Equation

An equation admits of description in two ways:—(1) It may be regarded purely as an algebraic expression, or (2) as a geometrical locus. In the first case there is obviously no limit to the number of unknowns and to the degree of the equation; and, consequently, this aspect is the most general. In the second case the number of unknowns is limited to three, corresponding to the three dimensions of space; the degree is unlimited as before. It must be noticed, however, that by the introduction of appropriate hyperspaces, i.e. of degree equal to the number of unknowns, any equation theoretically admits of geometrical visualization, in other words, every equation may be represented by a geometrical figure and every geometrical figure by an equation. Corresponding to these two aspects, there are two typical methods by which equations can be solved, viz. the algebraic and geometric. The former leads to exact results, or, by methods of approximation, to results correct to any required degree of accuracy. The latter can only yield approximate values: when theoretically exact constructions are available there is a source of error in the draughtsmanship, and when the constructions are only approximate, the accuracy of the results is more problematical. The geometric aspect, however, is of considerable value in discussing the theory of equations.

I. Simultaneous Equations.

Simultaneous equations which involve the second and higher powers of the unknown may be impossible of solution. No general rules can be given, and the solution of any particular problem will largely depend upon the student’s ingenuity. Here we shall only give a few typical examples.

1. Equations which may be reduced to linear equations.—Ex. To solve x(x − a) = yz, y (y − b) = zx, z (z − c) = xy. Multiply the equations by y, z and x respectively, and divide the sum by xyz; then

a	+	b	+	c	= 0
z		x		y

Multiply by z, x and y, and divide the sum by xyz; then

a	+	b	+	c	= 0
y		z		x

From (1) and (2) by cross multiplication we obtain

1	=	1	=	1	=	1	(suppose)
y (b² − ac)		z (c² − ab)		x (a² − bc)		λ

Substituting for x, y and z in x (x − a) = yz we obtain

1	=	3abc − (a³ + b³ + c³)	;
λ		(a² − bc) (b² − ac) (c² − ab)

and therefore x, y and z are known from (3). The same artifice solves the equations x² − yz = a, y² − xz = b, z² − xy = c.

2. Equations which are homogeneous and of the same degree.—These equations can be solved by substituting y = mx. We proceed to explain the method by an example.

Ex. To solve 3x² + xy + y² = 15, 31xy − 3x² − 5y² = 45. Substituting y = mx in both these equations, and then dividing, we obtain 31m − 3 − 5m² = 3 (3 + m + m²) or 8m² − 28m + 12 = 0. The roots of this quadratic are m = ½ or 3, and therefore 2y = x, or y = 3x.

Taking 2y = x and substituting in 3x² + xy + y² = 0, we obtain y² (12 + 2 + 1) = 15; ∴ y² = 1, which gives y = ±1, x = ±2. Taking the second value, y = 3x, and substituting for y, we obtain x² (3 + 3 + 9) = 15; ∴ x² = 1, which gives x = ±1, y = ±3. Therefore the solutions are x = ±2, y = ±1 and x = ±1, y = ±3. Other artifices have to be adopted to solve other forms of simultaneous equations, for which the reader is referred to J.J. Milne, Companion to Weekly Problem Papers.

II. Indeterminate Equations.

1. When the number of unknown quantities exceeds the number of equations, the equations will admit of innumerable solutions, and are therefore said to be indeterminate. Thus if it be required to find two numbers such that their sum be 10, we have two unknown quantities x and y, and only one equation, viz. x + y = 10, which may evidently be satisfied by innumerable different values of x and y, if fractional solutions be admitted. It is, however, usual, in such questions as this, to restrict values of the numbers sought to positive integers, and therefore, in this case, we can have only these nine solutions,

x = 1, 2, 3, 4, 5, 6, 7, 8, 9; y = 9, 8, 7, 6, 5, 4, 3, 2, 1;

which indeed may be reduced to five; for the first four become the same as the last four, by simply changing x into y, and the contrary. This branch of analysis was extensively studied by Diophantus, and is sometimes termed the Diophantine Analysis.

2. Indeterminate problems are of different orders, according to the dimensions of the equation which is obtained after all the unknown quantities but two have been eliminated by means of the given equations. Those of the first order lead always to equations of the form

ax ± by = ±c,

where a, b, c denote given whole numbers, and x, y two numbers to be found, so that both may be integers. That this condition may be fulfilled, it is necessary that the coefficients a, b have no common divisor which is not also a divisor of c; for if a = md and b = me, then ax + by = mdx + mey = c, and dx + ey = c/m; but d, e, x, y are supposed to be whole numbers, therefore c/m is a whole number; hence m must be a divisor of c.

Of the four forms expressed by the equation ax ± by = ±c, it is obvious that ax + by = −c can have no positive integral solutions. Also ax − by = −c is equivalent to by − ax = c, and so we have only to consider the forms ax ± by = c. Before proceeding to the general solution of these equations we will give a numerical example.

To solve 2x + 3y = 25 in positive integers. From the given equation we have x = (25 − 3y) / 2 = 12 − y − (y − 1) / 2. Now, since x must be a whole number, it follows that (y − 1)/2 must be a whole number. Let us assume (y − 1) / 2 = z, then y = 1 + 2z; and x = 11 − 3z, where z might be any whole number whatever, if there were no limitation as to the signs of x and y. But since these quantities are required to be positive, it is evident, from the value of y, that z must be either 0 or positive, and from the value of x, that it must be less than 4; hence z may have these four values, 0, 1, 2, 3.

If	z = 0,	z = 1,	z = 2,	z = 3;
Then	x = 11,	x = 8,	x = 5,	x = 2,
	y = 1,	y = 3,	y = 5,	y = 7.

3. We shall now give the solution of the equation ax − by = c in positive integers.

Convert a/b into a continued fraction, and let p/q be the convergent immediately preceding a/b, then aq − bp = ±1 (see Continued Fraction).

(α) If aq − bp = 1, the given equation may be written

ax − by = c (aq − bp);
∴ a (x − cq) = b (y − cp).

Since a and b are prime to one another, then x − cq must be divisible by b and y − cp by a; hence

(x − cq) / b = (y − cq) / a = t.

That is, x = bt + cq and y = at + cp.

Positive integral solutions, unlimited in number, are obtained by giving t any positive integral value, and any negative integral value, so long as it is numerically less than the smaller of the quantities cq/b, cp/a; t may also be zero.

(β) If aq − bp = −1, we obtain x = bt − cq, y = at − cp, from which positive integral solutions, again unlimited in number, are obtained by giving t any positive integral value which exceeds the greater of the two quantities cq/b, cp/a.

If a or b is unity, a/b cannot be converted into a continued fraction with unit numerators, and the above method fails. In this case the solutions can be derived directly, for if b is unity, the equation may be written y = ax − c, and solutions are obtained by giving x positive integral values greater than c/a.

4. To solve ax + by = c in positive integers. Converting a b into a continued fraction and proceeding as before, we obtain, in the case of aq − bp = 1,

x = cq − bt, y = at − cp.

Positive integral solutions are obtained by giving t positive integral values not less than cp/a and not greater than cq/b.

In this case the number of solutions is limited. If aq − bp = −1 we obtain the general solution x = bt − cq, y = cp − at, which is of the same form as in the preceding case. For the determination of the number of solutions the reader is referred to H.S. Hall and S.R. Knight’s Higher Algebra, G. Chrystal’s Algebra, and other text-books.

5. If an equation were proposed involving three unknown quantities, as ax + by + cz = d, by transposition we have ax + by = d − cz, and, putting d − cz = c′, ax + by = c′. From this last equation we may find values of x and y of this form,

x = mr + nc′, y = mr + n′c′,
or x = mr + n (d − cz), y = m′r + n′ (d − cz);

where z and r may be taken at pleasure, except in so far as the values of x, y, z may be required to be all positive; for from such restriction the values of z and r may be confined within certain limits to be determined from the given equation. For more advanced treatment of linear indeterminate equations see Combinatorial Analysis.

