roughly stated as follows. If x=y be an approximate value of any root, and y+h the correct value, then f(y+h) = 0, that is, h2 I 2 =0; and then, if h be so small that the terms after the second may be neglected, f(y)+hf'(v), =0, that is, h=-f(y)/f' (y), or the new approximate value is x=y-f(y)/f'(y); 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; y 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 y in order that the process may by successive repetitions actually lead to a certain real root of the equation; or that, y being an approximate value of a certain real root, the new value y-f(y) f'(y) may be a more approximate value. Referring to fig. 1, it is easy to see that if OC represent the assumed value y, 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 a-B Imaginary Theory. 7. It will be recollected that the expression number and the correlative epithet numerical were at the outset used in a wide to sense, as extending to imaginaries. This extension arises out confusion. But in fact a numerical equation of any order whatever has always a numerical root, and thus numbers (in the foregoing sense, number = quantity of the form a+ẞi) form (what real numbers do not) a universe complete in itself, such that starting in it we are never led out of it. There may very well be, and perhaps are, numbers in a more general sense of the term (quaternions are not a case in point, as the ordinary laws of combination are not adhered to), but in order to have to do with such numbers (if any) we must start with them. 8. The capital theorem as regards numerical equations thus is, every numerical equation has a numerical root; or for shortness (the meaning being as before), every equation has a root. Of course the theorem is the reverse of self-evident, and it requires proof; but provisionally assuming it as true, we derive from it the general theory of numerical equations. As the term root was introduced in the course of an explanation, it will be convenient to give here the formal definition. A number a such that substituted for x it makes the function x-Pix- Pn to be =0, or say such that it satisfies the equation f(x)=o, is said to be a root of the equation; that is, a being a root, we have -1 be a double root of the equation f(x)=0; and similarly f(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 f(x)=(x−a)aF(x) (a being a positive integer which may be =1, (x-a)a the highest power of x-a which divides f(x), and F(x) being of course of the order n-a), 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 f(x) will in this case be = (x−a)a (x−b)p(x) (8 a positive integer which may be = 1, (x-b) the highest power of x-b in F(x) or f(x), and Þ(x) being of course of the order n—a—ß). The original equation f(x) =o is in this case said to have a roots each =a, ẞ roots each b; and so on for any other factors (x-c), &c. We have thus the theorem-A numerical equation of the order n has in every case n roots, viz. there n (cos +i sin n =cosλ+i sin λ, a value of x is = p (cos +i sin). (cos The formula really gives all the roots, for instead of A we may write x = 。 (cos + + 2 n general all distinct, but which may arrange themselves in any sets of equal values), such that f(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 f(x) =0 has all its roots unequal. which has the n values obtained by giving to s the values 0, 1, 2 If the coefficients P1, P2, are all or any one or more of them n-1 in succession; the roots are, it is clear, represented by points imaginary, then the equation f(x) = 0, separating the real and imagin- lying at equal intervals on a circle. But it is more convenient to proary parts thereof, may be written F(x)+id(x)=0, where F(x), ceed somewhat differently; taking one of the roots to be 0, so that (x) are each of them a function with real coefficients; and it thusa, then assuming x=0y, the equation becomes y"-1=0, which appears that the equation f(x) =0, with imaginary coefficients, has equation, like the original equation, has precisely n roots (one of them not in general any real root; supposing it to have a real root a, this being of course 1). And the original equation x" -a -o is thus must be at once a root of each of the equations F(x) =0 and Þ(x) = 0. reduced to the more simple equation x-1=0; and although the But an equation with real coefficients may have as well imaginary theory of this equation is included in the preceding one, yet it is as real roots, and we have further the theorem that for any such proper to state it separately. equation the imaginary roots enter in pairs, viz. a+ẞi being a root, then a-Bi 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. 9. In the case of an equation with real coefficients, the question of the existence of real roots, and of their separation, has been already considered. In the general case of an equation with imaginary (it may be real) coefficients, the like question arises as to the situation of the (real or imaginary) roots; thus, if for facility of conception we regard the constituents a, ẞ of a root a+ẞi as the co-ordinates of a point in plano, and accordingly represent the root by such point, then drawing in the plane any closed curve or "contour," the question is how many roots lie within such contour. 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 f(x+y) 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 o), 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.1 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. 