R r_x - Rr
t_x = T -------- . ---------- (3)
R + r0 (r_x)²

Substituting successively r_x = r0 and r_x = R, we obtain
expressions for the stresses on the external and internal radii--

R - r0 R R - r0
t_R = T -------- and t_r0 = -T ---- --------
R + r0 r0 R + r0

Therefore, in a homogeneous hollow cylinder, in which the internal
stresses are theoretically most advantageous, the layer situated next to
the bore must be in a state of compression, and the amount of
compression relative to the tension in the external layer is measured by
the inverse ratio of the radii of these layers. It is further evident
that the internal stresses will obey a definite but very simple law,
namely, there will be in the hollow cylinder a layer whose radius is
sqrt(R r0), in which the stress is _nil_; from this layer the
stresses increase toward the external and the internal radii of the
cylinder, where they attain a maximum, being in compression in the
internal layers and in tension in the external ones.

The internal pressures corresponding to these stresses may be found by
means of very simple calculations. The expression for this purpose,
reduced to its most convenient form, is as follows:

R / R \ / r0 \
p_x = T -------- ( --- - 1 ) ( 1 - ----- ) (4)