6. We proceed to indeterminate problems of the second degree: limiting ourselves to the consideration of the formula y² = a + bx + cx², where x is to be found, so that y may be a rational quantity. The possibility of rendering the proposed formula a square depends altogether upon the coefficients a, b, c; and there are four cases of the problem, the solution of each of which is connected with some peculiarity in its nature.

Case 1. Let a be a square number; then, putting g² for a, we have y² = g² + bx + cx². Suppose √(g² + bx + cx²) = g + mx; then g² + bx + cx² = g² + 2gmx + m²x², or bx + cx² = 2gmx + m²x², that is, b + cx = 2gm + m²x; hence

x =	2gm − b	, y = √(g² + bx + cx²)=	cg − bm + gm²	.
	c − m²		c − m²

Case 2. Let c be a square number = g²; then, putting √(a + bx + g²x²) = m + gx, we find a + bx + g²x² = m² + 2mgx + g²x², or a + bx = m² + 2mgx; hence we find

x =	m² − a	, y = √(a + bx + g²x²) =	bm − gm² − ag	.
	b − 2mg		b − 2mg

Case 3. When neither a nor c is a square number, yet if the expression a + bx + cx² can be resolved into two simple factors, as f + gx and h + kx, the irrationality may be taken away as follows:—

Assume √(a + bx + cx²) = √{ (f + gx) (h + kx) } = m (f + gx), then (f + gx) (h + kx) = m² (f + gx)², or h + kx = m² (f + gx); hence we find

x =	fm² − h	, y = √{ (f + gx) (h + kx) } =	(fk − gh) m	;
	k − gm²		k − gm²

and in all these formulae m may be taken at pleasure.

Case 4. The expression a + bx + cx² may be transformed into a square as often as it can be resolved into two parts, one of which is a complete square, and the other a product of two simple factors; for then it has this form, p² + qr, where p, q and r are quantities which contain no power of x higher than the first. Let us assume √(p² + qr) = p + mq; thus we have p² + qr = p² + 2mpq + m²q² and r = 2mp + m²q, and as this equation involves only the first power of x, we may by proper reduction obtain from it rational values of x and y, as in the three foregoing cases.

The application of the preceding general methods of resolution to any particular case is very easy; we shall therefore conclude with a single example.

Ex. It is required to find two square numbers whose sum is a given square number.

Let a² be the given square number, and x², y² the numbers required; then, by the question, x² + y² = a², and y = √(a² − x²). This equation is evidently of such a form as to be resolvable by the method employed in case 1. Accordingly, by comparing √(a² − x²) with the general expression √(g² + bx + cx²), we have g = a, b = 0, c = −1, and substituting these values in the formulae, and also −n for +m, we find

x =	2an	, y =	a (n² − 1)	.
	n² + 1		n² + 1

If a = n² + 1, there results x = 2n, y = n² − 1, a = n² + 1. Hence if r be an even number, the three sides of a rational right-angled triangle are r, (½ r)² − 1, (½ r)² + 1. If r be an odd number, they become (dividing by 2) r, ½ (r² − 1), ½ (r² + 1).

For example, if r = 4, 4, 4 − 1, 4 + 1, or 4, 3, 5, are the sides of a right-angled triangle; if r = 7, 7, 24, 25 are the sides of a right-angled triangle.

III. Cubic Equations.

1. Cubic equations, like all equations above the first degree, are divided into two classes: they are said to be pure when they contain only one power of the unknown quantity; and adfected when they contain two or more powers of that quantity.

Pure cubic equations are therefore of the form x³ = r; and hence it appears that a value of the simple power of the unknown quantity may always be found without difficulty, by extracting the cube root of each side of the equation. Let us consider the equation x³ − c³ = 0 more fully. This is decomposable into the factors x − c = 0 and x² + cx + c² = 0. The roots of this quadratic equation are ½ (−1 ± √−3) c, and we see that the equation x³ = c³ has three roots, namely, one real root c, and two imaginary roots ½ (−1 ± √−3) c. By making c equal to unity, we observe that ½ (−1 ± √−3) are the imaginary cube roots of unity, which are generally denoted by ω and ω², for it is easy to show that (½ (−1 − √−3))² = ½ (−1 + √−3).

2. Let us now consider such cubic equations as have all their terms, and which are therefore of this form,

x³ + Ax² + Bx + C = 0,

where A, B and C denote known quantities, either positive or negative.

This equation may be transformed into another in which the second term is wanting by the substitution x = y − A/3. This transformation is a particular case of a general theorem. Let xn + Axn−1 + Bxn−2 ... = 0. Substitute x = y + h; then (y + h)n + A (y + h)n−1 ... = 0. Expand each term by the binomial theorem, and let us fix our attention on the coefficient of yn−1. By this process we obtain 0 = yn + yn−1(A + nh) + terms involving lower powers of y.

Now h can have any value, and if we choose it so that A + nh = 0, then the second term of our derived equation vanishes.

Resuming, therefore, the equation y³ + qy + r = 0, let us suppose y = v + z; we then have y³ = v³ + z³ + 3vz (v + z) = v³ + z³ + 3vzy, and the original equation becomes v³ + z³ + (3vz + q) y + r = 0. Now v and z are any two quantities subject to the relation y = v + z, and if we suppose 3vz + q = 0, they are completely determined. This leads to v³ + z³ + r = 0 and 3vz + q = 0. Therefore v³ and z³ are the roots of the quadratic t² + rt − q²/27 = 0. Therefore

v³ =	−½ r + √(1⁄27 q³ + ¼ r²); z³ = −½ r − √(1⁄27 q³ + ¼r²);
v =	3√{−½ r + √(1⁄27 q³ + ¼ r²) }; z = 3√{ (−½ r − √(1⁄27 q³ + ¼ r²) };
and y =	v + z = 3√{−½ r + √(1⁄27q³ + ¼ r²) } + 3√{−½ r − √(1⁄27 q³ + ¼ r²) }.

Thus we have obtained a value of the unknown quantity y, in terms of the known quantities q and r; therefore the equation is resolved.

3. But this is only one of three values which y may have. Let us, for the sake of brevity, put

A = −½ r + √(1⁄27 q³ + ¼ r²), B = −½ r − √(1⁄27 q³ + ¼ r²),

and put	α = ½ (−1 + √−3),
	β = ½ (−1 − √−3).

Then, from what has been shown (§ 1), it is evident that v and z have each these three values,

v = 3√A, v = α3√A, v = β3√A;
z = 3√B, z = α3√B, z = β3√B.

To determine the corresponding values of v and z, we must consider that vz = −1⁄3 q = 3√(AB). Now if we observe that αβ = 1, it will immediately appear that v + z has these three values,

v + z =  3√A +  3√B,
v + z = α3√A + β3√B,
v + z = β3√A + α3√B,

which are therefore the three values of y.

The first of these formulae is commonly known by the name of Cardan’s rule (see Algebra: History).

The formulae given above for the roots of a cubic equation may be put under a different form, better adapted to the purposes of arithmetical calculation, as follows:—Because vz = −1⁄3 q, therefore z = −1⁄3q × 1/v = −1⁄3 q / 3√A; hence v + z = 3√A − 1⁄3 q / 3√A: thus it appears that the three values of y may also be expressed thus:

y =  3√A − 1⁄3 q /  3√A
y = α3√A − 1⁄3 qβ / 3√A
y = β3√A − 1⁄3 qα / 3√A.

See below, Theory of Equations, §§ 16 et seq.

IV. Biquadratic Equations.

1. When a biquadratic equation contains all its terms, it has this form,

x4 + Ax³ + Bx² + Cx + D = 0,

where A, B, C, D denote known quantities.

