Separation of Laplace’s equation

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Abstract
The following results are established in this paper: (I) ∗ ∗ ^{**} For the Laplace equation Δ θ = 0 \Delta \theta = 0 in curvilinear co-ordinates ( u , v , w ) \left ( {u,v,w} \right ) in Euclidean space to be directly separable†into two equations, one for S S and one for Z Z , when the solution is θ = R ( u , v , w ) S ( u , v ) Z ( w ) \theta = R\left ( {u,v,w} \right )S\left ( {u,v} \right )Z\left ( w \right ) with fixed R R , it is necessary and sufficient that the surfaces w {w} = constant (1) be orthogonal to the surfaces u {u} = constant, v {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 R = 1 always and only in the first three cases, namely, when the surfaces w {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 R S Z RSZ , and hence, for the wave equation, R = 1 R = 1 automatically. (IV) For further separation of the equation found above for S S , when S = X ( u ) Y ( v ) S = X\left ( u \right )Y\left ( v \right ) so that the solution is now R X Y Z 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 R X ( u ) Y ( v ) Z ( w ) RX\left ( u \right )Y\left ( v \right )Z\left ( w \right ) , are only the well-known cases where this happens with R = 1 R = 1 , namely, degenerate ellipsoidal or paraboloidal co-ordinates (but see Sec. 8.2). (VI). In these cases, but only these, R = 1 R = 1 for the Laplace equation too. (VII) Co-ordinates for R S Z RSZ or R X Y Z RXYZ separation of the Laplace equation have the group property under inversion. (VIII) In all cases R R can be found by inspection of the linear element.

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