Propagation of spherical shock waves through self-gravitating polytropic gas spheres such as stars, caused by an instantaneous central explosion of finite energy E, is discussed theoretically. The problem is characterized by two lengths R0, L, where $R_0 = \left(\frac {E}{4\pi p_0}\right)^{1|3},\;\;\;\;\; L = \left(\frac {3C^2_0}{2\pi \rho_0G} \right)^{1|2}$p0 and C0 are the values of pressure, density and velocity of sound at the centre of the equilibrium pre-explosion state, and G is the constant of gravitation. R0 and L are scales connected with the power of the explosion and the dimensions of the star respectively, and their ratio A = R0/L has a fundamental significance. A solution especially suitable in the case of A = O(1) is developed in the form of power series in R/R0 (R is the distance between the shock front and the centre) by a method similar to that used in previous papers by the present author (1953, 1954). An approximation to this solution is carried out up to the term in R3. In particular, the velocity of the shock wave U is found to be $\frac {U}{C_0} = 1\cdot30 \left(\frac{R}{R_0}\right)^{-3|2} \{1 +0\cdot41A^2\left(\frac{R}{R_0}\right)^2 + 0\cdot 57\left(\frac{R}{R_0}\right)^3 +\ldot\}$ for the case of λ = 1.4, where λ is the ratio of specific heats.