CIRCULAR CYLINDERS UNDER CERTAIN PRACTICAL SYSTE.US OF LOAD. 225 
that no limiting value of c/rt (which should, of course, be independent of z) could be 
found, and the approximation need not necessarily hold. As a matter of fact, it is 
shown in § 29 to fail for particular cases. This is true d fortiori, if either series cease 
to be convergent at all. 
The same remarks apply in their entirety to the process of approximation given in 
§ 6, and further, to the approximate expressions given by Professor Pochhammer in 
his investigation on the bending of beams Crelle,’ vol. 81). 
§ 29. Special Case of Two Discontinuous Rings of Shear. 
Suppose that we have the following system of values for (jiZ :— 
(f)Z = T if c — e <c <c, 
(f)Z = 0 it — c e <2 < c — <?, 
(f)Z = — T if — c <2 < — c + c. 
so that we have a cylinder twisted by two equal and opposite rings of transverse 
shear extending over lengths e of the cylinder, near the ends. Then we find 
easily 
c„ = 
4T 
{ 2)1 + 1) TT 
( — 1 )" sin 
{2)1 + 1 ) Tre 
with the following values of the displacements and stresses : 
— y 
8Tc 
V = 
r(f) = 
/.( 2 «. + lp7r- 
7 {2n + 1) TT 
/ -I • 291 + 1 TTC . 2)1 + 1 TTS 
(— 1)”— Sin ---sm-—— 
12 (a) 2c 2c 
/ \„ \ip) • + 1TTC . 2 
)i lirz 
(f>z ='Ey:- 
4T 
0 {2n + 1) TT 
lAp) . 2 ;i + i Tre 2)i 1 ttz 
" , r sm -T-COS--- 
Ijla) 2c 2c 
. (131). 
Now it is easy to see that in this case the conditions for uniform convergency are 
satisfied, except at the boundary, and except with regard to the stress rf), whose 
approximate expression is not uniformly convergent, being in fact discontinuous for 
z= ±{c — e). 
At the boundary, {^)l\ (“) tends to unity with n, its approximate expression, 
when a is large, being 
=z: 1 + ^- + — + — + 
12 (a) ^ 2u ^ 8a2 ^ 8a« ^ 
(132). 
Hence V is always uniformly convergent and its approximate expression likewise, so 
for it the approximation, for sufficiently small values of cja, holds throughout. 
VOL. CXCVIII.—A. 2 G 
