80 
Q-. 
^IR. L. N. G. FILON ON AN APPROXIMATE SOLUTION FOR BENDING A 
, Z / , \ siiih mh + rnh cosh mh , 
“0 + r “inh -Mi+'-Mb 
— S (a„ + 
onj/ siiih mh sinh viy cos ??u 
sinh 2mh + ‘2mb 
" , cosh + 7?/&smhw& . , 
4- s {a, - IB,) ^ '^^■v 
— t {a„ — (B,) 
1 
S = — X (a„ -I- 
sinh '2mh — '2mh 
my cosh mh cosh my cos m-r 
sinh 2mh — 2mh 
vib cosh mh sinh my sin nw: 
Y (57). 
sinh 2mh + 2mh 
— t {a„ — IB,) 
^ . 7»7/sinh ??i& cosh my sin wa; ^ , . 
+ T smh2,;,(,+ 2,»f-+ r 
mh sinli mh cosh my sin mo: 
sinli 2'mh — 2mb 
my cosh mh sinh my sin mx 
sinh 2mh — 2mh 
§11. Approximate Values to which the Expressions o/‘§ 10 lead tvhen “is made 
very small. 
Tf7> is very small compared with a, so tliat, even for certain fairly Ingh values of m, 
mh is still small, we may expand the coefficients in (56) and (57) in powers of m,h, 
and also we may expand cosh my and sinh my in powers of my. Tliis is the method 
which has been employed by Pochh^kmmer (‘Crelle’s Journal,’ vol. 81). I have 
shown in a previous paper (“ On the Elastic Equilibrium of Circular Cylinders under 
Certain Practical Systems of Load,” ‘Phil. Trans.,’ A, vol. 198, pp. 147-233), that 
such an ajjproximation was valid provided that the original series and each of the 
approximate series obtained from the various terms in the expansion of the coefficients 
of cos mx, sin mx (wliich expansion is supposed carried out only to a limited number 
of terms) are absolutely and uniformly convergent for the region considered. 
Assuming that the values of a„, are such as to ensure that these conditions are 
satisfied, let us see what happens when, in the expressions for the displacements 
U and V, we neglect all terms of order greater than — 1 in m. 
We find 
U = 
2&«E7 
rn 
1 
A, + ^ 
^ \ "f iB], 
sm 70 X 
8m 
+ 
V j X' + / i- 
1 f'l + 1 ,,^2^2 
/ n. I / m~r\ 
yll -h ) sin mx 
t^ 
pn^h^ ( 1 + 
0 / .0 
m7^ 7^/- 
, «« - Ai 1 2 2 _ 
+ X - -7- y sm mx 
2n onAr'' 
fnrm I 1 + 
V 
a 
1 