10. A class of numerical equations which needs to be considered is that of the binomial equations x"-a=o(a=a+ßi, a complex number). 1 The earlier demonstrations by Euler, Lagrange, &c., relate to the case of a numerical equation with real coefficients; and they consist in showing that such equation has always a real quadratic divisor, furnishing two roots, which are either real or else conjugate imaginaries a+Bi (see Lagrange's Equations numériques). The equation x-1=0 has its several roots expressed in the form I, w, w2, . . . wh-1, where w may be taken =cos +i sin; in fact, 2πk 2πk 12 n 2π 2π n w having this value, any integer power w* is = cos +isin and all which follows in regard to algebraical equations is (with, it may i.e. regarding the coefficients P1, P2... Pn as given, then we assume the existence of roots a, b, c, .. such that p1=2a, &c.; or, regarding the roots as given, then we write P1, P2, &c., to denote the functions Za, Zab, &c. 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 2-px+9=0 has a root x = {p+ √ (p2-49)}, we mean that writing this value for x the equation becomes an identity, [+ √ (p2 - 4q) } ]2 — pl} {p+ √ (p2 −49}}]+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 2 The square root of a+Bi can be determined by the extraction of square roots of positive real numbers, without the trigonometrical tables. 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(a, B) 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. and P1 = 2a, p2 = Lab, &c. x" - Pixn−1+ . . . = x − a . x-b. &c. Observe that if, for instance, a=b, then the equations a” — p1an-1+...=0, bn-p1bn-1+...= =0 would reduce themselves to a single relation, which would not of itself express that a was a double root,—that is, that (x-a)2 was a factor of x"-p1x+, &c.; but by considering b as the limit of a+h, h indefinitely small, we obtain a second equation nan-1 - (n-1) pian-2+ =0, ... which, with the first, expresses that a is a double root; and then the whole system of equations leads as before to the equations Pi-Za, &c. But the existence of a double root implies a certain relation between the coefficients; the general case is when the roots are all unequal. We have then the theorem that every rational symmetrical function of the roots is a rational function of the coefficients. This is an easy consequence from the less general theorem, every rational and integral symmetrical function of the roots is a rational and integral function of the coefficients. In particular, the sums of the powers Za2, Za3, &c.; are rational and integral functions of the coefficients. The process originally employed for the expression of other functions Eaab, &c., in terms of the coefficients is to make them depend upon the sums of powers: for instance, Zaabß = ZaaΣaß — Σaa+8; but this is very objectionable; the true theory consists in showing that we have systems of equations being determined as above by an equation of the order n, any rational and integral function whatever of x, or more generally any rational function which does not become infinite in virtue of the equation itself, can be expressed as a rational and integral function of x, of the order n-1, the coefficients being rational functions of the coefficients of the equation. Thus the equation gives xn" a function of the form in question; multiplying each side by x, and on the right-hand side writing for x" its foregoing value, we have x+1, a function of the form in question; and the like for any higher power of x, and therefore also for any rational and integral function of x. The proof in the case of a rational non-integral function is somewhat more complicated. The final result is of the form (x)/(x) = I(x), or say (x)–(x)I (x) = 0, where ., I are rational and integral functions; in other words, this equation, being true if only f(x) = o, can only be so by reason that the left-hand side contains f(x) as a factor, or we must have identically (x) − y(x)I(x) = M(x)f(x). And it is, moreover, clear that the equation 4(x)/¥(x) = I(x), being satisfied if only f(x) =0, must be satisfied by each root of the equation. 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 (a),$(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 -Pix-1. =0, and y= Mox-1+M1x2+ ..., or, what is the same thing, from these two equations to eliminate x. This is in fact E. W. Tschirnhausen's transformation (1683). 14. In connexion with what precedes, the question arises as to the number of values (obtained by permutations of the roots) of given unsymmetrical functions of the roots, or say of a given set of letters: for instance, with roots or letters (a, b, c, d) as before, how many values are there of the function ab+cd, or better, how many functions are there of this form? The answer is 3, viz. ab+cd, ac+bd, ad+bc; or again we may ask whether, in the case of a given number of letters, there exist functions with a given number of values, 3-valued, 4-valued functions, &c. 2-valued functions; the product of the differences of the letters is It is at once seen that for any given number of letters there exist such a function; however the letters are interchanged, it alters only its sign; or say the two values are 4 and 4. And if P, Q are symmetrical functions of the letters, then the general form of such a function is P+QA; this has only the two values P+Q4, P-Q4. 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 3valued 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 theory is an extensive and important one, depending on the notions of substitutions and of groups (q.v.). 