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1+p2 /q. However, if we should wish, as before, to let dy = pdx,
dp = qdx, since dy is constant, we have 0 = pd2x + dp dx
and
d2x = -qdx2/p. Hence the given expression becomes -p 1+p2 /q.
III. Let the given expression be
yd2x - xd2y
,
dx dy
in which ydx is set constant. We let dy = pdx and dp = qdx; from
paragraph 268 we have d2x = -pdx2/y and
p2dx2
d2y = qdx2 - .
y
158 8. On the Higher Differentiation of Differential Formulas
When these are substituted into the given expression, it is transformed
into
xq xp
-1 - + .
p y
IV. Let the given expression be
dx2 + dy2
,
d2y
in which we let dx2 + dy2 be constant. Again we let dy = pdx,
dp = qdx, and from the preceding paragraph we have
qdx2
d2y = .
1+p2
2
Hence the given expression becomes 1+p2 /q.
From these examples it should be sufficiently clear, in any given case, the
way in which second and higher differentials should be eliminated when
any first differential is assumed to be constant.
271. Since second and higher differentials can be eliminated by introduc-
ing finite quantities p, q, r, s, etc., so that the whole expression is made
up only of the differential dx and the finite quantities p, q, r, s, etc., if
an expression reduced in this manner is given, we can again recover the
original form by substituting second and higher differentials for the letters
p, q, r, s, etc. Now in the same way, some first differential is assumed to
be constant, whether it be the one originally so assumed, or some other.
However, it could be that no differential was assumed to be constant while
it contains second and higher differentials and at the same time it has a
fixed signification. We have seen expressions of this kind above.
272. Now let any given expression contain the finite letters x, y, p, q, r,
etc. with one differential dx, in which
dy dp dq
p = , q = , r = , . . . .
dx dx dx
If we wish to eliminate these letters, in their place we introduce second and
higher differentials of x and y with no differential assumed to be constant.
Since
dx d2y - dy d2x
dp = ,
dx2
so that
dx d2y - dy d2x
q = ,
dx3
8. On the Higher Differentiation of Differential Formulas 159
this formula gives
dx2d3y - 3dx d2xd2y +3dy d2x2 - dx dy d3x
dq = ,
dx4
so that
dx2d3y - 3dx d2xd2y +3dy d2x2 - dx dy d3x
r = .
dx5
Furthermore, if the letter s, which indicates the value of dr/dx, is in the
expression, then
dx3d4y - 6dx2d2xd3y - 4dx2d2yd3x +15dx d2x2d2y
s =
dx7
10dx dy d2xd3x - 15dy d2x3 - dx2dy d4x
+ .
dx7
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.
I. Let the given expression be xd2y/dx2, 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,
dx d2y - dy d2x
q = .
dx3
The resulting expression is then equal to
xdxd2y - xdyd2x
,
dx3
and this involves no other constant differential.
160 8. On the Higher Differentiation of Differential Formulas
II. Let the given expression be
dx2 + dy2
,
d2x
in which dy is assumed to be constant. We let dy = pdx and dp = qdx,
so that the expression becomes -p 1+p2 /q, as in paragraph 270.
Since
dy dx d2y - dy d2x
p = and q = ,
dx dx3
we obtain the expression
dy dx2 + dy2
.
dy d2x - dx d2y
Here no differential is assumed constant, and this expression has the
same value as the one originally proposed.
III. Let the given expression be
yd2x - xd2y
,
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
xdxd2y - xdyd2x xdy
- 1 - +
dx2dy ydx
xdxdy2 - ydx2dy - yx dx d2y + yx dy d2x
= .
ydx2dy
IV. Let the given expression be
dx2 + dy2
,
d2y
in which we assume that the differential dx2 + dy2 is constant.
When we let dy = pdx and dp = qdx, there arises the expression
2
1+p2 /q as in paragraph 270. Now we set p = dy/dx and
dx d2y - dy d2x
q =
dx3
8. On the Higher Differentiation of Differential Formulas 161
with no differential being constant, and the expression
2
dx2 + dy2
dx2d2y - dx dy d2x
becomes equivalent to the proposed expression.
V. Let the given expression be dx d3y/d2y, in which the differential dx is
assumed to be constant. We let dy = pdx, dp = qdx, and dq = rdx.
Since d2y = qdx2 and d3y = rdx3, the given formula becomes rdx2/q.
Now for q and r we substitute those values that they receive when no
differential is assumed to be constant, that is,
dx d2y - dy d2x
q =
dx3
and
dx2d3y - 3dx d2xd2y +3dy d2x2 - dx dy d3x
r = .
dx5
We then obtain the following expression, which is equivalent to that
originally given:
dx2d3y - 3dx d2xd2y +3dy d2x2 - dx dy d3x
dx d2y - dy d2x
dx dx d3y - dy d3x
= - 3d2x.
dx d2y - dy d2x
274. If we consider these transformations more carefully, we can find a
more expeditious method in which it is not necessary to resort to the letters
p, q, r, etc. Depending on which differential in the formula is assumed to
be constant, different methods are used. First, suppose that the constant
differential is dx. When we have substituted pdx for dy and conversely
dy/dx for p, whenever the differentials dx or dy occur, they are retained
without alteration. However, wherever d2y occurs, after we have substituted
qdx2 and then for q we have written the value
dx d2y - dy d2x dy d2x
or d2y - ,
dx dx
the transformation is complete. Furthermore, if in the given expression d3y
occurs, since we have substituted rdx3, because of the value already found
for r, whenever d3y is found we write
3d2xd2y 3dy d2x2 dy d3x
d3y - + - .
dx dx2 dx
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
3/2
dx2 + dy2
dx d2y
and dx is set constant, when
dy d2x
d2y -
dx
is substituted for d2y, the new form with no constant differential is
3/2
dx2 + dy2
.
dx d2y - dy d2x
275. From this it is easily gathered that whenever in some expression the
differential dy is constant, then wherever we find d2x we should write
dx d2y
d2x -
dy
and for d3x we write
3d2xd2y 3dx d2y2 dx d3y
d3x - + - ,
dy dy2 dy
in order to obtain an equivalent expression in which no differential is set
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