We shall first consider pure biquadratics, or such as contain only the first and last terms, and therefore are of this form, x4 = b4. In this case it is evident that x may be readily had by two extractions of the square root; by the first we find x² = b², and by the second x = b. This, however, is only one of the values which x may have; for since x4 = b4, therefore x4 − b4 = 0; but x4 − b4 may be resolved into two factors x² − b² and x² + b², each of which admits of a similar resolution; for x² − b² = (x − b)(x + b) and x² + b² = (x − b√−1)(x + b√−1). Hence it appears that the equation x4 − b4 = 0 may also be expressed thus,

(x − b) (x + b) (x − b√−1) (x + b√−1) = 0;

so that x may have these four values,

+b,    −b,    +b√−1,    −b√−1,

two of which are real, and the others imaginary.

2. Next to pure biquadratic equations, in respect of easiness of resolution, are such as want the second and fourth terms, and therefore have this form,

x4 + qx² + s = 0.

These may be resolved in the manner of quadratic equations; for if we put y = x², we have

y² + qy + s = 0,

from which we find y = ½ {−q ± √(q² − 4s) }, and therefore

x = ±√½ {−q ± √(q² − 4s) }.

3. When a biquadratic equation has all its terms, its resolution may be always reduced to that of a cubic equation. There are various methods by which such a reduction may be effected. The following was first given by Leonhard Euler in the Petersburg Commentaries, and afterwards explained more fully in his Elements of Algebra.

We have already explained how an equation which is complete in its terms may be transformed into another of the same degree, but which wants the second term; therefore any biquadratic equation may be reduced to this form,

y4 + py² + qy + r = 0,

where the second term is wanting, and where p, q, r denote any known quantities whatever.

That we may form an equation similar to the above, let us assume y = √a + √b + √c, and also suppose that the letters a, b, c denote the roots of the cubic equation

z³ + Pz² + Qz − R = 0;

then, from the theory of equations we have

a + b + c = −P,    ab + ac + bc = Q,    abc = R.

We square the assumed formula

y = √a + √b + √c,

and obtain    y² = a + b + c + 2(√ab + √ac + √bc);

or, substituting −P for a + b + c, and transposing,

y² + P = 2(√ab + √ac + √bc).

Let this equation be also squared, and we have

y4 + 2Py² + P² = 4 (ab + ac + bc) + 8 (√a²bc + √ab²c + √abc²);

and since      ab + ac + bc = Q,

and   √a²bc + √ab²c + √abc² = √abc (√a + √b + √c) = √R·y,

the same equation may be expressed thus:

y4 + 2Py² + P² = 4Q + 8√R·y.

Thus we have the biquadratic equation

y4 + 2Py² − 8√R·y + P² − 4Q = 0,

one of the roots of which is y = √a + √b + √c, while a, b, c are the roots of the cubic equation z³ + Pz² + Qz − R = 0.

4. In order to apply this resolution to the proposed equation y4 + py² + qy + r = 0, we must express the assumed coefficients P, Q, R by means of p, q, r, the coefficients of that equation. For this purpose let us compare the equations

y4 + py² + qy + r = 0,
y4 + 2Py² − 8√Ry + P² − 4Q = 0,

and it immediately appears that

2P = p,    −8√R = q,    P² − 4Q = r;

and from these equations we find

P = ½ p,   Q = 1⁄16 (p² − 4r),   R = 1⁄64 q².

Hence it follows that the roots of the proposed equation are generally expressed by the formula

y = √a + √b + √c;

where a, b, c denote the roots of this cubic equation,

z³ +	p	z² +	p² − 4r	z −	q²	= 0.
	2		16		64

But to find each particular root, we must consider, that as the square root of a number may be either positive or negative, so each of the quantities √a, √b, √c may have either the sign + or − prefixed to it; and hence our formula will give eight different expressions for the root. It is, however, to be observed, that as the product of the three quantities √a, √b, √c must be equal to √R or to −1⁄8 q; when q is positive, their product must be a negative quantity, and this can only be effected by making either one or three of them negative; again, when q is negative, their product must be a positive quantity; so that in this case they must either be all positive, or two of them must be negative. These considerations enable us to determine that four of the eight expressions for the root belong to the case in which q is positive, and the other four to that in which it is negative.

5. We shall now give the result of the preceding investigation in the form of a practical rule; and as the coefficients of the cubic equation which has been found involve fractions, we shall transform it into another, in which the coefficients are integers, by supposing z = ¼ v. Thus the equation

z³ +	p	z² +	p² − 4r	z −	q²	= 0
	2		16		64

becomes, after reduction,

v³ + 2pv² + (p² − 4r) v − q² = 0;

it also follows, that if the roots of the latter equation are a, b, c, the roots of the former are ¼ a, ¼ b, ¼ c, so that our rule may now be expressed thus:

Let y4 + py² + qy + r = 0 be any biquadratic equation wanting its second term. Form this cubic equation

v³ + 2pv² + (p² − 4r) v − q² = 0,

and find its roots, which let us denote by a, b, c.

Then the roots of the proposed biquadratic equation are,

when q is negative,	when q is positive,
y = ½ (√a + √b + √c),	y = ½ (−√a − √b − √c),
y = ½ (√a − √b − √c),	y = ½ (−√a + √b + √c),
y = ½ (−√a + √b − √c),	y = ½ (√a − √b + √c),
y = ½ (−√a − √b + √c),	y = ½ (√a + √b − √c).

See also below, Theory of Equations, § 17 et seq.

In particular cases it is frequently possible to ascertain the number of the real roots, and to effect their separation by trial or otherwise, without much difficulty; but the foregoing was the general process as employed by Joseph Louis Lagrange even in the second edition (1808) of the Traité de la résolution des équations numériques;2 the determination of the limit δ had to be effected by means of the “equation of differences” or equation of the order ½ n(n − 1), the roots of which are the squares of the differences of the roots of the given equation, and the process is a cumbrous and unsatisfactory one.

First as to Horner and Lagrange. We know that between the limits β, α there lies one, and only one, real root of the equation; ƒ(β) and ƒ(α) have therefore opposite signs. Suppose any intermediate value is θ; in order to determine by Sturm’s theorem whether the root lies between β, θ, or between θ, α, it would be quite unnecessary to calculate the signs of ƒ(θ),ƒ′(θ), ƒ2(θ) ...; only the sign of ƒ(θ) is required; for, if this has the same sign as ƒ(β), then the root is between β, θ; if the same sign as ƒ(α), then the root is between θ, α. We want to make θ increase from the inferior limit β, at which ƒ(θ) has the sign of ƒ(β), so long as ƒ(θ) retains this sign, and then to a value for which it assumes the opposite sign; we have thus two nearer limits of the required root, and the process may be repeated indefinitely.

Horner’s method (1819) gives the root as a decimal, figure by figure; thus if the equation be known to have one real root between 0 and 10, it is in effect shown say that 5 is too small (that is, the root is between 5 and 6); next that 5.4 is too small (that is, the root is between 5.4 and 5.5); and so on to any number of decimals. Each figure is obtained, not by the successive trial of all the figures which precede it, but (as in the ordinary process of the extraction of a square root, which is in fact Horner’s process applied to this particular case) it is given presumptively as the first figure of a quotient; such value may be too large, and then the next inferior integer must be tried instead of it, or it may require to be further diminished. And it is to be remarked that the process not only gives the approximate value α of the root, but (as in the extraction of a square root) it includes the calculation of the function ƒ(α), which should be, and approximately is, = 0. The arrangement of the calculations is very elegant, and forms an integral part of the actual method. It is to be observed that after a certain number of decimal places have been obtained, a good many more can be found by a mere division. It is in the progress tacitly assumed that the roots have been first separated.

