CrcCULAE CYLINDERS UNDER CERTAIN PRACTICAL SYSTEMS OF LOAD. If)7 
The series on the left is divergent and log (cot j = oo if x = 0. We see, 
therefore, that, at the points 2 = ± dt wherever the shear rz changes dis- 
continuonsly, the stresses 2:2 and (f)(f> become infinite. 
The meaning of this in practice would be that, as the transition from the stressed 
to the unstressed surface becomes more abrupt, the tractions in the neighbourhood 
become dangerously large. And if the shear is applied by means of a projecting 
rim or collar of material, on which the pull is brought to bear, as in tig. 1, then 
this rim or collar must not project out of the material at a sharp angle, or in any 
way which tends to introduce a discontinuous tangential stress over the surftce 
of the cylinder. This is already recognised in practice ; test pieces, which are 
thicker at the ends than in the middle, being made in such a way that the transi¬ 
tion from the smaller to the larger diameter is gradual. 
The series containing 1/(2n -ft 1)" can also be evaluated in finite terms : 
V 
7 (2n -F If 
= " [ 
j 
C CO 
. 2ii -L Irre . 2>i + Irrb 2)i -f lirz 
sm ■-— sm ---COS -- 
2c 2c 2c 
. 2n + \izc . 2n + lirh . 2n + lirz 
sm-;- sm -;-sm 
{2n+l) 
-^\_{rz)r=adz 
0 from 2 = — c to — /> — c 
2c 
TT 
TGcS 
TT 
16c 
, {2 -f- 5 d-e) from 2 = — h — e to : = — h e 
tt'-c 
8c 
from 2 = c to z = J) — e 
(h-fe — z) from z — h — e to z = h -{■ e 
16c ' 
0 from z = h e to z = c. 
Thus we have only to calculate -pj, qj and to sum the corresponding series, the 
rest of tlie expressions for the stresses being reducible to finite terms. 
For y = 2 '3, I find the values of pj, qj to be given by : 
