144 
mi. L. N. G. FILOX ON AN APrUOXl.MATE SOLUTION FOII LFNDiNi; A 
.)■// • 
,I\j _ L _ 2L - 
il' 'IhVi ttVj f'" TT/i 1’'^ Ltt//J o siiih 2;( + 2i' 
l\, l'“ 1 — 2/' + C "" , /'//' . y 
cosli —- sm , a a 
h h 
H . . H If . XI.r. 
, 7 I ' • , , siiih : sni , dll 
(X + fL)7rl>J(, siiih 2(1 + 2/', 0 
Lif r® 1 - 2(/ + €--«■ . , vif . ar y 
« — 7 siiHi Usiii , dll. 
27r/L(.//-Jo siiili 2// + 2?', // h 
J / ' 0*5 ! 
1 "■ 1 "// • I 
■ --r, Cush - sill -dll 
TTijJr Jo siiili 2ii. + 2i' b h 
Putting in this y' = 0, 
ih: 
2 L 1 
//' = 0 
2/;E ttE 
Now the last integral may be written 
J®/ 1 — 2n + 
Jo \ siuh 2'U + 2u 
2\. r® 1 - 2/' + . n- . 
- 1 sin d ( 1 . 
ttIiE Jo si nil 2ii + 2ii h 
, _'U 1 \ . V..r , 
2a 
+ ^ if X is positive, 
and if x is negative, then — -tt/I must be written instead of + 77-/4. Ji is any 
positive constant. 
Now' the function 
1 _ 2)^-1- r~-" 1 
is such that /(O) =y'(x ) = 0, f (co ) = 0 and [ j/' (ti) \ da is linite. It follows 
J U 
f oo _ 
f {u) sin y da tends 
u ^ 0 
to zero as x increases. 
Hence, if x be positively increasing ( -) tends to U, and if x be negatively 
\ d'' ; _y'=U 
increasing 
iW 
tends to L/'IiE, as it should. 
d.c '^'=11 
The values of the integral, calculated for various values of the ratio x/b, ha^■e 
;'iven the following values for (, as compared with its value for a uniform 
\lLr I 
tension L U. 
