ference. If the preceding series be extended, it will be easy to prove, after what has been done in Art. 307, that the values formerly obtained will recur. As in the former case of the general problem, so here, each root may be derived from the first pair of the series; thus, denoting the first root, cos ir , It 1 ~± sin - . y/ — 1, by o or -, according as the upper or lower sign is taken, we evidently have, for the preceding series, the following equivalent expressions, viz.: a, a*, a5, . . . . a" \ 111 1 > when n is odd, a' «?<?'""' a* ) and a, a3, a8, a*-1 \ 111 1 \ when n is even. a' <?' o»J*-1 ) For further researches on the theory of binomial equations, the student may consult Lagrange's Traite de la Resolution des Equations Numeriques, Note 14; Legendre's Theorie des Nombres, Part V.; the Disquisitiones Arithmetics) of Gauss; Barlow's Theory of Numbers; and Ivory's article on Equa tions, in the Encyclopaedia Britannica. 309. We have already frequently had occasion to notice multiple values of radicals, without fixing the precise number which might exist, except for radicals of the second degree. It is time to introduce the following proposition: Every radical has as many values as there are units in its index., and has no more; in other xoords, every quantity has as many roots of a given degree as there are units in the index of that degree. If the given radical be represented by the general form y/ A, this radical designates evidently all the quantities, real or imaginary, which, raised to the power m, reproduce A; consequently they are merely the values of x in the equation xm=X. But we know, from the general theory of equations, that every equation of the m"1 degree has m values of the unknown quantity, which will each satisfy it; hence the proposition is proved. This will serve to explain some paradoxes. Let there be the expression ija V —1. By reducing the second radical to the index 4, it becomes y( — 1)-, and the given expression reduces to \/a, a result which might be supposed absurd, because, a being positive, the result represents a real quantity, while the proposed expression appears to be imaginary. There is here a confusion of ideas. If in the expression tya ^ — 1 the radical is an arithmetical determination, it is true that this expression is imaginary; but if \Ja be taken in all its generality, and we represent it by a' multiplied by the four roots of unity, or a', —a', a' —1, —a' V —1< we perceive that some of these values of V<z, multiplied by i/ — 1, cause this imaginary factor to disappear, and the proposed expression becomes real. I shall terminate this article by the explanation of a paradox which presents itself in the employment of fractional exponents. Let there be the expres 2 3 i sion a*. If the fraction § be simplified, the expression a' becomes a-. Then, in repassing to the radicals, we have Vai= V<*• This equality, however, is not wholly true, because the first member has four values, and the second but two. The difficulty may be presented in a general manner by placing <j°p =a», and in concluding from thence that To discover the cause of this error, we must remember that the fractional exponent is but a convention, by means of which we express in another way that the root of a certain power is to be extracted, and, therefore, this exponent must not be regarded in the light of an ordinary fraction. THE DETERMINATION OF THE IMAGINARY ROOTS OF EQUATIONS. 310. In what relates to the limits of roots at Art. 283 and following, real roots only were in view. We shall show here how the limits may be obtained for the moduli of all roots, whether real or imaginary. Let us consider the equation rra+PTn-1-r-Qx,,,-|'...=0 (1) in which P, Q may be real or imaginary. In order that a value of x may be a root, it is necessary that, after having substituted it in the result, the modulus should be zero. Call v the modulus of x, and p, q,... those of the coefficients P, Q. According to Art. 239, those of the terms of the equation will be vm, pvm~', yeTM-*,..., and that of the part Pxm_1-|-Qxm_s+ • • • can not surpass the sum pv">~>-\-qvm-'... Then, if we choose for v a value A such that we hnve C">_pK">-i_91)°>-« =0, or >0 .... (2) we are sure, by virtue of the article just cited, that the modulus of the first member of the equation (1) will not be less than the above difference ; and that from this point the modulus will not be zero, or, what is the same thing, the value substituted in place of x will not be a root of the equation. Every value of v above X will render this difference greater; then X is a superior limit of the moduli. The quantity A will be always easy to determine, because it will be sufficient to substitute in the difference (2) in place of v, increasing positive values until this difference becomes positive. If the coefficients P, Q ... are real, the moduli j>, q,... will be these coefficients themselves, but taken positively ; and if we designate the greatest of these values by N, we can take at once for the superior limit X=N + 1. To have an inferior limit, we make x=-, determine in the transformed in u y * the superior limit of the moduli of the roots, and finally divide unity by this limit. 