## 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

unmeaning

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

\(dy=(p+\lambda)dx\)

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

\(u=\varphi(x,\,y,\,z)\)

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']

\(\frac{d(u)}{dt}=\frac{du}{dx}\cdot\frac{dx}{dt}+\frac{du}{dy}[\cdot]\frac{dy}{dt}+\frac{du}{dz}\cdot\frac{dz}{dt}\)

Thus if \(u=xy^2\varepsilon^2\)

\(\frac{d(u)}{dt}=y^2\varepsilon^2\frac{dx}{dt}+2xy\varepsilon^2\frac{dy}{dt}+xy^2\varepsilon^2\frac{dz}{dt}\)

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

say

\( \varphi+\frac{d\varphi}{dx}\Delta x+\frac{d\varphi}{dy}\Delta y+\frac{d\varphi}{dz}\Delta z+M\)

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

say

\(\frac{d\varphi}{dx}\Delta x+\frac{d\varphi}{dy}\Delta y+\frac{d\varphi}{dz}\Delta z=0\)

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

\(\frac{d\varphi}{dx}dx+\text{&c}=-M\)

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

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