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Analytic formulas for spherical nanoshell (Au/silica) resonance wavelength and refractive index sensitivity were derived and compared with a numerical nonlinear theory. A universal scaling law was deduced in terms of a normalized thickness defined by the ratio of the shell thickness and its core diameter. The calculated figure of merit shows a maximum at an optimal value of the resonance wavelength and normalized thickness. The nonlinear theory of nanoshells improves the accuracy of the linear theory in the short wavelength regime (500 to 650 nm).