## Folios 115-118: AAL to ADM

Ockham Park

Sunday. 15^{th} Aug^{st}

['1841' added by later reader]

Dear M^{r} De Morgan. You must be beginning

to think me lost. I have been however hard at

work, with the exception of 10 days complete

interruption from company. I have now many

thing to enquire. First of all; can I spend

an evening with M^{r} De Morgan & yourself on

Tuesday the 24^{th}? On that day we go to Town to

remain till Friday, when we move down to Ashley

for 2 months at least. I would endeavour to

be e__arly__ in Gower S^{t}; before eight or not later

than eight. And I feel as if I should have

many mathematical things to discuss.

Now to my business :

1^{stly}: I send you a paper marked 1, containing my

development of two Integrals in page 116,

\( \int\frac{dx}{\sqrt{2ax-x^2}}=\sin^{-1}\left(\frac{x-a}{a}\right)\)

And \(\int\frac{dx}{\sqrt{2ax+x^2}}=\log(x+a+\sqrt{2ax+x^2})+\log 2\)

The __former one__ I think is plain enough, & I and

the book are quite agreed upon it. Not so with**[115v] ** the l__atter one__, & I begin to suspect the book.

I cannot make it anything but

\(\int\frac{dx}{\sqrt{2ax+x^2}}=\log(x+a+\sqrt{2ax+x^2})\)

or else\ \ \(=\log(\frac{x}{2}+a+\frac{\sqrt{2ax+x^2}}{2})+\log 2\)

I have tried various methods; but the

only one which I find hold good [\textit{sic}] __at all__, is

that applied in page 115 to \(\int\frac{dx}{\sqrt{a^2+x^2}}\), & which

seems clearly to bring out

my result above. By the bye I have a remark

to make on the Integration of \(\int\frac{dx}{\sqrt{a^2+x^2}}\) as

developped [\textit{sic}] in page 115.

Line 10 (from the bottom), you have \(xdx=ydy\) :

This is obvious, & similarly __I__ deduce in my paper

No 1, \((2a+x)dx=ydy\) . But I see no use in

what follows, ''and \(ydx+xdx=ydx+ydy\) ''.

It is equally o__bvious__ with the former equation,

but seems to me to have no purpose in bringing

out the results, which __I__ deduce as follows :

Since \(xdx=ydy\), we have \(\frac{dx}{y}=\frac{dy}{x}\)

Therefore by the Theorem of page 48, or at least ['by' inserted] a

Corollary of it, we have \(\frac{dx+dy}{x+y}=\frac{dx}{y}\), whence &c, &c.

And this is the method also which I have used**[116r] ** in developping [*sic*] \(\int\frac{dx}{\sqrt{2ax+x^2}}\) .

2^{ndly} : Page 113, lines 16 ['&c 17' inserted] from the bottom, you say ''The

''first form becomes impossible when \(x\) is greater than

''\( \sqrt{c}\), for in that case the Integral becomes the

''__Logarithm of a Negative Quantity__''. Now there are

surely c__ertain case__s in which n__egative quantities__

may be powers, & therefore may have Logarithms.

All the o__dd whole__ numbers may surely be the

Logarithms of __Negative__ Quantities.

\( (-a)\times(-a)=a^2\) But \((-a)\times(-a)\times(-a)=-a^3\)

or \((+a)\times(+a)=a^2\) \((+a)\times(+a)\times(+a)=a^3\)

\( 3\) is here surely the Logarithm of a N__egative__ Quantity.

Similarly a n__egativ__e quantity multiplied into itself

any o__dd__ number of times will give a __negati__ve result.

3^{rdly} : In the Paper marked 3, which I return

again ['for reference' inserted]; I perfectly understand the proof by means of the

Logarithms (added by you), why \(\frac{dy}{dx}\) can only \(=\frac{y}{x}\)

when \(y\) is either \(=x\), or \(=ax\) (\) a\) being Constant)

Y__our__ proof is perfect, but still I do not see that

mine was not sufficient, tho' derived from much

more ge__neral__ grounds.

My argument was as follows : Given us \(\frac{dy}{dx}=\frac{y}{x}\),

what conditions must be fulfilled in order

to make this equation __possible__? Firstly : I see that**[116v] ** since \(\frac{dy}{dx}\) means a Differential Co-efficient, which

from it's [*sic*] nature (being a L__imit__) is a __constant &__

f__ixed__ thing, \(\frac{y}{x}\) must also be a __constant & fixed__

quantity. That is \(y\) must have to \(x\) a __constant__

Ratio which we may call \(a\) .