Lagrange’s method (1767) gives the root as a continued fraction a + 1/b + 1/c + ..., where a is a positive or negative integer (which may be = 0), but b, c, ... are positive integers. Suppose the roots have been separated; then (by trial if need be of consecutive integer values) the limits may be made to be consecutive integer numbers: say they are a, a + 1; the value of x is therefore = a + 1/y, where y is positive and greater than 1; from the given equation for x, writing therein x = a + 1/y, we form an equation of the same order for y, and this equation will have one, and only one, positive root greater than 1; hence finding for it the limits b, b + 1 (where b is = or > 1), we have y = b + 1/z, where z is positive and greater than 1; and so on—that is, we thus obtain the successive denominators b, c, d ... of the continued fraction. The method is theoretically very elegant, but the disadvantage is that it gives the result in the form of a continued fraction, which for the most part must ultimately be converted into a decimal. There is one advantage in the method, that a commensurable root (that is, a root equal to a rational fraction) is found accurately, since, when such root exists, the continued fraction terminates.

6. Newton’s method (1711), as perfected by Fourier(1831), may be roughly stated as follows. If x = γ be an approximate value of any root, and γ + h the correct value, then ƒ(γ + h) = 0, that is,

ƒ(γ) +	h	ƒ′(γ) +	h²	ƒ″(γ) + ... = 0;
	1		1·2

and then, if h be so small that the terms after the second may be neglected, ƒ(γ) + hƒ′(γ) = 0, that is, h = {−ƒ(γ)/ƒ′(γ) }, or the new approximate value is x = γ − {ƒ(γ)/ƒ′(γ) }; and so on, as often as we please. It will be observed that so far nothing has been assumed as to the separation of the roots, or even as to the existence of a real root; γ has been taken as the approximate value of a root, but no precise meaning has been attached to this expression. The question arises, What are the conditions to be satisfied by γ in order that the process may by successive repetitions actually lead to a certain real root of the equation; or that, γ being an approximate value of a certain real root, the new value γ − {ƒ(γ)/ƒ′(γ) } may be a more approximate value.

Fig. 1.

Referring to fig. 1, it is easy to see that if OC represent the assumed value γ, then, drawing the ordinate CP to meet the curve in P, and the tangent PC′ to meet the axis in C′, we shall have OC′ as the new approximate value of the root. But observe that there is here a real root OX, and that the curve beyond X is convex to the axis; under these conditions the point C′ is nearer to X than was C; and, starting with C′ instead of C, and proceeding in like manner to draw a new ordinate and tangent, and so on as often as we please, we approximate continually, and that with great rapidity, to the true value OX. But if C had been taken on the other side of X, where the curve is concave to the axis, the new point C′ might or might not be nearer to X than was the point C; and in this case the method, if it succeeds at all, does so by accident only, i.e. it may happen that C′ or some subsequent point comes to be a point C, such that CO is a proper approximate value of the root, and then the subsequent approximations proceed in the same manner as if this value had been assumed in the first instance, all the preceding work being wasted. It thus appears that for the proper application of the method we require more than the mere separation of the roots. In order to be able to approximate to a certain root α, = OX, we require to know that, between OX and some value ON, the curve is always convex to the axis (analytically, between the two values, ƒ(x) and ƒ″(x) must have always the same sign). When this is so, the point C may be taken anywhere on the proper side of X, and within the portion XN of the axis; and the process is then the one already explained. The approximation is in general a very rapid one. If we know for the required root OX the two limits OM, ON such that from M to X the curve is always concave to the axis, while from X to N it is always convex to the axis,—then, taking D anywhere in the portion MX and (as before) C in the portion XN, drawing the ordinates DQ, CP, and joining the points P, Q by a line which meets the axis in D′, also constructing the point C′ by means of the tangent at P as before, we have for the required root the new limits OD′, OC′; and proceeding in like manner with the points D′, C′, and so on as often as we please, we obtain at each step two limits approximating more and more nearly to the required root OX. The process as to the point D′, translated into analysis, is the ordinate process of interpolation. Suppose OD = β, OC = α, we have approximately ƒ(β + h) = ƒ(β) + h{ƒ(α) − ƒ(β) } / (α − β), whence if the root is β + h then h = − (α − β)ƒ(β) / {ƒ(α) − ƒ(β) }.

Returning for a moment to Horner’s method, it may be remarked that the correction h, to an approximate value α, is therein found as a quotient the same or such as the quotient ƒ(α) ÷ ƒ′(α) which presents itself in Newton’s method. The difference is that with Horner the integer part of this quotient is taken as the presumptive value of h, and the figure is verified at each step. With Newton the quotient itself, developed to the proper number of decimal places, is taken as the value of h; if too many decimals are taken, there would be a waste of work; but the error would correct itself at the next step. Of course the calculation should be conducted without any such waste of work.

A number a such that substituted for x it makes the function x1n − p1xn−1 ... ± pn to be = 0, or say such that it satisfies the equation ƒ(x) = 0, is said to be a root of the equation; that is, a being a root, we have

an − p1an−1 ... ± pn = 0, or say ƒ(a) = 0;

and it is then easily shown that x − a is a factor of the function ƒ(x), viz. that we have ƒ(x) = (x − a)ƒ1(x), where ƒ1(x) is a function xn−1 − q1xn−2 ... ± qn−1 of the order n − 1, with numerical coefficients q1, q2 ... qn−1.

In general a is not a root of the equation ƒ1(x) = 0, but it may be so—i.e. ƒ1(x) may contain the factor x − a; when this is so, ƒ(x) will contain the factor (x − a)²; writing then ƒ(x) = (x − a)²ƒ2(x), and assuming that a is not a root of the equation ƒ2(x) = 0, x = a is then said to be a double root of the equation ƒ(x) = 0; and similarly ƒ(x) may contain the factor (x − a)³ and no higher power, and x = a is then a triple root; and so on.

Supposing in general that ƒ(x) = (x − a)αF(x) (α being a positive integer which may be = 1, (x − a)α the highest power of x − a which divides ƒ(x), and F(x) being of course of the order n − α), then the equation F(x) = 0 will have a root b which will be different from a; x − b will be a factor, in general a simple one, but it may be a multiple one, of F(x), and ƒ(x) will in this case be = (x − a)α (x − b)β Φ(x) (β a positive integer which may be = 1, (x − b)β the highest power of x − b in F(x) or ƒ(x), and Φ(x) being of course of the order n − α − β). The original equation ƒ(x) = 0 is in this case said to have α roots each = a, β roots each = b; and so on for any other factors (x − c)γ, &c.

We have thus the theorem—A numerical equation of the order n has in every case n roots, viz. there exist n numbers, a, b, ... (in general all distinct, but which may arrange themselves in any sets of equal values), such that ƒ(x) = (x − a)(x − b)(x − c) ... identically.

If the equation has equal roots, these can in general be determined, and the case is at any rate a special one which may be in the first instance excluded from consideration. It is, therefore, in general assumed that the equation ƒ(x) = 0 has all its roots unequal.

If the coefficients p1, p2, ... are all or any one or more of them imaginary, then the equation ƒ(x) = 0, separating the real and imaginary parts thereof, may be written F(x) + iΦ(x) = 0, where F(x), Φ(x) are each of them a function with real coefficients; and it thus appears that the equation ƒ(x) = 0, with imaginary coefficients, has not in general any real root; supposing it to have a real root a, this must be at once a root of each of the equations F(x) = 0 and Φ(x) = 0.

But an equation with real coefficients may have as well imaginary as real roots, and we have further the theorem that for any such equation the imaginary roots enter in pairs, viz. α + βi being a root, then α − βi will be also a root. It follows that if the order be odd, there is always an odd number of real roots, and therefore at least one real root.

