184 -MR. L. N. G. FTLON OX THE ELASTIC EQUILTBRIILM OF 
Tliis condition gives in terms of P, and hence by (63) 
\2Vq = P ( {(q + 2 (X + H-) 
a 
( 66 ). 
We have now to find such a value for P tliat the mean pressure on the plane 
ends is Q. 
TTCi 
'0 
[i^cdr 
J r» 
= 770 ' + (^ + 2 /a) U\) 
+ 277 X COS 
nir'i 
1 
c 
(X -|- 2ix]A,2 ~ XA, — x'^ rl^^dr fi- 2p,c| 
whence, apjfiying the well-known theorem, 
rJ.l 
{x)) = x%_,{x), 
we liave 
ttO'Q = 770^ (2X?/.,-, + (X -f 2/x) v\,) 
+ 277^ COS 
7i7rr: 
2C' 
(X -f- 2ja) A.t — XA^ — ^ 7 
f + 2A' f 
Usinn tire relation 
I 2 + 7 I 1 -I 0 
0 
and putting in for A^, A^ their values in terms of C, we find that the terms under 
the S vanish identically. Hence 
Q — — (2Xi/o + (X -f 2/x) 'U'q) 
(fiS). 
Now su])pose the distribution of stress is such that rr = 0 from : = — (c — e) to 
z = -]- (c — e) and rr = — P from 7 ; = — c to i 7 = — (c — e), and fi’om z = c — e 
to = c, we find 
, 2 ( — 1)'* . nire 
= — c/c a„ = — - - sm -- 
llTT C 
whence 
— _ s' 
. 7X11 (' 
Sin 
c r 
Hr 
1 n. n-TT- 7«H|f — (1 -f 7«hlf 
Wlien a is at all large, the terms of ^ are comparable with those of the series 
. ( 66 ). 
. 7177(1 
