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a

Thirdly, Nã x cot. 2u =

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a

Log. 6...... 0.7781513

2 log. 6...... 0.3890756

3 Log. 2.

0.30103002 Log. T.....

10.0000000 16x cot. 2u, whence Log. tan. 2..... 10.6901056 ilog. 6....

0.3890756 Therefore z= =78° 27' 47', and Log. 2....

0.3010300 -39° 13' 531":

Log. cot. 2u.

8.8301834 Secondly, (7* tan. ğz)=tan. u, Sum....

9.5202890 whence

Log.r.o.o

10.0000000 Log.pl 20.0000000

1.5202890 Log. tan. z.

9.9119523 Log. X.... Sum....

Consequently = .3313139, .3)29.91195231

the positive root of the equation, Log. tan. u. ... . .9.9706507

as required. Therefore u=43° 4' 55", and 2u=86° 7' 50".

2. Given 2-3x=1, which is an equation falling under the ir reducible case, to find its three roots. Here, a being =3, and b=1, we shall have, by taking the ra.

36 3 dius r=1, cos. 2=

= =.5=cos. 60°, whence

2a
x=2NX cos.

. 2 cos. 20°3+1.8793852,

3
2 x cos. (60° -2 cos. 40°5_1.5320888,
3

3
-20 X

-X cos. (600+ ) -2 cos. 80°54.3472964.

3 Therefore the three roots are 1.8793852, -1.5320888, and -.3472964..

And if the equation be 28_8x=-1, the three roots or values of x are ~1.8793852, 1.5320888, and .3472964.

Which are the negatives of the roots of the former equation. 1. Given 2_2x=-2, to find the root of the equation, or the

= value of x. See art. 94, p. 214.

Ans. X=- -1.7693. 2. Given 28_300x=-1000, to find the root of the equation, or the value of x. See index.

Ans. x=3.472964. 3. Given 23–9x=9, to find all the three roots of the equation, or values of x. Ans. 3.411474, -2.226682, and -1.184792.

4. Given 2–2–2x+1=0, to find all the three roots of the equation, or the three values of x. Ans. 2 cos. tn, -2 cos. qn, and 2 cos. 4n, where n=180°, and rad. = l.

a

Z

a

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a

=

(60o+=

98.

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(

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2. +2 1-3
= 2

2

This method of cubic equations by converging series, which in some cases will be found more convenient in practice than either of the former, consists in substituting the numeral parts of the given equations in the place of the literal one, in one of the following general formulæ to which it belongs; and then collecting as many terms of the series as are sufficient for determining the value of the unknown quantity to the degree of exactness required. l. Given x tax=b. 26

2.5 2782 2.5.8.11 2782 1., 2 =

2.5.8.11.14.17 2772 -)

6.9.12.15.18.21 278742

*+&c.}
26
2.5 2772

8.11 2762 14.17 2772
or te

20.23 2762 + V{2(278°44a")}{1+

3B

.
6.9 27++47) 4+ 12.15278440) B+ 18:21 278440C+24.276278440) D
In which case, as well as in the following ones, A, B, C, &c. denote the terms immediately preceding
or 2767_4a.
those in which they are first found. 2 Given 4-ax=+b, where 162 is supposed to be greater than ziyas

2 2782_4a 2.5.8 2762_4a

2.5.8.11.14 2762_423
3.6

:)
2762
2772 3.6.9.12.15.18

2772

)-&c.}, or
2 2762-4a? 5.8 2762-4a 11.14,2764a?. 17.20.2762-4a
3.6'

-)B
2762
9.12

-).
2762
15.18 2762 21.242762

)D-&c.
In which case the upper sign must be taken when b is positive, and the under sign when it is negative;
and the same for the first root in the two following cases.
3. Given 2013

#b, where 46 is supposed to be less than zla?, or 2762 X 4a'.

2 4a_2762 2.5.8 40?_2762 士

2.5.8.11.14 4a_2762 2772

3.6.9.12 126

-) + 2762 3.6.9.12.15.18

2762

)-&c.}, or
2 4a_2762
=

5.8 409_2762
)A-

11.14,4a3_2762 17.20 42_27/2.
2762
9.12
15.18

-)D+&c.
2762

21.24 2772
Which series answers to the irreducible case, and must be used when 4a is less than 276?. And if the
root thus found be put = r, the other two roots may be expressed as follows :

3.6.9.126

-) A

ar =

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2762)B+

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+ V (Ad°—270)

F

-)B

-)A+12.15

2.5

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2.5,4a_2762 2.5.8.11° 4a3_2762 2.5.8.11.14.17 4a_276% {1

6.96 9262

-)+ -)

-)*+&c,}
2762 6.9.12.15 2762 3.6.9.12.15.18.21 2762
N(423—2764) 2.5 40 -2762 8.11 423_2762 14.17 4a_2762 20.23,4a:—276
干。土
{1–

-)C+

)'D 99262 6.9 276?

2762
18.22 2762

25.27 2762
Where - r, or + fr, must be taken according as b is positive or negative; and the double signs # must
be considered as + in one case and in the other, as usual. 4. Given 203 — ax=+b, where 162 is still
supposed to be less than z'ya?, or 276=4a'.
26

2762 2.5.8.11 2772 2.5.8.11.14.17 2762
4. x=+2

{1 6.94–276

6.9.12.15.18.214~*—2782) + &c ^{2(4a276')}

6.9.12.15 40276 or} 2772 2.5, 2752 8.11 .2772 14.17

2782 20.23 2762
=F.

