Laplace's Equation--Bispherical Coordinates

In bispherical coordinates, Laplace's equation becomes

(1)

Attempt separation of variables by plugging in the trial solution

(2)

then divide the result by   to obtain

(3)

The function  then separates with

(4)

giving solution

(5)

Plugging  back in and dividing by  gives

(6)

The function  then separates with

(7)

giving solution

(8)

Plugging  back in and multiplying by  gives

(9)

so Laplace's equation is partially separable in bispherical coordinates. However, the Helmholtz differential equationcannot be separated in this manner.

REFERENCES:

Arfken, G. "Bispherical Coordinates ." §2.14 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 115-117, 1970.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 665-666, 1953.