## Folios 89-90: AAL to ADM

[Signature written sideways at the top of this page --- belongs at end of letter so transcribed there]

Ockham

Wed^{dy} 3^{d} Feb^{y}

Dear M^{r} De Morgan. I

have a question to put

respecting a __condition__ in

the establishment of the

conclusion

\( \frac{\varphi(a+h)}{\psi(a+h)}=\frac{\varphi^{(n+1)}(a+\theta h)}{\psi^{(n+1)}(a+\theta h)}\) in

page 69 of the Differential

Calculus. I have written

down, & enclose, my notions

on the steps of the reasoning

used to establish that**[89v] ** conclusion. So that you

may judge if I take in

the objects & methods of it.

The point I do not

understand, is why the

distinction is made, (&

evidently considered so

important a one), of ‘’\( \psi x\)

''being a function which has

''the property of __always\ __

''__increasing or always decreasing__,

''__from \(x=a\) to \(x=a+h\) __,

''in other respects fulfilling the

''conditions of continuity in

''the same manner as \(\varphi x\) ''.**[90r] ** For this, see page 68, lines

9, 10, 11, 12 from the top;

page 68, line 12 from the

bottom;

page 69, lines 7, 8 from the

bottom; &c

I see perfectly that this

condition must exist, & that

without it we could not

secure the denominators

(alluded to in page 68, line

13 from the bottom), being

all of one sign.

But what I do __not__

understand, is [something crossed out] why the

condition is not made**[90v] ** for \(\varphi\ x\) also. It appears

to me to be equally requisite

for this latter; because if

we do not suppose it,

how can we secure the

numerators \(\varphi(x+k\Delta x)-\)

\( -\varphi(x+\overline{k-1}\Delta x)\) being __all____of one sig__n; & unless they__are__ all of one sign, we

cannot be sure that they

will [something crossed out] when added,

so destroy one another as to

give us \(\varphi(a+h)-\varphi a\);

an expression essential to

obtain. I think I have

explained my difficulty, &

[something missing here?]

[the following written vertically on 89r]

believe me

Yours most truly

A. A. Lovelace

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