III. On the expression of the product of any two legendre’s coefficients by means of a series of Legendre’s coefficients

The expression for the product of two Legendre’s coefficients which is the subject of the present paper, was found by induction on the 13th of February, 1873, and on the following day I succeeded in proving that the observed law of formation of this product held good generally. Having considerably simplified this proof, I now venture to offer it to the Royal Society; and, for the sake of completeness, I have prefixed to it the whole of the inductive process by which the theorem was originally arrived at, although for the proof itself only the first two steps of this process are required. The theorem seems to deserve attention, both on account of its elegance, and because it appears to be capable of useful applications. As usual let Legendre’s wth. coefficient be denoted by Pn then Pn may be defined by the equation Pn = 1/2n∟n . dn/dμn(μ2 - 1)n It is well known that the following relation holds good between three consecutive values of the functions P, viz.