CIRCULAR CYLINDERS UNDER CERTAIN PRACTICAL SYSTEMS OE LOAD. 185 
which is equal to 
• (70). 
(70) will give the approximate value of C whenever vra/c is at all large. If a = 2c, 
this method of approximation will already be quite fair. 
We see therefore that if e tends to zero, ^ also tends to zero, hut ^je tends to 
become logarithmically infinite. 
Now from (68) 
PUr 
XQcq 
(X + 2^) ftp + (3X + 2 /a) 
(X + 2/a) 
- XQ 
Hence, since — ^/e tends to co when e tends to zero, Prtjj tends to zero when e 
tends to zero. 
And similarly, for any finite value of n, Pc/.^ tends to zero when e tends to zero. 
But if we write down the expressions for the stresses, they are : 
- ^ _ O I V («) 
(1 + 7-)IyU 
rr = Poo + S 
Pa„7al^ 
_ P Cifj -j- S 2T 2 
7““V - (1 + 7«^)Ii“ 
P«„7«Ti 
pi,(p) + lo(p)(2 
«T 
h /J 
nirz 
COS 
LV P 
+ l)I,(p)-I,(p) y^ + f- + p 
«Io . 1 
/ill yp 
cos 
nirz 
7«^Io^ - (1 + 7«P Id L\ Ii 7/ P 
h, , 1\P(6) /I 
+ - 
- - i;r(p) 
llTTZ 
cos 
7’Z — 2 
P«„7all_ 
7«'Io“ - (1 + 7«P Id . 
d„ 
p^Ap)-y^Ap) 
-^1 
nirz 
Sin 
(71). 
Now the above series are absolutely convergent for all values of r except ?■ = o, 
where indeed they are discontinuous. Leaving the neighbourhood of r = a out of 
account, we see that for points inside the material, when the space over which the 
constraining pressure acts is indefinitely reduced, LPa... = 0 and 
= -Q 
rz = rr = (fxf) = 0 ; 
therefore outside the rim, where plastic deformation may be expected to occur, the 
stresses are exactly the same as on the ordinary hypothesis. 
We come then to the conclusion that this method of preventing the ends from 
VOL. CXCVIII.—A. 2 B 
