102 
iMR, L. N. G. FIT.OX OX AX APPROXIMATE SOLUTIOX FOR EEXDIXG A 
‘lu sinli 'iu 
. ,, ^ , , sin sinh ' - du 
0 Rinlr 2it — 4:11- h h 
v.y 
2 ;rsiuli2;6 . ur. wf , 
r-— - — Sin -- cosh - - du 
0 sinh- 2u — 4u- o h 
W' [” _ 
irlr J 0 si 
MO/' p 2u cosh 2 m + sinli 2« ^ ^ '"V 
~^lJ J 
sinlF 2n — An- 
• liiO • 1 '*7 7 
8in — sinh riu . 
1) ii 
Tliese integrals remain convergent when we put y" = 0, but they are not convergent 
ill their present form for y" = '2h. 
If we expand now in powers of where x = sin (j)", y^ — i‘ cos ^ , we obtain 
the follovhno- series, which can easily lie sliown to be uniformly and absolutely 
convero'ent inside a circle of radius 2l> : 
sill v(f)' 
irh 
TT \X + jxl 1 
\ 
TT yU. 0 
(2l^+l)! 
Y ^ _ I -W' t(-iY 
U tt/i 7 ^ 
Hsr), 
+ 1 
2AY l_y 
TT r! " 
2W - /ry-'’ px' 
IT jX 0 
(2v)\ 
? = 
V y 1 FT' J- V /L!Y'' TV 
TM //' ^ / N ,, /y \' 7/^ ^ 4 ‘ TT' 
irlr 7 ^ ■ 
V : 
Q 
- mi2v4l(j) 
'rrh 7 
4AV 7 /7 
_V 
^2i' + 1)! 
H.2„.2+—^ S(-l) 11,... 
• (8S). 
S 
where 
_ 4W 7 /pV" 
+ 1 
ttV T^ ' \h] v' 
ttI) 1 \ h J (2i') ! 
117 = arbitrary constant depending upon tbe fixing conditions. 
ir 
’du 
IGr" 
du 
