Page 176 - Foundations Of Differential Calculus

8. On the Higher Differentiation of Differential Formulas 159
this formula gives

2
2
2 3
2 2
3
dx d y − 3dx d xd y +3dy d x − dx dy d x
dq = ,
dx 4
so that
2 3
2
2 2
2
3
dx d y − 3dx d xd y +3dy d x − dx dy d x
r = .
dx 5
Furthermore, if the letter s, which indicates the value of dr/dx,isinthe
expression, then
3 4
3
2 2 2
2 2
2 2
3
dx d y − 6dx d xd y − 4dx d yd x +15dx d x d y
s =
dx 7
3
2
2
4
2 3
10dx dy d xd x − 15dy d x − dx dy d x
+ .
dx 7
When these values are substituted for p, q, r, s, etc., into the given expres-
sion, that expression is transformed into another one that contains higher
differentials of x and y. Even though no first differential is assumed to be
constant, still the expression has a fixed signification.
273. In this way any formula for a higher differential in which some first
differential is assumed to be constant can be transformed into another form,
in which no differential is set equal to a constant, and in spite of this it
still has a fixed value. First, by means of the method already discussed, we
take the values dy = pdx, dp = qdx, dq = rdx, dr = sdx, etc., and the
higher differentials are eliminated. Then for p, q, r, s, etc., we substitute the
values just discovered and this transformation is illustrated by the following
examples.
2
2
I. Let the given expression be xd y/dx , in which we let dx be constant.
We would like to transform this into another form that involves no
constant differential. We let dy = pdx, dp = qdx, and, as seen before
in paragraph 270, the given expression becomes qx. Now for q we
substitute the value we obtain when no differential is constant, namely,
2
2
dx d y − dy d x
q = .
dx 3
The resulting expression is then equal to
2
2
xdxd y − xdy d x ,
dx 3
and this involves no other constant differential.