## Folios 104-105: AAL to ADM

Ockham

Monday. 22^{nd} Feb^{y}

Dear M^{r} De Morgan. The reply to o__ne__ of

my queries to you, dispatched on Friday, has

I believe just occurred to me. Probably

this letter will cross one from you tonight,

but the remaining points continue still

unsolved, so that I shall be equally glad

if I do receive an answer tomorrow morning.

The difficulty I have solved is

the one relating to the law for the Co-efficients

of ['the series for' inserted] \(\Delta u_n\) . I remarked that the law for the

Co-efficients of the Series for \(u_n\) being

ascertained, did not ascertain those for \(\Delta u_n\)

as a necessary consequence. But I see I am

wrong. If a Series is obtained for \(u_n\),**[104v--105r] ** we have only in order to obtain one for

\( \Delta u_n\), to take the Difference of __every term__

['of the' crossed out], (that is of the __variable__ part of every

term), of the Series for \(u_n\) . Thus,

\( u_n\) being \(=u+n\Delta u+n\frac{n-1}{2}\Delta^2u+\cdot\cdots+n\frac{n_1}{2}\Delta^{n-2}u+n\Delta^{n-1}u+\Delta^nu\)

\( \Delta u_n\) must \(=\Delta u+\Delta(n\Delta u)+\Delta\left(n\frac{n-1}{2}\Delta^2u\right)+\cdots+\Delta\left(n\frac{n-1}{2}\Delta^{n-2}u\right)+\Delta(n\Delta^{n-1}u)+\Delta(\Delta^nu)\)

\(=\Delta u+n\Delta^2u+n\frac{n-1}{2}\Delta^3u+\cdots+n\frac{n-1}{2}\Delta^{n-1}u+n\Delta^nu+\Delta^{n+1}u\)

Whence &c, &c. I think this is quite sufficiently obvious.

But I now have another query to put, in the place of the one I have just disposed

of, relating to the development in page 83,

\(\Delta u=amx^{m-1}+Ax^{m-2}+\cdots\cdot\cdot+Px+Q\)

and in which I cannot help thinking there is a mistake ['in the first Term' inserted]: I make out that

it ought to be

\(\Delta u=am\omega x^{m-1}+Ax^{m-2}+\cdots\cdot\)

But I enclose my developments and observations therefore, on a longer & more

convenient sheet. I will only add here, that we move to Town on

Thursday; and that I should much like to spend Sunday Evening with

M^{r} De Morgan & you, if this arrangement is suitable & agreeable to you. I**[105v] ** should arrive as usual, about 8 o'clock

I believe I shall have by the end of this

week several papers ready to discuss.

You see I do not __waste__ my time, at

any rate; and I only hope that I am

not the means of wasting y__ours__ either.

Believe me

Yours very truly

A. A. Lovelace

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