{1
{2(41—276?)} 6.944°—277) A+ 12.1542_275) B- 18.21'44°—277)C+24.274_*_2762)
D_&c., which series also answers to the irreducible case, and must be used when 4a? is greater than 276.
And if the root thus found be put =r, as before, the other two roots may be expressed thus:
°—2772 2 2762

2.5.8 2762

2.5.8.11.14 2772
Fa + {1+3.6'4—27

3.6.9.12 442—277)+
4
-2762

3.6.9.12.15.18(203_277)—

277) —&c. or 4a3_2762 2 2782

5.8 2762

11.14 2762 干。士 士。 21+ 2

2)A 4

*_)

3.6 4à°—2769 9.12'42°—2772) B+16.184 270)C-21.24422270)D+
Where the signs are to be taken as in the latter part of the preceding case.

Of Biquadratic Equations.
99. A biquadratic equation, as before observed, is one that rises to the fourth power, or that is of the gene-
ral form x*+axl+bx+ar+d=0, the root of which may be determined by means of the following formula,
substituting the numbers of the given equation, with their proper signs, in the places of the literal coeffi-
cients a, b, c, d. Rule I. Find the value of z in the cubic equation z+(face-152 ---d)z=rdpb+c+
da?)-4bac +8d) by one of the former rules; and let the root, thus determined, be denoted by r

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b

Then find the two values of x in each of the following quadratic equations, mat(fatı {ta+26r-})}) (r+$b+w{(r+15)—d} milja-^ta+2rjb)})a= -(r+16)-»{(r+16)-d}, and they will be the four roots of the biquadratic equation required. Or, if the equation be of the more commodious form, d'+bx+ +cx+d, to which it can always be reduced, by taking away its second term, it may be resolved thus :

RULE II. Find the value of z in: the cubic equation 28(136+d)z=11968+$c-hd, and let the root thus determined be denoted by r. Then find the two values of x, in each of the two following quadratic equations, 2 tv {2(r-b)}x= (r+b+v{(r++)—d}

{

and they 2^{2r-jor+b)-{(r+16) *

irtib_d will be the four roots of the biquadratic equation required.

1. Given the equation x4—10x+35x250c+24=0, to find its roots. Here as

Here a=-10, b=35, -50, and d=24; whence, by substituting these numbers in the cubic equation

23+(fac-16?_d)z=105b3+3(c+da?)-(ac+8d), we shall have the following reduced equation, 28 — fiz Boobs which being resolved, according to the rule before laid down for that purpose, gives z={(35+189–3)+(35—187—3)}.

t$ But, by the rule for binomial surds, given in Art. I, Case 2, in the former part of the work, (35+18N43)=i+ÎN_3, and (35—18/3)=-N-3;

wherefore z={tin–3+1N_3=. And if this number be substituted for r, —10 for a, 35 for b, and 24 for d, in the two quadratic equations, m++[latv{ta'+26r-3b)}]x==(r+16)+w{(r+16)—.d}, 2+[ja-{ta+21r-)}]x=-(+16)--{irtibd}, they will become, after reducing them to their most simple terms, po 3x==

-2, and 2*—7x=-12; from the first of which =

==2 or 1, and from the second x=IENI=2+1 =4 or 3; whence the four roots of the given equation are 1, 2, 3, and 4.

2. Given x'+12x--1750, to find the four roots of the equation. Here a=0,b=0, c==12, and d=-17:

Whence, by substituting these numbers in the equation 23(1262+d)zsidos+f0-bd, we shall have, after simplifying the result, z2 +177=18, where it is evident, by inspection, that 2= =1.

And if this number be substituted for r, O for b, and -17 for d, in the two quadratic equations mtw 2(1-1)=(+1)+(r+16)d},

{r—}}} rb{r X21-5)=(r+b^{(r+i)-d,

}

)^)

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they will become, after reducing them, in the usual manner, to their most simple forms, 772 173/2, and 1_21-3/2, which being resolved, according to the general rule, we shall have,

= 12+1+118)=2+w6+372), =-2-^(-it18)=-112-N-+372), I=+IN+N-1-18)=+12+N-5-32),

FİN--1-18)=+12-1-1-3/2), which are the four roots of the proposed equation. Where it is to be observed that the first two are real, and the two latter ima. ginary.

Rule III. The roots of any biquadratic equation of the form mtari+bx+c=0, may also be determined by the following, general formulæ first given by Euler, which are remarkable for their elegance and simplicity.

Find the three roots of the cubic equation zi+2az'+a_4c) =b', by one of the former rules, before given for this purpose; and let them be denoted by r', r", and p!". Then we shall have When 6 is positive,

When b is negative, -wp-^"'^"" tar'twi"two"" 2

2 -Wr'+wr"+w+""

+".

twr-wp-" 2

2 -twr-Wr"+w?!!

-Witwr"-W ""

2 twr'+Vr"-wp"

-Wr'-Vr"+"r"

-trip 2

2 101. If the three roots m', r", '", of the auxiliary cubic equation be all real and positive, the four roots of the proposed equation will, also, be real; and if one of these roots be positive, and the other two imaginary, or both of them negative, and equal to each other, two of the roots of the given equation will be real, and two imaginary; which are the only cases that produce real results.

3. Given x-25x+602—36=0, to find the four roots of the equation. Here a=-25, b=60, and =-36; whence, by substituting these values for their equals, in the cubic equation above given, we shall have 23_2X252° +(252+4x36)z=60%, or 250z? +7692=3600; the three roots of which last equation, as found by trial, or by one of the former rules, are 9, 16, and 25, respectively ; whence

= (-79-716-25)=(-3-4-5)

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