of the radius. This extension is thus represented--

(r0)² (r_x)² + R²
(t_x)¹ = P0 ------------ . ------------
R² - (r0)² (r_x)²

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)² + R² \
t_x = T ( 1 - ---------- · ------------- ) (2)
\ R0 + r0 (r_x)² /

The greatest value t_x = t0 corresponds to the surface of the bore
and must be t0 = -T, therefore

r0² + R²
--------------- = 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--