Secular Changes in Atomic "Constants"

With the modern definitions of the centimeter and second, the numerical value for the velocity of light is shown to be inversely proportional to ${\alpha }^2\beta$, where $\beta \equiv \frac{m}{M_p}$. The proportionality factor involves the two numbers adopted in the definitions, together with two pure numbers which have not actually been computed yet, but undoubtedly will be; each may contain small correction terms involving $\alpha$ and $\beta$, but apparently nothing else. Thus if $\alpha$ or $\beta$ is secularly variable, $c$ is inversely proportional only to ${\alpha }^2\beta$ in the first approximation; and if $\alpha$ and $\beta$ are both constant, $c$ is constant in all approximations. It is possible to formulate a similar kind of definition for the gram, and in that case $\hslash$ is independent of $\alpha$ and $\beta$, and $e^2$ is inversely proportional to $\alpha \beta$, in the first approximation; if $\alpha$ and $\beta$ (and perhaps some independent small corrections also) are constant, $\hslash$ and $e^2$ are constant in all approximations. If the centimeter, second, or gram is differently defined, but still in the same sort of atomic manner, the behavior of $c$, $\hslash$, and $e^2$ can be quite different, but their values will still be predictable in terms of $\alpha$ and $\beta$, and will be constant in all approximations if $\alpha$ and $\beta$ are. It is thus not permissible to make any arbitrary assumptions about them. Since the definitions are arbitrary, the behavior of $c$, $\hslash$, and $e^2$, though predictable, is secondary; the fundamental quantities are the dimensionless ones.