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In the limit of very thin shells, and using the scaling properties of the operator M , one can show [25, 27] that all four M operators are related to M = M$_{11}$ 

Equations (16) can then be arranged in a matrix form as 

By knowing the eigenvectors and the eigenvalues of M the charge densities µ$_{1}$ and µ$_{2}$ can be found by inverting the matrix in (18). For example, µ$_{1}$ is 

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where ˜ λ$_{k}$ = (˜ ε$_{k}$ − ε$_{0}$ ) / (˜ ε$_{k}$ + ε$_{0}$ ) and the equivalent permittivity ˜ ε$_{k}$ is defined for each eigenmode as: 

where δ = η $^{3}$− 1 ≪ 1. The distribution (20) is similar with the distribution (12) obtained for a homogeneous particle, except that λ has to be replaced for each mode with an equivalent quantity ˜ λ$_{k}$ . Equation (21) can be applied recursively for a multi-shelled structure. The strict separation of electric and geometric properties is weakened in this case, because the shape-dependent eigenvalue χ$_{k}$ appears now in the electric equivalent quantity ˜ λ$_{k}$ . 

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The polarizability of the shelled particle is obtained by using the distribution (20) to build the solution (4) of the Laplace equation and use it in equation (2), to get 

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