## Folios 144-145: AAL to ADM

Ashley-Combe

Porlock

Somerset

Sunday Mor^{g} . 28^{th }Aug^{st
} ['1842' added by later reader]

Dear M^{r} De Morgan. I am going on well; ['quite' inserted] as I

could wish. I have done much since I saw you;

& you will have a__ll__ the results of the last few days

in good time. I enclose you now two papers; one

on \(f=\frac{dv}{dt}\), the other on \(\int_a^{a'}f.dt\) .

You will have next those on \(v\frac{dv}{dt}=f\), and

\( v^2=2\int f.ds+C\) . This latter I think I have

succeeded in analysing to my mind.

I have ['now' inserted] two observations to make : [something crossed out]

1^{stly}: I think I have detected a slight error in one

of my former papers, that on \(t=\int\frac{ds}{v}\) . I return

it for reference. In order in the [something crossed out] Summation

[something crossed out]}

\( \left\{\frac{1}{\varphi s}+\frac{1}{\varphi(s+ds)}+\cdot\cdots\frac{1}{\varphi(2s)}\right\}ds\), to __end__ with \(\frac{1}{\varphi(2s)}\),

I should have b__egun__ with \(\frac{1}{\varphi(s+ds)}\) not with \(\frac{1}{\varphi s}\) .

If the time elapsed during the f__irst fraction__ of Space**[144v] ** (starting from \(s\) ) were ['made' inserted] \(=\frac{1}{\varphi s}\), then the time for the __last__

of the Fractions necessary to complete up to \(2s\), would

be \(\frac{1}{\varphi(2s-ds}\), and not \(\frac{1}{\varphi(2s)}\) which it __ought to be__.

I don't know that this affects the correctness of the

ultimate l__imit__ o__f the Summation__. But here, where

the Summation itself is made to represent a

hypothetical movement, it is clearly __wrong__.

The error is avoided in the former paper I had

written on \(s=\int v.dt\), which I likewise return to

refer to this Point.

2^{ndly} : In considering a priori the Integral \(\int f.ds\),

I am inclined still to adhere to my original

opinion (expressed in the pencil Memorandum I showed

you & ['which I' inserted] now return). I should premise that I now

mention this merely as a c__urious subject of investigation__,

not because it is concerned in the [something crossed out] papers I

am making out upon \(v^2=2\int f.ds+C\), in which I

have avoided the d__irect__ consideration of \(\int_a^{a'}f.ds\) .

I am disposed to contend that tho' \(ds\)

does here represent S__pace__, that still the __\( ds\) fraction__

o__f any one of the terms of the Summation__, say \(\varphi(a+n.ds)ds\)

means t__he same fraction of \(\varphi(a+n.ds)\) __ which __\( ds\) is of__**[145r] ** __a Unit of Space__; & therefore that since \(\varphi(a+n.ds)\)

represents F__orce__, (or ['__uniform__' inserted] __Acceleration of Velocity for \(1\) Second__

in operation during the performance of the length \(ds\) ),

the __\( ds\) fraction of this expression__ must represent the

['__\( ds\) part of this Force__ or the' inserted] __actual__ Acceleration for \(\frac{1}{ds}\) of a Second. I treat \(ds\) as

an __abstract quantity__. And so I conceive [something crossed out] \(dt\) must

be treated in \(s=\int v.dt\), ['\( ds\) ' inserted] in \(t=\int\frac{ds}{v}\), \(dt\) in \(\int f.dt\),

&c, &c.

I should tell you that I am much pleased with

the observation you added to __my inverse__ demonstration

of \(\int fx.\frac{dx}{dt}dt=\int fx.dx\), and that I q__uite____understand__ ['why' inserted] my proof can only be admissible on

the Infinitesimal Leibnitzian Theory. But t__his__

theory is to my mind the __only intelligible__ or

satisfactory one. In fact, (notwithstanding it's [\textit{sic}] __error__),

I should call it the only __true__ one.

By and bye, you will have some observations

of mine upon Differential Co-efficients & Integrals,

abstractly considered. I have been thinking much

upon them.

I am going on with Chapter VIII.

By the bye, I believe you will receive somehow tomorrow**[145v] ** a book (the 1^{st} Vol of Lamé's Cours de Physique)

in which there is a passage which I will write

to you about as soon as I find time.

I forgot to mention it to you on Thursday; &

so have ordered the Book to be sent to you, that

I might write about it sometime.

Believe me

Yours very truly

A. A. Lovelace

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