This equation was used in paper IV. to prove the equivalence of the formulæ (2) and (6) for any system governed by ordinary mechanics. The equation (7) further shows that if the relations (2) and (6) hold for a system of orbits, they will hold also for any small variation of these orbits for which the value of
is unaltered. If a hydrogen atom in one of its stationary states is placed in an external electric field and the electron rotates in a closed orbit, we shall therefore expect that
is not altered by the introduction of the atom in the field, and that the only variation of the total energy of the system will be due to the variation of the mean value of the potential energy relative to the external field.
In the former paper it was pointed out that the orbit of the electron will be deformed by the external field. This deformation will in course of time be considerable even if the external electric force is very small compared with the force of attraction between the particles. The orbit of the electron may at any moment be considered as an ellipse with the nucleus in the focus, and the length of the major axis will approximately remain constant, but the effect of the field will consist in a gradual variation of the direction of the major axis as well as the excentricity of the orbit. A detailed investigation of the very complicated motion of the electron was not attempted, but it was simply pointed out that the problem allows of two stationary orbits of the electron, and that these may be taken as representing two possible stationary states. In these orbits the excentricity is equal to 1, and the major axis parallel to the external force; the orbits simply consisting of a straight line through the nucleus parallel to the axis of the field, one on each side of it. It can very simply be shown that the mean value of the potential energy relative to the field for these rectilinear orbits is equal to
, where
is the external electric force and