## Folios 108-109: AAL to ADM

Ockham

Sunday. 6^{th} July

Dear M^{r} De Morgan. It is perhaps unfair of

me to write again with a batch of observations

& enquiries, before you have had time to reply

to the previous one. But I am so anxious to

get the p__resent__ matters off my mind, that I

cannot resist dispatching them by this post.

I have __two__ series of observations to send,

one relating to the passage from page 107, (line

8 from the bottom), to the last line of page 108;

the other to certain f__ormer__ passages in pages 99,

100 & 103, concerning which some questions have

suddenly occurred to me quite recently.

I shall begin with pages 107 & 108: I enclose

you __my__ development & explanation of \(\int\frac{x^ndx}{\sqrt{a^2-x^2}}\) up

to \(\int\frac{x^ndx}{\sqrt{a^2-x^2}}=-x^{n-1}\sqrt{a^2-x^2}+(n-1)a^2\int\frac{x^{n-2}dx}{\sqrt{a^2-x^2}}-(n-1)\int\frac{x^ndx}{\sqrt{a^2-x^2}}\)

from which you will judge if I understand it

so far. I should tell you that I have not yet

begun page 109.

I will now ask two or three questions : 1^{stly} : page 107,**[108v] ** (line 3 from the bottom): ''the diff. co of \(a^2-x^2\) being__\(-2xdx)\)__

&c''. This surely is incorrect; & you will see that

in __my__ development I have written it as I fancy

it should be ''being \(=(-2x)\), &c''

2^{ndly} : page 108, (lines 8, 9, 10 form the top) : ''By \(\int UdV\)

''we mean \(\cdots\cdots\cdots\cdots\cdot\) __p. 102, where__

''__the values of \(\Delta V\) in the several terms are__

''__different, but comminuent__.'' I do not see that

this is a case of page 102 rather than of page 100;

in other words, that the increments in this

Integration are ''__unequal__ __but comminuent__''.

3^{rdly} : the subtraction in line 15 from the top, of

\( (n-1)x^{n-2}\times dx\) for \(d.(-x^{n-1})\) appears to me quite

inconsistent with the i__nseparab__le i__ndivisible__

nature of a diff. co.

4^{thly} : Lines 9, 10 from the bottom, ''We have therefore

''&c \(\cdots\cdots\cdots\cdot\cdot\) that of \(\sqrt{a^2-x^2}x^{n-2}dx\) ''.

Admitted, most fully. But \(\int\sqrt{a^2-x^2}x^{n-2}dx\) does

not answer exactly to \(\int vdx\) or \(\int\sqrt{v}d2u\), and

therefore it appears to me that this Integration is

not strictly an example of lines 5, 6, 7 (from the bottom)

of page 107. You will remember that \(-x^{n-1}\) was \(=2V\),

therefore the \(x^{n-2}\) of \((\sqrt{a^2-x^2}x^{n-2})\) is equal to \((-1)\times\frac{2V}{x}\)

or \(\frac{-1}{x}.2V\) . So that another factor \(\frac{-1}{x}\) enters into the**[109r] ** expression which __was__, as I understand it, to answer

strictly to \(\int vdu\) or \(\int\sqrt{v}d2u\)

5^{thly} (line 5 from the bottom) page 108: I think there

is an Erratum. Surely \(\int\left(\frac{a^2x^{n-2}}{\sqrt{a^2-x^2}}-\frac{x^ndx}{\sqrt{a^2-x^2}}\right)\)

ought to be \(\int\left(\frac{a^2x^{n-2}dx}{\sqrt{a^2-x^2}}-\frac{x^ndx}{\sqrt{a^2-x^2}}\right)\)

I don't know if my pencil Sheet enclosed

will be very intelligible, for it is as I wrote

it down a__t the time__ quite roughly, & without

any very great amplitude or method.

I now proceed to my series of observations

relating to f__ormer__ pages, beginning with page 102,

(line 10 from the bottom)

''\( +\) less than \(nC\frac{\Omega^2}{2}\), or \(Ch\frac{\Omega}{2}\) '';

now in order to ['effect' inserted] the substitution of \(Ch\frac{\Omega}{2}\) for \(nC\frac{\Omega^2}{2}\)

the latter is resolved into \(C.n\Omega.frac{\Omega}{2}\), & ['for' inserted] \(n\Omega\) is

substituted \(h\) . But by the hypothesis & conditions,

\( h\) must be __less__ than \(n\Omega\) . Therefore it does not

necessarily follow that that which is proved l__ess__ than

\( nC\frac{\Omega^2}{2}\), is also l__ess__ than \(Ch\frac{\Omega}{2}\) . You see

my objection.

2^{ndly} . See __Note__ to page 102 : If the ''__completion__ of the ['first' inserted] Series''**[109v] ** in t__hi__s page is unnecessary, surely it is equally

unnecessarily in the first Series of page 100; for the

same observation applies to the l__atter__ as to the

f__ormer__, viz : t__hat the additional term is____comminuent with \(w\) __.

3^{rdly} . See page 99 (line 8 from the bottom) :

''\( \int_a^x\varphi x.dx=(x-a)a+\frac{(x-a)^2}{2}=\frac{x^2-a^2}{2}\) ''

This is another form of \(\int_a^{a+h}xdx=ha+\frac{h^2}{2}\) 8 lines

above, & of the __limit of the summation__ for \(\varphi x=x\) in

the previous page. And therefore it appears to me

that it ought to be

\(\int_a^xx.dx=(x-a)a+\frac{(x-a)^2}{2}=\frac{x^2-a^2}{2}\)

I do not see what business __\( \varphi x\) __ has.

Now a__t last__, I have done troubling you.

I am very anxious on __all__ these points.

With many apologies, believe me

Yours very truly

A. A. Lovelace

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