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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
there is a second differential of the variable x, d2x, 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
d2x = 0. However, if we held constant the differential of x2, that is, 2xdx,
or even xdx, since the differential of xdx is xd2x + dx2, this expression
is equal to zero, so that d2x = -dx2/x. Indeed, if the differential of some
power, for example nxn-1dx or xn-1dx, should be constant, then its second
differential satisfies
xn-1d2x +(n - 1) xn-2dx2 =0,
so that
(n - 1) dx2
d2x = - .
x
8. On the Higher Differentiation of Differential Formulas 147
Different values for d2x will be given if the differentials of other functions
of x are held constant. However, it is clear that the formulas in which d2x
occurs take on quite different values depending on whether in place of d2x
we write zero or -dx2/x or - (n - 1) dx2/x or some other expression of this
kind. For instance, if the given formula is x2d2x/dx2, then, because d2x and
dx2 are both infinitely small and homogeneous, the expression should have
a finite value. If dx is made constant, the expression becomes zero; if d.x2
is constant, it becomes -x; if d.x3 is constant, it becomes -2x; if d.x4 is
constant, it becomes -3x, and so forth. Hence, it can have no determined
value unless the differential of something is assumed to be constant.
253. This ambiguity of signification is present, for a similar reason, if the
third differential is present in some formula. Let us consider the formula
x3d3x
,
dx d2x
which also has a finite value. If the differential dx is constant, then the
formula takes the form 0/0, whose value we will soon see. Let d.x2 be
constant. Then d2x = -dx2/x and after another differentiation we obtain
2dx d2x dx3 3dx3
d3x = - + = ,
x x2 x2
since d2x = -dx2/x. Hence, for this reason, the given formula
x3d3x
dx d2x
becomes -3x2. However, if d.xn is constant, then
- (n - 1) dx2
d2x = ,
x
so that
2(n - 1) dx d2x (n - 1) dx3 2(n - 1)2 dx3 (n - 1) dx3
d3x = - + = + ,
x x2 x2 x2
(2n - 1) (n - 1) dx3
= .
x2
Hence for this reason we have
d3x (2n - 1) dx
=
d2x x
and
x3d3x
= - (2n - 1) x2.
dx d2x
148 8. On the Higher Differentiation of Differential Formulas
It follows that if n =1, or dx is constant, the value of the formula will be
equal to -x2. From this it is clear that if in any formula there occurs a
third or higher differential and at the same time it is not indicated which
of these differentials are taken to be constant, then that formula has no
certain value and can have no further significance. For this reason such
expressions cannot occur in the calculation.
254. In a similar way, if the formula contains two or more variables and
there occur differentials of the second or higher order, it should be under-
stood that it can have no determined value unless some differential is con-
stant, with the exception of some special cases that we will soon consider.
Since as soon as d2x is in some formula, due to the various differentials
that can be constant, the value of d2x always changes. The result is that it
is impossible that the formula should have a stated value. The same is true
for any higher differential of x and also for the second and higher differen-
tials of the other variables. However, if a formula contains the differentials
of two or more variables, it can happen that the variability arising from
one is destroyed by the variability of the others. It is for this reason that
we have that exceptional case that we mentioned, in which a formula of
this kind, involving second differentials of two or more variables, can have
a definite value, even though no differential is held constant.
255. The formula
yd2x + xd2y
dx dy
can have no fixed and stated signification unless one of the first differentials
is set constant. If dx is made constant, then we have
xd2y
.
dx dy
On the other hand, if dy is made constant, we have
yd2x
.
dx dy
It should be clear that these formulas need not be equal. If they were
necessarily equal, they would remain the same when any function of x is
substituted for y. Let us suppose that y = x2. When we set dx constant we
have d2y =2dx2, and the formula
xd2y
dx dy
becomes equal to 1. However, if dy, that is 2xdx, is set constant, then
d2y =2xd2x +2dx2 = 0, so that d2x = -dx2/x, and the formula
yd2x
dx dy
8. On the Higher Differentiation of Differential Formulas 149
becomes equal to -1 . Since we have this contradiction in a single case,
2
much less is it possible in general that
xd2y
,
dx dy
when dx is constant, is equal to
yd2x
,
dx dy
when dy is constant. Since the formula
yd2x + xd2y
dx dy
has no fixed meaning even though either dx or dy is constant, much less
will there be a fixed meaning if the differential of an arbitrary function of
either x of y or both is set equal to a constant.
256. Thus it appears that a formula of this kind cannot have a stated
value unless it is so made up that when for y or z or any function of x is
substituted, the second and higher differentials of x, namely d2x, d3x, etc.,
no longer remain in the calculation. Indeed, if after any such substitution
whatsoever in the formula there remains d2x or d3x or d4x, etc., the value
of this formula remains unsettled. This is because as different constants are
assigned, the differentials take on different meanings. The formula we have
just discussed,
yd2x + xd2y
,
dx dy
is of this kind. If this formula had a fixed value, no matter what y should
signify, the stated value should remain the same if y represents any func-
tion or x. But if we let y = x, the formula becomes 2xd2x/dx2, which is
undetermined due to the presence of d2x, so that it takes on various values
according to the various differentials that are made constant. This should
be sufficiently clear from the discussion in paragraph 252.
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