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4. If a,=a, ag=a+b, ag=a+26, 24 =a +36, the given cquation assumes by substitution the form ** +(4a +66)33+(6a2 +18ab+1162)x2 + (4a8 +180"b+ 22ab2 + 663).

+(a4 + 6a3b+11a? 12 + 6ab3)=d. The left side of this equation will become a complete square by the addition of bt. Let 64 be added to each side, and let the square root of each side be extracted,

then e? +(2a +36)x+(a” +3ab + b2)== V(d+b4), from these quadratics the four roots can be found. This method may be illustrated by the solution of the equations

(x+4)(x+6)(x+8)(x + 10)=48, and (x+3)(x + 7)(x+11)(* +15)=644. See example 2, under Exercise VII., p. 22 of Section VI. 5. Since the roots of x3 - p.c? +qr =0 are in harmonical progression.

1 1 1 Let

denote the three roots,
a-7' à

a+b
then !=the sum of the reciprocals of the roots=(1 6+a+a+b=30,
1

3r

3r the middle root

or qx - 3r=0 9

9 23 – px' +92then

= 0,

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Let x

a

qa-37

2

or 90° - (29 3r)qx+(23 3pqr +972)=0 is the equation which contains the roots

1 1 a-b' a+7 9. Since the roots of the given equation are in harmonical progression, for x substitute 2

then the equation becomes

Y

= 0, whose roots are in arithmetical progression. đ

d
Let x + 3y, x+y, X - y, x – 3y denote the roots of this equation,

b
then 4x
d'
ď

d'
from which by elimination may be found co. 4bd and c4 + 400ad=0.

ya + cy: by?

al +

+

с

a

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XXXIII. 7. Let a denote the possible root of the equation 23 - qx+p=0, then x=a, and 2-a=0); ..

=0, or x2 + ax +a? -q=0, the quadratic equation which contains the two impossible roots, and taking a=y+-, the required expression will be found.

8. The eqnation (x3 - 2r)3 – 2799x'=0 is an equation of nine dimensions, and includes the three equations

203 - 3qx+2r=0, x3 – 3qw« + 2r=0, and 33 - 3qwPx + 2r=0,

in which 1, w, w2 denote the three cube roots of unity. The first of these equations, 23 – 3qx+2r=0, is only another form of 23 – qx+r=0, obtained by writing 3q for q and 2r for r, when y=9 and =1, the two equa.

y tions will give values of y and %, from which

x=y+z={r+ (7* – 9°)}}}+{r^(12 – 93)}}".

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XXXIV. 1. See Art. 16, p. 20. 2. Let a.*,B2,92 denote the three values of e’ or y in ys -+2qy? +(2* - 48)y – go?=0,

as .. ? + ax +

4

a

=

the reducing cubic of Descartes' solution of the biquadratic 3* +972 +re+8=1,
then a?B?q?=r?, aby=r, and - 29=a? +82 +g?.
2? + ex+f=x2 + ex+ (2+e-). Let e=a one of the three valucs of c.
3

e.
=x2 + ax+(a?-B7-92 - 2By)=0,
. = {(8+), + = }(+),

X -
and x=3(-a+B+y) and 3(-2-3-y), two of the required roots.
In the same manner
22 - ex+g=x2 – cx+ (2+2+3)

te
=22 - - ax + 4(a? - B? -92 +2B9)=0,

and x2 2+*=+(8-9), ., 2-== }(B–»), .: x=f(a+B-7) and }(a-B+9) are the other two roots of the biquadratic in terms of the roots of the reducing cubic. 3. See Art. 16, p. 20. 7. See Art. 17, p. 22.

8. See Art. 17, p. 22. 12. See Art. 16, p. 20, and Art. 18, P. 23.

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XXXV. 2. Since a, B are two roots of the equation, x? – (a+b)x+aß=0, which is one of the quadratic equations of which the biquadratic is composed. Let 8 denote the roots of the other quadratic.

Since a +8+9+0=0, a +B=-(+8), and x? +(9+8)x+ go=0, or x? + (a+B) + 78=0 is the quadratic containing the other two roots, a+B

a+B
+

taß-
But q=+aBy+ab8 - ayo – Byg=+(4+8)a8-(a +B)78=(a+B)(aB -98),
9

at B
=..x=

9 a+B 3. The equations are p?q- p[r - s) - q ==0, and p’s+7(r--5)=0; and if r-s be eliminated, the resulting equation is pos+poq-q=0.

4. First suppose the four roots real ; since the last term of the equation is negative and equal to unity, the roots are of the form a, a-, b, -6-?; and it may be shewn that a+a-l=}(p+q), and 6-1-1=}\p-2).

5. First x*= - 4aRx+a“, add 2x?a? to each side of this equation, and x* + 2x?a?=2x?a? – 4a3x+a, complete the square of the left side,

i. X* + 2x?a? +at=2x’ao - 4aRx+ 2a4=2a? (x2 – 2ax +a?). Extract the square root of cach side, . . 202 +a?= + ao - a)V2, then c? +a? = + a(x - a) v2 and 2:+ a2 = ala-av2 are the two quadratic equations from which two values of a will be found to be possible and two impossible.

6. It is obvious from inspection that a, b, c respectively, satisfies the equation; and, therefore, three of the roots are known, that is x=a, x=b, x=c, and a- :- a=0, ic - b=1, * -c=0, .. (x – a)(x - 5)(2 – c)=0.

Next the given equation can be put into the following form :
a^{x? – (b+c)x+bc} 5*{2? – (a + c).x + ac} <*{x– (a+b)x+ ab}

+
(ab)(a -c)
(6-c)(b-a)

(c-a)(c-6)
84

ch

+

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+

2

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(+*+2+)-(p-1)(x+++)(9-p+1)(x2+*)-(4-9+p–1)

-(2+1)

.

