[134r]

 Ashley-Combe
         Monday. 8^th Nov^r ['1841' added by later reader]

 Dear M^r De Morgan.  I hope you intend to christen
the ''large boy'' by the name of Podge, with which I
am particularly pleased.
I am much obliged by your letter.  I send a corrected
version (now I believe quite right) of \(\frac{d^2u}{dx^2}-u=X\); on my
assumed supposition \(\frac{dK}{dx}\varepsilon^x+\frac{dK'}{dx}\varepsilon^{-x}=0\) .  As for my other
assumption \(K+K'=x^3\), it is so complicated a one that I
have not thought it worth while to pursue it's [sic] development.
I cannot think how I could be so negligent as to
forget that \(\varepsilon^{-x}\) is a function of \((-x)\) which is itself a
function of \(x\) .  A complete oversight; as indeed most
of the enquiries in my last letter seem to have been.
I should perhaps mention that lately [something crossed out] I have
had my mind a good deal distracted by some
circumstances of considerable annoyance & anxiety to
me; & I have certainly studied much less well &
more negligently in consequence.  Indeed the last few
weeks I have not at all got on as I wished and
intended; & I find that to force myself, (when
[134v] disinclined & distraite), beyond a certain point is
very disadvantageous.  So on these occasions I just
keep gently going, without however attempting very
much.  I am hoping now to get a good lift again
before long; as I think I am returning to a more
settled & concentrated state of mind.  I mention all
this as an excuse for some errors & over-sights which
I conceive are more likely just at present to creep
into my performances than would usually be the
case.  Now to business : Chapter VIII :
1.  I send you two Problems on hypotheses of my own,
intended as being worked out on the model of those
in page 150.  There are three different Hypotheses.
In the one where I obtain \(t=\frac{1}{\sqrt{2}}\int\frac{ds}{\sqrt{s^{-1}-a^{-1}}}\) I have
not attempted to develop this Integral further.
Perhaps I ought to have done so; but it was only my
object to get quite a general expression.
2.  Page 141; (lines 9, 10, from the bottom) : Series in page
116 (of Chapter VI), it was shown that \(\int\frac{ds}{\sqrt{2kx-x^2}}=\sin^{-1}\frac{x-k}{x}=\) 
\( =\left(v\sin^{-1}\frac{x}{k}\right)+\left(\frac{w}{2}\right)\), I do not see how it can be said (page 141)
that the Constant may have any value \(P\) .
3.  I have never succeeded in properly understanding
the Paragraph beginning page 134, ending page 135, on
which I before applied to you; & the paragraph of
page 148 -- (marked 2) -- has only added to my mistiness on
[135r] the subject.  There is something or other which I cannot
get at in the argument & it's [sic] objects.  That of page
135 seems very like another way of arriving at
Taylor's Theorem.  The expression taken in line 25 from
the top, I conclude to be arrived at as follows :
Having obtained \(\varphi a+\varphi'a.(x-a)\); a function agreeing
in value and diff-co with \(\varphi x\) when \(x=a\), let us now
find a function agreeing not only in these two points
but also in second diff-co with \(\varphi x\), when \(x=a\);
(the same conditions being continued of \(\varphi'a\), \(\varphi''a\) );
We see therefore that \(\varphi'a\) must be of the form

 \(\varphi''a.x+m\)   where  \(m=\varphi'a-\varphi''a.a\) 
Substituting this in \(\varphi a+\varphi'a.(x-a)\) we have

 \(\varphi a+(\varphi''a.x+m)(x-a)=\varphi a+(\varphi''a.\overline{x-a}+\varphi'a)(x-a)\) 

  \(=\varphi a+\varphi'a.(x-a)+\varphi''a(x-a)^2\) 
Similarly we may obtain \(\varphi a+\varphi'a.(x-a)+\varphi''a.(x-a)^2+\varphi'''a.(x-a)^3\) 
(By the bye I don't see how you get \(\frac{(x-a)^2}{2}\) and \(\frac{(x-a)^3}{2.3}\), instead
of \((x-a)^2\) and \((x-a)^3\) as I make it).
But I cannot perceive what all this is for; & (as
I mentioned below), paragraph 2 of page 148 has added
to my blindness. I am sorry to plague you again
about it.  On receiving your former reply, I felt none
the wiser; but determined to wait, thinking I
might see it as I went on, which is often the case
[135v] with difficulties.
I now proceed to some miscellaneous matters.
1.  I make nothing of the Irreducible Case in the
Penny Cyclopedia.  Is it perhaps for want of having
read Involution & Evolution?  I am puzzled quite
in the beginning of the Article.
2.  Article ''Negative & Impossible Quantities'' P. Cyclopedia -- page 137
''If the logarithm of two Units inclined at angles \(\theta\) and \(\theta'\) be
added, (the bases being inclined at angles \(\varphi\) and \(\varphi'\) ); the result
is the logarithm of a Unit inclined &c, &c''
I cannot develop this; but I enclose some remarks upon it.
3.  In the treatise you sent me on the ''Foundation of Algebra'',
I cannot make out ['in' inserted] the least [something crossed out] (page 5), about the general
solution of \(\varphi^2 x=ax\) .  I suspect I do not understand
the notation \(f^{-1}x\) .  I quite understand \(f^2 x\) or \(\varphi^2 x\),
\( f^n x\) or \(\varphi^n x\) .  Judging by analogy, from page 82 of the
Differential & Integral Calculus, (where \(\Delta^{-1}x\) is explained),
I conceive \(f^{-1}x\) or \(\varphi^{-1}x\) may mean ''the quantity
''which having had an operation \(f\) or \(\varphi\) performed
''with & upon it, is \(=x\) .''  But I have considered
much over the last half of this page 5, & I can't
understand it.
I have one or two other matters still to write about;
but they do not press; & this is plenty I think for
today.  Pray congratulate M^r De Morgan on the
arrival & prosperity of Podge.

 Yours most truly

 A. A. L.