without at the same time making f(x) vanish; for if it could, since those variations can never be restored, and since a variation must disappear for every passage through a real root, the total number of variations lost would surpass n, the degree of the equation, which is absurd, since there are but n derived functions in all. Whenever, therefore, variations disappear between values of r which do not include a root of f(x)=0, there is, corresponding to that occurrence, an equal number of imaginary roots of f(x)=0. Hence, if r=c produces a zero between two similar signs, or if it produces an even number of consecutive zeros either between similar or contrary signs, there will be respectively two, or p, imaginary roots corresponding; or if it produces an odd number q of consecutive zeros, there will be q±1 imaginary roots corresponding, according as they stand between similar or contrary signs; c, of course, not being a root of f(x)=0.

Р

OBSERVATION. Since the derivatives which follow any one ƒ'(x) may be supposed to arise originally from it, it is manifest that the same conclusions respecting the roots of f'(x)=0 may be drawn from observing the part of the series of derived functions

ƒ'(x), ƒ1+1(x), ...ƒa(x)

as were drawn respecting the root of f(x)=0 from the whole series.

269. Des Cartes's rule of signs is included in Fourier's theorem as a particular case.

For when, in the series formed by f(x) and its derived functions, we put x=∞, there are n variations; and when we put r=0, the signs of the series of functions become the same as those of the coefficients of the proposed equation

Let the number of variations in this series of coefficients =k, and therefore the number of permanences (supposing the equation complete) =n-k: if we make x=+∞, the signs of the functions are all positive, and the number of variations =0. Hence, between x∞ and r=0, the number of variations lost is n-k; therefore in a complete equation there can not be more than n―k negative roots, i. e., than the number of permanences in the series of coefficients; also, between r=0 and x=∞, the number of variations lost is k, whether the equation be complete or incomplete; hence in any equation there can not be more positive roots than k, i. e., than the number of variations in the series of coefficients, which is Des Cartes's rule of signs.

270. Fourier's theorem may also be presented under the following form: If an equation have m real roots between a and b, then the equation whose roots are those of the proposed, each diminished by a, has at least m more variations of signs than the equation whose roots are those of the proposed, each diminished by b.

The transformed equations would be

f(y+a)=0, f(y+b)=0;

and if these were arranged according to ascending powers of y, the coefficients would be the values assumed by f(x), f'(x), &c., when a and b are respectively written for r. Therefore, whatever number of variations of signs is lost in the series f(x), f'(x), &c., in passing from a to b, the same is lost in passing from one transformed equation to the other; but the series for a has at least m

more variations than that for b; therefore ƒ (y+a)=0 has at least m more variations than f(y+b)=0.

271. To apply this method to find the intervals in which the roots of f(x)=0 are to be sought, we must substitute successively for x, in the series formed by f(x) and its derived functions, the numbers

(-a and being the least negative and least positive number, which give respectively only variations and permanences), and observe the number of variations of sign in each result.

Let h and k be the numbers of variations of sign when any two consecutive terms in series (1), a and b, are respectively written for r; therefore h—k is the number of real roots that may lie between a and b : if this equals zero, f(x)=0 has no real root between a and b, and the interval is excluded; if h-k=1, or any odd number, there is at least one real root between a and b ; if h—k=2, or any even number, there may be two, or some even number, or none; the latter case will happen when, as explained above (Art. 268), some number between a and b makes two or some even number of variations vanish, without satisfying f(x)=0. Similarly, we must examine all the other partial intervals; and when two or more roots are indicated as lying in any interval, their nature must be determined by a succeeding proposition.

The two former of the following examples are extracted from Fourier's work.

f(x)=

EXAMPLE I.

x3-3x1- 24x2+ 95x2-46x-101=0

f'(x)= 5x1-12x3- 72x2+190x —46

f" (x)= 20x3-36x2-144x +190

ƒ'''(x)= 60x2—72x —144

ƒ1 (x)=120x −72

ƒ3 (x)=120.

Hence we have the following series of signs resulting from the substitutions of —10, -1, 0, &c., for x, in the series of quantities

Hence all the roots lie between 10 and +10, because five variations have disappeared; one root lies in each of the intervals -10 to -1, and −1 to 0, because in each of them a single variation is lost; no root lies between 0 and 1, because no variation is lost between those limits; and three roots may be sought between 1 and 10 (because three variations have disappeared), one of which is certainly real; it is doubtful whether the other two are real or imaginary.

OBSERVATION. When any value c of r makes one of the derived functions, f(x), vanish, we may substitute ch instead of c, h being indefinitely small; then all the other functions will have the same sign as when x=c, and the sign of f(ch) will depend upon that of ±hfm+1(c); i. e., it will be the

same or contrary to that of the following derivative, fm+1(c), according as his positive or negative, or according as we substitute a quantity a little less or a little greater than the value which makes f(x) vanish. The use of this remark will be seen in the following example.

Every value less than 0 gives results alternately + and —, therefore there is no real negative root; for x=0, we have a result zero placed between two similar signs, and therefore corresponding to it there is a pair of imaginary roots. There is no root between 0 and 1, but there may be two roots between 1 and 10.

