[84r]

 Ockham

 Sunday.  17^th Jan ^y 

Dear M^r De Morgan.  Many
thanks for your reply to my
enquiries.  I believe I now
understand about the limit
of \(\frac{\varphi(x+n\theta+\theta)-\varphi(x+n\theta)}{\theta}\) not
being affected by \(n\theta\) being a
gradually varying quantity.
I think your explanation of
it amounts to this :  that
provided [something crossed out] \((x+n\theta)\) varies only
towards a fixed limit, either
of increase or diminution; then
[84v] the result of the Subtraction
of \(\varphi(x+n\theta)\) from \(\varphi(x+n\theta+\theta)\) 
remains just the same as if,
(calling \((x+n\theta)=Z\) ), \(Z\) were
a fixed quantity.   Now
by the conditions of the Demonstration
in question, (in your pages
46 & 47), when a decrease}
takes place in \(\theta\), a certain
simultaneous increase takes
place in \(n\) .  That is to
say, suppose \(\theta\) has at any
one moment a certain value
corresponding to which \( n\()has
the value \( k\) .  If I alter
\( \theta\) to a lesser value \(\chi\), then
say that the corresponding
[85r] value of \(n\), necessary to fulfil
the constant condition \(n\theta=h\),
is not \(k\), but \(k+m=p\) .
What happens now? Why
as follows, I believe : there
were, before \(\theta\) became \(\chi\),
\( k\) fractions; there are now
\( k+m\), or \(p\) fractions.
In ['each of' inserted] the \(k\) former fractions,
[something crossed out]} \(Z\) will
have diminished, towards a
fixed limit ['of diminution' inserted] \(x\); in ['each of' inserted] the \(m\) 
new fractions introduced, \(Z\) 
will be greater than in the
old \(k\) fractions; but there
is a fixed limit of increase,
\( h\), which it can never pass,
[85v] up to the very last Term
of the Series of Fractions.
Therefore tho' the quantity
\( x+n\theta\) or \(Z\) varies necessarily
with a variation in the value
of \(\theta\), yet it varies within
fixed limits either of
diminution or increase, & thus
the result of the subtraction
\( \varphi(Z+\theta)-\varphi(Z)\) is not
affected.
I hope I have made
myself clear.  I think it is
now distinct & consistent in
my head.

I see that my proof of
the limit for the function \(x^n\) 
is a piece of circular argument,
[86r] containing the enquiry which
I was in fact aiming at
in the former paper, but
which required to be
separated from the confusion
attendant on my erroneous
statements on other points.
I merely return the old
paper with the present one,
because it might perhaps be
convenient to compare them.

On the other side
of the sheet containing the
remarks on \(\frac{a^\theta-1}{\theta}\), you
will find an enquiry
which struck me lately
quite by accident in
[86v] referring to some old
matters.
I ought to make many 
apologies I am sure for
this most abundant
budget. I am very
anxious about the matter
of the successive Differential
Co-efficients, & their
finiteness & continuity.  I
think it troubles my
mind more than any
obstacles generally do.  I
have a sort of feeling
that I ought to have
understood it before, &
[87r] that it is not a legitimate
difficulty.

 With many thanks,

 Yours most truly

 A. A. Lovelace