## Folios 134-135: AAL to ADM

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 q__uite 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 f__orce__ __myself__, (when**[134v] ** disinclined & __distraite__), __beyond a certain point__ is

very disadvantageous. So on these occasions I j__ust____keep gently going__, without however attempting very

much. I am hoping now to get __a good lif__t 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

g__et 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 w__hat__ 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 I__rreducible 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 o__peration__ \(f\) 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.

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