## Folios 121-122: AAL to ADM

Ockham.

Friday. [something crossed out] 21^{st} Aug^{st}

Dear M^{r} De Morgan. You have received safely I

hope my packet of y__esterday__, & my packet sent on

T__uesday__.

I now re-enclose you the paper marked 1. There is

another Integral added at the bottom. Also I have

altered one or two little minutiae in the development

of \(\int\frac{dx}{\sqrt{2ax+x^2}}\) above, which you had omitted to correct.

I __quite understand__ your observations upon it, & see

the mistake I had made; & which related to the

Differential \(dy\), and \(d(\varphi x.x)\)

If \(y=\varphi x.x\), then \(dy=d(\varphi x.x)=\frac{d(\varphi x.x)}{dx}dx=\)

\(=\left\{x.\frac{d(\varphi x)}{dx}+\varphi x.\frac{dx}{dx}\right\}dx=x.\left(\frac{d(\varphi x)}{dx}dx\right)+\varphi x.dx=\)

\(=x.d\varphi x+\varphi x.dx\)

Or if \(y^2=\varphi x.x\), then \(dy^2=x.d.\varphi x+\varphi x.dx\)

or \(\frac{d(y^2)}{dy}dy=x.d.\varphi x+\varphi x.dx\)

or \(2ydy=x.d.varphi x+\varphi x.dx\), and \(ydy=\frac{1}{2}x.d.\varphi x+\frac{1}{2}\varphi x.dx\) **[121v] ** This is all now r__ight__ in my head.

In \(\int\frac{dx}{\sqrt{2ax+x^2}}\) we arrive then in my corrected paper,

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

\(=\log\left(\frac{x}{2}+\frac{a}{2}+\frac{\sqrt{2ax+x^2}}{2}\right)+\log 2\)

which, as you observe ''again with the book all but

''the \(\log.2\), __which being a Constant, matters nothing__''.

Very true; but why did you then insist the

\( \log 2\) in page 116? it seems as if put in __on purpose__

to be effaced in the parenthesis (Omit the Constant).

And it might just as well have been \(\log 3\),

\( \log 4\), \(\log\,\)(anything in the world) .

As to my two papers marked 2 (& which I__again__ return, merely for the __convenience of reference__),

I see that in order to make them valid, as applying

each to __two separate & different velocitie__s, they should

be re-written (which is not worth while), & the

t__erms__ of the e__nunications__ altered as follows :

''If two quantities \(V\), \(V'\) be respectively equal to the

''Ratios \(\frac{S}{T}\), \(\frac{S'}{T'}\), and if \(V:V'=T:T'\), then the values

‘’\( S\), \(S'\) must be t__o each other__ as the __squares of \(T\), \(T'\) __

''are to one another'' &c, &c**[122r] ** __At last__ I believe I have it quite correctly.

As for \(\frac{dy}{dx}=\frac{y}{x}\), I see my fallacy about \(\frac{y}{x}\) being

a fixed quantity.

About page 113, ''__The first form becomes impossible__

''__when \(x\) is greater than \(\sqrt{c}\), for &c__'', I fancy I

[something crossed out] had a little misunderstood

the mathematical m__eaning__ of the words __impossible____quantity__. I have loosely interpreted it as being

equivalent to ''__an absurdity__'', or at least to

''__an absurdity__, __unless an extension be made in the__

''__ordinary meaning of words__''. And in this

instance I perceived that __if__ the __Logarithm be__

an __odd number__, there would be __no absurdity__

even __without extension in the meaning of terms__;

because that it would then merely imply a__negative Base__; which negative Base, would I think

be a__dmitted__ theoretically (tho' i__nconvenient__ __practically__)

on the common __beginner's__ instruction on the Theory of

Logarithms. Am I right?

By the bye this subject reminds me that I think

I find __a mistake__ in page 117, line 13 (from the top)

''(\( n\) an integer) \(\int_{-a}^{+a}x^ndx=0\) when __\( n\) is odd__, \(=\frac{2a^{n+1}}{n+1}\) __when \(n\) is even__''**[122v] ** It seems to me just the reverse, thus :

\( =0\) when __\( n\) is even__, \(=\frac{2a^{n+1}}{n+1}\) when __\( n\) is odd}__

I have it as follows :

\( \int_{-a}^{+a}x^ndx=\frac{a^{n+1}}{n+1}-\frac{(-a)^{n+1}}{n+1}=\frac{a^{n+1}-(-a)^{n+1}}{n+1}=\)

\(=\frac{a^{n+1}-a^{n+1}}{n+1}\) or \(0\) if \(n+1\) be even

(I __now__ see it __while working__; for if

\(n+1\) be even, __\( n\) __ must be __odd__.)

and vice versa.

So I need not trouble you upon this; as I have

solved my difficulty whilst stating it. I had only

looked at this Integral in ['a' inserted] great hurry, this morning.

I hope on Sunday to send you two

remaining papers I have to make out, on the

Accelerating Force subject;

upon \(f=\frac{dv}{dt}\), and \(v=\int fdt\)

I think I have been encouraged by your great

kindness, so as to give you really no Sinecure

just at this moment.

Yours most truly

A. A. Lovelace

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