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

Rh where $$D$$ represents the operator $$\Theta^2 \, d/d\Theta$$. Hence 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 whence 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 whence  and  whence  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