Page 163 - Foundations Of Differential Calculus

146 8. On the Higher Differentiation of Differential Formulas
III. Let
2 2 3/2
dx + dy
V = .
2
2
dx d y − dy d x
Then
2 2 2 2
3dx d x +3dy d y dx + dy
dV =
2
2
dx d y − dy d x
2 2 3/2 3 3
dx + dy dx d y − dy d x
− 2 .
2
2
(dx d y − dy d x)
Since these differentials are taken most generally, with no differential taken
to be constant, from these it is easy to derive the differentials that arise
when either dx or dy is held constant.
251. Since we are assuming that none of the differentials are constant,
we can give no law according to which the second differentials and those of
higher order can be determined, nor do they have a definite meaning. Hence
the formula for the second differential and those of higher order have no
determined value, unless some differential is assumed to be constant. But
even its signification will be vague and will change depending on which of
the differentials are held constant. There are, however, some expressions
that for second differentials, although no differential is held constant, still
have a determined signification. This always remains the same, no matter
which differential we decide to hold constant. Below we will consider very
carefully the nature of formulas of this kind, and we will discuss the way
in which these may be distinguished from those others that do not include
any determined values.
252. In order that we may more easily see the kind of formulas that
contain second or higher differentials, we consider first formulas containing
only a single variable. It will then be perfectly clear that if in such a formula
2
there is a second differential of the variable x, d x, and no differential is
held constant, then it is not possible for the formula to have a fixed value.
Indeed, if we decided that the differential of x should be constant, then
2 2
d x = 0. However, if we held constant the differential of x , that is, 2xdx,
2 2
or even xdx, since the differential of xdx is xd x + dx , this expression
2 2
is equal to zero, so that d x = −dx /x. Indeed, if the differential of some
n−1 n−1
power, for example nx dx or x dx, should be constant, then its second
differential satisfies
n−1 2 n−2 2
x d x +(n − 1) x dx =0,
so that

2 (n − 1) dx 2
d x = − .
x