| T0 - p0 = Tx - px | (1) |
| (T0 + p0)r02 = (Tx + px)rx2 | (2) |
| Tx - px = T1 - p1 | (3) |
| (Tx + px)rx2 = (T1 + P1)R2 | (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 px, we determine—
| Tx = | p0 r02(R2 + rx2) - p1 R2(rx2+r02) —————————————— (R2 + r02)rx2 |
This is the formula of Lame, from which, making p1=0, we obtain the expression in the text.
We must, however, remark that in a built-up hollow cylinder the compression of the metal at the surface of the bore may exceed the elastic limit. This cannot occur in the case of natural stresses.
In certain cases this, of course, may be an advantage, as, for instance, when the inner tube is under injurious initial stresses; but then, in order to be able to apply the necessary shrinkage, we must know the magnitude of these stresses.
When the inner tube is strengthened by means of wire, the initial or natural stresses in the latter may be neglected on account of its thinness; but when the thickness of the hoops is reduced, and the number of layers thereby increased, then the value of the initial stresses in these hoops is a very important factor with respect to the decrease or increase Of the powers of resistance of the gun.