## Folio 37: ADM to AAL

**[37r] ** My dear Lady Lovelace

If you look back to

page 48, you will there

see that

\( \frac{a+a'+a''+\cdots}{b+b'+b''+\cdots}\) always lies between

the greatest & least of \(\frac{a}{b}\) [,] \(\frac{a'}{b'}\) &c__whatever the signs__ of \(a\) [,] \(a'\) &c

may be, provided that \(b\), \(b'\)

&c are all of one sign. That

is the reason why \(\varphi x\) need not

continually increase or decrease

in the next chapter

The paper you have sent me

is correct. In page 70, the

reasons are given for**[37v] ** avoiding the common proof

of Taylor's Theorem, and 71 &c

contains the amended proof.

Of \(\frac{\varphi(a+h)}{\psi(a+h)}=\frac{\varphi'(a+\theta h)}{\psi'(a+\theta h}\) [bracket missing in last denominator] it

cannot only be said that

it turns out useful. A

beginner can hardly see

why a diff^{l} coeff^{t} itself

should be of any use

Yours truly

__ADeMorgan__

Feb^{y} 6/41

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