## Folios 110-111: AAL to ADM

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 h__ow__ 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 i__nterruption__ 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 s__paces might____be as the velocities__; ['& that' inserted] on merely abstract grounds

it could not be.

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