Separation of Laplace’s equation

The following results are established in this paper: (I)$^{**}$ For the Laplace equation $\Delta \theta = 0$ in curvilinear co-ordinates $\left ( {u,v,w} \right )$ in Euclidean space to be directly separable†into two equations, one for $S$ and one for $Z$, when the solution is $\theta = R\left ( {u,v,w} \right )S\left ( {u,v} \right )Z\left ( w \right )$ with fixed $R$, it is necessary and sufficient that the surfaces ${w}$ = constant (1) be orthogonal to the surfaces ${u}$ = constant, ${v}$ = constant and (2) be parallel planes, planes with a common axis, concentric spheres, spheres tangent at a common point, or one of the two sets of spheres generated by the co-ordinate lines when bicircular co-ordinates are rotated about the line joining the poles or about its perpendicular bisector. (II) We have $R = 1$ always and only in the first three cases, namely, when the surfaces ${w}$ = constant are parallel planes, planes with a common axis, or concentric spheres. (III) In these three cases, but only these, the wave equation separates in the sense $RSZ$, and hence, for the wave equation, $R = 1$ automatically. (IV) For further separation of the equation found above for $S$, when $S = X\left ( u \right )Y\left ( v \right )$ so that the solution is now $RXYZ$, it is necessary and sufficient that the co-ordinates be toroidal, or such that the wave equation so separates, or any inversions of these. (V) The co-ordinates where the wave equation so separates, that is, admits solutions $RX\left ( u \right )Y\left ( v \right )Z\left ( w \right )$, are only the well-known cases where this happens with $R = 1$, namely, degenerate ellipsoidal or paraboloidal co-ordinates (but see Sec. 8.2). (VI). In these cases, but only these, $R = 1$ for the Laplace equation too. (VII) Co-ordinates for $RSZ$ or $RXYZ$ separation of the Laplace equation have the group property under inversion. (VIII) In all cases $R$ can be found by inspection of the linear element.