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+

P aibe

1,400

crab +

+ lla - b)(a - c)* 76 - c)(6 – a) * ic-a)(c-4) ={\c+a)(c+b) + a2 +63 }*? – (a+b)(6+c)(c+a)e+abc(a+b+c), ::. **- {\c+a)(c+b) + a? +62}2? +(a+b)(b + c)(c+a)x - abc(a+b+c)=0 (1),

but (x – a)(x b)(2 – c), or x3 - (a+b+c)x2 +(ab +ac+bc)x - abc=0 (2). By dividing the equation (1) by the equation (2)

there results the equation sta+b+c=0, and consequently x=- - (a+b+c) is the fourth root of the equation. 7. Divide each term of the equation by x', and arrange the terms in pairs. Assume x+-=, and by the necessary substitutions, 23 – pza +(- 3s):+2ps – r=0 is the cubic equation.

8. The equation can be put into the form (29 – 1) – px(27 - 1)+q«c(25 - 1) – r23(x3 — 1)+sx"(ic – 1)=), which is obviously divisible by 2-1, ..x=1, one of the roots ; depressing the equation it becomes 28 – (P-1)x?+(9-p+1)x® – (r-9+p-1).5 +(8-r+q-p+1)x*--(r-9+p-1)2:3

+2-p+1)x? - (p-1)x+1=0. Dividing by x* and arranging the terms,

+P

+(s-r+q-+1)=0. 1

1
Let +
1

1
=%, then x? + =:? – 2, 2s+ =>3--37, 24 +- =* – 4:2 +2.

22
Making these substitutions and reducing the terms
24 - (P-1)z3 +(9-p-3)22 –(r+9+2p - 2)2+(8-p-atp+1)=0

is an equation of four dimensions. 9. The equation is (x' + 1) + x(x? +1) – 9x*(x3 +1)+3x3(x8 +1) – 8x*(*+1) = 0).

Obviously - 1 is a root of this equation, and x+1=0. Depressing the equation it becomes 28 – 90c6 + 12x5 – 20x4 +12.03 - 9x2 +1=0.

1
Divide by x4, arrange the terms, and let X+
then 24 - 13:2 + 122 +22–22=0, or z(z:– 13z +12) = 0,

1
... z=0, or +-=0, and x=+V -1,
and z: -13z + 12 = 0,
or a(z? — 1) – 12(3-1)=0, or (ö – 1)(22 +2– 12) = 0;
::8-1=0, or x+1+1=0;...= 1+1-3

2

1 and 22 +3 - 12=0, 23 +++

1_49

; ..z=3 and --4; 4'

3+VE •• + .-3, and x=

- 4, and x= -2+ V3. The nine roots are - 2V3, 3(3+V5), (1+1 -3), IV-1, -1.

10. Since 12 +mo+no=1, the given equation may be put under the form (2a + m2 + n2)u* – {{6% +c?)?? +(c* +a?)m2 +(a+62)n? }u?

+ 69c272 +c?a?m+a?b?n? =0, and resolved into

(vo-62)(02-CP)' + (v? - a?)(09-c?)m? +(v? - a')( - 62)=0, and then dividing by (v2 – 62)(09 - 62)(v? - c?), the form required is obtained.

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ELEMENTARY ALGEBRA,

WITH BRIEF NOTICES OF ITS HISTORY

In Twelve Sections, demy 8vo.

CONTENTS AND PRICES.

PRICE

pp. 56

..6d.

SECTION I. Of Algebra, pp. 32....

...6d. SECTION II. Of Algebra, continued, pp. 48 ...6d. SECTION III. Of Algebra, continued, pp. 48 ..... ...6d. SECTION IV. Addition and Subtraction, Multiplication and Division, pp. 48

...6d. SECTION V. Measures, Multiples & Fractions, pp. 48..6d. SECTION VI. Involution, Evolution and Surd Numbers,

...6d. SECTION VII. Simple Equations, pp. 48 .... SECTION VIII. Quadratic Equations, pp. 68.....

...6d. SECTION IX. Ratio, Proportion and Variation, pp. 48..6d. SECTION X. Arithmetical, Harmonical and Geometrical Progressions, pp. 52

...64. SECTION XI. Variations, Permutations and Combina

Binomial and Polynomial Theorems; The Theory of Probabilities, pp. 68 ...

...60. SECTION XII. Cubic and Biquadratic Equations, pp. 66..6d.

tions ;

LONDON: LONGMANS AND CO.

Each Section of the Algebra may be purchased separately; also the Twelve Sections together, done up in boards with cloth covers, at 68. 6d.

ELEMENTARY ARITHMETIC, WITH BRIEF NOTICES OF ITS HISTORY.

BY R. POTTS, M.A., TRINITY COLLEGE, CAMBRIDGE.

In Twelve Sections, demy Svo.

CONTENTS AND PRICES.

PRICE SECTION I. Of Numbers, pp. 28

3d. SECTION II. Of Money, pp. 52

6d. SECTION III. Of Weights and Measures, pp. 28 3d. SECTION IV. Of Time, pp. 24.....

3d. SECTION V. Of Logarithms, pp. 16

2d. SECTION VI. Integers, Abstract, pp. 40

5d. SECTION VII. Integers, Concrete, pp. 36

5d. SECTION VIII. Measures and Multiples, pp. 16 .. 2d. SECTION IX. Fractions, pp. 44

5d. SECTION X. Decimals, pp. 32

4d. SECTION XI. Proportion, pp. 32...

4d. SECTION XII. Logarithms, pp. 32

6d.

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NOTICE.

EACH section of the Arithmetic may be purchased separately; also the twelve sections together, done up in boards, with cloth

covers, at 48. 6d.

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