A general elementary linear system containing two inputs and two outputs is defined, and the behavior of a composite system consisting of a number of elementary systems connected in series is reviewed. In particular, the four proportionality coefficients relating the outputs of the composite system to its inputs have the same formal mathematical structure, independent of the number of elementary systems that are connected together. This composite linear system is then used to model scattering of an electromagnetic plane wave by a singly-coated sphere or a multi-layer sphere. Mirroring the behavior of a general linear system, the partial wave scattering amplitudes and their Debye series representation also have the same formal mathematical structure, independent of the number of layers of the sphere. Lastly, the interpretation of coherent multiple-scattering inside a multi-layer sphere in the frequency-domain is commented on.