[123r]

 Ashley Combe
          Thurs^dy Mor^g 
9^th Sep^r ['1841' added by later reader]

 Dear M^r De Morgan.  I have rather a large batch
now for you altogether :
1^ly : I am in the middle of the article on Negative &
Impossible Quantities; & I have a question to put on
page 134, (Second Column, lines 1, 2, 3, 4, 5 from the bottom)

\( (a+bk)^{m+nk}=\varepsilon^A\cos B+k\,\varepsilon^A.sin B\) &c
I have tried a little to demonstrate this Formula;
but before I proceed further in spending more time
upon it, I think I may as well ask if it is intended
to be demonstrable by the Student.  For you know I
sometimes try to do more than anyone means me to
attempt.  I have as yet only got thus far [something crossed] with
the above formula : If in \((a+bk)^{m+nk}\),
\( r\) is given \(=\sqrt{a^2+b^2}\), ['and' inserted] \(\tan\theta=\frac{b}{a}\); then \(\sin\theta=b\) 

  \(\cos\theta=a\) 
and \((a+bk)^{m+nk}=(\cos\theta+k.sin\theta)^{m+nk}=\) 

  \(=(\varepsilon^{k\theta})^{m+nk}=\varepsilon^{k(m\theta)}\times\left\{\varepsilon^{k(n\theta)}\right\}^k\) 

or \(=(\cos.m\theta+k.sin.m\theta)\times (\cos.n\theta+k.sin.m\theta)^k\), and
[123v] I dare say that from some of these transformations,
the Second Side of the given equation, with the
determination of \(A\) and \(B\), may be deduced.  But
it appears to me ['it must be' inserted] a very complicated process; &
therefore I should like to know before I undertook it,
that I was not wasting time ['in' deleted] doing so.
2^dly : I am plagued over page 135 of the Calculus.
It is not that there is any one thing in it which
I do not clearly see.  But it is the depth of the
whole argument which I cannot manage to discover.
I should say that whole argument from ''We now know &c''
page 134, to ''We can therefore take a function,
''which, for a particular value of \(x\), &c, &c'' page 135.

It seems to me all to be much ado about nothing;
and I do not see what is arrived at by means
of it [something crossed out].  A very complicated process appears to be
used in the 1^st Paragraph of page 135, to prove
that when \( h\) is small then the Increment in \(\varphi x\) 
is very nearly represented by \(\varphi'a+h\), which was
already shown in page 134.  And then suddenly in
the Second Paragraph the Formula \(\varphi a+\varphi'a(x-a)+\) 
\( +\varphi''a\frac{(x-a)^2}{2}\) is introduced, & I do not understand
\'{a} quoi bon the closing conclusion drawn from it.

3^dly : I am not sure that I agree to what you
say in preference (for ascertaining Maxima & Minima)
of the direct ascertainment of \ the value of \(\varphi'\) \) x\), over
[124r] the ordinary method.  Because it seems to me in
many cases impossible after you have determined \(0\) or
\( \alpha\) values of \(\varphi'x\), to determine further that the sign
does change at them & how it changes, unless by means
of the ordinary rule.  I have written out and
enclose an example from Peacock, in which unless
I had used the ordinary rule, after I had
determined \(0\) values for \(\varphi'x\), I should have been
at my wits' end how to bring out the conclusion.

4^thly : I send you a little Maxima & Minima
Theorem of my own, which occurred to me by accident;
It is for \(\varphi x=x^2-mx\) .  After proving it by the
Differential Calculus, I have given a direct proof
of another sort.  I merely wrote this ['direct proof' inserted], because it
[something crossed out] occurred to me; but it gave me a great deal of
trouble, & I think was rather a work of supererogation;
but I believe it is quite correct.  You will find
enclosed in the same sheet the demonstration of
''What is the number whose excess above it's [sic] Square
''Root is the least possible?'' (see page 133 of the Calculus);
and on the reverse side of this latter [something crossed out]
is the ''verification round the 4 Right Angles'' for the
continual increase together of \(x\) and it's [sic] tangent (See
page 132).  But here I have something further to
add.  In this Chapter VIII, we hear of Differential
Co-efficient which become \(=0\), or \(=\alpha\) .  In this very
[124v] instance, \(1+\tan^2x\) is alternately \(=1\), and \(=\alpha\) .
Now according to my previous ideas, the terms
Differential Co-efficient was only applied to some
finite quantity; and referring to pages 47, 48,
where one acquired one's first ideas of a Differential
Co-efficient, I think it is there clearly explained
that the term is only used with reference to a
finite limit.  But in this Chapter VIII, there
seems to be a considerable extension of meaning on
the subject.
^thly .  In page 132, it is very clearly deduced that
the Ratio of a [something crossed out] Logarithm to it's [sic] number is increasing
as long as \(x\) is \(<\varepsilon\), and afterwards decreases.

The proof is most obvious.  But, unluckily, the
conclusion seems to me to be contrary to the fact; at
least the first half of the conclusion, not the latter half.

On this principle : from the very nature of a
Logarithm, it is obvious that (\) x\) being \(>\varepsilon\) ), for
equal increments to the \(\log,x\), there will be
larger & larger Increments to \(x\) .  The one being in
arithmetical, the other in geometrical progression.
Therefore clearly the Ratio of the Logarithm to the
number, is a diminishing one.  But then the
same thing seems to me to apply [something crossed out] when
\) x<\varepsilon\) .  Surely there is then just the same
[125r] arithmetical & geometrical progression for equal
Increments of the Logarithms.  I suppose there is
some link that I have over-looked.

I send you two Integrations worked out.  They
are from Peacock.  I in vain spent hours over the
one marked 2, of which I could make nothing by
any method that I devised; until in despair, I
looked thro' your Chapter XIII to see if I could there
find any hints; & accordingly at page 277, I
found a general formula which included this
case.  But I do not believe I should ever have
hit upon it by myself.  The Integral marked 1,
might of course be proved also in the same way;
tho' ['my' crossed out] the method ['I have used' inserted] is sufficient in this instance.

I have written out no more papers on
Forces.  In fact there is only one more that is
left for me, viz: \(f=v\frac{dv}{ds}\) .  And for this I see
no occasion; for I am sure that I must thoro'ly
understand it, after all I have written.
I quite see ['the truth' inserted] your remarks on my having treated
Acceleration of Velocity as being identical with
Force; whereas, (as I now understand it), it is
simply the measure of Force, & our only way of
getting at expressions for this latter.  On the subject
[125v] of \(v^2=2\int fds+C\); I have considered it a great
deal; and any direct demonstration of it, after the
manner of my other papers, seems to me to be
quite impracticable.  Neither \(\int v.dv\), nor \(\int f.ds\) 
[something crossed out] now appear to me to have any actual proto-types in the
real motion.  Here then suggests itself to me the
question : ''then are there certain truths & conclusions
''which can be arrived at by pure analysis, & in
''no other way?''  And also, how far abstract
analytical expressions must express & mean
something real, or not.  In short, it has
suggested to me a good deal of enquiry, which
I am desirous of being put in the way of
satisfying.
By the bye, I fear that one little paper of mine
dropped out of the last packet.  It was a little
pencil memorandum on ['the meaning of' inserted] \(\int f.ds\); & there were
remarks upon it, (if you remember) in my accompanying
letter.  It bears upon the above question.
I could write it out again, if it has been lost.

 Is not this a budget indeed?

 Yours most truly

 A. A. Lovelace