Page 87 - Foundations Of Differential Calculus

70 4. On the Nature of Differentials of Each Order
3 3
differential, we let dq = rdx, so that d y = rdx . In like manner, if the
differential of this function r is sought, it will be dr = sdx, from which we
4 4
obtain the fourth differential d y = sdx , and so forth. Provided that we
can find the first differential of any function, we can find the differential of
any order.

133. In order that we may keep the form of these differentials, and the
method of discovery, in mind we present the following table: If y is any
function of x,

then and we let
dy = p dx, dp = q dx,
2
2
d y = qdx , dq = r dx,
3
3
d y = rdx , dr = s dx,
4
4
d y = sdx , ds = t dx,
5
5
d y = tdx , ....
Since the function p is known from y by differentiation, similarly we find
q from p, then r from q, then s, and so forth. We can find differentials of
any order, provided only that the differential dx remains constant.
134. Since p, q, r, s, t,... are finite quantities, in particular, functions
of x, the first differential of y has a finite ratio to the first differential of
x, that is, as p to 1. For this reason, the differentials dx and dy are said
2 2
to be homogeneous. Then, since d y has the finite ratio to dx as q to 1,
2 2 3 3
it follows that d y and dx are homogeneous. Similarly, d y and dx as
4 4
well as d y and dx are homogeneous, and so forth. Hence, just as first
differentials are mutually homogeneous, that is, they have a finite ratio, so
second differentials with the squares of first differentials, third differentials
with cubes of first differentials, and so forth, are homogeneous. In general,
n
the differential of y of the nth order, expressed as d y, is homogeneous with
n
dx , that is, with the nth power of dx.
135. Since in comparison with dx all of its powers greater than 1 vanish,
2
3
4
so also in comparison with dy all of the powers dx , dx , dx ,... vanish,
as well as the differentials of higher orders that have finite ratios with
2
3
2
4
these, that is, d y, d y, d y,... . In a similar way, in comparison with d y,
2
since this is homogeneous with dx , all powers of dx that are greater than
3
4
3
the second, dx , dx ,... , will vanish. Along with these will vanish d y,
4
5
5
3
4
4
d y,... . Furthermore, compared to d y,wehave dx , d y, dx , d y,... all
vanishing. Hence, given expressions involving differentials of this kind, it
is easy to decide whether or not they are homogeneous. We have only to
consider the differentials, since the finite parts do not disturb homogeneity.