This is solved theoretically by means of a theorem of A.L. Cauchy (1837), viz. writing in the original equation x + iy in place of x, the function ƒ(x + iy) becomes = P + iQ, where P and Q are each of them a rational and integral function (with real coefficients) of (x, y). Imagining the point (x, y) to travel along the contour, and considering the number of changes of sign from − to + and from + to − of the fraction corresponding to passages of the fraction through zero (that is, to values for which P becomes = 0, disregarding those for which Q becomes = 0), the difference of these numbers gives the number of roots within the contour.

It is important to remark that the demonstration does not presuppose the existence of any root; the contour may be the infinity of the plane (such infinity regarded as a contour, or closed curve), and in this case it can be shown (and that very easily) that the difference of the numbers of changes of sign is = n; that is, there are within the infinite contour, or (what is the same thing) there are in all n roots; thus Cauchy’s theorem contains really the proof of the fundamental theorem that a numerical equation of the nth order (not only has a numerical root, but) has precisely n roots. It would appear that this proof of the fundamental theorem in its most complete form is in principle identical with the last proof of K.F. Gauss (1849) of the theorem, in the form—A numerical equation of the nth order has always a root.3

But in the case of a finite contour, the actual determination of the difference which gives the number of real roots can be effected only in the case of a rectangular contour, by applying to each of its sides separately a method such as that of Sturm’s theorem; and thus the actual determination ultimately depends on a method such as that of Sturm’s theorem.

Very little has been done in regard to the calculation of the imaginary roots of an equation by approximation; and the question is not here considered.

The foregoing conclusions apply, viz. there are always n roots, which, it may be shown, are all unequal. And these can be found numerically by the extraction of the square root, and of an nth root, of real numbers, and by the aid of a table of natural sines and cosines.4 For writing

α + βi = √(α² + β²) {	α	+	β	i },
	√(α² + β²)		√(α² + β²)

there is always a real angle λ (positive and less than 2π), such that its cosine and sine are = α / √(α² + β²) and β / √(α² + β²) respectively; that is, writing for shortness √(α² + β²) = ρ, we have α + βi = ρ (cos λ + i sin λ), or the equation is xn = ρ (cos λ + i sin λ); hence observing that (cos λ/n + i sin λ/n )n = cos λ + i sin λ, a value of x is = n√ρ (cos λ/n + i sin λ/n). The formula really gives all the roots, for instead of λ we may write λ + 2sπ, s a positive or negative integer, and then we have

x = n√ρ ( cos	λ + 2sπ	+ i sin	λ + 2sπ	),
	n		n

which has the n values obtained by giving to s the values 0, 1, 2 ... n − 1 in succession; the roots are, it is clear, represented by points lying at equal intervals on a circle. But it is more convenient to proceed somewhat differently; taking one of the roots to be θ, so that θn = a, then assuming x = θy, the equation becomes yn − 1 = 0, which equation, like the original equation, has precisely n roots (one of them being of course = 1). And the original equation xn − a = 0 is thus reduced to the more simple equation xn − 1 = 0; and although the theory of this equation is included in the preceding one, yet it is proper to state it separately.

The equation xn − 1 = 0 has its several roots expressed in the form 1, ω, ω², ... ωn−1, where ω may be taken = cos 2π/n + i sin 2π/n; in fact, ω having this value, any integer power ωk is = cos 2πk/n + i sin 2πk/n, and we thence have (ωk)n = cos 2πk + i sin 2πk, = 1, that is, ωk is a root of the equation. The theory will be resumed further on.

By what precedes, we are led to the notion (a numerical) of the radical a1/n regarded as an n-valued function; any one of these being denoted by n√a, then the series of values is n√a, ωn√a, ... ωn−1 n√a; or we may, if we please, use n√a instead of a1/n as a symbol to denote the n-valued function.

As the coefficients of an algebraical equation may be numerical, all which follows in regard to algebraical equations is (with, it may be, some few modifications) applicable to numerical equations; and hence, concluding for the present this subject, it will be convenient to pass on to algebraical equations.

As already explained, the epithet algebraical is not used in opposition to numerical; an algebraical equation is merely an equation wherein the coefficients are not restricted to denote, or are not explicitly considered as denoting, numbers. That the abstraction is legitimate, appears by the simplest example; in saying that the equation x² − px + q = 0 has a root x = ½ {p + √(p² − 4q) }, we mean that writing this value for x the equation becomes an identity, [½ {p + √(p² − 4q) }]² − p[½ {p + √(p² − 4q) }] + q = 0; and the verification of this identity in nowise depends upon p and q meaning numbers. But if it be asked what there is beyond numerical equations included in the term algebraical equation, or, again, what is the full extent of the meaning attributed to the term—the latter question at any rate it would be very difficult to answer; as to the former one, it may be said that the coefficients may, for instance, be symbols of operation. As regards such equations, there is certainly no proof that every equation has a root, or that an equation of the nth order has n roots; nor is it in any wise clear what the precise signification of the statement is. But it is found that the assumption of the existence of the n roots can be made without contradictory results; conclusions derived from it, if they involve the roots, rest on the same ground as the original assumption; but the conclusion may be independent of the roots altogether, and in this case it is undoubtedly valid; the reasoning, although actually conducted by aid of the assumption (and, it may be, most easily and elegantly in this manner), is really independent of the assumption. In illustration, we observe that it is allowable to express a function of p and q as follows,—that is, by means of a rational symmetrical function of a and b, this can, as a fact, be expressed as a rational function of a + b and ab; and if we prescribe that a + b and ab shall then be changed into p and q respectively, we have the required function of p, q. That is, we have F(α, β) as a representation of ƒ(p, q), obtained as if we had p = a + b, q = ab, but without in any wise assuming the existence of the a, b of these equations.

The process originally employed for the expression of other functions Σaαbβ, &c., in terms of the coefficients is to make them depend upon the sums of powers: for instance, Σaαbβ = ΣaαΣaβ − Σaα+β; but this is very objectionable; the true theory consists in showing that we have systems of equations

p1	= Σa,
p2	=     Σab,
p1²	= Σa² + 2Σab,
p3	=        Σabc,
p1p2	=     Σa²b + 3Σabc,
p1³	= Σa³ + 3Σa²b + 6Σabc,

where in each system there are precisely as many equations as there are root-functions on the right-hand side—e.g. 3 equations and 3 functions Σabc, Σa²b, Σa³. Hence in each system the root-functions can be determined linearly in terms of the powers and products of the coefficients:

Σab	=     p2,
Σa²	= p1² − 2p2,
Σabc	=        p3,
Σa²b	=     p1p2 − 3p3,
Σa³	= p1³ − 3p1p2 + 3p3,

and so on. The other process, if applied consistently, would derive the originally assumed value Σab = p2, from the two equations Σa = p, Σa² = p1² − 2p2; i.e. we have 2Σab = Σa·Σa − Σa²,= p1² − (p1² − 2p2), = 2p2.

From the theorem that a rational symmetrical function of the roots is expressible in terms of the coefficients, it at once follows that it is possible to determine an equation (of an assignable order) having for its roots the several values of any given (unsymmetrical) function of the roots of the given equation. For example, in the case of a quartic equation, roots (a, b, c, d), it is possible to find an equation having the roots ab, ac, ad, bc, bd, cd (being therefore a sextic equation): viz. in the product

(y − ab) (y − ac) (y − ad) (y − bc) (y − bd) (y − cd)

the coefficients of the several powers of y will be symmetrical functions of a, b, c, d and therefore rational and integral functions of the coefficients of the quartic equation; hence, supposing the product so expressed, and equating it to zero, we have the required sextic equation. In the same manner can be found the sextic equation having the roots (a − b)², (a − c)², (a − d)², (b − c)², (b − d)², (c − d)², which is the equation of differences previously referred to; and similarly we obtain the equation of differences for a given equation of any order. Again, the equation sought for may be that having for its n roots the given rational functions φ(a), φ(b), ... of the several roots of the given equation. Any such rational function can (as was shown) be expressed as a rational and integral function of the order n − 1; and, retaining x in place of any one of the roots, the problem is to find y from the equations xn − p1xn−1 ... = 0, and y = M0xn−1 + M1xn−2 + ..., or, what is the same thing, from these two equations to eliminate x. This is in fact E.W. Tschirnhausen’s transformation (1683).

