94 MR. L. X. G. FILON ON AN APPROXIMATE SOLUTION FOR BENDING A 
formulae, we may distribute the integral sign among the terms of the series, provided 
that lioth the original and the resulting series are absolutely and uniformly 
convergent. This is easily seen to hold good for the series (78), and it will he shown 
later, in § 18, to he true of the resulting series, providing the points considered he 
inside a certain circle of convergence. 
Assuming for the moment this result, we obtain from (77) 
P., = 
4W / r' ,, (‘(.s 2p(j)' 4■^Y ^ ^ ^ / P cos v(f)' 
irh 0 \ 
I h 
{2v) . irh Q ^ 
cos V(f)' 
H. 
Q. = f(T) vr (.0 
1 . (79). 
rrh 
mx2vc}>' 4MY-,/ 
I/fj v: 
where 
9 \ 
~ )o (sinld 2h - hd “ I^) 
H, = 
* /id + iid + c~'^” 
0 siiili- 2n — 4?d 
.2i/ + 3 
IGid 
du 
r” u~''+^ 
II _ j 
J 0 sinlr 2u — 4id 
rn., = I 
- /„2r + .s ^ i^p. + 2 
a 
+ c"'^" 
sinld 2?/ — 4?f“ 
du 
(80). 
(^ > 0 ). 
§ 18. Convergeney of the Series of the last Section. 
Til order to justify the distrilnition of the integral sign over the separate terms of 
tlie series (78), we have to show that the series (79) are absolutely and uniformly 
convergent. 
Now the series are absolutely and uniformly convergent provided that the series 
. 
N ( — ) —is alisolutely and uniformly convergent. The convergeney ratio of this 
1) ! v\ 
1 
r H 
latter series = L V -^ • 
h J 1 V I 
?'=00 
Now, in order to find the a]iprnximate value of IT,, wlien v is large, let us consider 
the integral 
T _ u 
” Josinld 2u- ■ " 
write u = av 
I _ _ — rf + l 
" ~ Jo silild 2((r - 4«V ~ Jo 1 + 
4»2 
4r'‘ c. d V 