311. It has already been proved that imaginary roots always enter into equations in conjugate pairs of the form ni/3 y1—1. And this previous knowledge of the form which every root must take suggests a method for the actual determination of the proper numerical values for o and /? in any proposed caso. The method is as follows: Let r'+A^-ir"-'-! Ax+N=0 be an equation containing imaginary roots; then, by substituting a-|-/3\/ — i for x, we hnve (a+/3 V^Tr+An_,(a+/3A(o+/3 V^rl)+N==0; or, by developing the terms by the binomial theorem, and collecting the real and imaginary quantities separately, we have the form M+N Vlri=0, an equation which can not exist except under the conditions M = 0, N=0 (1) From these two equations, therefore, in which M, N contain only the quantities n, ,i, combined with the given coefficients, all the systems of values of a and 0 may be determined; and these, substituted in the expression «+jS V —1, will make known all the imaginary roots of the proposed equation; those that are real corresponding to 0=0. It is obvious from the theory of elimination as developed at page 157, and from the method of numerical solution explained in Art. 255, that the labor of deducing from this pair of equations the final equation involving only one of the unknowns a, 0, and of afterward solving the equation for that unkndwn, will in general be very laborious for equations above the third degree. Lagrange, by combining with the principle of this solution the method of the squares of the differences explained at Art. 278, avoids both the elimination and subsequent solution here spoken of. It is easy to see how this may be brought about if we have any independent means of determining one of the unknowns 0: for the adoption of these means would enable us to dispense with the elimination; and as the substitution of the value of 0 in both of the equations (1) would convert those equations into two simultaneous equations involving but one unknown quantity, their first members would necessarily have a common factor of the first degree in a, which, equated to zero, would furnish for a the proper value to accompany 0; and thus, instead of solving the final equation referred to, we should only have to find the common measure between the two polynomials M, N containing the unknown quantity a. Now corresponding to every pairof imaginary roots a-\-0 —1, a—0 <J —1, there necessarily exists, in the equation of the squares of the differences, a real negative root —4/?*; so that if all the negative roots of the latter equation be found, the quantity —4/J3 must appear among them; from which the value of/3 would be immediately obtained, and thence, by aid of the common measure as just explained, the corresponding value of a. But the equation of the squares of the differences may have a greater number of negative roots than there are pairs of imaginary roots in the proposed; which, however, can not happen except two non-conjugate imaginary roots have equal real parts, or except a real root be equal to the real part of an imaginary root. Lagrange discusses these peculiarities, and establishes the exactness and generality of the principle in question, as follows: When the real parts, a, y, ice, of the imaginaries a+0y/~l, Q-/3-/3I y+6 V-l, y-i V-l &c. &c. are unequal, as well when compared with one another as when compared with the real roots a, 6, c, kc, it is evident that the equation of the squares of the Bb differences can not have any other negative roots than those furnished by the several pairs of conjugate imaginary roots, and which are _4/J«, _4d», dec. All the other roots, not arising from the differences furnished by the real roots, a, b, c, Arc, will evidently be imaginary; those between the real and imaginary roots supplying the forms (o_a+/3 V^T)2, (a—a—/?■/—!)' (a—b+p-/^-iy, (u—6— 0 V—1)' &c. &c. and those between the non-conjugate roots the forms \{—r)+(P-i)S-i\', Ka-y)-(3-«i)V~i' U*-7)+(P+S)V-l\*, {(„_r)_(/}+<5)V_l}« so that in this case every negative root in the auxiliary equation will indicate a pair of imaginary roots in the proposed, and will, moreover, supply the value of the imaginary part. But if it happen that among the quantities a, y. Sec, there be found any equal among themselves, or equal to any of the quantities a, b, c. Sec., then the auxiliary equation will necessarily have negative roots, corresponding to which there can be no imaginary pair in the proposed equation. For let a = a, then the two imaginary roots (a—a+/3 \f —1)3> (a—<*■—C y/ —l)3 will become —p* and —(P, and, consequently, real and negative; so that if the proposed equation contain only two imaginary roots, a-|-/3 y/ — 1 and a—/9 y/ — 1, then, in the case of a = a, the equation of the squares of the differences will contain, besides the real negative root —AfP, the two —/P, —/?