The self-similar solution of the gasdynamic equations of a strong cylindrical shock wave moving through an ideal gas, with γ = cp/cv, is considered. These equations are greatly simplified following the transformation of the reduced velocity \[ U_1(\xi)\rightarrow U_1 = {\textstyle\frac{1}{2}}(\gamma + 1)(U+\xi). \] The requirement of a single maximum pressure, dζP = 0, leads to an analytical determination of the self-similarity exponent α(γ). For gases with γ < 2 + 3½ the slight maximum pressure occurs behind the shock front, nearing it as γ increases. For γ < 2 + 3½, this maximum ensues right at the shock front and the pressure distribution then decreases monotonically. The postulate of analyticity by Gelfand and Butler is shown to concur with the requirement dζP = 0. The saturated density of the gas left in the wake of the shock is computed and − U is shown to be the reduced velocity of sound at P = Pm.