[77r]

 Ockham

 Sunday. 10^th Jan ^y  

Dear M^r De Morgan.  I send
you the ['Series' crossed out] Analysis of the
new Series I received in your
letter yesterday morning.  I
believe I have made it out
quite correctly.  In fact, the
Verification at the end proves
this.  But, owing to a
carelessness in my ['first' inserted] inspection
of it, I have had the
trouble & advantage of analysing
two Series.  I glanced too
hastily at it, & did not
observe that the factors of the
[77v] Denominators (of the Co-efficients),
are not powers of \(2\), but
simply multiples of \(2\) .      
If you will open my Sheet,
you will find on the inside
my analysis of the Series I at
first mistook your's [\textit{sic}] for;
and I am not sorry this 
has happened.  I believe
both are correctly made out.       

You kindly request me
if I do not understand the
erasure in the former ['small' inserted] paper,
again to return it &c.  Now
I do not agree to it; & ['I' inserted] still
fancy that we are in fact
meaning exactly the same
thing, only that you are
[78r] speaking of the \( n\)^th Term, & I
of the \( n+1\) ^th.           For
convenience of reference I again
return the former large paper
(& at any rate H. M's
Post-Office will benefit).          
I quite understand that
\( \frac{1}{4}+\frac{1}{2}\sqrt{10^6+\frac{1}{4}+a}\) is less than
\( 501\) .  Therefore as \( n\) is
the next whole number above
this fractional expression, \(x=\) 
\( =501\) .  But \( n\) is not the
Term sought; the unknown
term to be determined being
by the conditions of the
Hypothesis & Demonstrations,
\( n+1\) , & therefore \(=502\) .
And if you will examine
[78v] your own ['former' inserted] Verification, you
will see that you there
determine the Term at which
Convergence begins, to be
\( A_{502}\), or the 502^nd Term,
which agrees with my
result \(n+1=502\) .      
I think it is quite clear
that we are both agreed,
but that you were not aware
at the moment you made
the erasure that I was
not speaking of the next
whole number above \(\frac{1}{4}+\frac{1}{2}\sqrt{10^6+\frac{1}{4}+a}\) 
but of the next but one above
it.        
So much for the three Series:

Now I must go to other
[79r] matters.  I am indeed sending
you a Budget.          

