}

which logarithms, multiplied by their respective numbers,
give the following products :-
1.999995025343512

both true to the last figure.
1.999985122662298
Therefore, the errors are 4974656488

and 14877337702 and the difference of errors 9902681214 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 = -.3.59728512354. 2. Given 2000, to find an approximate value of x.

Ans. x = 4.82782263. 3. Given (6x)* = 96, to find the approximate value of x.

Ans, X = 1.8826432. 4. Given wito 123456789; to find the value of x.

Ans. 8.6400268. 5. Given 2012 (2x — 2:2), to find the value of x.

Ans. X - 1.747933.

OF THE BINOMIAL THEOREM.

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:

m

т

(P + pa)"=p” [1 + 2 + (27") *+ (1.7")

)"
, &c.]

m т

2n

3n

DQ, &c.

4n 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.

n

term divided by the first, 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 (az + x)# into an infinite series.

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

a [1+ &c.]

)+

m mton

m mtnm tan

2t
2n
a +31

2m

Зп natal

at +3

n

m

2

n

ma
2 mm ton a- 20

m mtn m+2n lata 2a

3], a atal 2. vatal

2n In latal &c.

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(m-1) 1-2 m(n-1)(M—2) m-3: (a +x)" =a +-ma x+

22+
1.2

1.2.3
m(m-1)(m-2).. [m-(n-1)] ämnem

1.2.3.4 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.

m

x2

Х

т n

1 2
BQ=

Х X Х
2n
4 20

a

2.4a3
m
- 2n
1-4

323
CQ

X
3n

6
2.403

a2 2.4.6a5
m
3n
1 6 3203

x х:

3.5804
DQ

+
x

Е, 4n

8
2.4.6a5 aa

2.4.6.807
т
4n 1-8

3.5.24

3.5.7205
EQ
X

X 5n

10

2.4.6.807 a2 2.4.6.8.10a9 &c.

&c.

&c.
Therefore (a? + x)2 =
323
3.524

3.5.735
+

&c. 2.403 2.4.6a5 2.4.6.87 2.4.6.8.10 Where the law of formation of the several terms of the series is sufficiently evident.

1 2. It is required to convert

or its equal (a + b)--,

(a+b) into an infinite series.

m Here P = a, Q=

and
2, or m= -2,

2, and n=1:

n whence

22

十分

+

2a

X

Х

a

2n

m m

2
1
6

26
AQ

B, 12

a m n

2 - 1 26 b 352
BQ =

X
X

C,
2n

2

a
a

d4
2 -2 352

463
CQ
x X

D,
3n

3

a4
a

25
m
3n

2-3 403 b 564
DRE

Х

E,
4n

5
a

ab
&c.

&c.
1
1 26 362 463

574 Consequently,

+

&c. (a + b) a a3

a4 a5

aa 3. It is required to convert

or its equal aa

(az x}}' ( as - x) into an infinite series.

Here

1 p=a?, Q

and

or m ma2

- 1, and n= N

2

1

Х

т

1 1
AQ
X-X

B,
n

a

a2 2a3
m n
1--2 #

ec

3х2
BQ -
X

C.
2n

4

2a3 a2 2.4a
m
2n
1-4 3:2

3.5703
CQ
Х Х

D, 3n

6
2.425

aa 2.4.647
m
3n

1 -
3.523

C

3.5.724
DQ

Х
X

E, 4n

8 2.4.6a7 a2 2.4.6.80
&c.
&c.

&c.
Therefore,
1
1 1
3 2c2 3.5

003 3.5.7
+
+

+ L ? a 2.4 las 2.4.6

2.4.6.8 i

3

() +, de

十,

4. It is required to convert $79, or its equal (8 + 1)), into

n

in

in

2 = A,

m

n

n

an infinite series.

1

1 Here p= 8,Q

and

or m= 1 and n=3;

3'

Whence,

1 рп (8)3n= 8

1 2 1 1 AQ Xi? х

B,

23 3.22 m 1 3 1 1

1
BQ

X
x

4 = C,
an
6 3.22

3.6.24
т
2n
1 - 6

1
1

5
CR

Х 3n 9

23 3.6.9.27
n
3n 1 - 9

5
1

5.8
DQ =
X
х

E 12 3.6.9.27 23 3.6.9.12.210 m -4n 1-12

5.8

1

5.8.11 EQ=

Therefore, /9=
1
1
5
5.8

5.8.11
2+
+

+ 3.22 3.6.24 3.6.9.27 3.6.9.12.210 3.6.9.12.15.2139 &c.

5. It is required to convert 72, or its equal ✓ (1+1), into an infinite series.

1 1

3
3.5

3.5.7
.
+

+

&c. 2

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)}, into an infinite series.

1
1
5

5.8
Ans. 2 -

3.22 3.6.2 3.6.9.27 7. It is required to convert V 240, or its equal (243 — 3)), into an infinite series.

1
4
4.9

4.9.14 Ans. 3

&c. 5.33 5.10.37 5.10.15.311 5.10.15.20.315

2 를 8. It is required to convert (a + x)" into an infinite series.

202 3.23 3.5x4
1+
+

+ &c.
2α
2.4a

2.4.6a3 2.4.6.8a4

3.6.9.12.210 -, &c.

Ans. a

*{1

&c.

1

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

282 2.563 2.5.864
士

+, &c. За 3.6a2 3.6.903 3.6.9.12a4

b.

Ans. a

} 10. It is required to convert (a - b)# into an infinite series.

1+

1

b: 362

3.763 3.7.1184 Ans. a4 1

&c. 4a 4.8a2 4.8.123 4.8.12.1624

을 11. It is required to convert (a + x) into an infinite series. 13 x2 4203 4.724

4.7.10.25 1+

+

За 9a2 923 92.12a4 92.12.1505

Ans. a

+

2x

,.&c.

.

12. It is required to convert (1 - x)into an infinite series.

2.3x2 2.3.8x3 2.3.8.1324
Ans. 1
5 5.10 5.10.15 5.10.15.20

1 13. It is required to convert

or its equal

(a + x) (a + x) into an infinite series.

3х2 3.5003 3.5.724 Ans.

2.4.6a3

2.4.6.844