l;3-J Mil. L. K. U. FILUX UX AX AlTKUXIMATE SOLUTIOX FUJ; ]U-;XDIXG A 
U = 
. ' + ^ ) cosh >1 — — siiili (' 
' A ~|"/X 
+ •; 
-77 Jo 7' _ sink 'In + ' 2,11 
X / 
1 (x' + /X /X 
^coshycos ^ ~ y) 
^ + — ) sink u — ' cosk n 
Af- 
IT Jq H 
sink 2i(. — '2>i- 
. , I"! 1 ■’// 7 
siiili -cos ^ ” ' v , o.a 
h h A + ytx -ivo _l 
+ o 
L// 1 coslu? 
+ 
= 
27rZ^ jo sink '2u + 2ii. 
L// r* 3 r sink u 
a . , VH Ur 
—sinli , cos ^ a u 
1)0 
27rh J 0 A*- 
ti;/ yy 
—;-— cosh -7- cos -7 — , "o 
sink ' 2)1 — 2 ) 1 - 0 0 
du — X'lj, 
1 , 1 . , I 
-cosk a H— )' sink a : 
3 A + /X /X 
\(105). 
277 J 0 )' 
sink ‘2u + 2)i 
_ I - ^ 1 j A' + /X 
L// r“ 1 cosk ))j 
277?> Jo A*- *si 
L, f 
“ -'tt/- J 
> . , I'n . "y 7 
sum - sm - du 
1 h 0 
' ^ sink /'+ — /' cosk » ) 
/ 1 "H • c,.-; 
- ^ - cosh , sm — 
0 sink '2ii — 2)( o 0 
cosh ' f sin ' du 
0 b 
/- "a 
a' + /X fxj 
du 
0 /X sink 2)X + 2u 
L// r“ 1 sink /f. . , )'!! . , 
-r-^—;-T smii 7 — (/?/ + xV n 
277/' jo A'’ ■” 
J 
Xow, whoii ill \' AVC put y = <k we Imvc left 
^ — sink /'+—/' cosli n ) 
^ yin "- 
277 tki 1 " sink 2/' — 2/' b 
h 
A + /X /X ' 
du + xVA. 
L 
^ ^ - sink IX + - XX cosk xx ) 
t \A' + xx u 
277 J 0 L " 
L 
+ 27rJo 
sink 2xx — 2XX 
1 ^ XX.'- /x — sin XX,-- /' 
4- 
XX” 
_\A' + /X /X; 
Consider the first of these integrals, and let 
, , sink XX + - XX cosk xx, 
_ 1 \ ^ A A*- U 
A’o others, 
1 \ 3 
1 .9 
1 
/x/Txx” 
40xx i 7 t A' 
+u 
9 1 
. XX.' 1 
sm ,. y 
du. 
' U ~ A' + 
) f s>>‘ 
/(■») = 11 
sink2xx — 2xx 
1\ 3 1 ,9 1 ' 
^A' + /X A^) 4xx” 40xx \ix A' + /x 
Then /’(a) and its differential coefficients are finite and continuous for all values 
of n, and vani.sh for = 00 . /(a) itself = 0, vhen n = 0 and the integral 
I ; /'('/)] iy finite, denoting the ahs<j]ute ^'alne of /'(a). It is tlien 
