88

half sum

sum

sum

EXAMPLES.

7

1695 Er. 1. Find the roots of the equation ra +

44 12716 by tables of sines and tangents.

1695 Here p = 479

and the equation agrees with the 12716' 1695

1695 1st form. Also tan A = V and x = tan {A = 7. 12716'

12716 In logarithms thus :

Log 1695 = 32291697 Arith, com.

log 12716 = 5.8956495 sum + 10 = 19:1248192

9.5624096 log 88 = 1.9444827 Arith. com. log 7 = 9:1549020 10 = log tan A = 10•6617943 = log tan 77°42'31"); log tan A = 9.9061115 = log tan 38°51'15"};

= log vy, as above = 9:5624096

10 = log x = 1.4685211 = log .2941176. This value of x, viz •2941176, is nearly equal to

To find

17 whether that is the exact root, take the arithmetical compliment of the last logarithm, viz 0:5314379, and consider it as the logarithm of the denominator of a fraction whose numerator is unity: thus is the fraction found to be exactly, and this is manifestly equal to

As to the other root of 17 1695

339 the equation, it is equal to - 을

748
Ex, 2. Find the roots of the cubic equation
403

46
xt = 0, by a table of sines.
441

147

403 Here p =

the second term is negative, and

1470 4.p3> 2792: so that the example falls under the irreducible case. 3 x 46

414
Hence, sin 3A =
147

403 403
2v
3.441

1323 The three values of .x therefore, are

1612

1

3.4

5

12716

46

441) 9 =

1

1

х

441
403

Х

1612

sin A V 1928

1612 1323" 1612

sin (60°+a) V 1328

The 414 ...

The logarithmic computation is subjoined.

Log 1612 = 3.2073650 Arith. com. log 1323 = 6.8784402 sum 10 ... = 0.0858052

half sum = 0·0429026 const. log. Arith. com. const. log = 9.9570974 log

= 2.6170003 Arith. com. log 403 . = 7.3946950 log sin 3A · = 9.9687927 = log sin 68 32' 18"..

Log sin A = 9:5891206

const. log 0.0429026 1. sum – 10 = log x=- 1•6320232=log 4285714=logi.

= Log sin (60° - A) = 9•7810061

const. log : = 0.0429026 2. sum – 10 = log x= - 1.8239087 = log .6666666 = logi.

Log sin (60° +a)= 9.9966060

const. log · = 0.0429026 3. sum – 10 = log -* = 0.0395086 = log1.095238=logii. So that the three roots are }, }, and -2; of which the first two are together equal to the third with its sign changed, as they ought to be.

Ex. 3. Find the roots of the biquadratic x4 – 25x2 + 60.x 36 = (), by Euler's rule. Here a = 25, b = 60, and c = 36; therefore

225 f=

+9= and h 2'

16' Consequently the cubic equation will be 23 za +

= 0. The three roots of which are =p, and 2 = 4= 9, and z =

7* ; the square roots of these are vp= , vq = 2 or, vredi Hence, as the value of tb is negative, the four roots are 1st. X = 2 / 3 + 4/

1,
2d.

*-+{= 2,
3d.
+ +

3,
4th. t =
- Ž 쓸

-6. Ex. 4. Produce a quadratic equation whose roots shall be and

Ans. r? 3x + 3

x + = 0.

= Ex. 5. Produce a cubic equation whose roots shall be 2, 5, and 3. Ans. x3 4x2 11x + 30 = 0.

25

769

8=

625
16

25

225

769

Z 16

4

9

Z

25
4

.

a

Er. 6. Produce a biquadratic which shall have for the roots 1, 4, 5, and 6 respectively.

Ans. x4 6x3 21x2146x

120 = 0.

Ex. 7. Find x, when x2 + 347x = 22110.
Ans. x
Ex. 8. Find the roots of the quadratic x2
Ans. x =

10, x =

264

695

25

25

Ans. x = 5, x =

Ex. 9. Solve the equation 2

Ex. 10. Given x2

Ex. 14. Given x4

24113x=

Given + 24x3

x

3x-1=0.

Ex. 11. Find the roots of the equation x3 Ans. the roots are sin 70°, sin 50°, and sin 10°, to a radius 2; or the roots are twice the sines of those arcs as given in the tables.