This seems to me perfectly valid. And surely a

Differential Co-efficient is as f__ixed__ & __invariable__

in it's [\textit{sic}] nature as anything under the sun can be.

To be sure you may say that there is a d__ifferent__

Differential Co-efficient for every different i__nitial__

value of \(x\) taken to start from, thus :

\(\frac{d(x^2)}{dx}=2x\) if \(x=a\), \(\frac{d(x^2)}{dx}=2a\)

if \(x=b\), \(\frac{d(x^2)}{dx}=2b\)

And this is perhaps what invalidates my argument

above.

4^{thly} : In the two papers folded together & marked 2,

which I also again return for reference, I perfectly see

that tho' m__athematically__ correct. I was completely

wrong in my __application__. But my proofs __do__ apply

to any __two different & independent__ velocities, whatever

of __two different__ bodies, or of the s__am__e body moving

at two different __uniform__ ratio [\textit{sic}] at d__ifferent__ __epochs__.

Thus my paper (marked upon it 1^{st} __Paper__ proves**[117r] ** the following : that the __Spaces moved over__ at two

d__ifferent times__, in __virtue of the Velocity acquired__ __at____the end of each of those times__, (the i__mpelling cause__

being supposed to c__ease__ at the end respectively of each

time fixed on), would be to each other as the

squares of the t__imes__ fixed on. But I perfectly

see that this is quite a __different & independent__

consideration from that of the S__pace actually moved____over__ by a body impelled by an __accelerating__ __force__,

& how wholly inapplicable my ['former' inserted] view of it was.

I have been especially studying this

subject of [something crossed out] Accelerating Force, & believe that I now understand

it very completely. I found I could not __rest__ upon

it at all, until I made the __whole__ of the subject out

entirely to my satisfaction : I enclose you (marked

4) the first of a Series of papers I am making out

in the different parts ['of' inserted] it. This one is the more

g__eneral development__ of the p__articular case__ __of Gravitation__

in pages 27, 28; & my more especial object in it

has been the __identification__ of the results arrived at

in this __real application__, with the Mathematical

Differential Co-efficient.

I have worked most earnestly & incessantly at the

Application of the Differential & Integral Calculus to the**[117v] ** subject of Accelerating Force, & Accelerated Motion,

during the last 2 or 3 weeks. It has interested me

beyond everything. After making out (according to

my o__wn notions__) the two papers on \(v=\frac{ds}{dt}\), and

\( s=\int vdt\), (the first of which I now send, & the

Second you will have in a day or two), I attacked

your Chapter 8, pages 144, 145, worked out all the

Formulae there; & had excessive trouble with my

t__hird__ paper on \(t=\int\frac{ds}{v}\), (now successfully terminated);

and I am now on \(f=\frac{dv}{dt}\), page 146.

You will perhaps not approve my having thus __run____a little riot__, & __anticipated__. But I think it has

done me great good. And I am anxious to know

if I may read the rest of this Chapter 8, __before__

reading Chapter 7 on Trigonometrical Analysis; & if

I am likely to understand it all without having

read Chapter 7.

I shall probably write again tomorrow; or if not,__certainly__ I shall on Tues^{dy} .

We are very anxious to know if there is no time

between the 1^{st} Nov^{r} & the middle of Feb^{y}, when you &

Mr De Morgan (& family) would come & stay here

for as long as you __can__ & __would__ like. We should

be delighted if you would remain __2 or 3 week__s.**[118r] ** And if this should be impossible for y__ou__, perhaps still

you would bring Mr De Morgan & the children here,

& remain a few days; having t__hem__ to stay longer.

We b__oth of us__ assure you that it would be__no inconvenience__ whatever to us; but rather contrary

the greatest pleasure. And I am certain it

would do M^{r} De Morgan good to be here for

a time. Pray consider my proposal;

at any rate for her & the children, if y__our own__

avocations should make it impossible for you even.

Believe me

Yours most truly

A. A. Lovelace

## About this document

All Ada Lovelace manuscript images on the

Clay Mathematics Institute website are

© 2015 The Lovelace Byron Papers,

reproduced by permission of

Pollinger Limited. To re-use them in

any form, please apply to

katyloffman@pollingerltd.com.

The LaTeX transcripts of the letters

were made by Christopher Hollings

(christopher.hollings@maths.ox.ac.uk).

Their re-use in any form requires his

permission, and is subject to the

rights reserved to the owner of

The Lovelace Byron Papers.

Bodleian Library, Oxford, UK

Dep. Lovelace Byron