= Mae \ re - ae — 2 — sin 2¢) } + T,, tan ¢. 
Again, since the total load is 
27 ¢ 
2wr? le 2 (6+ @)dé 
0 
Lie : 
= {7-2 -sin 29 | 
a 
. R= R, = 20 f 2 = sin 26 f 
Fie. 10. 
The bending and twisting moments at a point x, distant 6, from OA are given by :— 
8) 
Mo, = M, cos 6, — (Ry - T,) sin 6, “[ wr? cos? (8 + ) sin (8, — ) dé. 
0 
a 
T,, =(T. — R,7r) cos 6 - M, sin6 + R,r - | wy? cos? (6 + @){1 — cos (8, — 6)}d6, 
0 
the last term in each case representing the moment, bending or twisting, about the 
point x, (fig. 10), of the load between A and a. 
On integrating these terms and writing 6 for 6,, the general expressions for M, 
and T, become :— 
3 
Mo=M, cos 6 ~ (Rar — T,,) sin 6+ = | (cos 6 — 1){cos 6 — sin® + sin 2¢ sin 6} + 2 sin? 6 cos* 6 | - (22) 
wis | 
: 9 
+ gin 6 , operas sa — cos) - cos? j 
Ts=(T, — Rr) cos 6- M, sin 0+ Ry — 3 . (23) 
3) 
= a — cos 6)? 