 I have been working hard
at the Differential Calculus,
& am putting together some
remarks upon Differential
Co-efficients (which in due
time will travel up to
Town for your approbation),
but in the progress of which
I am interrupted by a
slight objection to an old 
matter of Demonstration,
which did not occur to me
at the time I was studying
it before, & sent you a
paper upon it ['from Ashley' inserted].  In the
course of the observations I
[79v] am now writing, I have ['had' inserted]
occasion to refer to the old
['general' inserted]} Demonstration, (pages 46 & 47
of your Differential Calculus),
as to the finite existence of
a Differential Co-efficient
for all Functions of \(x\); &
a slight flaw, or rather what
appears to me a flaw, in the
conclusions drawn, has occurred
to me.  It is most clearly
proved that, \(\theta\) being supposed
to diminish without limit,
the Fractions \(Q_1\), \(Q_2\) &c
must have finite limit, for
some value or other at all
events of \(n\theta\) or \(h\) .  But the
fractions in question do not
[80r] appear to me to be strictly
speaking analogous to \(\frac{\Delta u}{\Delta x}\),
except the first of them \(\frac{\varphi(a+\theta)-\varphi a}{\theta}\) 
and the last of them \(\frac{\varphi(a+n\theta)-\varphi(a+\overline{n-1}\theta)}{\theta}\),
and for this reason.
In the expansion \(\frac{\Delta u}{\Delta x}\) or
\( \frac{\varphi(x+\theta)-\varphi x}{\theta}\), as \(\theta\) alters
\( x\) does not alter, but remains
the same.  In these fractions
on the contrary, which all
have the form \(\frac{\varphi(a+k\theta)-\varphi(a+\overline{k-1}\theta)}{\theta}\) 
and in which \(a+\overline{k-1}\theta\) [bar over \(k-1\) should have little downward-pointing hooks at the ends]
stands for the \(x\) of the
expression \(\frac{\Delta u}{\Delta x}\) or \(\frac{\varphi(x+\theta)-\varphi x}{\theta}\),
not only does \(\theta\) alter, but
from the conditions of the
Hypothesis & Demonstrations, \(\overline{k-1}\theta\) 
[80v] & consequently \(a+\overline{k-1}\ \theta\) must
likewise alter along with \(\theta\) .
There is therefore a double
alteration in value going on
simultaneously, which appears
to me to make the Case quite
a different one from that
of \(\frac{\Delta u}{\Delta x}\), & consequently to
invalidate all conclusions
deduced from the former with
respect to the latter.
The validity of the Conclusions
with respect to the fractions
\( Q_1\), \(Q_2\) &c, you understand
I do not question.  What I
question is the analogy between
these Fractions & the Fraction
\( \frac{\Delta u}{\Delta x}\) or \(\frac{\varphi(x+\theta)-\varphi x}{\theta}\) ['of' inserted] which
[81r] latter it is required to
investigate the Limits.
I also have another slight
objection to make, not to the
extent of Conclusions established
respecting the Fractions \(Q_1\), \(Q_2\) 
&c having finite limits,
but to the Conclusions on that
point not going far enough,
not going as far as they
might : ''either these are
''finite limits, or some increase
''without limit and the rest
''diminish without limit; if the
''latter, we shall have two
''contiguous fractions, one of which
''is as small as we please, and
''the other as great as we please,
''&c, &c, a phenomenon which
''which [\textit{sic}] can only be true when
[81v] ''\( Q_k\) is the fraction which is
''near to some singular value
''of the Fraction, & cannot be
''true of ordinary & calculable
''values of it &c.'' Now it
appears to me that in no
possible case could such a
phenomenon as this be true,
when we consider how the
fractions are successively
formed one out of the others
by the substitution of \(a+\theta\) for
\( a\), \(\theta\) too being as small as
we please.  I therefore think
it might have been concluded
at once that there must
always be finite limits to
the Fractions \(Q_1\), \(Q_2\) &c,
[82r] and this whatever \(k\) or \(n\theta\)
may be.  I suppose it is not
so, but I cannot conceive
the Case in which it could
be otherwise.       
I do not know if in writing
upon my two difficulties in
these pages 46, 47, 48, I have
expressed my objections (especially
in the former case of the
fractions \(Q_1\), \(Q_2\) not being
similar to \(\frac{\Delta u}{\Delta x}\) ) with the
clearness necessary to enable
you to answer them, or indeed
to apprehend the precise points
which I dispute.  It is not
always easy to write upon
these things, & at best one
must be lengthy.  I shall be
[82v] exceedingly obliged if you will
also tell ['me' inserted] whether a little
Demonstration I enclose as to
the Differential Co-efficient
of \(x^n\), is valid.  It appears
to me perfectly so; & if it is,
I think I prefer it to your's [\textit{sic}]
in page 55.  It strikes me
as having the advantage in
simplicity, & in referring to
fewer ['requisite' inserted] previous Propositions.

I have another
enquiry to make, respecting
something that has lately
occurred to me as to the
Demonstration of the Logarithmic
& Exponential Series in
your Algebra, but the real
truth is I am quite ashamed
[83r] to send any more; so will
at least defer this.          
I am afraid you will indeed
say that the office of my
mathematical Counsellor or
Prime-Minister, is no joke.

I am much pleased to
find how very well I stand
work, & how my powers of
attention & continued effort
increase.  I am never so
happy as when I am
really engaged in good
earnest; & it makes me most
wonderfully cheerful & merry
at other times, which is
curious & very satisfactory.    

What will you say when
[83v] you open this packet?     
Pray do not be very angry,
& exclaim that it really is
too bad.     

 Yours most trul

nbsp;A. A. Lovelace