Page 175 - Foundations Of Differential Calculus

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
2 2
dx + dy
2
d y ,
2
2

in which we let dx + dy be constant. Again we let dy = pdx,
dp = qdx, and from the preceding paragraph we have
2 qdx 2
d y = .
1+ p 2
2 2
Hence the given expression becomes 1+ p /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
2
2
dx d y − dy d x
dp = ,
dx 2
so that
2
2
dx d y − dy d x
q = ,
dx 3