as the equation whose roots are the squares of the differences of the roots of the proposed equation.

279. As an application of the foregoing principles, let us find the equation of the squares of the differences for the equation of the third degree. In the first place, I shall make the general remark, that equations (3) and (4) ought not to change when we augment, or when we diminish, by the same quantity all the roots of equation (1). Consequently, if the second term of a given equation be not wanting, we can cause it to disappear (Art. 273), and then find the equation of the differences for the transformed equation; we shall thus find the same equation as if we had not made the second term vanish, since the differences of the roots will be the same as before, while the calculations will be less complicated. This being premised, I will suppose that the equation of the third degree wants its second term, and has the form

Designate the given equation by f(x)=0, and the derived polynomials of f(x) by fi(x), ƒ2(x), ƒs(x)..........; the rule for finding the equation of the squares of the differences is to eliminate between the two equations

f(x)=x3+qx+r, fi(x)=3x2+q, f(x)=6x, fs(x)=6.

[B]

Substituting, therefore, these values in equations [B], we shall readily perceive that the elimination of x ought to be performed between equation [A] and the following equation,

We shall, therefore, arrange this equation with reference to x, and then eliminate x by proceeding as if we had to find the greatest common divisor of equations [A] and [C].

3x2+3qx+ y2+q__ [2(y2+q)x+y3+qy+3r 6(y2+q)x2+(y2+q)yx+2(y2+q) | 3x+3(y+qy-3r)

+6(y+q)+3(y+qy +3r).x

3(y+qy -3r)x+2(y2+q)2

6(y2+q) (y3+qy −3r)x+4(y2+q)3

6(y2+q)(y3+qy −3r)x+3(y3+qy+3r) (y3+qy-3r)

4(y2+q)-3(y +qy+3r) (y3+gy-3r).

In the last division we have multiplied twice by y2+q in order to render the divisions possible, but if we take y2+q=0, the divisor reduces to 3r, a quan tity in general differing from 0.

Making the last remainder equal to zero, and performing the operations indicated, the equation of the differences is

y+6qy'+9qy2+4q3+27r2=0;

taking y2=z, the equation of the squares of the differences becomes

z3+6qz2+9q2z+4q3+27r2=0.

For the equation x3-7x+7=0, we have q=-7, r=+7; and hence the equation in z becomes

23-42x2+441z-49=0.

BUDAN'S CRITERION

For determining the number of imaginary roots in any equation. 280. If the real positive roots of an equation, taken in the order of their magnitudes, be a1, а2, αз, α....an, where a, is the smallest, and if we diminish the roots of the equation by a number h greater than a1, but less than α, then the roots will be a-h, a-h, as-h, ...a-h, and the first of these will now be negative. But the number of positive roots is exactly equal to the number of variations of sign in the terms of the equation when the roots are all real; and as we have changed one positive root into a negative one, the transformed equation must have one variation less than the proposed equation.

Again, by reducing all the roots by k, a number greater than a2, but less than a3, we shall have two negative roots, a1-k, a2-k, in the transformed equation, and, therefore, we shall have two variations of sign less than in the proposed equation, for two positive roots have been reduced so as to become negative ones. Hence it is obvious, that if we reduce the roots by a number greater than a, all the positive roots will become negative, and the transformed equation, having all its roots negative, will have the signs of all its terms positive (Art. 259), and all the variations will have entirely disappeared.

We see, then, that if the roots of an equation be reduced until the signs of all the terms of the transformed equation be +, we have employed a greater number than the greatest positive root of that equation; and, therefore, its reciprocal must be less than the smallest real root of the reciprocal equation. Now, if we take the reciprocal equation, and reduce its roots by the reciprocal of the former number, we should have as many positive roots left in this transformed reciprocal equation as there were positive roots in the proposed equation, unless the equation has imaginary roots; hence the number of variations lost in the former case should be exactly equal to the number left in the latter, when the roots are all real; and, consequently, if this condition be not fulfilled, the difference of these numbers indicates the number of imaginary roots. To explain this reasoning more clearly, we shall suppose that an equation has three positive roots; as, for instance, 1, 2.5, and 3. Now if the roots of the proposed equation be reduced by 4, a number greater than 3, the greatest positive root, the three positive roots in the original equation will evidently be changed into three negative ones in the transformed one, and hence three variations must be lost. Again, the equation whose roots are the reciprocals of the proposed equation must have three positive roots, 1, 3, and; and it is evident that if we reduce the roots of the reciprocal equation by, the reciprocal of the former reducing number 4, we shall not change the character of the three positive roots, because is less than the least of them, and 1—},

-- are all positive; hence the three variations introduced by the three positive roots must still be found in the transformed reciprocal equation, and, therefore, three variations are left in the latter transformation, indicating no imaginary roots. The theorem may, therefore, be stated thus:

If, in transforming an equation by any number r, there be n variations lost, and if, in transforming the reciprocal equation by (the reciprocal of r), there be m variations left, then there will be at least n-m imaginary roots in the interval 0, r.

