606 



SCIENCE 



[N. S. Vol. LI. No. 1329 



spect to three mutually perpendicular planes 

 ■which pass through the nucleus. When one 

 electron approaches one of these planes it is 

 retarded by the repulsion of the electron on 

 the other side of the plane and is thus pre- 

 vented from passing through the plane. Al- 

 though each electron remains within a given 

 octant of the spherical region about the 

 nucleus, yet the momentum of the electron is 

 transfeiTed to the electrons in adjacent cells 

 across the cell boundaries. In this model the 

 momentum travels continuously around the 

 atom in a circular path, being relayed from 

 electron to electron. Thus even though the 

 electrons do not leave their respective cells, 

 the mathematical equations for their motion 

 are very closely related to those which apply 

 to the motions of electrons in circular orbits 

 about the nucleus. Lande's calculations lead 

 to the conclusions that this type of motion is 

 less stable than one in which all eight elec- 

 trons move in a single plane orbit. This ob- 

 jection can be overcome if we assume that 

 the angular momentum of each electron is 

 h/2'^ instead of the double value which is 

 usually assumed for the electrons in the 

 second shell. In fact, this conception gives 

 grounds for believing that all electrons in 

 their most stable positions in atoms, have 

 orbits corresponding to single quanta and it 

 is only because we have assumed coplanar 

 orbits that we have been led to the conclusion 

 that the outer orbits correspond to increasing 

 numbers of quanta. 



This model of Lande's has suggested to me 

 that there should be a similar interrelation- 

 ship between the two electrons of the helium 

 atom, and also of the hydrogen molecule, and 

 of the pair of electrons constituting the chem- 

 ical bond. 



I assume that the two electrons have no 

 velocity components i)erpendieidar to the 

 plane which passes through the nucleus and 

 the two electrons. The motion is thus con- 

 fined to a single plane. The two electrons, 

 however, are assumed to rotate about the 

 nucleus in opposite directions, and in such a 

 way they are always located symmetrically 

 with respect to a line passing through the 



nucleus. Consider for example that this line 

 of symmetry is horizontal and that one 

 electron is located directly above the nucleus 

 at a unit distance, and is moving horizontally 

 to the right. Then the other electron will be 

 located at an equal distance below the nucleus 

 and will move in the same direction and with 

 the same velocity. If there were no forces 

 of repulsion between the two electrons, and 

 if we choose the proper velocities, it is clear 

 that the two electrons might move in a 

 single circular orbit about the nucleus, but in 

 opposite directions of rotation. This would 

 require, however, that the electrons should 

 pass through each other twice in each com- 

 plete revolution. When we take into account 

 the mutual repulsion of the electrons, we see 

 that their initial velocities will suffice to 

 carry them only within a certain distance of 

 each other, and they will then tend to return 

 in the general direction from which they 

 came. With properly chosen initial condi- 

 tions the electrons will return back exactly 

 on the paths in which they advanced and 

 will then pass over (towards the left) to the 

 other side of the nucleus and complete the 

 second half of an oscillation. Each electron 

 has its own orbit which never crosses the line 

 of symmetry. The orbit however does not 

 consist of a closed curve, but a curved line 

 of finite length along which the electron 

 oscillates. 



Unfortunately the equations of motion for 

 this three-body problem are difficult to handle 

 and I have only been able to determine the 

 motion by laborious numerical calculations 

 involving a series of approximations. These 

 however, can be carried to any desired degree 

 of accuracy. By four approximations I have 

 been able to calculate the path and the 

 velocities, etc., to within about one tenth 

 per cent. It is to be hoped that a general 

 solution of this special type of three-body 

 problem may be worked out, if indeed one is 

 not already known to those more familar with 

 this type of problem. 



The results of this calculation show that 

 the path of each electron is very nearly an are 

 of an eccentric circle, extending 7Y° 58' each 



