70 
MR. L. N. G. FILON ON AN APPROXIMATE SOLUTION FOR BENDING A 
From these, by simple integration, 
(V + 3p-) + (V + p) = (f)' (^)'] 
cJri ~~ J 
}> where (f)' (i), x iv) arbitrary functions. 
whence 
or 
= 4, F) -X'(»)5 + 4 2.-) 
= I(^ - 4, 
T = [•#> (0 - A' (>))] + i, [4>(0 + fx' e)] + r (>)) 
w = hf (f) + X 0)] - [’)f (^) - X ('^)] + G (?) 
where F ( 77 ), G (^) are again arbitraiy functions. 
Hence U and V can be found almost immediately. Writing 
r(>))- 4 ^^t 72 ;A(’)) = F>( 0 . ■ 
(19) . 
( 20 ) , 
we have 
u=(f) + X (01 + ^ (f) - (’>» + i T. (»)+G.(f)) . (21), 
^ = 8if^Sx(i)-^(f)S-i,G£y(x'(.)+f(f))+hG,(f)-F,(.)J .(22), 
4:^ X “}~ 
from which we obtain easily 
p = 4'VVhvV'^' (P + x 'm + in w (?) - x" m + mG'. (?) + (^), 
Q = 
^ J Af / C\ I + 
(?) + x '(>))5 - 2(V4 2^)‘>W''(P - x"(’;)) - f^G',)?) - /.F, (,), 
4 (X + 2 /x) 
X' + yLt ■ / , ^ 
® <?)+x''(A+i ,;:ffg (f (?) - x'(’i))+/-oG'i (?) - F/(,)j, 
and these last may be put into the simpler form 
I"" = b j: + o/w'tlx2/;7b7,) {i>iO + xiv) > +/^ biCli (f) + F^ (>7)} . (23), 
X' + 2 ixdx ' 2 (X' + 2 /j.)'^ dx dy 
dx 
