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 prin ciple identical with Gauss s last proof (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 determina tion 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 deter mination 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. 18. A class of numerical equations which needs to be considered is that of the binomial equations x n -a = (a = a + {3i, a complex number). The foregoing conclu sions apply, viz., there are always n roots, which, it may be shown, are all unequal. And these can be found numeri cally 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. 2 For writin there is always a real angle X (positive and less than 2-rr), sucli that its cosine and sine are = p=-= - and - -p--- V* 2 +/3 2 __ y^+# respectively; that is, writing for shortness v / 2 + /3 2 = p, we have a + /3i = p (cos X. + i sin X), or the equation is x n = /A X n p (cos X + i sin X); hence observing that ( cos - +i sin )
n nj
sm - n -/ x = cosX + i sin X, a value of x is = ?/p(cos - N f n The formula really gives all the roots, for instead of X we may write X + 2s-rr, s a positive or negative integer, anrl then we have
+ 2.V7T . A + 2.S7T
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 8, so that 6 n = a, then assum ing x=> Oy, the equation becomes y n - 1 = 0, which equation, like the original equation, has precisely n roots (one of them being of course =1). And the original equation x n - a = is thus reduced to the more simple equation
- "-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 x n -1 = has its several roots expressed 1 The earlier demonstrations by Euler, Lagrange, &c., relate to the case of a numerical equation with I cal coefficients ; and they consist in showing that such equation has always a real quadratic divisor, fur nishing two roots, which are either real or else conjugate imaginaries a -*- 0i (see Lagrange s Lquations Numeriqiics). 2 The square root of o + 0i fan be determined by the extraction of square roots of positive real numbers, without the trigonometrical tables. 503 in the form 1, w, w 2 , . . . to"" 1 , where w may be taken = cos -M* sin : in fact, w having this value, any integer n n power to* is = cos + i sin . and we thence have (o>*) n = n n cos 2-irk + i sin 2-Trk, = 1, that is, a>* is a root of the equation. The theory will be resumed further on. By what precedes, we are led to the notion (a numerical) i of the radical regarded as an ^-valued function; any one of these being denoted by %/a, then the series of values s /a, w Z/a, w"" 1 %fa , or we may, if we please, use %/a instead of a* as a symbol to denote the w-valued function. As the coefficients of an algebraical equation may be numerical, all which follows in regard to algebraical equa tions is (with, it may be, some few modifications) appli cable to numerical equations; and hence, concluding for the present this subject, it will be convenient to pass on to algebraical equations. II. We consider secondly algebraical equations (19 to 34). 19. The equation is x n -p i x n ~ l + . . . p n = 0, and we here assume the existence of roots, viz., we assume that there are n quantities a, b, c . . (in general all of them different, but which in particular cases may become equal in sets in any manner), such that or looking at the question in a different point of view, and starting with the roots a, b, c . . as given, we express the product of the n factors x a, x-b,..in the foregoing form, and thus arrive at an equation of the order n having the n roots a, b, c . , In either case we have 2>i = 2<7, p. 2 2&, . . p,i abc . ; i.e., regarding the coefficients p v p 2 . . p n as given, then we assume the existence of roots a, b, c, . . such that p 1 = 2<7, &c. ; or, regarding the roots as given, then we write p v p 2 , &c., to denote the functions 2, ~^ab, &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 re stricted to denote, or are not explicitly considered as denot ing, numbers. That the abstraction is legitimate, appears by the simplest example; in saying that the equation y? px + q = has a root x = ^(p+ /p 2 -4q), we mean that writing this value for x the equation becomes an identity, {l(p + v^J 2 - 4q)} 2 - P{ (P + *J P" ~ 4q)} + q = ; and the verification of this identity in nowise de pends 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 indepen dent of the roots altogether, and in this case it is un doubtedly valid ; the reasoning, although actually con ducted by aid oif the assumption (and, it may be, most
easily and elegantly in this manner), is really independent