of
, which is unchanged by differentiation? Accordingly; let us assume as a general expression that
.,
(in which the coefficients
,
,
, etc. will have to be determined), and differentiate it.
.
Now, if this new expression is really to be the same as that from which it was derived, it is clear that
must
; that
; that
; that
, etc.
The law of change is therefore that
.
If, now, we take
for the sake of further simplicity, we have
.
Differentiating it any number of times will give always the same series over again.
If, now, we take the particular case of
, and evaluate the series, we shall get simply