126
A PERMANENT DISTRIBUTION IN WHICH
|
(407)
|
Differentiating (405), we get
|
(408)
|
where
denotes the least value of
consistent with the external coördinates. The last term in this equation represents the part of
which is due to the variation of the lower limit of the integral. It is evident that the expression in the brackets will vanish at the upper limit. At the lower limit, at which
, and
has the least value consistent with the external coördinates, the average sign on
is superfluous, as there is but one value of
which is represented by
. Exceptions may indeed occur for particular values of the external coördinates, at which
receive a finite increment, and the formula becomes illusory. Such particular values we may for the moment leave out of account. The last term of (408) is therefore equal to the first term of the second member of (407). (We may observe that both vanish when
on account of the factor
.)
We have therefore from these equations
|
|
or
|
(409)
|
That is: the average value in the ensemble of the quantity represented by the principal parenthesis is zero. This must