Page 179 - Foundations Of Differential Calculus

162 8. On the Higher Differentiation of Differential Formulas
When this is done, the given expression is transformed into a different one
that involves no constant differential. For example, if the given expression
is
2 2 3/2
dx + dy
2
dx d y
and dx is set constant, when
2
2 dy d x
d y −
dx
2
is substituted for d y, the new form with no constant differential is
2 2 3/2
dx + dy
.
2
2
dx d y − dy d x
275. From this it is easily gathered that whenever in some expression the
2
differential dy is constant, then wherever we find d x we should write
2
2 dx d y
d x −
dy
3
and for d x we write
2
3
2 2
2
3 3d xd y 3dx d y dx d y
d x − + 2 − ,
dy dy dy
in order to obtain an equivalent expression in which no differential is set
constant. However, if in the given expression ydx is assumed constant, then
according to paragraph 268, we have
2
2 pdx 2 2 2 p dx 2
d x = − and d y = qdx − .
y y
2
2
In place of d x we should everywhere write −dx dy/y and in place of d y
we should everywhere write
2 2
2 dy d x dy
d y − − .
dx y
Since the higher differentials seldom occur in this business, we will progress
2

no further. However, if in the given expression the differential dx + dy 2
is assumed constant, since in paragraph 269 we obtained
2 pq dx 2 2 qdx 2
d x = − 2 and d y = 2 ,
1+ p 1+ p
2
for d x we should everywhere write
2 2
2
dy d x − dx dy d y ,
2
dx + dy 2