[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