## Folios 84-87: AAL to ADM

Ockham

Sunday. 17^{th} Jan ^{y}

Dear M^{r} De Morgan. Many

thanks for your reply to my

enquiries. I b__eliev__e I now

understand about the limit

of \(\frac{\varphi(x+n\theta+\theta)-\varphi(x+n\theta)}{\theta}\) not

being affected by \(n\theta\) being a

gradually varying quantity.

I think your explanation of

it amounts to this : that__provided__ [something crossed out] \((x+n\theta)\) varies only

towards a f__ixed__ li__mit__, either

of increase or diminution; then**[84v] ** the result of the Subtraction

of \(\varphi(x+n\theta)\) from \(\varphi(x+n\theta+\theta)\)

remains just the same as if,

(calling \((x+n\theta)=Z\) ), \(Z\) were

a f__ixed__ q__uantity__. Now

by the co__ndition__s of the Demonstration

in question, (in your pages

46 & 47), when a __decrease__}

takes place in \(\theta\), a certain

simultaneous i__ncrease__ takes

place in \(n\) . That is to

say, suppose \(\theta\) has at any

one moment a certain value

corresponding to which __\( n\()__has

the value __\( k\) __. If I alter

\( \theta\) to a lesser value \(\chi\), then

say that the corresponding**[85r] ** value of \(n\), necessary to fulfil

the constant condition \(n\theta=h\),

is not \(k\), but \(k+m=p\) .

What happens now? Why

as follows, I believe : there

were, before \(\theta\) became \(\chi\),

\( k\) fractions; there are now

\( k+m\), or \(p\) fractions.

In ['each of' inserted] the \(k\) fo__rme__r fractions,

[something crossed out]} \(Z\) will

have d__iminished__, towards a

f__ixed__ l__imit__ ['of diminution' inserted] \(x\); in ['each of' inserted] the \(m\)

new fractions introduced, \(Z\)

will be g__reater__ than in the

old \(k\) fractions; but there

is a f__ixed__ __limit__ of i__ncrease__,

\( h\), which it can never pass,**[85v] ** up to the very l__ast__ Term

of the Series of Fractions.

Therefore tho' the quantity

\( x+n\theta\) or \(Z\) varies necessarily

with a variation in the value

of \(\theta\), yet it varies within__fixed limits__ either of

diminution or increase, & thus

the result of the subtraction

\( \varphi(Z+\theta)-\varphi(Z)\) is not

affected.

I hope I have made

myself clear. I think it is

now distinct & consistent in__my__ __head__.

I see that my proof of

the limit for the function \(x^n\) __is__ a piece of ci__rcula__r argument,**[86r] ** containing the enquiry which

I was in fact aiming at

in the former paper, but

which required to be

separated from the confusion

attendant on my erroneous

statements on other points.

I merely return the old

paper with the present one,

because it might perhaps be

convenient to compare them.

On the other side

of the sheet containing the

remarks on \(\frac{a^\theta-1}{\theta}\), you

will find an enquiry

which struck me lately

quite by accident in**[86v] ** referring to some old

matters.

I ought to make many

apologies I am sure for

this most abundant

budget. I am very

anxious about the matter

of the successive Differential

Co-efficients, & their

finiteness & continuity. I

think it troubles my

mind more than any

obstacles generally do. I

have a sort of feeling

that I o__ught__ to have

understood it before, &**[87r] ** that it is not a le__gitimat__e

difficulty.

With many thanks,

Yours most truly

A. A. Lovelace

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