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 and if no other force intervenes except the binding forces, the shape of that electron, when it is given a uniform velocity, may be such that the ideal electron corresponds to a sphere, except the case where the binding is such that the volume is constant, in conformity with the hypothesis of.

We are led in this way to pose the following problem: what additional forces, other than the binding forces, are necessary to intervene to account for the law of or, more generally, any law other than that of ?

The simplest hypothesis, and the first that we should consider, is that these additional forces are derived from a special potential depending on the three axes of the ellipsoid, and therefore on θ and on r; let F(θ, r) be the potential; in which case the action will be expressed:

$$J=\int\left[H+F(\theta,r)\right]dt$$

and the equilibrium conditions are written:

If we assume r and θ are connected by r = bθm, we can look at r as a function of θ, consider F as depending only on θ, and retain only the first equation (8) with:

$$H=\frac{\varphi}{bk^{2}\theta^{m}},\quad\frac{dH}{d\theta}=\frac{-m\theta}{bk^{2}\theta^{m+1}}+\frac{\varphi^{\prime}}{bk^{3}\theta^{m}}$$

For k = θ we need equation (8) to be satisfied; which gives, taking into account equations (7):

$$\frac{dF}{d\theta}=\frac{ma}{b\theta^{m+3}}+\frac{2}{3}\frac{a}{b\theta^{m+3}}$$

where:

$$F=\frac{-a}{b\theta^{m+2}}\frac{m+\frac{2}{3}}{m+2}$$

and in the hypothesis of, where m = -1:

$$F=\frac{a}{3b\theta}.$$

Now suppose that there is no connection and, considering r and θ as independent variables, retain the two equations (H); it follows:

$$H=\frac{\varphi}{k^{2}r},\quad\frac{dH}{d\theta}=\frac{\varphi^{\prime}}{k^{3}r},\quad\frac{dH}{dr}=\frac{-\varphi}{k^{2}r^{2}},$$

Equations (8) must be satisfied for k = θ, r = bθm; which gives: