## Folios 112-114: AAL to ADM

Ockham Park

Sunday. 11^{th} July ['1841' added by later reader]

Dear M^{r} De Morgan. I enclose you a paper

(marked No 1) from which I think you will see that

I now quite understand the real relationship

between \(\int\frac{x^ndx}{\sqrt{a^2-x^2}}\) and \(\int\sqrt{a^2-x^2}x^{n-2}dx\); & that

I [something crossed out] am now aware I wanted to apply to the l__atter__ what is

not intended to be directly applied to it at all;

& that ['my' inserted] __getting both \(du\) and \(dx\) __ in, was a

complete puzzle & blunder. For where a few lines

previously \((-\overline{n-1})\int\sqrt{a^2-x^2}x^{n-2}dx\) is substituted

for \(\int\sqrt{a^2-x^2}\times(-\overline{n-1})x^{n-2}dx\), \(du\) ceases of

course to enter under the __Integrated quantity__, [something crossed out] since

it has been decomposed & otherwise distributed.

I am still occupied on pages 108, 109, 110,

& hope to complete to page 112 during this week. I

find this part requires studying with great care.

I think you anticipated this.

I must now thank you very much for your two

letters; & will proceed to notice one or two points**[112v] ** in your replies to my enquiries.

I see that in objecting to what I called the__division__ of \(\frac{dV}{dx}\), when \(dV\) is substituted for

\( \frac{dV}{dx}dx\), I took a completely wrong view of the

matter. It does so happen that the expression

(derived from a separate & distinct Theorem) which

we may substitute for \(\frac{dV}{dx}dx\) c__oincides__ in

f__orm__ with what we may call the __numerator__ \(dV\)

of the diffco. But the \(dV\) that is substituted is

not therefore d__erived__ from \(\frac{dV}{dx}\), at least ['not directly or' inserted] from

the decomposition of that which is indecomposible [*sic*].

I return again my former paper (marked No 2.)

with a clearer explanation of what I i__ntended__ to

convey by the term e__quivalent__; a term which it seems

I had no business to use in the application which I

['there' inserted]} meant to make of it.

I enclose (marked No 3) my answer to your ''Try

''to prove the following. It is o__nly__ when \(y=ax\)

''(\( a\) being constant) that \(\frac{dy}{dx}=\frac{y}{x}\) '' I do not feel

quite sure that my p__roof__ __is__ a __proof__. But I think

it is too.

Now about \(v=gt\) and \(s=\frac{1}{2}gt^2\); a subject

which t__roubles__ me not a little.

Is the following a correct development of the note in

Useful Knowledge Mechanics? I re-copy the notes first;**[113r] ** ''\( V=\frac{dS}{dT}=gT\) . Hence \(dS=gT.dT\), which being

''integrated gives \(S=\frac{1}{2}g.T^2\) ''

[something crossed out on two lines]

The Integral of \(\frac{dS}{dT}dT\) or of \(gT.dT\) will

obviously give us \(S\); & we know that \(\int gT.dT=\)

\( =\frac{1}{2}gT^2+C\), (by formula of page 104 of the Calculus).

But it appears to me that the statement

above ''Hence \(dS=gT.dT\) '' is an __unnecessary__

intermediate step :

It is true that \(\int\frac{dS}{dT}dT=\int dS\),

that is providing we extend the theorem

\(\int fx\frac{dx}{dt}dt=\int fx.dx\)

to the case when \(fx=1\), which I conclude it is

allowable to do, since \(1\) may be considered a

function of __anything__, I imagine; thro' the formula

\( \frac{fx}{fx}=1\) . But tho' t__rue__, yet the above ['clause' inserted] appears

to me ['an' inserted] unnecessary introduction.

I am not sure that I have explained myself well.

With respect to this formula

\( \frac{1}{2}gt^2\), & it's [*sic*] derivation & application; I have

referred as you desired to pages 27, 28, & have**[113v] ** fully refreshed my memory upon them. But I

do not feel this helps me much. In this first

place the process is the co__nverse__ of that __I__ enquired

upon. \(S\) is t__here__ give, & __\( V\) __ is to be d__erived__

from __\( S\) __. __My__ position was; \(V\) given, & \(S\) to

be derived from \(V\) .

I understand the process of pages 27, 28, considered

as a distinct & separate thing. But I do not

i__dentify__ it with Differentiation or Integration.

I, (knowing by abstract rules & theorems) that

\( 2x\) is the diffco of \(x^2\), see that the limit \(2t\)

which comes out, might be perfectly well expressed

by \(\frac{d(t^2)}{dt}\) . And that we may put the __result__

of the Differentiation of \(t^2\), and the __result__ of all

the reasoning of pages 27, 28, i__ndifferently__ o__ne for__

t__he othe__r. But I only see it as I see that

in the processes \(12\div 4=3\) \(1+2=3\) we

might indifferently put the r__esults__ (__3__, in both cases)

one for the other. There may, for anything I yet

see or understand, be as little connection between

the abstract process of Differentiation and the

Stone-falling process, as between the above processes

of __Division & Addition__, which latter tho' their re__sults__

agree, cannot be identified, or one made to represent**[114r] ** the other.

I apprehend [something crossed out] you will perhaps answer me

here, that I must wait patiently for Chapter 8,

in which (page 143) I see something very like an

explanation of all I want. At the same time

I think it better to express fully my difficulties.

I am very anxious to see your Comments on

my two papers ['sent the other day' inserted] upon \(\frac{1}{2}gt^2\) . For I do not see

where the flaw in them can be; & yet I suppose

there is one. It is some comfort in the confusions

& puzzles one makes, that they are always

exceedingly a__musing__ to me, __after__ they are cleared

away. And this is at least s__ome__ compensation

for the plague of them b__efor__e.

With many thanks,

Yours most truly

A. A. Lovelace

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