Page 183 - Foundations Of Differential Calculus
P. 183
166 8. On the Higher Differentiation of Differential Formulas
3
for d y. In this way the formula
2 2 3/2
− dx + dy
,
2
dx d y
in which dx is set constant, is transformed into
2 2 3/2
dx + dy
,
2
dy d x
in which dy is set constant.
280. If, on the other hand, a formula in which dy is set constant is to be
2
transformed into another in which dx is constant, then for d x we have to
substitute
2
−dx d y
,
dy
3
and for d x the expression
2 2
3
3dx d y − dx d y .
dy 2 dy
2 2
In a similar way, if a formula in which dx + dy is set constant is to be
2
transformed into another in which dx is constant, then for d x we write
2
−dx dy d y ,
2
dx + dy 2
2
and for d y we write
2 2
dx d y .
2
dx + dy 2
However, if a formula in which dx is assumed to be constant is to be
2
2
transformed into another in which dx + dy is to be constant, since
2
2
2
2
dx + dy is constant, we have dx d x + dy d y = 0 and
2
2 dy d y
d x = − .
dx
2
2
This value is given to d x, and for d y we write
2 2 2 2 2
2 dy d y dx + dy d y
d y + = .
dx 2 dx 2
Hence the formula
2 2 3/2
− dx + dy ,
2
dx d y
2 2
in which dx is constant, is transformed into another in which dx + dy
is set constant, which is
2
−dx dx + dy 2 .
2
d y