Read before the New York Mathematical Society, June 3, 1893.
1. There is a type of equations that remain unaltered when x is changed into – 1/X.
From the character of the roots, and from the relation that such equations bear to those named by the text-writers "reciprocal equations," the suggestive name for this type is negative-reciprocal equations.
In order to preserve a clear distinction, "reciprocal equations" will herein be referred to as positive-reciprocal equations.
To find the conditions inherent in negative-reciprocal equations.—Let the proposed equation be substitute – 1/x for x, and divide by pn, whence we have.
2. If n be odd, the identity of equations (2) and (1) requires.
Therefore, negative-reciprocal equations of odd degree will be dismissed from further consideration, as involving imaginary coefficients. This is in contrast with the, case of positive-reciprocal equations, in which the coefficients are real whether n be odd or even.
Taking, in turn, the two values of pn, we obtain two forms of negative-reciprocal equations, thus:
3. From the preceding article, we see that to have a negative-reciprocal equation, with real coefficients, the necessary conditions are: (1st) The degree of the equation must be even; (2d) the coefficients of the corresponding terms, counting from the beginning and end of the first member of the equation (f(x) = 0), must be equal numerically, and (3d) must be alternately alike and unlike in sign, giving rise to two forms of the equation, (I.) when the extreme terms have the same sign, and (II.) when they have opposite signs; (4th) the middle term = must be absent if the equation be of the first form and m be odd, or if of the second form and m be even; for then the conditions giving equations (3) and (4) require pm= -pm therefore pm= 0.
4. From the identity of the proposed equation with that derived by substituting for x, it is evident that the negative-reciprocal of any root, as a root itself, will satisfy the equation. This would indicate that the roots occur in pairs, of the form
5. For convenience of treatment, negative-reciprocal equations of the dimensions 2m will be divided into two classes, as follows:
I. The equation will be of the first class, (a1) if m be even and the first and last terms have the same sign, or (a2) if m be odd and the extreme terms have different signs.
II. The equation will be of the second class, (b1) if m be odd and the extreme terms have the same sign, or (b2) if m be even and the extreme terms have different signs; and in this class the middle term must be absent.
From equation (3), substituting 2m for n, the equation of the second class (b1), when m is odd.
In equation (5) the sums of the odd powers of x2 and 1, and the differences of the even powers, occur as factors so as to render the equation divisible by x3 + 1.
Similarly, from equation (4), for (b2) when m is even, and, as before, we have the sums of the odd powers, and the differences of the even powers of x2 and 1.
Therefore, every negative-reciprocal equation of the second class contains the factor x2 + 1 = 0, from which we derive the two isolated roots each being the negative-reciprocal of itself. Since these roots are imaginary, they occur only together, and as single roots in the equation of the second class; unlike the case of isolated roots in positive-reciprocal equations, in which 1 or — 1 may be a root, thus giving an odd-degree reciprocal equation; or they may occur together, giving an even-degree positive-reciprocal equation, of the second class as commonly distinguished.
Dividing a negative-reciprocal equation of the second class by x2 + 1, to remove the isolated roots, the character of the roots makes the quotient a negative-reciprocal equation. Moreover, the reduced equation will be of the first class, since the signs of the extreme terms of the quotient remain the same as in the given equation of the second class, while m changes from odd to even, or vice versa; the condition (b1) becoming (a1), or (b2) becoming (a2).
6. The divisor x2 + 1 put equal to zero gives the simplest form of negative-reciprocal equations of the second class, corresponding to m = 1.
The simplest form of negative-reciprocal equations of the first class is the quadratic x2 - 1 = 0; in which the middle term has zero for its coefficient, though such condition is not required, the next simplest equation of this class taking the form x2 — Ax — 1 = 0, in which A = a –where a is a root.
It may be of interest to note that while x2 + 1 = 0 and x2 — 1 = 0 are both negative-reciprocal equations, they are, at the same time, positive-reciprocal equations: that though x2 + 1 = 0, as a factor, enters every negative-reciprocal equation of the second class, yet its roots are positively reciprocal: that though x2 — 1 = 0, as a factor, enters every positive-reciprocal equation of the second class of an even degree, yet its roots are negatively reciprocal: that x2 + 1 = 0 as a negative-reciprocal equation is of the first class, giving a pair of negatively reciprocal roots, 1 and —1; but as a positive-reciprocal equation it is of the second class, giving the two roots as isolated: that x2 + 1 = 0 as a positive-reciprocal equation is of the first class, giving a pair of positively reciprocal roots, but as a negative reciprocal equation it is of the second class, giving the two roots as isolated.
7. A negative-reciprocal equation of the first class can be depressed to an equation of half its dimensions.
With unity for the coefficient of the first term, the given equation will be in one of two forms (Article 5, I.); thus (a1) when m is even,
(a2) when m is odd,
Dividing equations (7) and (8) by xm, and collecting in pairs the terms that are equidistant from the beginning and end, we have—when m is even, when m is odd.
From the last two equations, we see that every term containing x to an even power is connected with the reciprocal of the same power by the positive sign, and that every odd power of x is connected with its reciprocal by the negative sign; and we may make use of two general relations, A and B, for deducing the depressed equation in terms of powers of
By A we have from (12), and then from (11),
It is seen that each of the expressions for xn may be derived directly from the two immediately preceding, by multiplying the nearest expression by z and to the product adding the next preceding. These expressions differ from the corresponding ones in positive-reciprocal equations, only in having the signs of all the terms positive.
