any point of the thickness of the tube is a constant quantity,
and that the sum of these two stresses is inversely proportional
to the square of the radius of the layer under consideration. Let
r0, R, and r_x be the respective radii, p0, p¹, and p_x the
corresponding pressures, and T0, T¹, and T_x, the tensions, then
we have:

T0 - p0 = T_x - p_x (1)
(T0 + p0) r0² = (T_x + p_x) (r_x)² (2)
T_x - p_x = T¹ - p¹ (3)
(T_x + p_x)(r_x)² = (T¹ + P¹)R² (4)

if the radii are known and p and p¹ be given, then deducing from
the above equations the values T0 and T¹, and also the variable
pressure p_x, we determine--

p0 r0²(R² + (r_x)²) - p¹ R²((r_x)² + r0²)
T_x = ------------------------------------------
(R² + r0²) (r_x)²

This is the formula of Lame, from which, making p¹ = 0, we obtain
the expression in the text.]

For these reasons, and in order to increase the power of resistance of a
cylinder, it is necessary to obtain on the inner layer a state of
initial compression approaching as nearly as possible to the elastic
limit of the metal. This proposition is in reality no novelty, since it
forms the basis of the theory of hooped guns, by means of which the
useful initial stresses which should be imparted to the metal throughout
the gun can be calculated, and the extent to which the gun is thereby