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

(256)

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

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

(257)

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

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

(258)

whence

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

(259)

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

So from equation (254) we have

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

(260)

whence

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

(261)

and

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

(262)

whence

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

(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 ${\displaystyle u-{\overline {u}}}$ any or all of the operators

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

(264)