120 MR. L. N. G. FILON ON AN APPROXIMATE SOLUTION FOR BENDING A 
V = - 
W 
r 
1 1 \ Vj 
T i I —-1—) shill u + cosh M- 
^ VV + /A /i/ 
’TT J Q U 
+ 
I 
sinli u 
sinh 2 u + ' 2 u 
ul . , uy ux j 
COS — sinh -i- cos — ciu 
loo 
v.l T uy nx , 
, , ^ cos — cosh — cos — cnt 
/i,7r5Josinh 2u + 2u I o '> 
-— + — ) cosh li + — sinli u 
\\' + IJL _ yj __ 
sinh 2?'. — 2u 
W 
vr J 0 ! V' 
. vl v.y . v.x 
sillcosh A- sin v 
h lb 
^ (97). 
+ 
y, y ) 4 / 6 ^ j 
+ 
Wv/ cosh?f. . id . , uy . 11.x , 
-r sin ^ smh — sin —du 
yirl J 0 sinh 2?<. — 2 u I I ^ 
P = 
2M1 
irl J 
sinh u — 10 cosh lo vd ux , v.y , 
cos — cosh — du 
sinh 2it + 2 io 
cos 
I 
u sinh 10 id lox . , uy , 
-I -cos — cos 7- sinh — du 
•nlr Jo sinh 2 it + 2 it lb b 
2W cosh 10 — u sinh 10 . id . ux . , uy , 
-— Sin -- sill — sinh — du 
0 sinh 2 it —2 it b b b 
2 w r 
irb j, 
u cosh it 
. ul . ux , uy , 
. . , ^ sni — sin 7 cosh 7- du, 
ttIu Jo sinh 2it — 2 to b b b 
0 = 
2M^ 
ttI j 
r 
'' Jo sinhi 
sinh it + it cosh u 
0 sinh 2it + 2it 
10 sinh it 
7tt ux , uy 7 
cos — cos — cosh 7- du 
b b b 
ul ux . , uy , 
COS 7- cos 7- sinh -f- du 
2it + 2it b b b 
2 W f cosh it + 10 sinh 10 . ul . ux . , uy , 
- '-sin — sin -7 sinh -y- du 
lo sinh 2it —2it b b b 
+ 
TtI j 
2 \Yy 
10 cosh ?t .id . ux . utj 
7- Sin — sin 7- cosh 7- du, 
O,, ^7 h h 
s 
Trlr Jo sinh 2it — 2it 
10 cosh it 
2M^ r_ 
ttI Jo sinh 2it + 2it 
b b 
uy 
id . ux . , ■«!/ 7 
COS 7 Sill -7 sinh ~ du 
b b b 
24 ¥i/ 1'"’ 7t sinh it 
ul , UK , uy 7 
, cos 7 sill 7- cosh J- du 
ttI- Jo sinh 2it + 2 to b b b 
2AV 
. id ux . uy 7 
, . , , , Sin 7 cos 7- cosh 7- du 
irb Jo sinh2it —2it b b b 
10 sinh u 
li 2 it — 
u cosh to 
1 
ttIu Jo sinh 2it 
. ul ux . , uy 7 
Sin 7 cos 7- sinh '■ du 
2 to h b b 
lx ! 
! du 
i 
( 98 ). 
Now, as before, these expressions may he expanded in powers of r about the 
origin. In tliis case they will he found to have a radius of convergence .^/P-1-6". 
Or they may be expanded about either point of concentrated loading, when they 
will liave a radius of convergence 2 ^Id + Ir, or they may he split up as follows : 
