of these anomalies is of course zero. The natural measure of such anomalies is the square root of their average square. Now

${\displaystyle {\overline {(\epsilon -{\overline {\epsilon }})^{2}}}={\overline {\epsilon ^{2}}}-{\overline {\epsilon }}^{2},}$

(204)

identically. Accordingly

${\displaystyle {\overline {(\epsilon -{\overline {\epsilon }})^{2}}}=\Theta ^{2}{\frac {d{\overline {\epsilon }}}{d\Theta }}=-\Theta ^{3}{\frac {d^{2}\psi }{d\Theta ^{2}}}.}$

(205)

In like manner,

${\displaystyle {\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{2}}}=\Theta ^{2}{\frac {d{\overline {\epsilon }}_{q}}{d\Theta }}=-\Theta ^{3}{\frac {d^{2}\psi _{q}}{d\Theta ^{2}}}.}$

(206)

${\displaystyle {\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{2}}}=\Theta ^{2}{\frac {d{\overline {\epsilon }}_{p}}{d\Theta }}=-\Theta ^{3}{\frac {d^{2}\psi _{p}}{d\Theta ^{2}}}={\tfrac {1}{2}}n\Theta ^{2}.}$

(207)

Hence

${\displaystyle {\overline {(\epsilon -{\overline {\epsilon }})^{2}}}={\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{2}}}+{\overline {(\epsilon _{p}-{\overline {\epsilon }}_{p})^{2}}}.}$

(208)

Equation (206) shows that the value of ${\displaystyle d{\overline {\epsilon }}_{q}/d\Theta }$ can never be negative, and that the value of ${\displaystyle d^{2}\psi _{q}/d\Theta ^{2}}$ or ${\displaystyle d{\overline {\eta }}_{q}/d\Theta }$ can never be positive.^[1] To get an idea of the order of magnitude of these quantities, we may use the average kinetic energy as a term of comparison, this quantity being independent of the arbitrary constant involved in the definition of the potential energy. Since

1. ↑ In the case discussed in the note on page 54, in which the potential energy is a quadratic function of the ${\displaystyle q}$'s, and ${\displaystyle \Delta _{\dot {q}}}$ independent of the ${\displaystyle q}$'s, we should get for the potential energy

   ${\displaystyle {\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{2}}}={\frac {1}{2}}n\Theta ^{2},}$

   and for the total energy

   ${\displaystyle {\overline {(\epsilon -{\overline {\epsilon }})^{2}}}=n\Theta ^{2}.}$

   We may also write in this case,

   ${\displaystyle {\frac {\overline {(\epsilon _{q}-{\overline {\epsilon }}_{q})^{2}}}{({\overline {\epsilon }}_{q}-\epsilon _{a})^{2}}}={\frac {2}{n}},}$

   ${\displaystyle {\frac {\overline {(\epsilon -{\overline {\epsilon }})^{2}}}{({\overline {\epsilon }}-\epsilon _{a})^{2}}}={\frac {1}{n}}.}$