Page 182 - Foundations Of Differential Calculus

8. On the Higher Differentiation of Differential Formulas 165
fixed significance and does not depend on the variability of second differen-
tials. However, if we obtain a different expression, then the given expression
2 2
has unsettled significance. Thus, if the given expression is yd x − xd y,in
which no differential is set constant, we investigate whether the significance
is unsettled or fixed. We let dx be constant, so that the expression becomes
2
−xd y. Now, with the first rule of the previous paragraph, we substitute
2
2 dy d x
d y −
dx
2
for d y and obtain
2
2 xdy d x
−xd y + .
dx
Since these two expressions do not agree, this indicates that the given
expression has no fixed and stated significance.
278. In a similar way if the given general expression is of the type
2 2
Pd x + Qdxdy + Rd y,

it is possible to define a condition under which the expression will have a
fixed value even though no differential is assumed to be constant. When dx
2
is set constant, the expression becomes Qdxdy + Rd y. Now this is once
more transformed into another form, so that its signification remains the
same, even though no differential is thought to be constant. In this way we
obtain
2
2 Rdy d x
Qdxdy + Rd y − ,
dx
which agrees with the given expression, provided that Pdx+Rdy = 0. Only
in this case will the value of the expression be fixed. Indeed, if P is not
equal to −R dy/dx or if R is not equal to −P dx/dy, the given expression
2
2
Pd x+Qdxdy+Rd y has no fixed value. Its signification will be unsettled
and vary depending on which differential is assumed to be constant.
279. Using these principles it will be easy to convert a differential ex-
pression in which some differential is set constant into another form in
which a different differential is assumed to be constant. We reduce the first
expression to the form that involves no constant differential. Once this is
accomplished, we set the other differential constant. Thus if in the pro-
posed expression the differential dx is assumed to be constant, and this is
transformed into another that involves a constant dy, in the above formulas
2
2
3
3
instead of d y and d y, since dy is constant, we would let d y =0, d y =0,
2
but the desired result is obtained if for d y we substitute
3
2
2
−dy d x 3dy d x dy d x
and −
dx dx 2 dx