[110r]

 Ockham

 Monday.  6^th July

 Dear M^r De Morgan.  Since dispatching my letter
yesterday, I remember that I have not even
quite fully & correctly stated the whole points
of difference ['between' inserted] \(\int\sqrt{a^2-x^2}x^{n-2}dx\) and \(\int\sqrt{v}d2u\) .  I
think I stated that \(\int\sqrt{a^2-x^2}x^{n-2}dx=\int\sqrt{v}d2u.\frac{-1}{x}\),
that in other words the 1^st side differs from
\( \int\sqrt{v}d2u\) in containing a factor \(\left(-\frac{1}{x}\right)\) .  But
it differs also in containing \(dx\) as well,
which in writing yesterday I omitted I believe
to notice.  So that \(\int\sqrt{a^2-x^2}x^{n-2}dx=\int\sqrt{v}d2u.\frac{(-1)}{x}.dx\) 
or the 1^st side differs from \(\int\sqrt{v}d2u\) in
containing \(-\frac{1}{x}.dx\) .  Is not this what I ought
to have stated?  Or is there still any confusion?

I also wish to observe upon
what I wrote on Friday as to the application
of the Differential & Integral Calculus to \(\frac{gt^2}{2}\),
[110v] that I am aware this formula ['\((e=\frac{gt^2}{2}\) ' inserted] can be
derived from \(V=gt\), by the simple Theory of
algebraical proportion; but that I was anxious
to know how it is derived in the other way.

I will with your leave ['(which I do not wait for)' inserted], send you
my paper making it out on the doctrine of
Proportions.
You must tell me if I presume too much
on your kindness to me.  I am so
engaged at present with my mathematical
& scientific plans & pursuits that I can
think of little else; & perhaps may be a
plague & bore to my friends about [something crossed out] these
subjects; for after my interruption from
Paris & London pursuits & occupations, my
whole heart is with my renewed studies; &
every minutia even is a matter of the greatest
interest.

 Believe me

 Yours most truly

 A. A. Lovelace

[111r] [something crossed out] You ['will receive' inserted] two papers on \(e=\frac{gt^2}{2}\) tomorrow
evening, or Wed^dy . One of them is to show the
absurdity of the supposition that the spaces might
be as the velocities; ['& that' inserted] on merely abstract grounds
it could not be.