The effect of a number of thick radiation shields of infinite thermal conductivity and of a type realizable in practice is analyzed by matrix algebra, and a method for calculating the temperatures of the various shields is also given. For identical thin shields it is found that the ratio of the power required to maintain a radiating surface of emissivity ε0 at a given temperature with N shields to the power required to maintain the same temperature without shields is given by [1+ε0(f+f2+f3+···fN)(1εk+1εl−1)]−1,where εk, εl are the emissivities of the two sides of a shield and f is the fraction of the radiation leaving the inside of the nth shield which is incident upon the n−1th shield.