*, both negative and equal. We thus see that when the equation of the squares of the differences has three negative roots, of which two are equal to one another, the proposed moy have either three pairs of imaginary roots, or but a single pair. If the proposed contains four imaginary roots, a-\-j3y/— 1, a—fly/— 1, y-{-i V — 1, y—Sy/ — 1, then the equation of the squares of the differences must contain the two negative roots —4,32 and —4<P; if a = a. it must also contain the two equal negative roots —fP, —jf; and if, moreover, y=b, it must contain, in addition to these, the negative pair —i3, —ft; and lastly, if a=y, the four imaginary roots j(a-y)+(/3_(J)V3I}2, \(a-y)-(P-S)y/~^[\' i(»-r)+V-ip. {(«-y)-(/3+<5) V-i}« will be converted into the two negative pairs -{3-6)\ -(()-*)>; -(f3+6)\ -{P+6)*. Hence we may deduco the following conclusions, viz.: (1) When all the real negative roots of the equation of the squares of the differences are unequal, then the proposed will necessarily have so many pairs of imaginary roots. If in this case we call any one of these negative roots —w, we shall have V w 0=——; and if this value be substituted for (t in the two equations (1), and the operation for the common measure of their first members bo carried on till we arrive at a remainder of the first degree in a, the proper value of a will be obtained by equating this remainder to zero. Thus, each negative root, —w, will furnish two conjugate imaginary roots, a-)-/3 y/ —1, and a—j} —1. (2) If among the negative roots of the equation of the squares of the differences equal roots are found, then each unequal root, if any such occur, will, as in the preceding case, always furnish a pair of imaginary roots. Each pair of equal roots may, however, give either two pairs of imaginary roots or no imaginary roots, so that two equal roots will give either four imaginary roots or none; three equal roots will give either six imaginary roots or two; four equal roots will give either eight imaginary roots, or four, or none; and so on. Suppose two of the negative roots, —w, —to, are equal; then putting, as above, (3=—^-, we shall substitute this value of/? in the two polynomials (1), and shall carry on the process for the common measure between these polynomials till we arrive at a remainder of the second degree in a; since the polynomials must have a common divisor of the second degree in a, seeing that the equations (1) must have two roots in common, on account of the double value of/3. Equating, then, this quadratic remainder to zero, we shall be furnished with two values for a: these may be either both real or both imaginary. In the former case call the two values a' and a"; we shall then have the four imaginary roots In the second case, the values of a being imaginary, contrary to the conditions by which the fundamental equations (1) are governed, we infer that to the equal negative roots —w, —w, there can not correspond any imaginary roots in the proposed equation. If the equation of the squares of the differences have three equal negativr V f roots, —to, —to, —to, then, putting, as before, /?=—we should operate on the polynomials (1), for the common measure, till we reach a remainder of the third degree in a; this remainder, equated to zero, will furnish three values of a, which will either be all real, or one real and two imaginary. Iu the first case six imaginary roots will bo implied: in the second only two; the imaginary values of a being always rejected, as not coming within the conditions implied in (1). It follows from the above, and from what has been established in Art. 259, that there are at least as many variations of sign in the equation of the squares of differences as there are combinations of two real roots in the proposed equation. Also, it must have at least as many permanences of sign as there are pairs of conjugate imaginary roots in the proposed equation; or, in other words, it can not have a less number of permanences of sign than half the number of imaginary roots in the proposed equation. Hence we may infer, that if the equation of the squares of the differences have its terms alternately positive and negative, there can be no imaginary root in the proposed equation. The foregoing principles are theoretically correct; but the practical application of them, beyond equations of the third and fourth degrees, is too laborious for them to become available in actual computation. We give the following illustration of them from Lagrange. |