IV. On some definite integrals, and a new method of reducing a function of spherical co-ordinates to a series of spherical harmonics

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Abstract
The following investigation deals with some definite integrals which are useful when it is desired to express a function of two angular variables by means of a series of spherical surface harmonics. An important theorem concerning these integrals leads to a method which considerably reduces the arithmetical labour involved in the reductions, and secures in practice the advantage of obtaining the numerical values of the coefficients of lower degrees independently of those of higher degrees. The zonal harmonic of degree n is denoted by P n and defined as usual by P n = 1/2 n 1·2 . . . n dμ n ( μ 2 - 1) n .