1(5G 
MR. L. N. Cl. FILON OX THE ELASTIC EQUILIBRIUM OF 
Now let us write 
6yciTJ — (47 + 2) LI, — 27 «I,^ 1 . '2n + Ittc . 'An + IttJ 
—! — I --—— -=--mil-sill - 
7 «H,/ - (1 + 7 « 2 )Ii 2 {'An +1) 26- 2c 
In + 1 
+ 
2 (1 - 7) 
(27^ + 1)2 
. ‘An + Ittc . ‘An + lirh , 
Sin-—sill--. . 
( 51 ), 
27 al ,2 + 4 (1 — 7 )L)I, — 2(1 — 7 )« 1(,2 ^ , 2n + lire . ‘An + lir^' 
—i -±-^-!-i---gill -- (.'111 - 
7 « 2 y _ (1 + 7 ^ 2 ) Q3 (2/i + 1) 2c 2c 
47 
1 
7 2 /; + 1 
+ 
(2 - 27 ) (07 — 1 ) 
7“ 
(271 + 1)2 
2/1 + lire . An + lirb , , /_ x 
Sill-^-.Sill-^- hiU ■ (o'A), 
Jc 
2 c 
so that j>,/, qj are comparable Avith the terms of tlie series X — + fp ’ AA'hich coiiA'erge 
fairly rapidly. We see therefore that .'i.-; and are made up of tAAm kinds of terms 
(«), terms of the form — %/jJ cos ~ 1 ' Kgj , AA'hich are abso- 
ir Ac ir 
ic 
liitely and uniformly convergent series, and (6) series, in Avhich the coefficients are 
the ajjproximate expressions found aboA^e. Of the series (6), those AAdiich haA-e 
terms containing l/(2n + 1)^ or l/(2n + 1)® are absolutely and uniformly convergent. 
Th is, hoAvever, is not the case Avith the series formed by taking the leading terms in 
the approximation, auz. :— 
48^ 8 . ( 2/1 + l) 7 rc . ( 2/1 -t- l)irh {‘An + l)ir:~ 
^ Sill -—- Sill --- COS-. “ — 
and 
ir 0 ( 2/1 + 1 ) 
48* 47-2 
. ( 2/1 + I) TTC . ( 2/1 l)irl) {‘An l)irz 
ir 0 ^{An + 1 ) 
For the series 
i 1 
sill 
2 c 
sill 
cos 
zc 
. 2/1 + liri' . ‘An + lirh ‘An + lir: 
Sin-;-sin---cos ■ 
0 ( 2/1 + 1 ) 2 c 2 c 
may be liroken uji into the sum of four other series, thus : 
1 ^ 1 2/1 -f Itt , 7 I \ I 1 1 
* - (Smi ^ 2 r (*-'' + «)+ 4 - 
0 (An + 1) 
An + 177^ , ^ 
COS -T- (s + 0 — e) 
_i V 
^ 0 {An + 1) 
An lir / 1 ; I \ 1 
COS - - -- (z+b + c) - i 
1 
0 (2 /1 - f - 1 ) 
2/1 + Itt , 
cos - - (z 
- h - e). 
Noav it is easy to sIioav that 
7(2/7+ I) 
A\here 1 x 1 is the numerical value of./ 
Y cos ( 2/1 -h 1) X = 7 log cot 