EXAMPLE III.

f(x)=x-6x+40x360x2-x-1=0.

Here there is no root <-1; there is one, and there may be three, be tween -1 and 0; there is one root between 0 and 1, and there may be two roots between 2 and 3.

272. The above process will determine the intervals in which the roots are to be sought, but not always their nature; when an even number of roots is indicated, they may all turn out to be impossible. The series of magnitudes between -∞ and ∞, to be substituted for x in the derived functions, has been divided into intervals of two sorts, each contained by assigned limits, a and b. The first sort of interval is one within which no root is comprehended, i. e., the limits of which give the same number of variations of signs in the series of derived functions. The second sort is one within which roots may lie, i. e., where the number of variations resulting from the substitution of b is less than the number resulting from the substitution of a, in the series of derived functions. This second sort of interval has two subdivisions, viz., cases where the indicated roots do really exist, and others where they are imaginary. When we have ascertained that a certain number of roots may lie between a and b, we may substitute c (a quantity between a and b) in the series of derived functions, and if any variations disappear, our interval is broken into two others; if no variations disappear, we may increase or diminish e, and make a second substitution, and it may still happen that no variation is lost, and so on continually; and we may be left, after all, in a state of uncertainty, whether the separation of the roots is impossible because they are imaginary, or only retarded because their difference is extremely small. This uncertainty is relieved by taking the interval so small as to be sure to include the real roots, if they exist.

One method of arriving at the proper interval is by means of the so-called equation of the squares of the differences of the roots of the given equation, which we shall hereafter have occasion to deduce. This process is tedious in practice; and as our object in unfolding the method of Fourier was to pursue it only so far as it threw light upon the general theory of equations, we shall here leave it.

We should now introduce the theorem of Budan, but it requires a transformation which we have not yet exhibited, and we therefore take this opportunity to complete a subject, one proposition of which (Art. 251) we have already had occasion to anticipate.

TRANSFORMATION OF EQUATIONS.

PROPOSITION I.

273. To transform an equation into another whose second term shall be removed. Let the proposed equation be

and by Art. 245 we know that the sum of the roots of this equation is —A1; therefore, the sum of all the roots must be increased by A, in order that the transformed equation may want its second term; but there are n roots, and A1

hence each root must be increased by and then the changed equation will

n

have its second term absent. If the sign of the second term of the proposed equation be negative, then the sum of all the roots is +A1; and in this case

A1

we must evidently diminish each root by and the changed equation will

n

then have its second term removed. Hence this

RULE.

Find the quotient of the coefficient of the second term of the equation divided by the highest power of the unknown quantity, and decrease or increase the roots of the equation by this quotient, according as the sign of the second term is negative or positive.

EXAMPLES.

(1) Transform the equation r3-6x2+8x-2=0 into another whose second term shall be absent.

Here A-6, and n=3; .. we must diminish each root by or 2.

.. y3-4y-20 is the changed equation.

And since the roots are diminished, we must have the relation x=y+2. (2) Transform the equation 1—16x3-6x+15=0 into another whose second term shall be removed.

(3) Transform the equation x+15x1+12x3-20x+14x-25=0 into another whose second term shall be absent.

(4) Change the equation x2+ax+b=0 into another deficient of the second

term.

(5) Change the equation +ax2+bx+c=0 into another wanting the second term.

274. To transform an equation into another whose roots shall be the recipro cals of the roots of the proposed equation.

1

.....

A-1¤+A=0 be the proposed equa

1

1 =; then x=-, and by writing for r in the proposed equa'y' y

tion, and put y=x

tion, multiplying by y", and reversing the order of the terms, we have the equation

Any”+An—1y”—1+ An-2y”—2+ . . . · A1⁄2y2+A1y+a=0,

....

whose roots are the reciprocals of the roots of the proposed equation.

The transformation is then effected by simply changing the order of the coefficients of the given equation.

Corollary 1.-Hence an equation may be transformed into another whose roots shall be greater or less than the reciprocals of the roots of the proposed equation, simply by reversing the order of the coefficients, and then proceeding as in the Proposition to Art. 251.

Corollary 2.-If the coefficients of the proposed equation be the same, whether taken in reverse or direct order, then it is evident that the transformed equation will be the same as the original one; and, therefore, the roots of such equations must be of the form

Corollary 3.-If the coefficients of an equation of an odd degree be the same whether taken in direct or inverse order, but have contrary signs, then, also, the roots of the transformed equation will be the same as the roots of the proposed equation; for, changing the signs of all the terms, the original and transformed equations will be identical, and the roots remain unchanged when the signs of all the terms are changed. And this will likewise be the case in an equation of an even degree, provided only the middle term be absent, in order that the transformed equation, with all its signs changed, may be identical with the original equation.

Equations whose coefficients are the same when taken either in direct or reverse order, are, therefore, called recurring equations, or, from the form of the roots, reciprocal equations.

Corollary 4.-If the sign of the last term of a recurring equation of an odd degree be+, one of the roots of such equation will be -1; and if the sign