223 
CIRCULAR CYLINDERS UNDER CERTAIN PRACTICAL SYSTEMS OF LOAD. 
Tr - . 
V = “ - 2 77 Sill kz 
fxd" 0 k-1 
^ cc 
r<i = — Y sill k^ 
fp 0 
4-7’ 00 U 
cf>z =^„t~ COS kz 
a- 0 a; 
Now if 
we see that 
( )„ = xjj (z) , 
r(f) = r'xjj (z) /tr, 
cf>z = (4r/a}\xfj{z)dz, 
<■=—fffefVc)*- 
/a-CJo JU ' ' 
Now if M be the total torsion moiiieiit up to any cross-section, 
M = 27roy-j i/i [z) clz , 
# - M , 
7r(k 
v/r = angle turned through by a radius = 0 say. 
Therefore 
0 = , 
TTLLCr J /. 
Therefore 
Therefore 
irfue 
torsion at the point = t. 
2 M 
irfM^ 
M = /X X .p X T and (f)Z =■ jxrr . 
dd 
(130). 
But these are the actual formula! connecting the torsion Avith the applied couple 
and with the shear across the section for a circular cylinder. 
We see, then, that the usual formulm continue to hold, to the first approximation, 
when the forces applied to the surface of the cylinder vary Avith 2 , provided aa'o define 
our torsion-couple at any section (much as the bending moment at any section of a 
beam is defined), as the couple of all the external applied forces to one side of that 
cross-section. 
It is interesting to note also that, to this approximation, there is no distortion of 
