Page 166 - Foundations Of Differential Calculus

8. On the Higher Differentiation of Differential Formulas 149
1
becomes equal to − . Since we have this contradiction in a single case,
2
much less is it possible in general that
2
xd y
,
dx dy
when dx is constant, is equal to
2
yd x
,
dx dy
when dy is constant. Since the formula
2 2
yd x + xd y
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
2 3
substituted, the second and higher differentials of x, namely d x, d x, etc.,
no longer remain in the calculation. Indeed, if after any such substitution
2 3 4
whatsoever in the formula there remains d x or d x or d x, 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,
2
2
yd x + xd y
,
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-
2 2
tion or x. But if we let y = x, the formula becomes 2xd x/dx , which is
2
undetermined due to the presence of d x, 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.
257. From this there arises a doubt as to the existence of any formulas
that contain two or more second, or higher, differentials that still have the
property that when arbitrary functions of one of the variables are substi-
tuted for the other variables, then the second differentials are eliminated.
We propose this doubt in order to present a formula that has this pre-
cise property. By this investigation we will more easily see the force of the
question. I say that the following formula has this remarkable property:
2
2
dy d x + dx d y .
dx 3
Indeed, no matter what function of x we substitute for y, the second dif-
ferentials always vanish completely.