Page:Elementary Principles in Statistical Mechanics (1902).djvu/109

${\overline {(\epsilon +D)(u-{\overline {u}})}}=0,$

(256)

where $D$ represents the operator $\Theta ^{2}\,d/d\Theta$.
Hence

${\overline {(\epsilon +D)^{h}(u-{\overline {u}})}}=0,$

(257)

where $h$ is any positive whole number. It will be observed, that since $\epsilon$ is not function of $\Theta$, $(\epsilon +D)^{h}$ may be expanded by the binomial theorem. Or, we may write

${\overline {(\epsilon +D)u}}=({\overline {\epsilon }}+D){\overline {u}},$

(258)

whence

${\overline {(\epsilon +D)^{h}u}}=({\overline {\epsilon }}+D)^{h}{\overline {u}}.$

(259)

But the operator $({\overline {\epsilon }}+D)^{h}$, although in some respects more simple than the operator without the average sign on the $\epsilon$, cannot be expanded by the binomial theorem, since ${\overline {\epsilon }}$ is a function of $\Theta$ with the external coördinates.

So from equation (254) we have

${\overline {{\bigg (}{\frac {A}{\Theta }}+{\frac {d}{da}}{\bigg )}(u-{\overline {u}})}}=0,$

(260)

whence

${\overline {{\bigg (}{\frac {A}{\Theta }}+{\frac {d}{da}}{\bigg )}^{h}(u-{\overline {u}})}}=0;$

(261)

and

${\overline {{\bigg (}{\frac {A}{\Theta }}+{\frac {d}{da}}{\bigg )}u}}={\bigg (}{\frac {\overline {A}}{\Theta }}+{\frac {d}{da}}{\bigg )}{\overline {u}},$

(262)

whence

${\overline {{\bigg (}{\frac {A}{\Theta }}+{\frac {d}{da}}{\bigg )}^{h}u}}={\bigg (}{\frac {\overline {A}}{\Theta }}+{\frac {d}{da}}{\bigg )}^{h}{\overline {u}}.$

(263)

The binomial theorem cannot be applied to these operators.

Again, if we now distinguish, as usual, the several external coördinates by suffixes, we may apply successively to the expression $u-{\overline {u}}$ any or all of the operators

$\epsilon +\Theta ^{2}{\frac {d}{d\Theta }},\quad A_{1}+\Theta {\frac {d}{da_{1}}},\quad A_{2}+\Theta {\frac {d}{da_{2}}},\quad \mathrm {etc.}$

(264)