## Folios 146–148: AAL to ADM

Ashley-Combe

Wed^{dy}. 16^{th} Nov^{r}

['1842' or '1847' added by later reader]

[CH: this might date from earlier than 1842, as it contains material that seems to fit better with the earlier letters]

Dear M^{r} De Morgan. I am very much obliged

for your long letter. The Formula in Peacock

comes out quite correct now that I have

written diff. co of \((b-x)^4=4(b-x)^3\times(-1)=-4(b-x)^3\) .

It is odd that notwithstanding the caution you

gave me in Town on this very point, I should

have fallen into the trap. There is nothing like

one's own blunders after all for instruction.

I do not however understand why example (19)

page 4, has not come out wrong also in

my working out. I enclose a copy of my

solution, and it appears to me it o__ught__ to

be wrong, because I surely should have had

diff.\ co of \((1-x)^4=4(1-x)^3\times(-1)=-4(1-x)^3\), whereas

I have diff.\ co of \((1-x)^4=4(1-x)^3\) . __ __

On looking over my development again very

carefully, I am inclined to think that my solution**[146v] ** \(\frac{(1+x)^2}{(1-x)^5}\times(7+x)\), comes out right only because

I have managed to make a__nother__ blunder of

a sign in the course of the proofs, which has

corrected the first blunder. I therefore now

write on the o__the__r side of the paper, what I

think it should be. __ __

The note in page 2 I do not imagine to be of

any consequence. It is on ''rendering the Differentiation

''of complicated Functions sometimes much easier'' by

means of three Theorems from Maclaurin's Fluxions.

Certainly had I thought a little

more upon what I read some weeks ago, before

I wrote my last letter to you, I should not

have sent the question about \(du=\varphi(x)\times dx\) [flourishes at tops of stems of 'd's, here and after].

I ['must have' inserted] forgot exactly what a Differential Co-efficient

means, when I did so. But how is it then

that in your 1^{st} Chapter of the Differential Calculus''

there is no mention of the multiplication by \(dx\) ?''

I conclude that the real D__ifferential__ C__o-efficient__

is \(\frac{du}{dx}=\varphi(x)\), and that Peacock's solutions are''

not strictly speaking Differential Co-efficients. ''

I think pages 13 to 15 of your Elementary**[147r] ** Illustrations bear considerably upon the observations

in your letter, do they not? __ __

Your explanation of Euler's proof of the Binomial

Theorem is perfectly satisfactory to me. Unluckily

I have not any Book here which contains the

Theory of Combinations. I wanted to refer to

this when reading page 215, as I have forgotten

it in it's [\textit{sic}] particulars. However this can very

well wait a short time, & I have only to take

the Formula for Combinations for granted meanwhile.

The necessity of the truth of \((1+x)^n\times(1+x)^m=(1+x)^{n+m}\)

for __all__ values of \(n\) and \(m\), since it is true

when they are whole numbers, I shall probably

see more clearly at some further time. __ __

I can explain exactly what my

difficulty is in Chapter X. ''For instance, if we

''know that \(\varphi(xy)=x\times\varphi y\), supposing this always

''true, it is true when \(y=1\), which gives \(\varphi(x)=\)

''\( =x\times\varphi(1)\) . But \(\varphi(1)\) is an independent quantity,

''made by writing \(1\) instead of \(y\) in \(\varphi(y)\) . Let us

''call it \(c\) &c. ''

It is this substitution of \(1\) and of \(c\), and

c__onsequent__ ascertainment of the form which will**[147v] ** satisfy the equation, which is all dark to me.

It is ditto in lines 12, 13, & 14 from the top.

I understand quite well I believe from

''We have seen that if \(\varphi x=c^x\) &c'', all through

the next page.

That I do not comprehend at all the means of

deducing from a Functional Equation the form

which will satisfy it, is I think clear from

my being quite unable to solve the example

at the end of the Chapter ''Shew that the equation

''\( \varphi(x+y)+\varphi(x-y)=2\varphi x\times\varphi y\) is satisfied

''by \(\varphi x=\frac{1}{2}(a^x+a^{-x})\) ''. I have tried

several times, substituting first \(1\) for \(x\), then

\( 1\) for \(y\) . but I can make nothing whatever

of it, and I think it is evident there is

something that has preceded, which I have

not understood. The 2^{nd} example given for

practice ''Shew that \(\varphi(x+y)=\varphi x+\varphi y\) can

''have no other solution than \(\varphi x=ax\) '', I

have not attempted. __ __

I have a question to ask upon page 229.

''By extracting a sufficiently high root of \(z\), we**[148r] ** ''can bring \(z^m\) as near to \(1\) as we please, or

''make \(z^m-1\) as small as we please; that is

''(page 187) \(z^m-1\) __may be made as nearly equal__

''__to the sum of the whole series as we please__''. __ __

I cannot find what it is that is referred

to in page 187; and Secondly, it appears to me

somewhat of a contradiction that a quantity

\( z^m-1\) which can certainly be made a__s__ __small____as we please__ by the diminution of \(m\), should

become __as near__ __as we please__ to a f__ixed__

limit or sum (the \(\log z\) I conclude is the

sum of the series, referred to), since by continued

diminution the quantity \(z^m-1\) may become a

great deal l__ess__ than the sum of the Series, &

keep receding from it.

To return to Chapter X, there is one other

thing in it that I do not understand. Page

205, lines 5, 6, 7 from the bottom. It seems

to me fallacious to substitute first one value

\( 0\), for a letter; & then another value, let \(y=-x\),**[148v] ** in the same equation & in a manner at the

same time. How can the two suppositions

consist together at all. __ __

I go on well with the Trigonometry, &

have nearly finished the Number & Magnitude.

I think there is another Erratum in page

34 of the Trigonometry, line 13 from the bottom

\(=\frac{OM}{ON}\cdot\frac{ON}{OP}-\frac{NR}{NP}\cdot\frac{NP}{NO}\) &c

should be \(-\frac{NR}{NP}\cdot\frac{NP}{OP}\)

I am really ashamed to send you such

troublesome letters. __ __

Believe me

Yours most truly

A. A. Lovelace

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