130 
ME. L. N. G. FILOX OX AX APPEOXIMATE SOLUTIOX FOE BEXDIXG A 
v = 
1 1 
, ^ ^ -cosh ml + — ml) sinh ml 
- -^- '-■' _ X —^ ^ ^ ^- - -siiih my sin mx 
16/l(,(M + /i) (d) ,^i-i2ccm 
sinh 2ml + 2inh 
\j A “i“ ^ 
^ sinh mh + — ml cosh ml 
sinh 2ml — 2ml 
cosh my sin mx 
X Ely cosh ml cosh my . 
-P X ■ ■-;- ^ ' sin mx 
, 3c yu. Sinh 2ml + 2ml 
X L 1 y sinh w//sinli 1/0 / . . a ■ rt 
+ X - ■ , ^ , - —n + A.X- + C, 
‘ ,,^1 2a jji sinh 2vil — 2ml 
V = 
].-• L 4 CDsli left — 2//i//sinli///// , 
— X -. ^ r ,-;r'; — cosh wi/siii mix 
4cA ,,,^1 2a sinh 2ml + 2ml 
X Iv 4 sinh »?/> — 2»hi cosh 7/i/^ , , 
— X ---- 'in'll sin mx 
,=] 2c sinh 2)nl) — 2mI 
=” L 2///y cosh w6 sinli 7/iy . X L 2icy sinh////> cosh/c y . 
N* --^^ sin ),}x — ^-^-^'-Ri 
5,3,1 2a sinh 2ml + 2ml 
5,=i 2c sinh 2ml — 2mI 
smmx, 
Q 
L 2/«&sinh OT?) cosh »iy . X h 2//i5 cosh v/hi sinh »o/ . 
— X-sin mx — X - . , ^ ^— 7 —'- sin mx 
5,=i 2c sinh 2ml + 2ml 
5, = i 2« sinh 2ml — 2ml 
“ L 2 m?/cosh 7 ?i&sinli?Hy . , X L 2 i»c/sinh i/i?/cosh 7/0/ 
+ X - '.A, ^ 7.0 7 ’ + X - Sin mx. 
5,=i 2a sinh 2ml + 2ml 
,,=i 2c sinh 2ml — 2ml 
s = 
Lc L X E 27cy cosli 7 h 77 cosh 7//7/ 
- + 4- X -- ■ , ^ cos mx 
4-ah ' 4c 5,=i 2a sinh 2ml + 2ml 
X E 2 m y sinh ml sinli m y 
4- X - — •. - cos mx 
‘5,=! 2c smh 2ml — 2rnl 
” 1. cosli y/Ey — 7 h A sinli 7c/y . , 
- 1 - X -;-;-— sinli my cos mx 
‘ ,,^1 a sinh 2/H?y + 2ml 
X h sinh ml — ml cosli ml , 
4- X - ■ , ^ ;— cosh mij cos mx 
,,^i c sinh 2ml — 2ml 
> (103). 
where rn = inr/a, and A, B, C are arliitrary constants to lie determined from the 
fixing- conditions. 
Now if the fixing conditions are 
(I.) That the displacement of the origin Is to be 7 .ero ; 
(ii.) That the extremities of the axis are to remain on the .same horizontal line, tlien 
C 
B 
= 0. 
_ X 
n=i 2 am 
--— + —) co.sh ml) — — ml sinli ml 
E\A'4-/X yU,/ fl 
sinli 2rnl + 2ral 
A = 0 ; 
