+

31.23 w(a + ab +63) + &c.

2- 2-3
log, x=

&c.
1
2

3

n 1 lo

2 1 n(n-1)

1
+

+
3 1:2 1.2

13
n(n-1) 1

n(n-1)(n-2) 1 +

* + &c. 1. 2 2

1. 2 3 log. (x+a+b)=log (a - a +b) + log (x + a) + log.(x+6)

- log. (x - a)-log(x — b) ab(a+b)

1 ab(a+b)

+ X (a2 + ab + b)

(Delambre, Introd. aux Tables de Borda.)

n

B

*2

+ &c.}.

log, x = 2 log. (x— 1) – log. (x - 2)

2

1

+ &c. 3(2002 – 4 &– 1)3

log, « = Sm( - 1)m-1

.*---*

m

& log, (1 + v)=(1 + x) –". S.mm*5,(-1)m-

p-1.(m— p+1) ab(a + b) v 25.m G

— «(a2 + ab + 5%)

2m - 1

log. (x + 2) – 2 log. (x + 1) + 2 log. (x– 1) - log. (x - 2)

2

2
- 2
+

a
23

28 - 3«

log. (x+5) – log. (x + 4) – log. (w + 3) + 2 logo

- log. (x-3) - log. (2 — 4) + log. (20-5)
72

1

72
3 - 25 x° +72

+

2

- log. (x+6) + 2 log. (20+5) – log: (x + 3) – log: (x + 2)
+ 2 log. 0 -- log. (x - 1)
18

18
+10.003 +252_18

+ &c. 20* + 10.03 +250?– 18

(Lavernède, Ann. de Math.)

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

log 10 (10 x + a) = 1+ log x

+ flogue) (1702- (102) + $(163) + &c.}.

(Cagnoli, Trig.)

CONTINUED FRACTIONS.

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

a

which

may
for convenience be thus expressed:
1 1 1

1
5

Ci+ Cg +' Cg +

{...-C + by inverting the order of the terms.

+ cn

n is a finite quantity only when is rational.

If p, q are any numbers whatever < Am, Bon, respectively,

a

form an increasing series, each term of which is <

a

form a decreasing series, each term of which is >

ő

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 c, 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.)

b + b + b, + &c. To reduce

to a continued fraction: dia ta, + ag + &c. viding the denominator by the numerator and taking one term of the quotient, we have

Assume a, - ,=c, - 6.0,=c, &c. = &c.,