Elliptic co-ordinates

The parabolic co-ordinates used in the separation method to determine the motion of an electron in the hydrogen atom under the influence of an electric field are a special case of elliptic co-ordinates. The latter are the appropriate separation variables for the more general problem of the motion of a particle attracted to two fixed centres of force by forces obeying Coulomb s law. If one centre of force be displaced to an infinite distance, with an appropriate simultaneous increase in the intensity of its field, we get the case of the Stark effect at the same time the elliptic eo-ordinates become parabolic. 

If the distance apart of the fixed points Fx and Fa is 2c, the elliptic co-ordinates of a point f, ij, distant rx and ra from the fixed points, are given by the equations... 

PhilUpson 82) showed that it is important which co-ordinate representation is used to describe the electronic wave function. He investigated the force constant calculation of diatomic molecules, using confocal elliptic co-ordinates 86, 87) t, rjt, pi) where pi is the azimuthal angle of the i th electron around the intemuclear axis, R and... 

Goodisman (89) used the Hartree-Fock method to calculate the variation of the force on the nuclei in H2 withR, where the variation wave function is of the James-Coolidge t5q>e in confocal elliptic co-ordinates. The normal Hartree-Fock method (without configuration interaction)... 

It thus represents all surfaces with constant mean curvature, and, consequently, all the shapes of equilibrium which would be rqtpropriate to liquid without gravity but it could not, up to now, be integrated except in certain cases, and, among the infinitely many surfaces which satisfy it, there is only one score, to my knowledge, which one has either the equation in final co-ordinates, or the determination by elliptic functions, or by geometrical generation. 

Figure 4. Discretised Wheel and Elliptical Contact Co-ordinate Systems...

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