Page 85 - Foundations Of Differential Calculus

68 4. On the Nature of Differentials of Each Order
undergoes no increment or decrement, it has no differences. Strictly speak-
ing, only variable quantities have differentials, but we say that constant
quantities have differentials of all orders equal to 0, and hence all powers
2
of dx vanish. Since the differential of dx, that is d x, is equal to 0, the
differential dx can be thought of as a constant quantity; as long as the
differential of any quantity is constant, then that quantity is understood
to be taking on equal increments. Here we are taking x to be the quantity
whose differential is constant, and thus we estimate the variability of all
the functions on which the differentials depend.

127. We let the first differential of y be pdx. In order to find the second
differential we have to find the differential of pdx. Since dx is a constant
and does not change, even though we write x + dx for x, we need only find
the differential of the first quantity p. Now let dp = qdx, since we have seen
that the differential of every function of x can be put into this form. From
what we have shown for finite differences, we see that the differential of np
is equal to nq dx, where n is a constant quantity. We substitute dx for the
2
constant n, so that the differential of pdx is equal to qdx . For this reason,
2 2
if dy = pdx and dp = qdx, then the second differential d y = qdx , and
so it is clear, as we indicated before, that the second differential of y has a
2
finite ratio to dx .
128. In the first chapter we noticed that the second and higher differences
cannot be determined unless the successive values of x are assumed to follow
some rule; since this rule is arbitrary, we have decided that the best and
easiest rule is that of an arithmetic progression. For the same reason we
cannot state anything certain about second differentials unless the first
differentials, by which the variable x is thought to increase constantly,
follow the stated rule. Hence we suppose that the first differentials of x,
I
II
namely, dx, dx , dx ,... , are all equal to each other, so that the second
differentials are given by
2 I 2 I II I
d x = dx − dx =0, d x = dx − dx =0, ....
Since the second differentials, and those of higher order, depend on the
order by which the differentials of x are mutually related, and this order
is arbitrary, first differentials are not affected by this, and this is the huge
difference between the method for finding first differentials and those of
higher order.
I
II
III
IV
129. If the successive values of x, namely, x, x , x , x , x ,... ,do
not form an arithmetic progression, but follow some other rule, then their
II
I
first differentials, namely, dx, dx , dx ,... , will not be equal to each other,
2
and so we do not have d x = 0. For this reason the second differentials
are functions of x with a different form. If the first differential of such a
function y is equal to pdx, to find the second differential it is not enough to
multiply the differential of p by dx, but we must also take the differential