Therefore, to obtain the maximum resistance in the cylinder, the value t_x of the initial stress will be determined by the difference (T - t'_x),[*need to check the prime with library or work out the equations] and since P0 is given by Equation (1), then / r0 (r_x) squared + R squared \ t_x = T ( 1 - ---------- . ------------- ) (2) \ R0 + r0 (r_x) squared / The greatest value t_x = t0 corresponds to the surface of the bore and must be t0 = -T, therefore r0 squared + R squared --------------- = 2 r0 (R + r0) whence P0 = T sqrt(2) = 1.41 T. From the whole of the preceding, it follows that in a homogeneous cylinder under fire we can only attain simultaneous expansion of all the layers when certain relations between the radii obtain, and on the assumption that the maximum pressure admissible in the bore does not exceed 1.41 U. Equation (2) may be written thus-- R r_x - Rr t_x = T -------- . ---------- (3) R + r0 (r_x) squared Substituting successively r_x = r0 and r_x = R, we obtain expressions for the stresses on the external and internal radii-- R - r0 R R - r0 t_R = T -------- and t_r0 = -T ---- -------- R + r0 r0 R + r0 Therefore, in a homogeneous hollow cylinder, in which the internal stresses are theoretically most advantageous, the layer situated next to the bore must be in a state of compression, and the amount of compression relative to the tension in the external layer is measured by the inverse ratio of the radii of these layers. It is further evident that the internal stresses will obey a definite but very simple law, namely, there will be in the hollow cylinder a layer whose radius is sqrt(R r0), in which the stress is _nil_; from this layer the stresses increase toward the external and the internal radii of the cylinder, where they attain a maximum, being in compression in the internal layers and in tension in the external ones. The internal pressures corresponding to these stresses may be found by means of very simple calculations. The expression for this purpose, reduced to its most convenient form, is as follows: R / R \ / r0 \ p_x = T -------- ( --- - 1 ) ( 1 - ----- ) (4) R + r0 \ r_x / \ r_x /