[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 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) r0 squared = (T_x + p_x) (r_x) squared     (2)
T_x — p_x = T¹ — p¹                (3)
(T_x + p_x)(r_x) squared = (T¹ + P¹)R squared            (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 r0 squared(R squared + (r_x) squared) — p¹ R squared((r_x) squared + r0 squared)
T_x = ------------------------------------------
(R squared + r0 squared) (r_x) squared

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 strengthened determined.  The stresses which arise in a hollow cylinder when it is formed of several layers forced on one upon another, with a definite amount of shrinkage, we call the stress of built-up cylinders, in order to distinguish them from natural stresses developed in homogeneous masses, and which vary in character according to the conditions of treatment which the metal has undergone.  If we conceive a hollow cylinder made up of a great number of very thin layers—­for instance, of wire wound on with a definite tension—­in which case the inner layer would represent the bore of the gun, then the distribution of the internal stresses and their magnitude would very nearly approach the ideally perfect useful stresses which should exist in a homogeneous cylinder; but in hollow cylinders built up of two, three, and four layers of great thickness, there would be a considerable deviation from the conditions which should be aimed at.