Page 168 - Foundations Of Differential Calculus

8. On the Higher Differentiation of Differential Formulas 151
has a fixed value, even though no differential is assumed to be constant, all
the more easily we can furnish a demonstration. Let y be any function or x,
and then dy is its differential, so that dy = pdx, where p is some function
of x. The differential of p will have the form dp = qdx and q is a function
2 2 2
of x. Since dy = pdx, by differentiation we have d y = pd x + qdx and
2 2 2 2 3 3
dy d x − dx d y = pdxd x − pdxd x − qdx = −qdx .

In this expression, since there is no second differential, it has a fixed value
and
2
2
dy d x − dx d y
dx 3 = −q.
No matter how y depends on x, the second differentials in this formula
vanish. For this reason its value is quite fixed, although in other respects
it may be unsettled.

259. Although we have here supposed that y is a function of x, never-
theless the truth of the assertion remains true even if y does not depend
on x at all. While we substitute for y an arbitrary function, and whatever
kind it might be we do not determine, we attribute to y no dependence
on x. Meanwhile, with no mention of a function, a demonstration can be
given. No matter what quantity y might be, whether it depends on x or
not, its differential dy will be homogeneous with dx, so that dy/dx will be
some finite quantity p. The differential of p that we take when x goes to
x + dx and y to y + dy will be fixed, and have no dependence on the law
of second differentials. Hence, since dy/dx = p,wehave dy = pdx and
2 2
d y = pd x + dp dx, so that
2 2 2
dx d y − dy d x = dp dx ,
and this value is not unsettled, since it contains only first differentials.
This property is consistent whether any differential is taken as constant,
whatsoever it might be, or even if no differential is held constant.
2
2
260. The formula dy d x − dx d y has a fixed signification even though it
contains second differentials, which can be thought of as destroying each
other. Any expression in which there are no other second differentials be-
2
2
sides the formula dy d x − dx d y likewise has a fixed meaning. Now, if we
2
2
let dy d x − dx d y = ω and if V is a quantity formed from x, y, their first
differentials dx, dy, and ω, then V will have a fixed value. Since in the
first differentials dx and dy there is no reason for uncertainty as to the law
by which the successive values change as x increases, and in ω the second
differentials cancel each other, so that the quantity V is not uncertain but
fixed. Thus the expression
2 2 3/2
dx + dy
2
2
dx d y − dy d x