55, x=

55

12-2

Ex. 12. Find the real root of x3

XC

6 = 0.

N

Ans. 3x sec 54° 44′ 20′′. Ex. 13. Find the real root of 25x3 + 75x 46 0. Ans. 2 cot 74° 27′ 48′′.

8x3 12x2+84x 63 1

find x by quadratics. Ans. x2+√7±√11+√7. Ex. 15. Given x4 +36x3-400x23168x + 7744=0, xa to find x, by quadratics. Ans. 11+/209.

Ex. 16.

139

25

481860, to find x. Ans. x 20, x = 24093.

x =

402.

325

6

--

114.x2 24x+10, to 19714, x=2±√5.

65

12*

O, to

CHAPTER IX.

ON THE NATURE AND PROPERTIES OF CURVES, AND THE

CONSTRUCTION OF EQUATIONS.

SECTION I.

a

:

a

Nature and Properties of Curves. Der. 1. A curve is a line whose several parts proceed in different directions, and are successively posited towards different points in space, which also may be cut by one right Iine in two or more points.

If all the points in the curve may be included in one plane, the curve is called a plane curve; but if they cannot all be comprized in one plane, then is the curve one of double curzature.

Since the word direction implies straight lines, and in strictness no part of a curve is a right line, some geometers prefer defining curves otherwise : thus, in a straight line, to be called the line of the abscissas, from a certain point let a line arbitrarily taken be called the abscissa, and denoted (commonly) by r: at the several points corresponding to the different values of x, let straight lines be continually drawn, making a certain angle with the line of the abscissas: these straight lines being regulated in length according to a certain law or equation, are called ordinates; and the line or figure in which their extremities are continually found is, in general, a curve line. This definition however is not free from objection ; for a right line may be denoted by an equation between its abscissas and ordinates, such as y = ax + b.

Curves are distinguished into algebraical or geometrical, and transcendental or mechanical.

Def. 2. Algebraical or geometrical curves, are those in which the relations of the abscissas to the ordinates can be denoted by a common algebraical expression : such, for example, as the equations to the conic sections, given in the first chapter of this volume.

Def. 3. Transcendental or mechanical curves, are such as cannot be so defined or expressed by a pure algebraical equawion; or when they are expressed by an equation, having one

of

a

of its terms a variable quantity, or a curve line. Thus, y = log x, y = A.sin x, y = A.cos x, y = A*, are equations to transcendental curves; and the latter in particular is an equation to an exponential curve.

Def. 4. Curves that turn round a fixed point or centre, gradually receding from it, are called spiral or radial curves.

Def. 5. Family or tribe of curves, is an assemblage of several curves of different kinds, all defined by the same equation of an indeterminate degree; but differently, according to the diversity of their kind. For example, suppose an equation of an indeterminate degree, a"-'x =y": if m=2,

" then will ax = - yo; if m = 3, then will aʻx = y; if m= 4, then is a’x = 34 ; &c: all which curves are said to be of the same family or tribe.

Def. 6. The axis of a figure is a right line passing through the centre of a curve, when it has one : if it bisects the ordinates, it is called a diameter.

Def. 7. An asymptote is a right line which continually approaches towards a curve, but never can touch it, unless the curve could be extended to an infinite distance.

Def. 8. An abscissa and an ordinate, whether right or oblique, are, when spoken of together, frequently termed coordinates.

ART. 1. The most convenient mode of classing algebraical curves, is according to the orders or dimensions of the equations which express the relation between the co-ordinates. For then the equation for the same curve, remaining always of the same order so long as each of the assumed systems of co-ordinates is supposed to retain constantly the same inclination of ordinate to abscissa, while referred to different points of the curve, however the axis and the origin of the abscissas, or even the inclination of the co-ordinates in different systems, may vary; the same curve will never be ranked under different orders, according to this method. If therefore we take, for a distinctive character, the number of dimensions which the co-ordinates, whether rectangular or oblique, form in the equation, we shall not disturb the order of the classes, by changing the axis and the origin of the abscissas, or by varying the inclination of the co-ordinates.

2. As algebraists call orders of different kinds of equations, those which constitute the greater or less number of dimensions, they distinguish by the same name the different kinds of resulting lines. Consequently the general equation of the first order being = a + Br + vy; we may refer to the