negative: then substituting, instead of r and its powers, their values resulting from the hypothesis that x=z+m; so shall there arise an equation, whose roots shall be greater or less than the roots of the primitive equation, by the assumed quantity m.

Corol. The principal use of this transformation is, to take away any term out of an equation. Thus, to transform an equation into one which shall want the second term, let m be so assumed that nm a = 0, or m = n being the index of the highest power of the unknown quantity, and a the coefficient of the second term of the equation, with its sign changed then, if the roots of the transformed equation can be found, the roots of the original equation may also be found, because x = +

a

THEOREM IV.

Every Equation may be Transformed into Another, whose Roots shall be Equal to the Roots of the First Multiplied or Divided by a Given Quantity.

x

f'

1. Let the equation be 23+ az2 + bx + c = 0: if we put fzx, or z = the transformed equation will be 3 + fax2 +fbx +ƒ3c = 0, of which the roots are the respective products of the roots of the primitive equation multiplied into the quantity f.

h

k

x

g

By means of this transformation, an equation with fractional quantities, may be changed into another which shall be free from them. Suppose the equation were z3 + + d + 1/ = 0: multiplying the whole by the product of the denominators, there would arise ghkz3+hkaz2 + gkbz + ghd = 0: then assuming ghkz = x, or z = the transformed equa. would be x3+hkax + g2k2hbx‍+g3k3h3d=0. The same transformation may be adopted, to exterminate the radical quantities which affect certain terms of an equation. Thus, let there be given the equation z3 + az2 √ k+ bz+c√k: make z✔k r; then will the transformed equation be 3+ akx2 + bkx + ck2 0, in which there are no radical quantities.

=

8hk

2. Take, for one more example, the equation z3 + az2 + bx + c = 0. Make = x; then will the equation be

f

axa

C

transformed to "+++= 0, in which the roots

fr

are

are equal to the quotients of those of the primitive equations divided by f.

It is obvious that, by analogous methods, an equation may be transformed into another, the roots of which shall be to those of the proposed equation, in any required ratio. But the subject need not be enlarged on here. The preceding succinct view will suffice for the usual purposes, so far as relates to the nature and chief properties of equations. We shall therefore conclude this chapter with a summary of the most useful rules for the solution of equations of different degrees, besides those already given in the first volume.

I. Rules for the Solution of Quadratics by Tables of Sines and Tangents.

1. If the equation be of the form x2 + px = 9:

2

Make tan A= ✔g; then will the two roots be,

x = +tan

x =

2

q: then will

xcot A√9.

3. For quadratics of the form x2 + px = — q.

Make sin A =

2

√9: then will

cot A√9.

x = - tan Ag 4. For quadratics of the

Make sin A =

x = +tan A√9 In the last two cases, if

x=

x2

form 2 — px = — q•

✔g: then will

2

and consequently the values of x.

The logarithmic application of these formulæ is very simple. Thus, in case 1st. Find A by making

Then log x = S+ log tan A +
-(log cot A+ log q

Note. This method of solving quadratics, is chiefly of use when the quantities p and q are large integers, or complex fractions.

II. Rules for the Solution of Cubic Equations by tables of Sines, Tangents, and Secants.

1. For cubics of the forin x3 + px ±q=0.

demde of sin 3 should exceed unity, s would , and the equation would fall in what is called Paucible case of cubics. In that case we must make

2005A= 2p: and then the three roots would be x = sin A. 2√ √ P.

x=sin (60° - A). 2√ p.

x = ± sin (60° + a) . 2 √ {p.

If the value of sin в were 1, we should have в = 90°, tar A = 1; therefore A = 45°, and x = 2√÷p. But this would not be the only root. The second solution would give 1: therefore a = ; and then

cosec 3A

90°

3

Here it is obvious that the first two roots are equal, that their sum is equal to the third with a contrary sign, and that this third is the one which is produced from the first solution*.

In these solutions, the double signs in the value of x, relate to the double signs in the value of q.

N. B. Cardan's Rule for the solution of Cubics is given in the first volume of this course.

The tables of sines, tangents, &c, besides their use in trigonometry, and in the solution of the equations, are also very useful in finding the value of algebraic expressions where extraction of roots would be otherwise required. Thus, if a and b be any two quantities, of which a is the greater. Find x, *,

m

log 2+ log cos y
log 2+ log sin ly.

log (a+b)" ➡ · [log a + log cos t + log tan 45° ±‡t)].

n

The first three of these formule will often be useful, when two sides of a right-angled triangle are given, to find the third.

