T = P0 ------- · -------------
R²-r0² (r_x)²

From which we see that T is a maximum when r_x = r0, i.e., for
the layer immediately next to the bore of the cylinder. Calling t0
the internal stress in this layer, and T0 the stress resulting from
the action inside the bore of the pressure P0, and allowing that the
sum of both these quantities must not exceed the elastic limit U of the
material, we have--T0 = U - t0. And for this value of T0, the
corresponding pressure inside the bore will be

R² - r0²
P = (U - t0) ----------.
R² + r0²

This pressure increases with the term (U - t0). With t0 positive,
i.e., when the internal stresses in the thickness of the hollow
cylinder are such that the metal of the layers nearest to the bore is
in a state of tension and that of the outer layers in a state of
compression, then the cylinder will have the least strength when t0
has the greatest numerical value. Such stresses are termed injurious
or detrimental stresses. With t0 negative, the strength of the
cylinder increases with the numerical value of t0, and those stresses
which cause compression in the layers nearest to the bore of the
cylinder and tension in the outer layers are termed beneficial or
useful stresses.

[Footnote 1: Lame holds that in a homogeneous tube subjected to
the action of two pressures, external and internal, the
difference between the tension and the compression developed at