15. Returning to equations, we have the very important theorem that, given the value of any unsymmetrical function of the roots, e.g. in the case of a quartic equation, the function ab+cd, it is in general possible to determine rationally the value of any similar function, such as (a+b)3+(c+d)3. a The a priori ground of this theorem may be illustrated by means of 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 deterb to be 1, 2 and c, d to be 3, 4 (viz. a=1, b = 2 or a=2,b=1, mines a, 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)+(c+d) to be = 370, and not any other value; that is, (a+b)+(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 ab 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, t=abcd, y1 = (a+b)3 +(c+d)3, then y1+y2+ys, tryi+t2y2+tзys, ti2yı +122y+ts2ys will be respectively will be known. Suppose for a moment that t1, t2, ts are all known; then the equations being linear in y1, y2, ys these can be expressed rationally in terms of the coefficients and of t1, t, ts; that is, yi, y2, ys will be known. But observe further that y1 is obtained as a function of t1, t2, ts symmetrical as regards to, ts; it can therefore be expressed as a rational function of ħ, and of t2+ts, tats, and thence as a rational function of t1 and of t1+t+ts, tit2+tit3+t2t3, ttats; 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 tą or tз) being known, y1 will be rationally determined. 16. We now consider the question of the algebraical solution of equations, or, more accurately, that of the solution of equations by radicals. In the case of a quadric equation x2-px+9=0, we can by the assistance of the sign √() or ( ) 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= x = } { p± √ (p2 — 49)}, 3ab=q, in its original form, but use only the derived equation and consequently · √ (r2 — 2793)}, a3 = }{r+√ (r2 — 24q3)}, b3= 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 I-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 44 x = √ [}{r + √(x2 - 2 + q3) } ] + } q÷√ [} {r + √(x2 - Aq3)}], this may depend on an expression of the form (y+di) where y and d are real numbers (it will do so if r2-93 is a negative number), and then we cannot by the extraction of any root or roots of real positive numbers reduce (y+di) 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 b ̧ the extraction of a cube root, but) to a cubic of the form 4x3-3x=6 corresponding to the equation 4 cos30-3 cos 0=cos 30 which serve. to determine cos when cos 30 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. 18. A quartic equation is solvable by radicals. and it is to be remarked that the existence of such a solution depends on the existence of 3-valued functions such as ab+cd of the four roots (a, b, c, d): by what precedes ab+cd is the root of a cubic equation, which equation is solvable by radicals: hence ab+cd can be found by radicals; and since abcd is a given function, ab and cd can then be found by radicals. But by what precedes, if ab be known then any similar function, say a+b, is obtainable rationally; and then from the values of a+b and ab we may by radicals obtain the value of a or b, that is, an expression for the root of the given quartic equation: the expression ultimately obtained is 4-valued, corresponding to the different values of the several radicals which enter therein, and we have thus the ex and the equation is on this account said to be algebraically solvable, or more accurately solvable by radicals. Or we may by writing =-p+z reduce the equation to 22= (p2-49), 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" () 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=p√(p2-49), sub-pression by radicals of each of the four roots of the quartic stituting for p, q their given numerical values, we obtain for x an expression of the form x=a+Bi±√(y+di), where a, 8, 7, 8 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 √(r+di) It will be understood from the foregoing explanation as to the should have a value=c+di, where c and d are real numbers cal-quartic how in the next following case, that of the quintic, the question culable by the extraction of a root or roots of real numbers; however of the solvability by radicals depends on the existence or nonthe case is (what there was no a priori right to expect) that √(y+di) existence of k-valued functions of the five roots (a, b, c, d, e); the has such a value calculable by means of the radical expressions fundamental theorem is the one already stated, a rational function √ {√ (y2+82) ±y} and hence the algebraical solution of a numerical of five letters, if it has less than 5, cannot have more than 2 values, quadric equation does in every case give the numerical solution. The that is, there are no 3-valued or 4-valued functions of 5 letters: and case of a numerical cubic equation will be considered presently. by reasoning depending in part upon this theorem, N. H. Abel (1824) 17. A cubic equation can be solved by radicals. 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. 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 — q3, a quadric equation solvable by radicals, and giving a3 — b3 = √ (r2 — § q), a 2-valued function of the coefficients: combining this with a3+b =r, we have a3 = } {r + √ (r2 — »*;q3)}, a 2-valued function: we then have a by means of a cube root, viz. equation. But when the quartic is numerical the same thing happens as in the cubic, and the algebraical solution does not in every case give the numerical one. 