Page 170 - Foundations Of Differential Calculus

8. On the Higher Differentiation of Differential Formulas 153
method of dealing with them be explained. Soon, now, we will show the
method by means of which second and higher differentials can always be
eradicated.
264. If an expression contains a single variable x and its higher differ-
2
4
3
entials d x, d x, d x, etc. occur in the expression, then it can have no
fixed meaning unless some first differential is set constant. Thus, let t be
3
2
that variable whose differential dt is set constant. Then d t =0, d t =0,
4
d t = 0, etc. We let dx = pdt, and p will be a finite quantity whose differ-
ential is not affected by the unsettled signification of second differentials;
furthermore, dp/dt will be a finite quantity. Let dp = qdt, and in a similar
way dq = rdt, dr = sdt, etc. Here q, r, s, etc. are finite quantities with
fixed signification. Since dx = pdt,wehave
2 2
d x = dp dt = qdt ,
3 2 3
d x = dq dt = rdt ,
4 3 4
d x = dr dt = sdt ,
....
2 3 4
If these values are substituted for d x, d x, d x, and so forth, the whole
expression will contain only finite expressions and the first differential of
dt, nor will there be any unsettled signification.

265. If x were a function of t, then in this way the quantity x could be
completely eliminated, so that only the quantity t and its differential dt
would remain in the expression. However, if t were a function of x, then
x would also be a function of t. Nevertheless, this quantity x with its first
differential dx can be retained in the calculation, provided that after the
substitutions previously made for t and dt, the values expressed by x and
n
dx are restored. In order that this might become clearer, we will let t = x ,
n
so that the first differential of x will be held constant. Since dt = nx n−1 dx,
n−1
we have p =1/ nx and
− (n − 1) dx n−1
dp = = qdt = nqx dx,
nx n
so that
− (n − 1)
q =
2 2n−1
n x
and
(n − 1) (2n − 1) dx n−1
dq = = rdt = nrx dx.
2 2n
n x
From this it follows that
(n − 1) (2n − 1)
r =
3 3n−1
n x