[89r]

 [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 sign; & 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