123 
bea:\i of rectangulak cross-section under any system of load. 
AY 
r sill 2i>(}) r"° (1 — + i; ~“) c 
2v )!'I 
— ^ 1/ \ I, 
(2i/) 
(sinli 2i/ + 2i>) 
nl 
cos du 
h 
+ 
AY - co^2^0 r (1 _ 4^, _ c-2«) Q-n _ ^,i 
7tI) 7 y (2^) ^ -L (sink 2u — 2u) 
sin — du 
0 
AAO/ 
^ Trft- 7 U 
• \ Si'+l ol,, -1 
sill 2z.+ 1 c6 r «2>'+2 (1 + - c-2«) rr 
(2c+ 1)1 Jo 
uJ 
+ 
W// * j r Y^+i cos 2v-\- l (f) a~''+-{l + 4c- + c~~") e~'‘ . /T , 
,, A I — ) -:— I -—-sin - cm, 
0 
sink 2u + 2tt 
(1 + 4c 
sink 2u 
cos -~du 
0 
u 
ttY 0 7v (Sc + 1)! Jo 
where r- = x" + ij^, x — y tan ff). 
U, Y^ may be broken up in like manner and the parts Uo, wliich remain finite 
and continuous everywhere can be expanded in the same way. 
We sliall rec[uire also the series for U, Y", P, Q, S in powers of r, deduced directly 
from the expressions ( 98 ). They are 
AA k / 1 1 ,. . , i’ v3t'+l ypj Oc+l (j) 
u— — ’’ ^ ^ r - u 
TT 0 \A. + /i ~ " 
(2c + 1)! 
YA\// 
X) 
—\i! ^ Q / 
irhij, 7 \ h 
r d" sill 2v(f) AA(// ^ ( L V'' 
(2c) I Trhfi'^ ^ \ l) J (2c)! 
j_ ^ 1 Cl 1 Q \ / ci-is 2c+1 (f) 
TT 7 ' A-' + + fM ' h J (2c + 1) 1 
v=-^A 
71' 0 1 '.A + /!■ jX 
^ +ho-.+ M’> 
-2V+1 
I Wv l r_ Y cos 2c0 W// ^ 
0 ^ vC' ( 2 c)' 7r?i/u. 7 h 
cos 2c + I (f) 
h I ( 2 z/ + 1 ) ’ 
r 7" ‘Sin 2i'0 
( 2 .-)! 
AY k 
_S' 
TT T 
A + + ixj ^ ^ IX ~ 
/ j. ,2i'+i n,, 9 
sin 2 c +■ 1 (j) 
(2„ + 1); 
p _ _ i/r, ,, Ur Y’ cos 2,4, 2W,, J / y-rl 
• -'TO "(2s): - (3„+,); 
irh 7 
2 AY / 
- f - s,.) ( 
f d'd'' sin 2c<f) 2A\Y ^ ^ ( > 
hj (2c)! ' "■“ 77/13 7 Hi 
cos 2 c +1 
( 2 c - 
/■/■\2'c+lfiin 2y+i^ 
( 2 c + 1 )! ’ 
2 AY 
/' 7 . Y" P*® 7 , /' r '\~''+1 cos 2 c +1 0 
( 2 c)! 7 r />2 7 '^ 2 .+ j 
2AY 
irh 1 
; -L ^ A / 7 -^0 I 2AY//“ , , 
i.-l + ( J ^ ^ bo,+ i ( 
( 2 c + 1 ) 
g _ V r< / 7Y‘'^k2c(f) _ 2AA0/ ” /jp Y'+i sin 2c+1 (f > 
7 ^^ 1 \ h) (2c)! 7r&^ 7 \ /i / 
_ ; Q (L t PP ^ Q / ^ 
’ ' (2c)! + 
R 2 
irh 7 
(2c + 1)! 
i.. 2v +1 (2Qy 2v + \ y) 
(2c + 1)! 
> (101). 
hj (2c+1)! 
' r sin (2c + Vjcp 
I ) 