For there are as many positive roots in the interval 0, r of the direct equation as there are between and of the reciprocal equation; hence, if n, the number of variations lost in the transformation of the direct equation by r, be greater than m, the number of variations left in the transformation of the reciprocal equation by, there will be a contradiction with respect to the character of a number of the roots, equal to the difference n―m. Hence these roots are imaginary.

Here two variations are lost in the transformation of the direct equation, and no variations are left in the transformation of the reciprocal equation; therefore this equation has at least two imaginary roots; and it has only two, for the sign of the absolute term is negative, implying the existence of two real roots, the one positive and the other negative. (See Art. 248, Pr. VIII., Cor. 5.)

EXAMPLE.

To find the number and situation of the real roots of the equation 4+3 +x2+3x-100=0 by Budan's method.

If the roots of this equation be all real, the permanences and variation indicate three negative roots and one positive root.

In the transformation by 2, one variation is left, and, in transforming by 3, there is no variation left; therefore the positive root is between 2 and 3. (2) For the negative roots.

Here two variations are lost in the direct transformation, and no variations are left in the reciprocal transformation; therefore the two roots in the interval 0 and -1 are imaginary.

1-1+1-3-100 (3 1-1+ 1—3—100 (4

281. In any equation, if we have a cipher-coefficient, or term wanting, and if the cipher-coefficient be situated between two terms having the same sign, there will be two imaginary roots in that equation.

Let the order of the signs be

++―0-+.

and for 0 writing + or — we have either

++-+-+-−, or ++−−−+

In the former of these we find two permanences and five variations, and in the latter we have four permanences and only three variations; hence, if the roots are all real, we must, in the former case, have five positive and two negative roots, and in the latter, three positive and four negative roots (Art. 259); hence we have two roots, both positive and negative, at the same time, and, therefore, these two roots can not be real roots. These two roots, which involve the absurdity of being both positive and negative at the same time, must, therefore, be imaginary roots.

In nearly the same manner it may be shown that

(1) If between terms having like signs, 2n or 2n-1 cipher-coefficients intervene, there will be 2n imaginary roots indicated thereby.

(2) If between terms having different signs, 2n+1 or 2n cipher-coefficients intervene, there will be 2n imaginary roots indicated thereby.

EXAMPLE.

The equation -x3+6x2+24=0 has two imaginary roots, for the absent term is preceded and succeeded by terms having like signs; and the equation 3±1, having the coefficients 10±0±1, has also two imaginary roots.

EXAMPLES FOR PRACTICE.

(1) How many imaginary roots are in the equation

x2+x3—2x2+2x−1=0?

(2) Has the equation ra—2x2+6x+10=0 any imaginary roots ?

THE LIMITS OF THE ROOTS OF EQUATIONS.

282. The limits of any group of roots of an equation are two quantities between which the whole group lies; thus, and 0 are limits of the positive roots of every equation, and 0 and - of the negative roots. But in practice we are required to assign much closer limits than these, usually the two con

secutive whole numbers between which each root lies, so that the inferior limit is the integral part of the included root. This may be effected without knowing any of the roots of the equation, as will be seen in the following propositions. The roots spoken of in this section are the real roots.

SUPERIOR AND INFERIOR LIMITS OF THE ROOTS.

283. The greatest negative coefficient increased by unity is a superior limit of the positive roots of an equation.

Let p be the greatest negative coefficient; then any value of r which makes

x-p(x1+x2+ ··· +x2+x+1) positive,

...

or f(x) positive, because in the latter, all the terms after will not generally be negative, and of the negative terms not one is greater than the corresponding term in the former expression.

Since, therefore, p+1 and every greater number, when substituted for x, will make ƒ(r) positive, the numerical value of the greatest negative coefficient increased by unity is a superior limit of the positive roots.†

284. In any equation, if p‚a"— be the first term which is negative, and —p the greatest negative coefficient, 1+ Vp is a superior limit of the positive

roots.

Any value of r which makes

x">p(x2¬'+xa¬r−1+ +x+1)>p

...

will of course make "+pia"-1+P+... positive.

9

x">P 1—1, or x2-'(x−1)>p, or if (x−1)—1(x−1)= or >p, or (x−1) = p・

Since, therefore, 1+ √p and every greater number gives a positive result, 1+Vp is a superior limit.

This method may be employed when the first term is followed by one or more positive terms.

EXAMPLE.

+11x2-25x-61-0.

Here r=3, and a limit of the positive roots is

1+61, or 5, taking the next higher integer.

285. If each negative coefficient, taken positively, be divided by the sum of

* See (Art. 23).

This is commonly known as Maclaurin's limit.