## Folios 77-83: AAL to ADM

Ockham

Sunday. 10^{th} Jan ^{y}

Dear M^{r} De Morgan. I send

you the ['Series' crossed out] Analysis of the

new Series I received in your

letter yesterday morning. I

believe I have made it out

quite correctly. In fact, the

Verification at the end proves

this. But, owing to a

carelessness in my ['first' inserted] inspection

of it, I have had the

trouble & advantage of analysing__two__ Series. I glanced too

hastily at it, & did not

observe that the factors of the**[77v] ** Denominators (of the Co-efficients),

are not p__owers__ of \(2\), but

simply __multiples__ of \(2\) . __ __

If you will open my Sheet,

you will find on the i__nside__

my analysis of the Series I at

first mistook your's [\textit{sic}] for;

and I am not sorry this

has happened. I believe

b__oth__ are correctly made out. __ __

You kindly request me

if I do not understand the

erasure in the former ['small' inserted] paper,

again to return it &c. Now

I do __not__ agree to it; & ['I' inserted] still

fancy that we are in f__act__

meaning exactly the same

thing, only that y__ou__ are**[78r] ** speaking of the __\( n\)__^{th} Term, & __I__

of the __\( n+1\) ^{th}__.

__For__

convenience of reference I again

return the former large paper

(& at any rate H. M's

Post-Office will benefit).

I quite understand that

\( \frac{1}{4}+\frac{1}{2}\sqrt{10^6+\frac{1}{4}+a}\) is

__less__than

\( 501\) . Therefore as

__\( n\)__is

__the next whole number__above

this fractional expression, \(x=\)

\( =501\) . But

__\( n\)__is

__not__the

Term sought; the unknown

term to be determined being

by the conditions of the

Hypothesis & Demonstrations,

__\( n+1\)__, & therefore \(=502\) .

And if you will examine

**[78v]**your own ['former' inserted] Verification, you

will see that you there

determine the Term at which

Convergence begins, to be

\( A_{502}\), or the 502

^{nd}Term,

which agrees with

__my__

result \(n+1=502\) .

I think it is quite clear

that we are both agreed,

but that you were not aware

at the moment you made

the erasure that I was

not speaking of the

__next__

__whole number__above \(\frac{1}{4}+\frac{1}{2}\sqrt{10^6+\frac{1}{4}+a}\)

but of the

__next but one__above

it.

So much for the three Series:

Now I must go to other**[79r] ** matters. I am indeed sending

you a Budget. __ __

I have been working hard

at the Differential Calculus,

& am putting together some

remarks upon Differential

Co-efficients (which in due

time will travel up to

Town for your approbation),

but in the progress of which

I am interrupted by a

slight objection to an old

matter of Demonstration,

which did not occur to me

at the time I was studying

it before, & sent you a

paper upon it ['from Ashley' inserted]. In the

course of the observations I**[79v] ** am now writing, I have ['had' inserted]

occasion to refer to the old

['general' inserted]} Demonstration, (pages 46 & 47

of your Differential Calculus),

as to the finite existence of

a Differential Co-efficient

for all Functions of \(x\); &

a slight flaw, or rather what__appears__ to __me__ a flaw, in the

conclusions drawn, has occurred

to me. It is most clearly

proved that, \(\theta\) being supposed

to diminish without limit,

the Fractions \(Q_1\), \(Q_2\) &c__must__ have finite limit, for__some__ value or other at all

events of \(n\theta\) or \(h\) . But the

fractions in question do not**[80r] ** appear to me to be strictly

speaking analogous to \(\frac{\Delta u}{\Delta x}\),

except the f__irs__t of them \(\frac{\varphi(a+\theta)-\varphi a}{\theta}\)

and the la__st__ of them \(\frac{\varphi(a+n\theta)-\varphi(a+\overline{n-1}\theta)}{\theta}\),

and for this reason.

