Page 169 - Foundations Of Differential Calculus

152 8. On the Higher Differentiation of Differential Formulas
has a fixed value although it seems to be contaminated by second differ-
entials. In addition, since the numerator is homogeneous with the denomi-
nator, it has a finite value, unless by chance it becomes infinitely large or
infinitely small.
2
2
261. Just as the formula dx d y − dy d x has a fixed value, as has been
2
2
shown, so also if a third variable z is added, these formulas dx d z − dz d x
2
2
and dy d z − dz d y have fixed values. Hence, expressions in three variables
x, y, and z, provided that there are no second differentials except these, then
the expression will be fixed, just as if they contained no second differential
at all. It follows that this expression
2 2 2 3/2
dx + dy + dz
,
2
2
2
(dx + dz) d y − (dy + dz) d x +(dx − dy) d z
although it does contain second differentials, still it keeps a fixed signifi-
cation. In a similar way it is possible to exhibit formulas containing many
variables in which second differentials do not prevent the formulas from
having a fixed significance.
262. Except for formulas of this kind, all others that contain second dif-
ferentials will give uncertain signification, and for this reason they have no
place in calculations. On the other hand, a first differential may be defined
to be constant. As soon as any first differential is assumed to be constant,
all expressions, no matter how many variables they may contain and no
matter what differentials higher than the first may be present, obtain fixed
significance, and they are no longer excluded from calculations. For exam-
ple, if dx is assumed to be constant, the second differential of x and all
higher differentials vanish. Whatsoever functions of x may be substituted
for the other variables y, z, and so forth, their second differentials through
2 3
dx , their third through dx , and so forth, will be determined. In this way
the ambiguity is removed from the second differentials. The same thing
is true if the first differential of some other variable or function is made
constant.
263. From this it follows that second and higher differentials never enter
into a calculation, and because of their unsettled signification they are
completely unsuitable for analysis. Now, when second differentials seem to
be present, either some first differential is assumed to be constant, or this is
not the case. In the first case, the second differentials simply vanish, since
they are determined by the first differential. In the latter case, unless they
cancel each other, the signification is unsettled, and for this reason they
have no place in analysis. On the other hand, if they cancel each other,
they only seem to be present, and only finite quantities with their first
differentials are to be considered really present. However, since they very
frequently only seem to be used in calculations, it was necessary that the