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Formulae for the longitudinal shielding effectiveness of N thin, closely spaced, concentric cylinders of high permeability material have been developed and experimentally tested. For shields which cannot be oriented, or which change their orientation in the ambient field, the shielding effectiveness for longitudinal fields is generally the limiting criterion and no design formulae have previously been published for more than two shields. A simple diagrammatical method of writing the shielding formula is presented. Use of these equations is demonstrated by application to the design of magnetic shields for hydrogen maser atomic clocks. Examples of design tradeoffs such as size, weight, and material thickness are discussed. Experimental data on three sets of shields fabricated by three manufacturers are presented.