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 = 1}p+√(p2-49)}, substituting for p and q their values, and observing that (a+b)2—4ab = (a−b)2, this becomes x=(a+b+ √(a-b), the value being a or b according as the radical is taken to be +(ab) or (a−b). So in the cubic equation x3-px2+qx-r=o, if the roots are a, b, c, and if w is used to denote an imaginary cube root of unity, w2+w+ 1=0, then writing for shortness p=a+b+c, L=a+wb+w2c, M = a+b+wc, it is at once seen that LM, L+M3, and therefore also (L3-M3)2 are symmetrical functions of the roots, and consequently rational functions of the coefficients: hence }{L3+M3+v (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, L, L; and LM divided by the same cube root has therefore the values M, M, M; whence finally the expression } [p+√ {} (L3+M3 + √ (L3 − M3)2)}+LM÷√ {}L3+M3+√ (L3—M3)2)}] has the three values (p+L+M), (p+wL+w2M), }(p+w2L+wM); 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-ro, it will readily be seen that the two solutions are identical, and that the function 2-93 under the radical sign must (by aid of the relation po 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. 20. The formulae for the cubic were obtained by J. L. Lagrange (1770-1771) from a different point of view. Upon examining and comparing the principal known methods for the solution of algebraical equations, he found that they all ultimately depended upon finding a "resolvent " equation of which the root is a+wb+w2c+w3d+..., w being an imaginary root of unity, of the same order as the equation; e.g. for the cubic the root is a+wb+w2c, w an imaginary cube root of unity. Evidently the method gives for L3 a quadric equation, which is the "resolvent equation in this particular case. 2Sπ 2Sπ 22. It has already been shown how the several roots of the equation x-1=0 can be expressed in the form cos + i sin but the question is now that of the algebraical solution (or solution by radicals) of this equation. There is always a root = 1; if w be any other root, then obviously w, w2, ... wh-1 are all of them roots; x^-1 contains the factor x-1, and it thus appears that w, w2,... w1 are the n-I roots of the equation ... xn-1+xn−2+ +x+1=0; we have, of course, w1+w7-2+. +w+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 a be an imaginary root of x3-1=0 (or root of x2+x+1=0), and ẞ an imaginary root of x3—1=0 (or root of xa+x3+x2+x+1=0), then w may be taken = aß; the successive powers thereof, aß, a22, p3, aßt, a2, Bß, aß2, a2 ß3, B4, a, a2ß, B2, aß3, a2ß1, are the roots of x11+x13+...+x+1=0; and x-1=0. the solution thus depends on the solution of the equations x3-1=0 In the latter case, if a be an imaginary root of x3-10 (or root of x2+x+1=0), then the equation x3- I =0 gives x3=1, a, or a2; x3=1 gives x=1, a, or a2; and the solution thus depends on the solution of the equations x3- I=0,x3-a=0, x3-a2 = 0. The first equation has the roots I, a, a2; if 8 be a root of either of the others, say if 33 = a, then assuming w=8, the successive powers are B, B2, a, aß, aß2, a2, a2ß, a2ß2, which are the roots of the equation x+x+...+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) in place of w, we have r, r2, r3,... -1 as the (n-1) roots of the reduced equation x2-1+x^2+.. +x+1=0; then not only rn-I =0, but also -1+rn-2+...+r+1=0. ... 23. The process of solution due to Karl Friedrich Gauss (1801) depends essentially on the arrangement of the roots in a certain For a quartic the formulae present themselves in a somewhat order, viz. not as above, with the indices of r in arithmetical different form, by reason that 4 is not a prime number. Attempt-progression, but with their indices in geometrical progression; ing to apply it to a quintic, we seek for the equation of which the root is (a+b+w2c+w3d+w1e), w an imaginary fifth root of unity, or rather the fifth power thereof (a+wb+w2c+w3d+w1e)5; this is a 24-valued function, but if we consider the four values corresponding to the roots of unity w, w2, w3, w1, viz. the values (a+wb+w2c+w3d+w1e)5, any symmetrical function of these, for instance their sum, is a 6-valued function of the roots, and may therefore be determined by means of a sextic equation, the coefficients whereof are rational functions of the coefficients of the original quintic equation; the conclusion being that the solution of an equation of the fifth order is made to depend upon that of an equation of the sixth order. This is, of course, useless for the solution of the quintic equation, which, as already mentioned, does not admit of solution by radicals; but the equation of the sixth order, Lagrange's resolvent sextic, is very important, and is intimately connected with all the later investigations in the theory. ... ... the prime number n has a certain number of prime roots g, 21. It is to be remarked, in regard to the question of solv- The theory was further developed by Lagrange (1808), who, applying his general process to the equation in question, x"1+ x2+...+x+1=o(the roots a, b, c... being the several powers that the function (a+wb+w2c+...)" was in this case a given of r, the indices in geometrical progression as above), showed function of w with integer coefficients. x2+x+1=0, it is very interesting to compare the process of solution Reverting to the before-mentioned particular equation x+x+ with that for the solution of the general quartic the roots whereof are a, b, C, d. Take ω, a root of the equation w1-1=0 (whence w is = 1, or i, at pleasure), and consider the expression (a+wb+w2c+w3d)^, the developed value of this is = -I, i, a1+b+c1+d1+6(a2c2+b2d2)+12(a2bd+b2ca+c2db+d2ac) +w {4(a3b+b3c+c3+d3a) +12(a2cd+b2da+c2ab+d2bc)} |