## Folios 152-153: AAL to ADM

Ockham Park

Surrey

Dear M^{r} De Morgan. I am indeed extremely

obliged to you for all your late communications.

In two or three days more, I shall have

several observations & to send you in reply

to some of them.

My object in writing today, is to make another

enquiry concerning the substitution of \(\varphi_1(a+h)-\varphi_1a\)

for \(\varphi(a+h)-\varphi a\) in page 100, for I perceive

on carefully examining the passage, that I do

not quite understand it

\(\varphi_1x=\varphi x+C\), which ['last side' inserted] means t__he Primitive Function__ ['of \(\varphi'x\) ' inserted]

and the P__rimitive Function__ means the F__unction____which differentiated gives \(\varphi'x\) __

Therefore \(\varphi_1(a+h)=\varphi(a+h)+C\)

And \(\varphi_1a=\varphi a+C\)

Consequently \(\varphi_1(a+h)-\varphi_1a=\varphi(a+h)+C-(\varphi a+C)\)

\(=\varphi(a+h)-\varphi a\)

**[152v] ** This is __my__ version of it. But y__ou__ tell me,

\( \varphi_1(a+h)=\varphi(a+h)+C\)

\(\varphi_1a=\varphi a\), (which ought to be __I__ say

\(\varphi_1a=\varphi a+C\) )

From which we should have,

\(\varphi_1(a+h)-\varphi_1a=\varphi(a+h)+C-\varphi a\)

Consequently \(\varphi_1(a+h)-\varphi_1a\) is n__ot__ equal

to \(\varphi(a+h)-\varphi a\) as is __required____to be proved__, but is \(=\varphi(a+h)-\varphi a+C\) .

I c__annot__ unravel this at all.

Second^{ly} : [something crossed out] I do not see why the __Indefinite____Integral__ o__nly__ is \(=\varphi x+C=\) Primitive Function.

of \(\varphi'x\)

The argument at the top of page 101 seems

to [something crossed out] me to apply eq__ually__ to the __Definite Integral__

As follows : It is proved that

\(\varphi_1(a+h)-\varphi_1a=\int_a^{a+h}\varphi'x.dx\)

\( \varphi_1a\) is just as much h__ere__ an __arbitrary Constan__t

as it is in \(\varphi_1x-\varphi_1a=\int_a^x\varphi'x.dx\)

Therefore \(\int_a^{a+h}\varphi'x.dx=\varphi_1(a+h)+C_1\)

\(=\varphi(a+h)+C+C_1\)

\(=\varphi(a+h)+\text{an arbitrary Constant}\)

**[153r] ** just as with \(\varphi_1x-\varphi_1a=\int_a^x\varphi'x.dx\)

Thirdly : With respect ['to' inserted] the assumption that when

\( a\) is arbitrary, then any function of \(a\), say \(\varphi a\),

is also a__rbitrary__ or __may be anything we please__,

seems to me not always valid.

For instance if \(\varphi a=a^0\), it must be

always \(=1\) . We may assume \(a=\) anything we

like, but \(\varphi a\) will not in this case be

arbitrary.

It is curious how many little things I [something crossed out] discover in

this Chapter, which in looking back upon

them, I find I have only __half-understood__.

I shall be exceedingly obliged, if you

can answer t__hese__ points s__oon__; I think a

word almost may explain them, & they rather

annoy me.

Believe me, with many thanks

Yours very truly

A. A. Lovelace

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