Page 164 - Foundations Of Differential Calculus

8. On the Higher Differentiation of Differential Formulas 147
2
Different values for d x will be given if the differentials of other functions
2
of x are held constant. However, it is clear that the formulas in which d x
2
occurs take on quite different values depending on whether in place of d x
2 2
we write zero or −dx /x or − (n − 1) dx /x or some other expression of this
2 2 2 2
kind. For instance, if the given formula is x d x/dx , then, because d x and
2
dx are both infinitely small and homogeneous, the expression should have
2
a finite value. If dx is made constant, the expression becomes zero; if d.x
3 4
is constant, it becomes −x;if d.x is constant, it becomes −2x;if d.x 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
3 3
x d x
2
dx d x ,
which also has a finite value. If the differential dx is constant, then the
2
formula takes the form 0/0, whose value we will soon see. Let d.x be
2 2
constant. Then d x = −dx /x and after another differentiation we obtain
2
3 2dx d x dx 3 3dx 3
d x = − + = ,
x x 2 x 2
2 2
since d x = −dx /x. Hence, for this reason, the given formula
3 3
x d x
2
dx d x
2
n
becomes −3x . However, if d.x is constant, then
2 − (n − 1) dx 2
d x = ,
x
so that
2
2
3 2(n − 1) dx d x (n − 1) dx 3 2(n − 1) dx 3 (n − 1) dx 3
d x = − + 2 = 2 + 2 ,
x x x x
(2n − 1) (n − 1) dx 3
= .
x 2

Hence for this reason we have
3
d x = (2n − 1) dx
2
d x x
and
3 3
x d x 2
2
dx d x = − (2n − 1) x .