## Folios 142-143: AAL to ADM

Ashley-Combe

Sunday. 21^{st} Nov^{r} ['1841' inserted by later reader]

Mr Dear M^{r} De Morgan. [something crossed out]

I said __Wed ^{dy}__. At least I

meant to do so. On T

__uesda__y I have already an

engagement in the morning. Perhaps you have

written T

__uesda__y by mistake. But of you cannot

come on Wed

^{dy}, then I must put off my T

__uesday's__

engagement, that I may see you t

__hen__. If it is the

same to you however, I should much prefer

__Wed__.

^{dy}Can you kindly give me one line tomorrow to

say which it is to be. I shall get ['it' inserted] in the evening in

S^{t}James' Sq^{re} . Now I proceed to business :

1^{stly} : You have mistaken my i__ntentions__ I think about

the formulae of pages 155, 156. My enclosures 1 & 2

will explain.

2^{ndly} . Enclosure 3 contains the demonstration of ''Exercise''

page 159

3^{dly} . Enclosure 4 \( . . . . . . . . . . \) ''Exercise''

page 158

4^{thly} : About the Constant in page 141 : I still am**[142v] ** unsatisfied. I perfectly understand that ''__any value''__

consists with everything __in page 141__. The principle

is I conceive exactly the same as that by which in

page 149, \(y\) is made \(=a+\sin.x\) instead of \(y=\sin x\) .

I only mean that this result seems i__nconsistent__

with __page 116__ when it is shown that the Constant

must \(=\frac{w}{2}\) .

5thly : page 161, (line 14 from the top):

\(\varphi''(x+\theta h,y+k)-\varphi''(x+\theta h,y)=\varphi_1^{('')}(x+\theta h,y+vk).k\)

\(v<1\)

Why is __\( v\)__ introduced at all?

I have as follows :

\( \frac{\varphi''(x+\theta h,y+k)-\varphi''(x+\theta h,y)}{k}=\varphi_1^{('')}(x+\theta h,y)\)

__if__ __\( k\) diminishes without limit__; (\( k\) being \(=\Delta y\) )

or \(\varphi''(x+\theta h,y+k)-\varphi''(x+\theta h,y)=\varphi_1^{('')}(x+\theta h,y)k\)

But I do not see how __\( v\)__ comes in.

6^{thly} : I have several remarks to make altogether

on the Article O__peration__. I will only now subjoin

two. I believe on the whole that I understand the

Article very well.

See page 443, at the top, 2^{nd} Column) :

\(\varphi^2+2\varphi\psi+\psi^2\), or \((x^2)^2+2(x^3)^2+(x^3)^3\)

should be __it appears to me__ \(\varphi^2+2\varphi\psi+\psi^2\), or \((x^2)^2+2x^3.x^3+(x^3)^2\)

or \((x^2)^2+2(x^3)^2+(x^3)^2\)

\(=(x^2)^2+3(x^3)^2\)

**[143r] ** I only allude to \((x^3)^3\), instead of \((x^3)^2\) as __I__ make it.

See page 444, at the bottom, (\) 2^\textup{nd}\) column) :

''Where \(B_0\), \(B_1\), &c are the values of \(fy\) and its

''successive diff-co's [*sic*] __when \(y=0\)__, &c, &c''

Surely it should be __when \(y=1\)__.

The same as when immediately afterwards, (see page

445, 1^{st }column, at the top), in developping [*sic*] \((2+\Delta)^{-1}\varphi x\);

\( B_0\), \(B_1\) &c are the values of \(fy\) & its Co-efficients__when \(y=2\)__, &c, &c.

I have referred to __Numbers of Bernoulli__

& to __Differences of Nothing__; in consequence of

reading this Article __Operation__. And find that

I must read that on __Series__ also.

I left off at page 165 of the Calculus; &

suppose that I may now resume it; (when I return

here that is).

I will not trouble you further in this letter.

But I have a f__ormidable list__ of s__mall matters__

down, against I see you.

Yours most sincerely

A. A. Lovelace

## About this document

All Ada Lovelace manuscript images on the

Clay Mathematics Institute website are

© 2015 The Lovelace Byron Papers,

reproduced by permission of

Pollinger Limited. To re-use them in

any form, please apply to

katyloffman@pollingerltd.com.

The LaTeX transcripts of the letters

were made by Christopher Hollings

(christopher.hollings@maths.ox.ac.uk).

Their re-use in any form requires his

permission, and is subject to the

rights reserved to the owner of

The Lovelace Byron Papers.

Bodleian Library, Oxford, UK

Dep. Lovelace Byron