[144r]

 Ashley-Combe
       Porlock
              Somerset
Sunday Mor^g .  28^th Aug^st 
                              ['1842' added by later reader]

  
Dear M^r De Morgan.  I am going on well; ['quite' inserted] as I
could wish.  I have done much since I saw you;
& you will have all the results of the last few days
in good time.  I enclose you now two papers; one
on \(f=\frac{dv}{dt}\), the other on \(\int_a^{a'}f.dt\) .
You will have next those on \(v\frac{dv}{dt}=f\), and
\( v^2=2\int f.ds+C\) .  This latter I think I have
succeeded in analysing to my mind.
I have ['now' inserted] two observations to make : [something crossed out]
1^stly: I think I have detected a slight error in one
of my former papers, that on \(t=\int\frac{ds}{v}\) .  I return
it for reference.  In order in the [something crossed out] Summation
[something crossed out]}
\( \left\{\frac{1}{\varphi s}+\frac{1}{\varphi(s+ds)}+\cdot\cdots\frac{1}{\varphi(2s)}\right\}ds\), to end with \(\frac{1}{\varphi(2s)}\),
I should have begun with \(\frac{1}{\varphi(s+ds)}\) not with \(\frac{1}{\varphi s}\) .
If the time elapsed during the first fraction of Space
[144v] (starting from \(s\) ) were ['made' inserted] \(=\frac{1}{\varphi s}\), then the time for the last
of the Fractions necessary to complete up to \(2s\), would
be \(\frac{1}{\varphi(2s-ds}\), and not \(\frac{1}{\varphi(2s)}\) which it ought to be.
I don't know that this affects the correctness of the
ultimate limit of the Summation.  But here, where
the Summation itself is made to represent a
hypothetical movement, it is clearly wrong.
The error is avoided in the former paper I had
written on \(s=\int v.dt\), which I likewise return to
refer to this Point.
2^ndly : In considering a priori the Integral \(\int f.ds\),
I am inclined still to adhere to my original
opinion (expressed in the pencil Memorandum I showed
you & ['which I' inserted] now return).  I should premise that I now
mention this merely as a curious subject of investigation,
not because it is concerned in the [something crossed out] papers I
am making out upon \(v^2=2\int f.ds+C\), in which I
have avoided the direct consideration of \(\int_a^{a'}f.ds\) .

I am disposed to contend that tho' \(ds\) 
does here represent Space, that still the \( ds\) fraction
of any one of the terms of the Summation, say \(\varphi(a+n.ds)ds\) 
means the same fraction of \(\varphi(a+n.ds)\) which \( ds\) is of
[145r] a Unit of Space; & therefore that since \(\varphi(a+n.ds)\) 
represents Force, (or ['uniform' inserted] Acceleration of Velocity for \(1\) Second
in operation during the performance of the length \(ds\) ),
the \( ds\) fraction of this expression must represent the
['\( ds\) part of this Force or the' inserted] actual Acceleration for \(\frac{1}{ds}\) of a Second.  I treat \(ds\) as
an abstract quantity.  And so I conceive [something crossed out] \(dt\) must
be treated in \(s=\int v.dt\), ['\( ds\) ' inserted] in \(t=\int\frac{ds}{v}\), \(dt\) in \(\int f.dt\),
&c, &c.
I should tell you that I am much pleased with
the observation you added to my inverse demonstration
of \(\int fx.\frac{dx}{dt}dt=\int fx.dx\), and that I quite
understand ['why' inserted] my proof can only be admissible on
the Infinitesimal Leibnitzian Theory.  But this
theory is to my mind the only intelligible or
satisfactory one.  In fact, (notwithstanding it's [\textit{sic}] error),
I should call it the only true one.
By and bye, you will have some observations
of mine upon Differential Co-efficients & Integrals,
abstractly considered.  I have been thinking much
upon them.
I am going on with Chapter VIII.
By the bye, I believe you will receive somehow tomorrow
[145v] a book (the 1^st Vol of Lamé's Cours de Physique)
in which there is a passage which I will write
to you about as soon as I find time.
I forgot to mention it to you on Thursday; &
so have ordered the Book to be sent to you, that
I might write about it sometime.

Believe me

 Yours very truly

 A. A. Lovelace