17 



301 



and to higher powers of P\ For the system under consideration a separation of 

 variables can be obtained if parabolic coordinates are used to describe the 

 position of the electron in space'). If x, y, z, are the coordinates of the electron in 

 a system of rectangular Cartesian coordinates with the origin at the nucleus and 

 with the z-axis parallel to the direction of the external electric force, these para- 

 bolic coordinate's may be defined by 



z = -~ 0- + ly = l/i^e'>. (43) 



Ç and 7j are two parameters defining the two paraboloids of revolution which have 

 their common focus at the nucleus and their common axis parallel to the z-axis 

 and which pass through the electron, while (p is the angular distance between Ihe 

 arz-plane and the plane containing the z-axis and the electron. Denoting in the 

 usual way the differential coefficients , .... by x, . . . ., the kinetic energy 



of the system will be given by 



so that the momenta, wiiich are canonically conjugated to the coordinates ç, tj and 

 (p, are given by 



ÔT mç + Wi ÔT m$-\-r;. BT ^ . 



Denoting the distance Vx'^ -\-y'^ ^ Z' of the electron from the nucleus by r, the 

 potential energy of the system will be represented by 



P = + eFz = - + e(ç-;7)F, 



so that the total energy E, expressed as a function of p^, p^, p^^, c, rj, ç, which 

 enters in the Hamillonian equations of motions (2) of the system, will be given by 



The Hamilton-Jacobi partial differential equation will be obtained by introducing 



dS dS ÔS . , ..• ^, • r iu .1 



Pi ^ p p = and bv putlmg the expression for the energv thus 



Ç ' C7j ^ 0(p ~ 



obtained equal to a constant a^: 



Effecting in this equation a separation of variables we find 



') P. Epstein, Ann. d. Phys. L., p. 489 (1916). 



1). K. D. VIdensk. Selsk. Skr., naturvldensk. og niathem. Afd.. 8. Hække. III. 3. 39 



