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 is only 3,262 atmospheres, or 72 per cent. of the resistance of a homogeneous cylinder. By increasing the number of layers or rows of hoops shrunk on, while the total thickness of metal and the caliber of the gun remains the same, we also increase the number of layers participating equally in the total resistance to the pressure in the bore, and taking up strains which are not only equal throughout, but are also the greatest possible. We see an endeavor to realize this idea in the systems advocated by Longridge, Schultz, and others, either by enveloping the inner tubes in numerous coils of wire, or, as in the later imitations of this system, by constructing guns with a greater number of thin hoops shrunk on in the customary manner. But in wire guns, as well as in those with a larger number of hoops--from four to six rows and more--the increase in strength anticipated is acknowledged to be obtained in spite of a departure from one of the fundamental principles of the theory of hooping, since in the majority of guns of this type the initial compression of the metal at the surface of the bore exceeds its elastic limit.[3] We have these examples of departure