In the expansion \(\frac{\Delta u}{\Delta x}\) or

\( \frac{\varphi(x+\theta)-\varphi x}{\theta}\), as \(\theta\) alters__\( x\) __ does n__ot__ alter, but remains

the same. In these fractions

on the contrary, which all

have the form \(\frac{\varphi(a+k\theta)-\varphi(a+\overline{k-1}\theta)}{\theta}\)

and in which \(a+\overline{k-1}\theta\) [bar over \(k-1\) should have little downward-pointing hooks at the ends]

stands for the \(x\) of the

expression \(\frac{\Delta u}{\Delta x}\) or \(\frac{\varphi(x+\theta)-\varphi x}{\theta}\),

not only does \(\theta\) alter, but

from the conditions of the

Hypothesis & Demonstrations, \(\overline{k-1}\theta\) **[80v] ** & consequently \(a+\overline{k-1}\ \theta\) must

likewise alter along with \(\theta\) .

There is therefore a d__ouble__

alteration in value going on

simultaneously, which appears

to me to make the Case quite

a different one from that

of \(\frac{\Delta u}{\Delta x}\), & consequently to

invalidate all conclusions

deduced from the f__ormer__ with

respect to the l__atter__.

The validity of the Conclusions

with respect to the fractions

\( Q_1\), \(Q_2\) &c, you understand

I do not question. What I

question is the analogy between

these Fractions & the Fraction

\( \frac{\Delta u}{\Delta x}\) or \(\frac{\varphi(x+\theta)-\varphi x}{\theta}\) ['of' inserted] which**[81r] ** l__atte__r it is required to

investigate the Limits.

I also have another slight

objection to make, not to the

e__xtent__ of Conclusions established

respecting the Fractions \(Q_1\), \(Q_2\)

&c having finite limits,

but to the Conclusions on that

point not going far enough,

not going as far as they

might : ''either these are

''finite limits, or some increase

''without limit and the rest

''diminish without limit; if the

''latter, we shall have two

''contiguous fractions, __one of which__

''__is as small as we please, and__

''__the other as great as we please__,

''&c, &c, a phenomenon which

''which [\textit{sic}] can only be true when**[81v] ** ''\( Q_k\) is the fraction which is

''near to some s__ingular__ value

''of the Fraction, & cannot be

''true of ordinary & calculable

''values of it &c.'' Now it

appears to me that in __no__

p__ossible__ case c__ould__ such a

phenomenon as this be true,

when we consider how the

fractions are successively

formed one out of the others

by the substitution of \(a+\theta\) for

\( a\), \(\theta\) too being as small as

we please. I therefore think

it might have been concluded

at once that there __must__

a__lways__ be fi__nite__ l__imits__ to

the Fractions \(Q_1\), \(Q_2\) &c,**[82r] ** and this __whatever \(k\) or \(n\theta\) __

may be. I suppose it is not

so, but I cannot conceive

the Case in which it c__ould__

be otherwise. __ __

I do not know if in writing

upon my two difficulties in

these pages 46, 47, 48, I have

expressed my objections (especially

in the f__ormer__ case of the

fractions \(Q_1\), \(Q_2\) not being

similar to \(\frac{\Delta u}{\Delta x}\) ) with the

clearness necessary to enable

you to answer them, or indeed

to apprehend the precise points

which I dispute. It is not

always easy to write upon

these things, & at best one

must be lengthy. I shall be**[82v] ** exceedingly obliged if you will

also tell ['me' inserted] whether a little

Demonstration I enclose as to

the Differential Co-efficient

of \(x^n\), is valid. It appears

to me perfectly so; & if it is,

I think I prefer it to your's [\textit{sic}]

in page 55. It strikes me

as having the advantage in

simplicity, & in referring to

fewer ['requisite' inserted] p__revious__ Propositions.

I h__ave__ a__nother__

enquiry to make, respecting

something that has lately

occurred to me as to the

Demonstration of the Logarithmic

& Exponential Series in

your Algebra, but the real

truth is I am quite as__hamed__**[83r] ** to send any more; so will

at least defer this. __ __

I am afraid you will indeed

say that the office of my

mathematical Counsellor or

Prime-Minister, is no joke.

I am much pleased to

find how very well I stand

w__ork__, & how my powers of

attention & continued effort

i__ncrease__. I am never so

happy as when I am

really engaged in good

earnest; & it makes me most

wonderfully cheerful & merry

at o__ther__ times, which is

curious & very satisfactory. __ __

What w__ill__ you say when**[83v] ** you open this packet? __ __

Pray do not be v__ery__ angry,

& exclaim that it really is

too bad. __ __

Yours most trul

nbsp;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