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Folios 168-170: ADM to AAL

[168v] [In De Morgan's hand] When an equation involves only two
variables, it is easy enough to write all
differential equations so as to contain nothing
but differential coefficients; thus

\( y=\log x\frac{dy}{dx}=\frac{1}{x}\) 

If we prefer to write \(dy=\frac{dx}{x}\), it
must be under a new understanding.
By \(\frac{dy}{dx}\), we mean the limit of \(\frac{\Delta y}{\Delta x}\), not
any value which \(\frac{\Delta y}{\Delta x}\) ever can have, but
that which it constantly tends towards,
as \(\Delta x\) is diminished without limit.
But in \(dy=\frac{dx}{x}\), we cannot by \(dy\) 
and \(dx\) mean limits, for the limits are
zeros; and \(0=\frac{0}{x}\), though very true, is
What then do we mean by this
When \(y=\log x\), \(dy=\frac{dx}{x}\) 
we mean that \(\Delta y=\frac{\Delta x}{x}\), as \(\Delta x\) diminishes
without limit, not only diminishes without
limit, but diminishes without limit as
compared with \(\Delta x\) or \(\Delta y\) .  So that, if we
call it \(a\), or if

[168r] \(\Delta y-\frac{\Delta x}{x}=a\) 
Then \(a\) is useless, and we might as
well write \(0\) .  For since the processes of
the differential calculus always terminate
in taking limits of ratios and since

\( \frac{\Delta y}{\Delta x}-\frac{1}{x}=\frac{a}{\Delta a}\) 
(or some transformation of this sort) must
come at last, our limiting equation
must be

  \(\frac{dy}{dx}-\frac{1}{x}=\text{Limit of }~\frac{a}{\Delta x}=0\) 
The truth of every equation differentially
written, as \(dy=p\,dx\), is always
absolutely speaking, only approximate:
but the approximation is relatively
closer and closer . Understand it as
if it were

where \(\lambda\) diminishes with \(dx\), so that
the error made in \(dy\) by writing \(dy\) 
\( =p\,dx\), namely \(\lambda\,dx\), not only diminishes
with \(dx\), but becomes a smaller and
smaller fraction of \(dx\) : because
\( \lambda\) diminishes without limit
[169v] All this is conveniently signified in
the language of Leibnitz, namely, that
when \(dx\) is infinitely small, \(dy-p\,dx\) 
is as nothing (or infinitely small)
when compared with \(dx\), or \(dy\) is
(relatively to its own value) infinitely
near to \(p\,dx\) .

 The differential might easily be
avoided when there are only two variables,
and even when there are more, provided
we only want to use one independent
variable at a time . Thus

may give the equations

  \(\frac{du}{dx}=P, \frac{du}{dy}=Q, \frac{du}{dz}=R\) 
But when we want to make
\( x\), \(y\), and \(z\), all vary together,
we have no notion of a differential
coefficient attached to this simulta-
neous variation, unless we suppose some
one new variable on which \(x\), \(y\), and \(z\) 
all depend, and the variation of which
sets them all varying together.
[169r] If this new variable be \(t\), and
if \(x\), \(y\), and \(z\) be severally functions
of \(t\), we have then
[in margin: 'See chapter on Implicit differentiation']

Thus if \(u=xy^2\varepsilon^2\) 

But observe that this makes \(x\), \(y\), and \(z\),
(which we want to be independent of one
another) really functions of one another:
thus if \(x=t^2\), \(y=\log t\), we must
have \(y=\log\sqrt{x}\) .  We might it is
true avoid this by the following suppo-
sition.  Let \(x\), \(y\), and \(z\), instead of being
given functions of \(t\), be unassigned and
arbitrary functions, which we can always
make whatever functions we please.  We can
then really hold \(\frac{dx}{dt}\) \(\frac{dy}{dt}\) \(\frac{dz}{dt}\) to be indepen-
dent of one another, for it is always in our
power to assign them any values we like.
But this method would be awkward, and
would put continual impediments in our
way.  It is better therefore to avoid that
[170r] notation which while it makes the
first step by supposing relations to exist
between \(x\), \(y\), and \(z\), immediately contradict
that supposing by making these relations mean
any relations.

If in \(\varphi(x,\,y\,z)\) we suppose \(x\), \(y\), and \(z\) 
to be simultaneously altered into \(x+\Delta x\), \(y+\Delta y\),
\( z+\Delta z\), then \(\varphi(x,\,y\,z)\) takes the value

  \(\varphi(x+\Delta x,\,y+\Delta y,\,z+\Delta z)\) 
which may be expounded as follows

\( \varphi+\frac{d\varphi}{dx}\Delta x+\frac{d\varphi}{dy}\Delta y+\frac{d\varphi}{dz}\Delta z\)
\(+A\Delta x\Delta y+B\Delta y\Delta z+C\Delta z\Delta x\)
 \(+D\overline{\Delta x}^2+E\overline{\Delta y}^2+F\overline{\Delta z}^2\)
 \(+\) &c &c


\( \varphi+\frac{d\varphi}{dx}\Delta x+\frac{d\varphi}{dy}\Delta y+\frac{d\varphi}{dz}\Delta z+M\) 
If it be required that \(\varphi=\) constant, or \(\varphi=c\) 
we must have

  \(\frac{d\varphi}{dx}\Delta x+\frac{d\varphi}{dy}\Delta y+\frac{d\varphi}{dz}\Delta z+M=0\) 
Now if we were to leave out \(M\), and

  \(\frac{d\varphi}{dx}\Delta x+\frac{d\varphi}{dy}\Delta y+\frac{d\varphi}{dz}\Delta z=0\) 
we should of course commit an error:
but it is one the magnitude of which
relatively to \(\Delta x\), for instance, diminishes
[170v] without limit as the increments \(\Delta x\), \(\Delta y\),
\( \Delta z\), are diminished without limit.  The
considerations already given apply here again :
because all the terms contain [sic] in \(M\), diminish
without limit as compared with those which
are not [something crossed out] contained in \(M\) .  This rejection of all terms
When therefore I say that except those of the

 \(\varphi=c\)  first order is always
gives \(\frac{d\varphi}{dx}.dx+\frac{d\varphi}{dy}[.]dy+\frac{d\varphi}{dz}.dz=0\)   accompanied and
I should, if asked whether this equation\,marked by
is absolutely true, answer no . If then  writing \(dx\) for
asked why I write it, I should answer \(\Delta x\), \(dy\) for \(\Delta y\),
that it leads to truth , and for this &c.
reason that it is more and more nearly
true as \(dx\) &c are diminished : not because
\( \frac{d\varphi}{dx}dx+\text{&c}\) diminishes in that case, though
undoubtedly it does so; but because it
diminishes as compared with \(dx\), &c.  Hence,
when we form ratios and take their limits,
it matters nothing, as to the results we obtain,
whether we write


or \( \frac{d\varphi}{dx}dx+\text{&c}=0\) 

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