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2.5

8.11

12.15

+

2762

1
[2(41 - 2762)] ]

6.9 4.1 – 2772) A +
2712
14.17 2762

2762
403 2772 18.21 (442–2772)

2)e+

24.27 4¢* - 2772)

20.23

B

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2

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3.

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4

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2 =

{ 1+25 ()2

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&c.}. But

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B

2
2.5

2.5.8
2s

3.6 12s

3.6:9 2s, 3.6.9.12 13 &c.

And consequently, by taking the latter of these series from the former, and making the first term of the result a multiplier, we shall have

2.5 10 2.5.8.11 76 2.5.8.11.14.17 b
2+

4+
6s
6.9
6.9.12.15
6.9.12.15.18.21

2762 &c. But since s =V (162— ,, as), we shall have

2. 2762-4a3 2762

2762

2051

20

and 4a3 276212s 4a3

-2762 ,

6s 20

26 63/(h2 - 1,3) V {2(4a3-2762)

Whence, if these values be substituted for their equals in the last series, there will arise the above expression for the first root of the equation. And if we put the root thus found, or its equivalent expression 十

a3- 36 we shall have, according to the formula before given for the other two roots,

V-3 士 I

13. 2

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1

于2

"]}. Or, taking, as before, v (?12 — 24, ba) =s, and simplifying

V the result, x=

2
2s

2s Whence, by extracting the roots of the righthand side of this equation, there will arise 6

2
2.5 5

2.5.8
1+1
2.

2s
3.6

3.6:9 \2s 3.6.9.12 &c.

2 76 2.5 16

2.5.8 6

3.62s 3.6.9 2s 3.6.9.12 &c.

And, consequently, if the latter of these series be added to the former, we shall have, by making the first term of the result a multiplier,

2 b
2.5.8 b

2.5.8.11.14 x=7&s3V-3 1.

) 2

3.6 (

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3.6.9.12 2s 3:6.9:12.15.18

2762-423 (

4.27

423)

we shall

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2

2.5.8

D-, &c., which series also answers to the irreducible case, and must be used when 2a’ is greater than 2762. And if the root thus found, be put =r, aş before, the other two

403 — 2762 roots may be expressed thus: x = F2

{1+

4 2762

2772

2.5.8.11.14 3.6 (4* – 2762) – 3.6.9.12 (40 3.6.9.12 4. – 2772)2 +

3.6.9.12.15.18 2762

tc.}. Or, 403 Ꮞ - 27b2 2 2762

5.8

3.6 4Q 2762 9.12
2732

11.14
2752

17.20 2762
(

( 40 2769

2762 21.24 443 -- 2762 Dt, &c. Where the signs are to be taken as in the latter part of the preceding case.

2752): –, &c.

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7270{

{1+

А

2

с

)

EXAMPLES,

3

[2(2762 +4a')]

1 Given 23 + 6x 2, to find the value of x.

Here a=6, and b = 2, whence
2772
27 x 4

1 1

and
2762 + 4a3 27 X 4 + 4 X 216 1 +8 9
26

4

3/[2(4 X 27 + 4 X 216)]
4
4 2/81

3/648

Consequently, 2 (27 + 8 X 27) 63/9 27

27
by formula 1, we shall have
1

1.0000000 (A)
2.5 1
5:

,0205761 (R)
6.9

9 8.11 1 x

.0011177 (c) 12.159 14.17 1

.0000782 (D) 18.21

X

А

B

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C

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2s

2762

2762 also have (

&c., and consequent2762-4a3 4a3 2762 1 를

423 -- 2762 ly, soy-3=(( +2763)

4 Hence, if these values be substituted for their equals in the above series, the result will give the above expressions for the two remaining roots of the equation.

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2. Given 23

12;

12 whence 2v

2

9x = 12, to find the real value of x.
Here a = 9 and 1

2762 403 27 X144-4X272
2 6 and
2762

27 X144 36 1

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144

144 4
Consequently, by formula 2, we shall have
1

1,0000000 (A)
2 1
X 1

-,0277778 (B)
3.64
5.8 1
Xi(B)

-,0025720 (0)
9.12^4
11.14 1

-.0003667 (D) 15,1844 17.20 1 X-(D)

--,0000619 (E) 21.2444 23.26 1 X; (e)

--.0000114 (F) 27.30^4 29.32 1 Х

--,0000022 (6) 33.36°4

X:()

Sum

-.0307920

Comp.

.9692080

Log. 969208

1.9864137

Log. 2 V6, or Log. > 48

0:5604137

No. 3.522334

:5478274
therefore x= 3.522334.
3. Given 23 12x = 16, to find the three values of x.

Here a= 12 and b= 15;
b
15

403 -- 2782
Whence 2 V = 20 3/ 60 and

2
2

2762
4.123 - 27.152 256 - 225 31
27.152

225

225 Consequently, by formula 3, we shall have 1

+1.0000000 (A)
2 31

+0.0153086 (B)
3.6 225
5. 8 31

X225
9.12225

-0.0007812 (c)
11.14 31
+ X

+0.0000614 (D)
15.18~225
17.20 31
Х

--0.0000057 (E)
21.24*225
23.26 31
X

+0.0000006 (F)
27.30? 225

+36X225

А

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D

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No. 3.971962

.5990051 Therefore the affirmative value of x or first root, r= 3.971962. ✓(4a - 2762) ✓837 ✓ (9 X 93)

✓ 93 Again,

926 9V 450 93 450 33450 403 2767 31 and 2762

225

Hence,
+1

1.0000000 (A)
2.5 31
Х

-.0255144 (B)
6.9° 225

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Result

-1.5760365 Or

-2.3959255 Whence the three roots or valucs of X 3.971962 - 1.5760365, and -2.3959255.

are

4. Given 203 68 2, to find the three values of x. 26

4
Here

V [2(4.42 2762)] V [2(4.68 - 27.4)]
4
4 - 2 249

2762

and 3V [2(4.8 - 4)] 637 337 21

493_-2762 4.27

1 1 4.63 - 27.4 8-1 7

Hence, by the formula 4, we shall have
1

1.0000000 (1)
2.5 1

- 0264550 (B)
6.9°ng
8.11 1
+ X-B

+.0018476 (C)
12.15 7
14.17 1
X

--.0001662 (D)
^7

X-A

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