156 
ME. L. N. G. FILON ON THE ELASTIC EQINLIBRIUM OF 
Also the conditions of the j)roblein require that u shall he an even function of 2 
and IV an odd function of 2 . Hence 
«! = — — 7r/2 , = Co = 0 . 
Integrating (15) and (16) we have 
n = X (^■) + - “I — cos kzl^ {I'v) — y; cos h..r\{Jir) + ^ 
sin kz 
cos kz 
‘T 
Ii {h' 
ir = 
sin l'z\ (kr) + 
y sin kz.r\^ [kr) -j- y 
To find the relations between the constants we must sulistitute in equations (1) and 
(3). AVe then find the following relations : 
,H;^(r)=0. . . .(17), D2d(2) = 0. . . .(18), 
(A^ — Ao) (X fi- /x) k' + 2/'{Cj (X + 'Ijx) — /xEj — (A + ^) = 0 . . (19), 
(A^ - A.) (X + /x) B + 2X:{Ci (X + /x) + /xC. - (X + 2/x) EJ 0 . . (20), 
= Co = C, say ... (21), E^ = Eo = E, say . . . (22). 
In virtue of equations (21) and (22), (19) and (20) reduce to the single equation 
(A, - Ao) (X + /x) k + 2 (X + 2/x) (C - E) 0.(23). 
Also from (17) and (18) 
X (’’) = ^ V ’ ^ ( 2 :) = , 
remembering that w is odd in s and that u is not to he Infinite when r — 0. 
For the stresses rr, zz, rz we find from (4), after some obvious reductions, 
rn = 2 (X + /x) n.jj -j- Xiq, 
+ 
- ^ 
+ 2^ 
{(2X fi- 3^) A^ 4- cos kz 
+ 2/x|(A, - |)-j^^osy (:2 + E:sinH (^Io(H) - j - CVIi(H)cos H 
2Xi?o + (X -}- 2/x) xiq 
+ 2/x) Ao — XAj ~ ^ “1“ + t) )7 [■ ^0 cos h 
_ + 2/x{CrIj [kr] cos kz — El,, {kr) z sin h 
(24). 
• (^5). 
' j 
rz 
/xS[(A^ q- Ao) Ij {kr) sin kz -f- 2Crl|, {kr) sin kz + 2EIj {kr) z cos kz] . . . (26), 
