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

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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 .

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