8. The forms assumed by the depressed equations.—Taking 2m successively as 2, 4, 6,... in equations (8) and (7) in turn, m alternately odd and even, retaining, of course, only the proper number of terms, equations (10) and (9) give the following:
When m = 1, from (10), z + pm = 0; whence x2 + pmx — 1 = 0, as given directly by (8).
In equations (9) and (10), then, each binomial may be expressed in terms of z and the resulting equations will be of the mth degree.
9. To obtain a general expression for Zn = xn +in terms of powers of
The sum of the nth powers of the roots of the equation
x2 - px + q = 0............(19)
has been shown (Todhunter's Theory of Equations, Arts. 260, 261) to be
Suppose q = —1, then the quadratic equation (19) is a negative-reciprocal equation, and its roots are of the form a Therefore, in this case, and from equation (20), changing the notation, for our purpose, by putting x for a, and z for p, we have it being understood that no powers with negative exponents are included, and the general term being
10.-RECAPITULATION.
The character of the roots in the two classes of negative-reciprocal equations may be stated as follows:
I. In equations of the first class, if a be a root then must be another root, and there exists no root that does not observe this law. Therefore the roots all occur in pairs of negative reciprocals; as,
II. In the second class of these equations, and must be two imaginary, isolated roots, while all the remaining roots occur in pairs of negative-reciprocals; as . . . and in this class the middle term of the equation must be wanting.
Any negative-reciprocal equation of the second class can be reduced to one of the first class, by dividing by x2+ 1; and any negative-reciprocal equation of the first class can be depressed to an equation of half the dimensions.
For recognition of the class of a given negative-reciprocal equation.—I. It will be of the first class if the extreme terms have like signs and the exponent of the highest power of x be of the form 4n; or, if the extreme terms have unlike signs and the greatest exponent be of the form 4n + 2. II. The equation will be of the second class if the extreme terms have unlike signs, the greatest exponent being of the form 4n; or, have like signs, the form of the greatest exponent being 4n + 2.
II. If the term "recurring equations" had not been given—as another name—to positive-reciprocal equations, it would seem appropriate to make that the inclusive term for the two types of reciprocal equations without discrimination—the "reciprocal equations" of the text-books, and those discussed in this paper.
The statement is hazarded here that positive-reciprocal equations will be met with no oftener than will negative-reciprocal equations—if so often. Yet the latter, so far as known to the writer, have not been noticed as a class.
In a particular direction of investigation of loci, various equations of this type have occurred, requiring solution by the writer. They have been, for the most part, of the fourth or the sixth degree, but also of higher degrees. Immediate recognition of the character of the equation would have been the means of saving much time in finding the roots.
Examples.
I. y6 + Ay3 — Ay + 1 = 0; the expression of a law giving a problem met with. Dividing by y2
It is seen that if A
Divide by y5 and rearrange,
assume z = y and find each binomial in terms of z, from equations (14) to (17), or from (21); then
From the first factor, (y2 – 1)2 = 0, whence the four roots 1, -1, 1, and -1.
From the second factor, Az + 2 = 0, y2 y = 1, whence the roots .
From the third factor, s = -A whence
It is seen that the roots by (3) and (4) will be real or imaginary under the same conditions as in Example I., the six other roots of equation (1) being real.
Equation (1) presents the singular case of the depressed equation being a reciprocal equation. It is of the first class of the positive type, therefore may be solved by depressing, thus and, again, we have the singular case of the second depressed equation being a reciprocal equation—of the first class of the negative type.
Taking the positive sign of the radical, two values of z are imaginary whence four values of y are imaginary.
Taking the negative sign of the radical in (3.), from which we obtain for the values of y, giving four real roots.
IV. x8 – 3x6 + 3x2 – 1 = 0 .... (1)
a negative-reciprocal, and a positive-reciprocal equation of the second class; therefore x2 + 1 and x2 – 1 are divisors, giving the four roots. Dividing equation (1) by x4 — 1, we have
x4 – 3x2 + 1 = 0; .... (2)
which, as a negative-reciprocal, and a positive-reciprocal equation of the first class, may be solved by assuming as well as by treating it as a quadratic in x2.
It is seen that by (5) the roots do not immediately show the simplest forms, as by (3) or (4) when treated as those of either type of reciprocal equations, and grouped as of the form a,; b, where b = or of the form a, b, where b = -a. By (5) the roots directly take the form ±a and ±b, where b = .
V. y8 – A2y6 + 2(A2 + 1)y1 – A2y2 + 1 = 0.
A combined positive-reciprocal and negative-reciprocal equation, met with in the solution of a real case of roots required. Divide by y4, and rearrange, whence we find four y's of the same values as those in Example I., and four other values equal numerically to the first four, respectively, but with opposite signs.
To have a recurring equation that combines in one the two types of reciprocal equations, it is obvious that the terms containing odd powers of the unknown quantity must have the coefficient zero; that if a be a root, then —a must be another root; that though and as roots, must satisfy the equation, they may not be additional roots when a = 1 or , though otherwise they will be roots necessary to the forming of the equation.
VI. For comparison, two solutions of a given equation are as follows:
1st solution, cited from Todkunter's Algebra, Article 333, (3).-
We have now ordinary quadratics, namely,
From the former we shall obtain
2d solution (as a negative reciprocal equation).—Divide the given equation (1) by x2 and rearrange.