## Folios 12-13: ADM to AAL

**[12r] **My dear Lady Lovelace

You have got through the

matter about which you write

better than I should have expected.

I have finished what you sent

as you will see

With regard to the curve, I drew

it as containing every possible sort

of singular point. Its equation would

be enormously complex

There must be an infinite number

of different equations which belong

to a curve of a similar form, but

the question 'given the more general**[12v] ** form of a curve, required the

equations which may belong to

such form' is a very difficult

one.

I will merely give you a glimpse

Required an equation to a

curve such that it passes

through the following points \(P\) \(Q\) \(R\)

[diagram in original] at \(P\) let \(x=a\), \(y=A\)

\( Q\) \(x=b\) \(y=B\)

\( R\) \(x=c\) \(y=C\)

[the next formula and the following line of text stretch across 12v and 13r]

\( y=A\frac{(x-b)(x-c)}{(a-b)(a-c)}+B\frac{(x-c)(x-a)}{(b-c)(b-a)}+C\frac{(x-a)(x-b)}{(c-a)(c-b)}+\left\{\begin{matrix}{\tiny {\rm any\: function\: of}\: x\: {\rm which}} \\{\tiny \rm does \:not become\: infinite}\\ {\tiny {\rm when}\: x=a, \:{\rm or}\: b,\: {\rm or} \: c }\end{matrix}\right\}\times (x-a)(x-b)(x-c)\)

Here is an infinite number of equations which you will find to satisfy the conditions

I have to thank you for very good

partridges received from Ockham

With kind remembrances to Lord Lovelace

I am Yours very truly

__ADeMorgan__

I have heard of Lady Byron by

M^{r} Phitton [?] who left her safe

at Fountainebleu

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