Page 178 - Foundations Of Differential Calculus

8. On the Higher Differentiation of Differential Formulas 161
with no differential being constant, and the expression

2 2 2
dx + dy
2 2
2
dx d y − dx dy d x
becomes equivalent to the proposed expression.
3
2
V. Let the given expression be dx d y/d y, in which the differential dx is
assumed to be constant. We let dy = pdx, dp = qdx, and dq = rdx.
2
2
2
3
3
Since d y = qdx and d y = rdx , the given formula becomes rdx /q.
Now for q and r we substitute those values that they receive when no
differential is assumed to be constant, that is,
2
2
dx d y − dy d x
q =
dx 3
and
2 2
2 3
2
3
2
dx d y − 3dx d xd y +3dy d x − dx dy d x
r = .
dx 5
We then obtain the following expression, which is equivalent to that
originally given:
2 3 2 2 2 2 3
dx d y − 3dx d xd y +3dy d x − dx dy d x
2
2
dx d y − dy d x
3 3
dx dx d y − dy d x 2
= − 3d x.
2
2
dx d y − dy d x
274. If we consider these transformations more carefully, we can find a
more expeditious method in which it is not necessary to resort to the letters
p, q, r, etc. Depending on which differential in the formula is assumed to
be constant, different methods are used. First, suppose that the constant
differential is dx. When we have substituted pdx for dy and conversely
dy/dx for p, whenever the differentials dx or dy occur, they are retained
2
without alteration. However, wherever d y occurs, after we have substituted
2
qdx and then for q we have written the value
2
2
2
dx d y − dy d x 2 dy d x
or d y − ,
dx dx
3
the transformation is complete. Furthermore, if in the given expression d y
3
occurs, since we have substituted rdx , because of the value already found
3
for r, whenever d y is found we write
3
2 2
2
2
3 3d xd y 3dy d x dy d x
d y − + − .
dx dx 2 dx