It is at once seen that for any given number of letters there exist 2-valued functions; the product of the differences of the letters is such a function; however the letters are interchanged, it alters only its sign; or say the two values are Δ and −Δ. And if P, Q are symmetrical functions of the letters, then the general form of such a function is P + QΔ; this has only the two values P + QΔ, P − QΔ.

In the case of 4 letters there exist (as appears above) 3-valued functions: but in the case of 5 letters there does not exist any 3-valued or 4-valued function; and the only 5-valued functions are those which are symmetrical in regard to four of the letters, and can thus be expressed in terms of one letter and of symmetrical functions of all the letters. These last theorems present themselves in the demonstration of the non-existence of a solution of a quintic equation by radicals.

The a priori ground of this theorem may be illustrated by means of a numerical equation. Suppose that the roots of a quartic equation are 1, 2, 3, 4, then if it is given that ab + cd = 14, this in effect determines a, b to be 1, 2 and c, d to be 3, 4 (viz. a = 1, b = 2 or a = 2, b = 1, and c = 3, d = 4 or c = 3, d = 4) or else a, b to be 3, 4 and c, d to be 1, 2; and it therefore in effect determines (a + b)³ + (c + d)³ to be = 370, and not any other value; that is, (a + b)³ + (c + d)³, as having a single value, must be determinable rationally. And we can in the same way account for cases of failure as regards particular equations; thus, the roots being 1, 2, 3, 4 as before, a²b = 2 determines a to be = 1 and b to be = 2, but if the roots had been 1, 2, 4, 16 then a²b = 16 does not uniquely determine a, b but only makes them to be 1, 16 or 2, 4 respectively.

As to the a posteriori proof, assume, for instance,

t1 = ab + cd,   y1 = (a + b)³ + (c + d)³,
t2 = ac + bd,   y2 = (a + c)³ + (b + d)³,
t3 = ad + bc,   y3 = (a + d)³ + (b + c)³;

then y1 + y2 + y3, t1y1 + t2y2 + t3y3, t1²y1 + t2²y2 + t3²y3 will be respectively symmetrical functions of the roots of the quartic, and therefore rational and integral functions of the coefficients; that is, they will be known.

Suppose for a moment that t1, t2, t3 are all known; then the equations being linear in y1, y2, y3 these can be expressed rationally in terms of the coefficients and of t1, t2, t3; that is, y1, y2, y3 will be known. But observe further that y1 is obtained as a function of t1, t2, t3 symmetrical as regards t2, t3; it can therefore be expressed as a rational function of t1 and of t2 + t3, t2t3, and thence as a rational function of t1 and of t1 + t2 + t3, t1t2 + t1t3 + t2t3, t1t2t3; but these last are symmetrical functions of the roots, and as such they are expressible rationally in terms of the coefficients; that is, y1 will be expressed as a rational function of t1 and of the coefficients; or t1 (alone, not t2 or t3) being known, y1 will be rationally determined.

In the case of a quadric equation x² − px + q = 0, we can by the assistance of the sign √( ) or ( )1/2 find an expression for x as a 2-valued function of the coefficients p, q such that substituting this value in the equation, the equation is thereby identically satisfied; it has been found that this expression is

x = ½ {p ± √(p² − 4q) },

and the equation is on this account said to be algebraically solvable, or more accurately solvable by radicals. Or we may by writing x = −½ p + z reduce the equation to z² = ¼ (p² − 4q), viz. to an equation of the form x² = a; and in virtue of its being thus reducible we say that the original equation is solvable by radicals. And the question for an equation of any higher order, say of the order n, is, can we by means of radicals (that is, by aid of the sign m√( ) or ( )1/m, using as many as we please of such signs and with any values of m) find an n-valued function (or any function) of the coefficients which substituted for x in the equation shall satisfy it identically?

It will be observed that the coefficients p, q ... are not explicitly considered as numbers, but even if they do denote numbers, the question whether a numerical equation admits of solution by radicals is wholly unconnected with the before-mentioned theorem of the existence of the n roots of such an equation. It does not even follow that in the case of a numerical equation solvable by radicals the algebraical solution gives the numerical solution, but this requires explanation. Consider first a numerical quadric equation with imaginary coefficients. In the formula x = ½ {p ± √(p² − 4q) }, substituting for p, q their given numerical values, we obtain for x an expression of the form x = α + βi ± √(γ + δi), where α, β, γ, δ are real numbers. This expression substituted for x in the quadric equation would satisfy it identically, and it is thus an algebraical solution; but there is no obvious a priori reason why √(γ + δi) should have a value = c + di, where c and d are real numbers calculable by the extraction of a root or roots of real numbers; however the case is (what there was no a priori right to expect) that √(γ + δi) has such a value calculable by means of the radical expressions √{√(γ² + δ²) ± γ}; and hence the algebraical solution of a numerical quadric equation does in every case give the numerical solution. The case of a numerical cubic equation will be considered presently.

Taking for greater simplicity the cubic in the reduced form x³ + qx − r = 0, and assuming x = a + b, this will be a solution if only 3ab = q and a³ + b³ = r, equations which give (a³ − b³)² = r² − 4⁄27 q³, a quadric equation solvable by radicals, and giving a³ − b³ = √(r² − 4⁄27 q³), a 2-valued function of the coefficients: combining this with a³ + b³ = r, we have a³ = ½ {r + √(r² − 4⁄27 q³) }, a 2-valued function: we then have a by means of a cube root, viz.

a = 3√[½ {r + √(r² − 4⁄27 q³) }],

a 6-valued function of the coefficients; but then, writing q = b/3a, we have, as may be shown, a + b a 3-valued function of the coefficients; and x = a + b is the required solution by radicals. It would have been wrong to complete the solution by writing

b = 3√[½ {r − √(r² − 4⁄27 q³) } ],

for then a + b would have been given as a 9-valued function having only 3 of its values roots, and the other 6 values being irrelevant. Observe that in this last process we make no use of the equation 3ab = q, in its original form, but use only the derived equation 27a³b³ = q³, implied in, but not implying, the original form.

An interesting variation of the solution is to write x = ab(a + b), giving a³b³ (a³ + b³) = r and 3a³b³ = q, or say a³ + b³ = 3r/q, a³b³ = 1⁄3 q; and consequently

a³ =	3⁄2	{r + √(r² − 4⁄27 q³) }, b³ =	3⁄2	{r − √(r² − 4⁄27 q³) },
	q		q

i.e. here a³, b³ are each of them a 2-valued function, but as the only effect of altering the sign of the quadric radical is to interchange a³, b³, they may be regarded as each of them 1-valued; a and b are each of them 3-valued (for observe that here only a³b³, not ab, is given); and ab(a + b) thus is in appearance a 9-valued function; but it can easily be shown that it is (as it ought to be) only 3-valued.

In the case of a numerical cubic, even when the coefficients are real, substituting their values in the expression

x = 3√[½ {r + √(r² − 4⁄27 q³) }] + 1⁄3 q ÷ 3√[½ {r + √(r² − 4⁄27 q³) }],

this may depend on an expression of the form 3√(γ + δi) where γ and δ are real numbers (it will do so if r² − 4⁄27 q³ is a negative number), and then we cannot by the extraction of any root or roots of real positive numbers reduce 3√(γ + δi) to the form c + di, c and d real numbers; hence here the algebraical solution does not give the numerical solution, and we have here the so-called “irreducible case” of a cubic equation. By what precedes there is nothing in this that might not have been expected; the algebraical solution makes the solution depend on the extraction of the cube root of a number, and there was no reason for expecting this to be a real number. It is well known that the case in question is that wherein the three roots of the numerical cubic equation are all real; if the roots are two imaginary, one real, then contrariwise the quantity under the cube root is real; and the algebraical solution gives the numerical one.

