Since the wave and diffusion equations can be reduced to the Helmholtz equation by separation of the time-term, while the Poisson equation can be reduced to the Laplace by change of variable, only three, (a), (b), (g), of the foregoing equations require individual treatment. If the assumption, = Ul(u,). U2(u2)... Uo(u,9 permits the reduction of the partial differential equation to a set of n ordinary differential equations, we say that the equation is simply separable. Robertson's method (1), a which he used in obtaining conditions for simple separability of the Schr6dinger equation, will be extended in this paper to the Helmholtz and Laplace equations. A more general type of separability is also possible. If the assumption, g (u2) ... u,,(,,,) = R(u,, u ... permits separation into n ordinary differential equations, and if R # const, the equation is said to be R-separable. R-separation of the Laplace equation has been employed by C. Neumann (2) (1862), Wangerin (3) (1875), Darboux (4) (1876), B6cher (8) (1891), and others.
A brief treatment is given by Morse and Feshbach (6). The present paper extends the Robertson method to obtain necessary and sufficient conditions for R-separability of the Helmholtz and Laplace equations.