log, (x+a+b)=log,(a− a + b) + log,(x + a) + log.(x+b}

(Delambre, Introd. aux Tables de Borda.)

♣ log, (1+x) = (1 + x) ̄”. S„xTM S ̧ ( − 1)m-r

m

7° 1. (m − r + 1)`

log, (x+2)- 2 log, (+1) + 2 log, (x-1)-log, (≈ → 2)

log, (x+5)-log, (x+4) — log, (x+3)+2 log,≈

=2

−log (–3)– log ( −4)+log (–5)

— log, (x+6) + 2 log, (x+5)—log, (x+3)—log, (x+2)

18

18

3

=2 {~2+10x2+252-18 + + (+103 + 253-18) + &c.}.

(Lavernède, Ann. de Math.)

To find the logarithm of any number from the tabulated logarithm of that number without the last digit;

CONTINUED FRACTIONS.

(30.) The general form of a continued fraction is

n is a finite quantity only when is rational.

b

2

If p, q are any numbers whatever < A, B, respectively,

m>

then

A2m+2A2m+1, &c.
B2m+2 B2m+1

Acm +2

B2 m + 2

form an increasing series, each term of which is <

a

=n, then

A2m-1+A2 m
B2m-1+ B2 m

A2m-1+2 Acm
Bem-1+2 Bem

&c.

form a decreasing series, each term of which is >

a

No rational fraction whose denominator lies between the denominators of any two adjacent terms of these series can be inserted between those terms.

To approximate to the value of a fraction, whose numerator and denominator are high numbers: obtain c1, C, C3, &c. by actual division, and the series of converging fractions may be obtained from equation (1).

A continued fraction may frequently be simplified by the introduction of negative quotients; for

(G. Chap. xxxi; G. A. 12-20; E. Add. Art 1; Lagrange, Equ. Num. Chap. vi, Art. 1, 3.)

viding the denominator by the numerator and taking one term of the quotient, we have