The irreducible case is solvable by a trigonometrical formula, but this is not a solution by radicals: it consists in effect in reducing the given numerical cubic (not to a cubic of the form z³ = a, solvable by the extraction of a cube root, but) to a cubic of the form 4x³ − 3x = a, corresponding to the equation 4 cos³ θ − 3 cos θ = cos 3θ which serves to determine cosθ when cos 3θ is known. The theory is applicable to an algebraical cubic equation; say that such an equation, if it can be reduced to the form 4x³ − 3x = a, is solvable by “trisection”—then the general cubic equation is solvable by trisection.

It will be understood from the foregoing explanation as to the quartic how in the next following case, that of the quintic, the question of the solvability by radicals depends on the existence or non-existence of k-valued functions of the five roots (a, b, c, d, e); the fundamental theorem is the one already stated, a rational function of five letters, if it has less than 5, cannot have more than 2 values, that is, there are no 3-valued or 4-valued functions of 5 letters: and by reasoning depending in part upon this theorem, N.H. Abel (1824) showed that a general quintic equation is not solvable by radicals; and a fortiori the general equation of any order higher than 5 is not solvable by radicals.

19. The general theory of the solvability of an equation by radicals depends fundamentally on A.T. Vandermonde’s remark (1770) that, supposing an equation is solvable by radicals, and that we have therefore an algebraical expression of x in terms of the coefficients, then substituting for the coefficients their values in terms of the roots, the resulting expression must reduce itself to any one at pleasure of the roots a, b, c ...; thus in the case of the quadric equation, in the expression x = ½ {p + √(p² − 4q) }, substituting for p and q their values, and observing that (a + b)² − 4ab = (a − b)², this becomes x = ½ {a + b + √(a − b)²}, the value being a or b according as the radical is taken to be +(a − b) or −(a − b).

So in the cubic equation x³ − px² + qx − r = 0, if the roots are a, b, c, and if ω is used to denote an imaginary cube root of unity, ω² + ω + 1 = 0, then writing for shortness p = a + b + c, L = a + ωb + ω²c, M = a + ω²b + ωc, it is at once seen that LM, L³ + M³, and therefore also (L³ − M³)² are symmetrical functions of the roots, and consequently rational functions of the coefficients; hence

½ {L³ + M³ + √(L³ − M³)²}

is a rational function of the coefficients, which when these are replaced by their values as functions of the roots becomes, according to the sign given to the quadric radical, = L³ or M³; taking it = L³, the cube root of the expression has the three values L, ωL, ω²L; and LM divided by the same cube root has therefore the values M, ω²M, ωM; whence finally the expression

1⁄3 [p + 3√{½ (L³ + M³ + √(L³ − M³)²) } + LM ÷ 3√{½ L³ + M³ + √(L³ − M³)²) }]

has the three values

1⁄3 (p + L + M), 1⁄3 (p + ωL + ω²M), 1⁄3 (p + ω²L + ωM);

that is, these are = a, b, c respectively. If the value M³ had been taken instead of L³, then the expression would have had the same three values a, b, c. Comparing the solution given for the cubic x³ + qx − r = 0, it will readily be seen that the two solutions are identical, and that the function r² − 4⁄27 q³ under the radical sign must (by aid of the relation p = 0 which subsists in this case) reduce itself to (L³ − M³)²; it is only by each radical being equal to a rational function of the roots that the final expression can become equal to the roots a, b, c respectively.

22. It has already been shown how the several roots of the equation xn − 1 = 0 can be expressed in the form cos 2sπ/n + i sin 2sπ/n, but the question is now that of the algebraical solution (or solution by radicals) of this equation. There is always a root = 1; if ω be any other root, then obviously ω, ω², ... ωn−1 are all of them roots; xn − 1 contains the factor x − 1, and it thus appears that ω, ω², ... ωn−1 are the n-1 roots of the equation

xn−1 + xn−2 + ... x + 1 = 0;

we have, of course, ωn−1 + ωn−2 + ... + ω + 1 = 0.

It is proper to distinguish the cases n prime and n composite; and in the latter case there is a distinction according as the prime factors of n are simple or multiple. By way of illustration, suppose successively n = 15 and n = 9; in the former case, if α be an imaginary root of x³ − 1 = 0 (or root of x² + x + 1 = 0), and β an imaginary root of x5 − 1 = 0 (or root of x4 + x³ + x² + x + 1 = 0), then ω may be taken = αβ; the successive powers thereof, αβ, α²β², β³, αβ4, α², β, αβ², α²β³, β4, α, α²β, β², αβ³, α²β4, are the roots of x14 + x13 + ... + x + 1 = 0; the solution thus depends on the solution of the equations x³ − 1 = 0 and x5 − 1 = 0. In the latter case, if α be an imaginary root of x³ − 1 = 0 (or root of x² + x + 1 = 0), then the equation x9 − 1 = 0 gives x³ = 1, α, or α²; x³ = 1 gives x = 1, α, or α²; and the solution thus depends on the solution of the equations x³ − 1 = 0, x³ − α = 0, x³ − α² = 0. The first equation has the roots 1, α, α²; if β be a root of either of the others, say if β³ = α, then assuming ω = β, the successive powers are β, β², α, αβ, αβ², α², α²β, α²β², which are the roots of the equation x8 + x7 + ... + x + 1 = 0.

It thus appears that the only case which need be considered is that of n a prime number, and writing (as is more usual) r in place of ω, we have r, r², r³,...rn−1 as the (n − 1) roots of the reduced equation

xn−1 + xn−2 + ... + x + 1 = 0;

then not only rn − 1 = 0, but also rn−1 + rn−2 + ... + r + 1 = 0.

In the most simple case, n = 5, the equation to be solved is x4 + x³ + x² + x + 1 = 0; here 2 is a prime root of 5, and the order of the roots is r, r², r4, r³. The Gaussian process consists in forming an equation for determining the periods P1, P2, = r + r4 and r² + r³ respectively;—these being such that the symmetrical functions P1 + P2, P1P2 are rationally determinable: in fact P1 + P2 = −1, P1P2 = (r + r4) (r² + r³), = r³ + r4 + r6 + r7, = r³ + r4 + r + r², = −1. P1, P2 are thus the roots of u² + u − 1 = 0; and taking them to be known, they are themselves broken up into subperiods, in the present case single terms, r and r4 for P1, r² and r³ for P2; the symmetrical functions of these are then rationally determined in terms of P1 and P2; thus r + r4 = P1, r·r4 = 1, or r, r4 are the roots of u² − P1u + 1 = 0. The mode of division is more clearly seen for a larger value of n; thus, for n = 7 a prime root is = 3, and the arrangement of the roots is r, r³, r², r6, r4, r5. We may form either 3 periods each of 2 terms, P1, P2, P3 = r + r6, r³ + r4, r² + r5 respectively; or else 2 periods each of 3 terms, P1, P2 = r + r² + r4, r³ + r6 + r5 respectively; in each ease the symmetrical functions of the periods are rationally determinable: thus in the case of the two periods P1 + P2 = −1, P1P2 = 3 + r + r² + r³ + r4 + r5 + r6, = 2; and the periods being known the symmetrical functions of the several terms of each period are rationally determined in terms of the periods, thus r + r² + r4 = P1, r·r² + r·r4 + r²·r4 = P2, r·r²·r4 = 1.

Reverting to the before-mentioned particular equation x4 + x³ + x² + x + 1 = 0, it is very interesting to compare the process of solution with that for the solution of the general quartic the roots whereof are a, b, c, d.

