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Folios 104-105: AAL to ADM



 Monday.  22nd Feby 

 Dear Mr De Morgan.  The reply to one 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
Mr 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 yours either.
Believe me

 Yours very truly

 A. A. Lovelace

About this document

Date of authorship: 

22 Feb 1841

Holding institution: 

Bodleian Library, Oxford, UK


Dep. Lovelace Byron

Box 170