1. It is known that any polynomial in μ. can be expanded as a linear function of Legendre polynomials . In particular, we have
The earlier coefficients, say A0, A2, A4 may easily be found by equating the coefficients of μp+q, μp+q-2, μp+q-4 on the two sides of (1). The general coefficient A2k might then be surmised, and the value verified by induction. This may have been the method followed by Ferrers, who stated the result as an exercise in his Spherical Harmonics (1877). A proof was published by J. C. Adams . The proof now to be given follows different lines from his.
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