which logarithms, multiplied by their respective numbers, give the following products :

1.999995025343512

1.999985122662298 both true to the last figure.

Therefore, the errors are

4974656488

and

14877337702

9902681214

and the difference of errors

Now, since only 6 additional figures are to be obtained, we may omit the three last figures in these errors; and state thus as difference of errors 9902681: difference of sup. 1:: error 4974656: the correction 502354, which, united to 3,59728, gives us the true value of x = - 3.59728502354.* 2. Given a2 = 2000, to find an approximate value of x. Ans. x4.82782263. 3. Given (6x)* = 96, to find the approximate value of x. Ans. x 1.8826432.

The binomial theorem is a general algebraical expression, or formula, by which any power, or root of a given quantity, consisting of two terms, is expanded into a series; the form of which, as it was first proposed by Newton, being as follows:

When P is the first term of the binomial, q the second

* The correct answer to this question has been first given by Doctor Adrain, in his edition of Hutton's Mathematics, who plainly proves that Hutton's answer, which is the same as Bonnycastle's, is incorrect. See Hutton's Mathematics, Vol. I. p. 263, N. Y. Edition.-ED.

term divided by the first,

m

n

the index of the power, or root,

and A, B, C, &c., the terms immediately preceding those in which they are first found, including their signs + or

Which theorem may be readily applied to any particular case, by substituting the numbers, or letters, in the given example, for P, Q, m, and n, in either of the above formulæ, and then finding the result according to the rule.*

EXAMPLES.

1. It is required to convert (a2 + x)* into an infinite series. or m = 1, and n = 2;

х m 1

a2, n

Here Pa2, Q = —

* This celebrated theorem, which is of the most extensive use in algebra and various other branches of analysis, may be otherwise expressed as follows:-

It may here also be observed, that if m be made to represent any whole, or fractional number, whether positive or negative, the first of these expressions may be exhibited in a more simple form,

m-1 m(m-1) m-2 m(m—1)(m—2) m-3

m

(a + x) = a + ma

x+

a

a

[m — (n − 1)] am-nöm

23

Where the last term is called the general term of the series, because if 1, 2, 3, 4, &c., be substituted successively for n, it will give all the rest.

Where the law of formation of the several terms of the series is sufficiently evident.

4. It is required to convert /9, or its equal (8 + 1)3, into

(8)3n

8
2

1/4

1 - 3 1

Whence,

B,

1

1

BQ =

X

2n

6

3.22 23

3.6.2

5. It is required to convert √2, or its equal ✓ (1 + 1), into an infinite series.

1

Ans. 1+2

1

3.5.7

+

+

&c.

3
3.5
2.4 2.4.6 2.4.6.8 2.4.6.8.10'

6. It is required to convert 5/7, or its equal (8 — 1)3,

into an infinite series.

1

Ans. 2

3.22

4

5.8

&c.

3.6.2 3.6.9.27 3.6.9.12.210

7. It is required to convert 5/240, or its equal (243 — 3)3,

into an infinite series.

1

Ans. 3

4

4.9

4.9.14

&c.

5.33 5.10.37 5.10.15.311 5.10.15.20.315'
1/1404

8. It is required to convert (ax)" into an infinite series.

3x3 3.524 2.4a2 2.4.6a2 2.4.6.8a1

1

9. It is required to convert (a+b)3 into an infinite series.

2.5.864

2

OC

+ &c.

2a

c.} }

Ans. at{1

1.

&c.

}

3.6a2 3.6.9a3 3.6.9.12a, &c.}

10. It is required to convert (a

1
4

b) into an infinite series.

3.7.11b4

11. It is required to convert (a + x) into an infinite series.

12. It is required to convert (1 - x)5 into an infinite series. 2x 2.3x2 2.3.8x3 2.3.8.13x*