IV. On the equation of Laplace’s functions, &c

Abstract

The equation d ² u / dx ² + d ² u / dy ² + d ² u / dz ² = 0, transformed to polar coordinates by putting x = r sin θ cos ϕ , y = r sin θ sin ϕ , z = r cos θ , becomes, as is well known, d ² u / dθ ² + cot θ du / dθ + 1/(sin θ ) ² d ² u / dϕ ² + r ² d ² u / dr ² + 2 r du / dr = 0, which may be written in the form {(sin θ d / dθ ) ² + ( d / dϕ ) ² + (sin θ ) ² r d / dr ( r d/dr + 1)} u = 0; . . . . (1.) and if it be assumed that u = u ₀ + u ₁ r + u ² r ² + . . . . ., then by substituting this value of u in either of the two equations last written, that u _n is a function of θ and ϕ satisfying the equation {(sin θ d / dθ ) ² + ( d / dϕ ) ² + n ( n + 1)(sin θ ) ² } u _n =0, . . . . . (2.) which, under a slightly different form, is commonly called the Equation of Laplace’s Functions .