158 
MR. L, N. G. FILON ON THE ELASTIC EQUILIRPJUM OF 
Now f{z) being an even function, we can expand it in a Fouiuer’s series between 
the limits i c in the form 
2n-\- 
2c ~ 
2n + Itts , 
COS - ^ clz 
2c 
/(-) - /(c) = cos 
0 
where «» = M i/(/“/(c)} 
C J _ (• 
and being an odd function can be expanded in tlie form 
00 4 - 1 77 -* 1 
^{z)=% K sin 
0 
• ■ (38). 
h. = i 1^' 
C J — c 
2n + lirz , 
Sm - - dz 
2c 
where 
Now since {rrl ^ „ = /r//.), (= pi// i^:) 
we have, comparing (3G) and (37) with (38) and (39), 
(39). 
3X- + 2fi 
\ 
«b=/(c) 
Ai ( - ! I + 7} Iq («) + b + f / ) + A/- > 1 - 7I lo («) - y«Ii («)) = 
Ai (Ii (a) - 7“lo (“) ) + Ao (Ii (a) + yalo (a) ) = 
whence 
_ j (“) + 7«%(«)) + + (1 - 7)«Io(«)) . 
^ ' 7«M,^(«)-(l + 7«blF(«) • • • t h 
A 1 («) — («)) + ^«((1 + 7 ) ^h)(^) — (- + («)) 
7«3y («) - (1 + 7«-) O («) ■ ■ ^ 
W’a = 
-' 1 ^/( 0 ) 
00. 
and therefore from (35) and (27) 
C = 
(271 + l)7r j rr„a7b(«) + (7“Io («) ~ 7b(*)) 
2c ~ 7a2I(,2(a) — (1 + 7^2) 
(^3), 
w.„ 
2(47-1) 
/(c) 
(44). 
