Page 177 - Foundations Of Differential Calculus

160 8. On the Higher Differentiation of Differential Formulas
II. Let the given expression be

2 2
dx + dy ,
2
d x
in which dy is assumed to be constant. We let dy = pdx and dp = qdx,
2
so that the expression becomes −p 1+ p /q, as in paragraph 270.
Since
2
2
dy dx d y − dy d x
p = and q = ,
dx dx 3
we obtain the expression
2 2
dy dx + dy
2
2
dy d x − dx d y .
Here no differential is assumed constant, and this expression has the
same value as the one originally proposed.
III. Let the given expression be

2
2
yd x − xd y
,
dx dy
in which the differential ydx is assumed to be constant. We let dy =
pdx, dp = qdx, and as we saw in paragraph 270, this expression is
transformed into
xq xp
−1 − + .
p y
When we do not assume any differential to be constant, the expression
is transformed into
2 2
xdxd y − xdy d x xdy
− 1 − +
2
dx dy ydx
2
2
2
2
xdxdy − ydx dy − yx dx d y + yx dy d x
= .
2
ydx dy
IV. Let the given expression be
2
dx + dy2 ,
2
d y
2

in which we assume that the differential dx + dy 2 is constant.
When we let dy = pdx and dp = qdx, there arises the expression
2 2
1+ p /q as in paragraph 270. Now we set p = dy/dx and
2
2
dx d y − dy d x
q =
dx 3