## Folios 35-36: ADM to AAL

**[35r] ** My dear Lady Lovelace

We shall be happy to

see you on Monday Evening, and

Lord Lovelace too if he be not

afraid of the algebra

Your points in your letter are

are [should be 'I'] think, clear enough in

your own head. A little addition

however may be made as follows.

You are not [something crossed out] to think that

because \(x\) must be diminished

without limit to prove a conclusion

that conclusion is only true for

small values of \(x\), or for \(x=0\) .

For example suppose I know that

\((a+x)(a-x)=P+Qx+Rx^2\)

but of \(P\) \(Q\) and \(R\) __I only know that__

__they are independent of \(x\) __. What

**[35v] ** therefore they are for any

one value of \(x\), they are for

any other. I find them thus

Since the preceding is by hypothesis

true for __all__ values of \(x\), and

since altering \(x\) does not alter

\( P\) \(Q\) or \(R\), I take \(x=0\)

to begin with

\(a^2=P\) when \(x=0\)

but \(a\) and \(P\) are independent

of \(x\), therefore what relation

exists when \(x=0\) exists always

or \(a^2=P\)

Let \(x=a\)

\(0=P+Qa+Ra^2\)

Let \(x=-a\)

\(0=P-Qa+Ra^2\)

[something crossed out] subtract \(2Qa=0\) or \(Q=0\)

Here are two values of \(x\) made use

**[36v] ** of.

Add

\(2P+2Ra^2=0\)

\(R=-\frac{P}{a^2}=-1\)

whence

\((a+x)(a-x)=a^2+0.x-x^2\)

\(=a^2-x^2\)

__if__ it must be of the form

\(P+Qx+Rx^2\)

We are much obliged by your

invitation to Ockham, but

I am closely tied up by

lectures & other things. Even

at such times as Xtmas I am

generally very busy

With kind remembrances to

Lord Lovelace I am

Yours very truly

__ADeMorgan__

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