Page 90 - Foundations Of Differential Calculus

4. On the Nature of Differentials of Each Order 73

dp = dx rdx, so that if we place the symbol before both sides, we

have p = dx rdx. Finally, we have dy = dx dx rdx and so y =

dx dx rdx.
142. Since the differential dy is an infinitely small quantity, its integral y
2
is a finite quantity. In like manner the second differential d y is infinitely
less that its integral dy. It should be clear that a differential will vanish in
the presence of its integral. In order that this relation be better understood,
the infinitely small can be categorized by orders. First differentials are said
to be infinitely small of the first order; the infinitely small of the second
order consist of differentials of the second order, which are homogeneous
2
with dx . Similarly, the infinitely small that are homogeneous with dx 3
are said to be of the third order, and these include all differentials of the
third order, and so forth. Hence, just as the infinitely small of the first
order vanish in the presence of finite quantities, so the infinitely small of
the second order vanishes in the presence of the infinitely small of the first
order. In general, the infinitely small of any higher order vanishes in the
presence of an infinitely small of a lower order.

143. Once the orders of the infinitely small have been established, so that
the differential of a finite quantity is infinitely small of the first order, and
so forth, conversely, the integral of an infinitely small of the first order is a
finite quantity. The integral of an infinitely small of the second order is an
infinitely small of the first order, and so forth. Hence if a given differential is
infinitely small of order n, then its integral will be infinitely small of order
n − 1. Thus, just as differentiating increases the order of the infinitely
small, so integrating lowers the order until we come to a finite quantity. If
we wished to integrate again finite quantities, then according to this law
we obtain quantities infinitely large. From the integration of these we get
quantities infinitely greater still. Proceeding in this way we obtain orders
of infinity such that each one is infinitely greater than its predecessor.

144. It remains to give something of a warning about the use of symbols
in this chapter, lest there still be any ambiguity. First of all, the symbol
for differentiation, d, operates on only the letter that comes immediately
after it. Thus, dx y does not mean the differential of the product xy, but
rather the product of y and the differential of x. In order to minimize the
confusion we ordinarily would write this with the y preceding the symbol
d,as ydx, by which we indicate the product of y and dx.If y happens to be
a quantity preceded by a symbol indicating either a root √ or a logarithm,
√
2
then we usually place that after the differential. For instance, dx a − x 2
√
2
2
signifies the product of the finite quantity a − x and the differential dx.
In like manner, dx ln (1 + x) is the product of the logarithm of the quantity
2 √
1+ x and dx. For the same reason d y x expresses the product of the
2 √
second differential d y and the finite quantity x.