Using the notation of Byerly (1959, pp. 252253), Laplace's equation can be reduced to
(1)

where
(2)
 
(3)
 
(4)
 
(5)
 
(6)
 
(7)

In terms of , , and ,
(8)
 
(9)
 
(10)

Equation (◇) is not separable using a function of the form
(11)

but it is if we let
(12)
 
(13)
 
(14)

These give
(15)
 
(16)

and all others terms vanish. Therefore (◇) can be broken up into the equations
(17)
 
(18)
 
(19)

For future convenience, now write
(20)
 
(21)

then
(22)
 
(23)
 
(24)

Now replace , , and to obtain
(25)

Each of these is a Lamé's differential equation, whose solution is called an ellipsoidal harmonic of the first kind. Writing
(26)
 
(27)
 
(28)

gives the solution to (◇) as a product of ellipsoidal harmonics of the first kind .
(29)