Take ω, a root of the equation ω4 − 1 = 0 (whence ω is = 1, −1, i, or −i, at pleasure), and consider the expression

(a + ωb + ω²c + ω³d)4,

the developed value of this is

=	a4 + b4 + c4 + d4 + 6 (a²c² + b²d²) + 12 (a²bd + b²ca + c²db + d²ac)
+ω	{4 (a³b + b³c + c³ + d³a) + 12 (a²cd + b²da + c²ab + d²bc) }
+ω²	{6 (a²b² + b²c² + c²d² + d²a²) + 4 (a³c + b³d + c³a + d³b) + 24abcd}
+ω³	{4 (a³d + b³a + c³b + d³c) + 12 (a²bc + b²cd + c²da + d²ab) }

that is, this is a 6-valued function of a, b, c, d, the root of a sextic (which is, in fact, solvable by radicals; but this is not here material).

If, however, a, b, c, d denote the roots r, r², r4, r³ of the special equation, then the expression becomes

r4	+ r³ + r + r² + 6 (1 + 1)	+ 12 (r² + r4 + r³ + r)
	+ ω {4 (1 + 1 + 1 + 1)	+ 12 (r4 + r³ + r + r²) }
	+ ω²{6 (r + r² + r4 + r³)	+ 4 (r² + r4 + r³ + r) }
	+ ω³{4 (r + r² + r4 + r³)	+ 12 (r³ + r + r² + r4) }

viz. this is

= −1 + 4ω + 14ω² − 16ω³,

a completely determined value. That is, we have

(r + ωr² + ω²r4 + ω³r³) = −1 + 4ω + 14ω² − 16ω³,

which result contains the solution of the equation. If ω = 1, we have (r + r² + r4 + r³)4 = 1, which is right; if ω = −1, then (r + r4 − r² − r³)4 = 25; if ω = i, then we have {r − r4 + i(r² − r³) }4 = −15 + 20i; and if ω = −i, then {r − r4 − i (r² − r³) }4 = −15 − 20i; the solution may be completed without difficulty.

A more general theorem obtained by Abel is as follows:—If the roots of an equation of any order are connected together in such wise that all the roots can be expressed rationally in terms of any one of them, say x; if, moreover, θx, θ1x being any two of the roots, we have θθ1x = θ1θx, the equation will be solvable algebraically. It is proper to refer also to Abel’s definition of an irreducible equation:—an equation φx = 0, the coefficients of which are rational functions of a certain number of known quantities a, b, c ..., is called irreducible when it is impossible to express its roots by an equation of an inferior degree, the coefficients of which are also rational functions of a, b, c ... (or, what is the same thing, when φx does not break up into factors which are rational functions of a, b, c ...). Abel applied his theory to the equations which present themselves in the division of the elliptic functions, but not to the modular equations.

The fundamental theorem is the Proposition I. of the “Mémoire sur les conditions de résolubilité des équations par radicaux”; viz. given an equation of which a, b, c ... are the m roots, there is always a group of permutations of the letters a, b, c ... possessed of the following properties:—

1. Every function of the roots invariable by the substitutions of the group is rationally known.

2. Reciprocally every rationally determinable function of the roots is invariable by the substitutions of the group.

Here by an invariable function is meant not only a function of which the form is invariable by the substitutions of the group, but further, one of which the value is invariable by these substitutions: for instance, if the equation be φ(x) = 0, then φ(x) is a function of the roots invariable by any substitution whatever. And in saying that a function is rationally known, it is meant that its value is expressible rationally in terms of the coefficients and of the adjoint quantities.

For instance in the case of a general equation, the group is simply the system of the 1.2.3 ... n permutations of all the roots, since, in this case, the only rationally determinable functions are the symmetric functions of the roots.

In the case of the equation xn−1 ... + x + 1 = 0, n a prime number, a, b, c ... k = r, r g, r g² ... r g n−2, where g is a prime root of n, then the group is the cyclical group abc ... k, bc ... ka, ... kab ... j, that is, in this particular case the number of the permutations of the group is equal to the order of the equation.

This notion of the group of the original equation, or of the group of the equation as varied by the adjunction of a series of radicals, seems to be the fundamental one in Galois’s theory. But the problem of solution by radicals, instead of being the sole object of the theory, appears as the first link of a long chain of questions relating to the transformation and classification of irrationals.

Returning to the question of solution by radicals, it will be readily understood that by the adjunction of a radical the group may be diminished; for instance, in the case of the general cubic, where the group is that of the six permutations, by the adjunction of the square root which enters into the solution, the group is reduced to abc, bca, cab; that is, it becomes possible to express rationally, in terms of the coefficients and of the adjoint square root, any function such as a²b + b²c + c²a which is not altered by the cyclical substitution a into b, b into c, c into a. And hence, to determine whether an equation of a given form is solvable by radicals, the course of investigation is to inquire whether, by the successive adjunction of radicals, it is possible to reduce the original group of the equation so as to make it ultimately consist of a single permutation.

The condition in order that an equation of a given prime order n may be solvable by radicals was in this way obtained—in the first instance in the form (scarcely intelligible without further explanation) that every function of the roots x1, x2 ... xn, invariable by the substitutions xak + b for xk, must be rationally known; and then in the equivalent form that the resolvent equation of the order 1.2 ... (n − 2) must have a rational root. In particular, the condition in order that a quintic equation may be solvable is that Lagrange’s resolvent of the order 6 may have a rational factor, a result obtained from a direct investigation in a valuable memoir by E. Luther, Crelle, t. xxxiv. (1847).

Among other results demonstrated or announced by Galois may be mentioned those relating to the modular equations in the theory of elliptic functions; for the transformations of the orders 5, 7, 11, the modular equations of the orders 6, 8, 12 are depressible to the orders 5, 7, 11 respectively; but for the transformation, n a prime number greater than 11, the depression is impossible.

The general theory of Galois in regard to the solution of equations was completed, and some of the demonstrations supplied by E. Betti (1852). See also J.A. Serret’s Cours d’algèbre supérieure, 2nd ed. (1854); 4th ed. (1877-1878).

26. The modern work, reproducing the theories of Galois, and exhibiting the theory of algebraic equations as a whole, is C. Jordan’s Traité des substitutions et des équations algébriques (Paris, 1870). The work is divided into four books—book i., preliminary, relating to the theory of congruences; book ii. is in two chapters, the first relating to substitutions in general, the second to substitutions defined analytically, and chiefly to linear substitutions; book iii. has four chapters, the first discussing the principles of the general theory, the other three containing applications to algebra, geometry, and the theory of transcendents; lastly, book iv., divided into seven chapters, contains a determination of the general types of equations solvable by radicals, and a complete system of classification of these types. A glance through the index will show the vast extent which the theory has assumed, and the form of general conclusions arrived at; thus, in book iii., the algebraical applications comprise Abelian equations, equations of Galois; the geometrical ones comprise Q. Hesse’s equation, R.F.A. Clebsch’s equations, lines on a quartic surface having a nodal line, singular points of E.E. Kummer’s surface, lines on a cubic surface, problems of contact; the applications to the theory of transcendents comprise circular functions, elliptic functions (including division and the modular equation), hyperelliptic functions, solution of equations by transcendents. And on this last subject, solution of equations by transcendents, we may quote the result—“the solution of the general equation of an order superior to five cannot be made to depend upon that of the equations for the division of the circular or elliptic functions”; and again (but with a reference to a possible case of exception), “the general equation cannot be solved by aid of the equations which give the division of the hyperelliptic functions into an odd number of parts.” (See also Groups, Theory of.)

Bibliography.—For the general theory see W.S. Burnside and A.W. Panton, The Theory of Equations (4th ed., 1899-1901); the Galoisian theory is treated in G.B. Matthews, Algebraic Equations (1907). See also the Ency. d. math. Wiss. vol. ii.