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]

In order to understand more thoroughly the difference of the law of distribution of useful internal stresses as applied to homogeneous or to built-up cylinders, let us imagine the latter having the external and internal radii of the same length as in the first case, but as being composed of two layers—­that is to say, made up of a tube with one hoop shrunk on under the most favorable conditions—­when the internal radius of the hoop = sqrt(R v0) or 118.7 mm., Fig. 2, has been traced, after calculating, by means of the usual well known formulae, the amount of pressure exerted by the hoop on the tube, as well as the stresses and pressures inside the tube and the hoop, before and after firing.  A comparison of these curves with those on Fig. 1 will show the difference between the internal stresses in a homogeneous and in a built-up cylinder.  In the case of the hooped gun, the stresses in the layers before firing, both in the tube and in the hoop, diminish in intensity from the inside of the bore outward; but this decrease is comparatively small.  In the first place, the layer in which the stresses are = 0 when the gun is in a state of rest does not exist.  Secondly, under the pressure produced by the discharge, all the layers do not acquire simultaneously a strain equal to the elastic limit.  Only two of them, situated on the internal radii of the tube and hoop, reach such a stress; whence it follows that a cylinder so constructed possesses less resistance than one which is homogeneous and at the same time endowed with ideally perfect useful initial stresses.  The work done by the forces acting on a homogeneous cylinder is represented by the area a b c d, and in a built-up cylinder by the two areas a’ b’ c’ d’ and a” b” c” d".  Calculation shows also that the resistance of the built-up cylinder