108 
MR. L. N. G. FILON ON AN APPROXIMATE SOLUTION FOR BENDING A 
We might repeat this process any finite number of times. It will be found that 
siiih u + 1 cosh 1 . even function of u, the integrated terms will in all cases 
sinh 2;^ + 2». 
2AV^' 
Vf 
anish at Ijoth limits, and v^e obtain Q = X an integral which is not infinite when 
TTcU 
X is large. Therefore we see that Q diminishes faster than any finite inverse power 
of X, however high. This seems to suggest an exponential law. 
24. New Form o f Expansion for the Pressure on the Rigid Plane. 
Consider tlie integral 
We have 
1 1 
sinh 2» + 2h sinh '2v sinh~ 2u 
-r sinli u + n cosh n , 
I = . , ^ „ cos uz dv. 
Jo smh 2a A 2// 
J_ /_ iv-i J_ (_ ly 
' sinh" 2it ' ' oinli” 
sinh” 2n (sinh 2« + 2«) ’ 
Sul)stitute in 1, we find 
1 — J() “h J1 + • • • “h J/ "T • • • J/i-i “h Ip/j 
wliere 
j. = (- ly 
ih = (-1)" 
(2«)'- 
(sinh u + u cosh it) cos uz du, 
cos uz du. 
Jo sinh’"''^ 2tt 
r” {2it)“ (sinli » + « cosh ?;) 
Jo sinli” 2u sinh 2h + 2u 
Now 
r / \ -.o v?''(sinh u + « cosh 'H) , 
-h = - ly- 2-''+' - — TT^ COS uz du 
‘ \ / I ph-+-[>u (] _ g-tey+l 
( 1 - c-^'<y 
Let us assume that in tliis we may expand (l — in ascending powers of 
(‘ This will he justified later. 
— V 
whence 
(1 — 
s = 0 
r ! 
J, = (-!)'• 2-'- If % 
.'o s=0 
(s + 1) . . . (s + r) 
r ! 
[e — g (4.5+2<-+3)i!| du 
+ (- l)- 2 =' f 
u 
V • • • (-^ + _ 0 j-g (4s + 2 i-+l )0 _j_ ^ (43+2/-+B)1!| QQg ^^2 du. 
s=0 I' • 
The cases r even and r odd have to be treated separately. Consider first r even 
and = 2^, and let K,. and L,. denote the first and second integrals in the last written 
expression for J,.. Then owing to the vanishing factor we may take the S in K,. as 
going back to 6' = — t, or, putting s' = *’ -f- t, 