III. Solution

III. Solution of Biquadratic Equations.

Let the proposed biquadratic be r* + 2px3 = qx2 + rx + s. Now (22+px + n)2 = x2 + 2px3 + (p2+2n)x2 +2pnx+n2: if therefore (p + 2n) x2 + 2pnx + n2 be added to both sides of the proposed biquadratic, the first will become a complete square (x2+ pr+n), and the latter part (p2 + 2n + g) x2 + (2pm + r)x + n2+s, is a complete square if 4(p + 2n +q)·(n2+s)=2pn + r2; that is, multiplying and arranging the terms according to the dimensions of n, if 8n3 +4qn2 + (8s - 4rp)n + 4qs + 4p2s — 2 = 0. From this equation let a value of n be obtained, and substituted in the equation (x2 + px + n)2 = (p2 + 2n + q).x2 + (2pn + r)x + n3 + s; then extracting the square root on both sides

x2+px+n=± {√(p2+2n+q)x‍+√(n2‍+s) { when 2pn+r is positive; orx2+px+n=± {} √(p2+2n+q).x— √(n2+s) { “is negative. when 2pm+r And from these two quadratics, the four roots of the given biquadratic may be determined*.

Note. Whenever, by taking away the second term of a biquadratic, after the manner described in cor. th. 3, the fourth term also vanishes, the roots may immediately be obtained by the solution of a quadratic only.

A biquadratic may also be solved independently of cubics, in the following cases:

1. When the difference between the coefficient of the third term, and the square of half that of the second term, is equal to the coefficient of the fourth term, divided by half that of the second. Then if p be the coefficient of the second term, the equation will be reduced to a quadratic by dividing it by ripx.

2. When the last term is negative, and equal to the square of the coefficient of the fourth term divided by 4 times that of the third term, minus the square of that of the second: then to complete the square, subtract the terms of the proposed biquadratic from (x2± px)2, and add the remainder to both its sides.

3. When the coefficient of the fourth term divided by that of the second term, gives for a quotient the square root of the last term: then to complete the square, add the square of half the coefficient of the second term, to twice the square

This rule, for solving biquadratics, by conceiving each to be the difference of two squares, is frequently ascribed to Dr. Waring; but its original inventor was Mr. Thomas Simpson, formerly Professor of Mathematics in the Royal Military Academy.

root

root of the last term, multiply the sum by 2, from the product take the third term, and add the remainder to both sides of the biquadratics.

4. The fourth term will be made to go out by the usual operation for taking away the second term, when the difference between the cube of half the coefficient of the second term and half the product of the coefficients of the second and third term, is equal to the coefficient of the fourth term.

IV. Euler's Rule for the Solution of Biquadratics.

Let x-ax2- bx c = 0, be the given bìquadratic equation wanting the second term. Take ƒ = 4a, g = +α2 + 1c, and hb, or h= b; with which values of f, g, h, form the cubic equation, z3 —ƒz2 + gz - h = 0. Find the roots of this cubic equation, and let them be called p, q, r. Then shall the four roots of the proposed biquadratic be these following: viz.

When bis positive.

1. x =

2. x =

3. x =−√p + √ q = √r. 4. x = √p -√q+√r.

Note 1. In any biquadratic equation having all its terms, if of the square of the coefficient of the 2d term be greater than the product of the coefficients of the 1st and 3d terms, or of the square of the coefficient of the 4th term be greater than the product of the coefficients of the 3d and 5th terms, or of the square of the coefficient of the 3d term greater than the product of the coefficients of the 2d and 4th terms; then all the roots of that equation will be real and unequal: but if either of the said parts of those squares be less than either of those products, the equation will have imaginary

roots.

2. In a biquadratic + + ax3 + bx2 + cx + d = 0, of which two roots are impossible, and d an affirmative quantity, then the two possible roots will be both negative, or both affirmative, according as a3 - 4ab8c, is an affirmative or a negative quantity, if the signs of the coefficients a, b, c, d, are neither all affirmative, nor alternately and +*.

Various general rules for the solution of equations have been given by Demoivre, Bezout, Lagrange, &c; but the most universal in their application are approximating rules, of which a very simple and useful one is given in our first volume.

EXAMPLES.