In order to represent more clearly the distribution of stresses and pressures in the metal of a homogeneous ideally perfect hollow cylinder, let us take, as an example, the barrel of a 6 in. gun--153 mm. Let us suppose T = 3,000 atmospheres; therefore, under the most favorable conditions, P0 = 1.41 T, or 4,230 atmospheres. From Equation (1) we determine R = 184.36 mm. With these data were calculated the internal stresses and the pressures from which the curve represented in Fig. 1 is constructed. The stresses developed under fire with a pressure in the bore of 4,230 atmospheres are represented by a line parallel to the axis of the abscissae, since their value is the same throughout all the layers of metal and equal to the elastic limit, 3,000 atmospheres. If, previous to firing, the metal of the tube were free from any internal stresses, then the resistance of the tube would be R squared - r squared_0 P0 = U ----------- , R squared + r squared_0 or 2,115 atmospheres--that is, one-half that in the ideally perfect cylinder. From this we perceive the great advantage of developing useful initial stresses in the metal and of regulating the conditions of manufacture accordingly. Unless due attention be paid to such precautions, and injurious stresses be permitted to develop themselves in the metal, then the resistance of the cylinder will always be less than 2,115 atmospheres; besides which, when the initial stresses exceed a certain intensity, the elastic limit will be exceeded, even without the action of external pressures, so that the bore of the gun will not be in a condition to withstand any pressure because the tensile stress due to such pressure, and which acts tangentially to the circumference, will increase the stress, already excessive, in the layers of the cylinder; and this will occur, notwithstanding the circumstance that the metal, according to the indications of test pieces taken from the bore, possessed the high elastic limit of 3,000 atmospheres. [Illustration: Fig. 1]