Page 165 - Foundations Of Differential Calculus

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
2
equal to −x . 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.
2
Since as soon as d x is in some formula, due to the various differentials
2
that can be constant, the value of d x 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
2
2
yd x + xd y
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
2
xd y
.
dx dy
On the other hand, if dy is made constant, we have
2
yd x
.
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
2
substituted for y. Let us suppose that y = x . When we set dx constant we
2
2
have d y =2dx , and the formula
2
xd y
dx dy
becomes equal to 1. However, if dy, that is 2xdx, is set constant, then
2 2 2 2 2
d y =2xd x +2dx = 0, so that d x = −dx /x, and the formula
2
yd x
dx dy