x in each, and write the signs of the results in a parallel column. In passing down the column of signs, whenever two consecutive ones are alike, there is said to be a permanence; and whenever two consecutive ones are unlike, there is said to be a variation. If we now suppose a to increase by inappreciably small increments from ∞ to +∞, we shall find in succession values that will reduce V

--

or some of its

no value of x There may be

derivatives to 0, but from principle 1° will reduce two consecutive ones to 0. two cases: first, when V reduces to 0, and secondly, when one of its derivatives reduces to 0.

1

First. Suppose that any value of x, as x = k, reduces V to 0; then is a real root of the equation V = 0; it follows from principle 3° that for the value of next preceding k, V and V1 have contrary signs, and there is a variation; but for the value of x next following k, V and V1 have the same signs, and there is a permanence: hence, every time that the value of 2 passes a real root of the equation V = 0, there is a variation lost, or converted into a permanence. Secondly. Suppose that any value of x, as x = m, reduces one of the derivatives of V, as Vn, to 0; then is m a root value of V; it follows from principle 2° that V-1 and Va+1 have contrary signs for this value. of x; it follows from principle 4° that V, changes sign when the value of x passes from the value immediately preceding m to the value immediately following m; it also follows from principles 2° and 4° that V-1 and V+1 do not change sign for these values of x.

n

Vn

Now, for the value of x immediately preceding m, V-1 and V2+1, have contrary signs, and V, must have the same sign as one of them; there is therefore, one corresponding variation, and but one: for xm Vn-1 and Vn+1 have contrary signs, and V, is 0; there is therefore, one corresponding variation, and but one: for the value of x next following m, V-1 and Vn+1 have contrary signs, and V, must have the same sign as one of them; there is therefore, one corresponding variation, and but one: hence, whenever the value of x passes a root value of any derivative of V, there is no variation either lost or gained.

We therefore conclude that there is one variation lost in the column of signs whenever a passes a real root of the equation V = 0, and that there is no variation either lost or gained under any other circumstances; hence,

If we make x = -coin V and its derivatives, and write the corresponding signs in a column, and then if we make x = + < in the same expressions and write the corresponding signs in a second column, the number of variations in the first column, diminished by the number of variations in the second column will be the number of real roots in the equation V = 0.

This is STURM'S theorem.

When we make x = - ∞, or x = +, in V, or in any of its derivatives, the value of the first term in each will be infinitely great with respect to all the following terms, and consequently, ' the sign of each result will be the same as the sign of its first

term.

Illustration.

Let it be required to find the number of real roots.

Here, we have, V = x4

12x2 + 12x 3; finding the first derivative of V, and suppressing the factor, +4, we have,

we find, for a remainder, +3, and changing the = 2x2 3x + 1; multiplying

V1 = x3 6x + 3; dividing V by V1, - 6x2 + 9x 3; suppressing the factor, sign of the result, we have, V2 V1 by 4, and dividing by V2, we have, for a remainder, — 17x + 9; changing the sign, we have, V3 = 17x — 9; multiplying V2 by 289, and dividing by V3, we find, for a remainder, - 8; changing the sign, we have, V1 = 8; writing these expressions in a column, and substituting -, and then +, for x, we have the results indicated below.

In the first case, there are 4 variations, and in the second, there are no variations; hence, all four of the roots are real.

Sturm's Theorem also enables us to determine the places of the real roots. If we substitute for x, in the above expressions, any two numbers whatever, the number of variations corresponding to the less, diminished by that corresponding to the greater, will give the number of real roots between the two numbers. Let us begin by making x equal to 0, 1, 2, &c., until we get as many permanences, as when x; then make x equal to 1, 2, &c., until we get as many vari

ations, as when x = . Taking the same example as before, we have the following results:

110++

--

++++ | ∞

+++++

Hence, we conclude that two of the roots lie between 0 and — 1; one between 3 and 4; and one between 3 and 4. Here, we have found the values of the roots, to within less than 1; the method of completing the approximation, will be explained hereafter.

In the preceding case, we have seen that +4 gives the same number of permanences, as +; hence, no real roots lie between +4 and +; we have also seen, that 4 and, give the same number of variations; hence, no real root lies between them. The values, − 4 and + 4, are called the limits of the roots of the given equation; the former being the inferior, and the latter, the superior limit.

If we consider the positive roots alone, 0 and 4 are the limits; if we consider the negative roots alone, - 4 and 3 are the limits. In the same manner, the limits of the positive and negative roots of any equation may be found. It is often useful to determine these limits, especially when seeking the entire roots of an equation, by the process of article 206.

EXAMPLES.

1. Find the number, the places, and the limits of the real roots of the equation, x3 + x2 + x − 100 = 0.

Hence, there is one real root lying between 4 and 5, which are the limits of the root.

2. Find the number, places, and limits of the real roots of the equation,

Ans. There are 4 real roots; one between 0 and 1, one between 2 and 3, one between 5 and 6, and one between 1 and 0. The limits are -1 and 6.

3. Find the number, places, and limits of the real roots of the equation, x3 23.x 24 = 0.

Ans. 3 real roots; one between 5 and 6, one between -1 and - 2, and one between 4 and 5. The limits are, − 5 and + 6.

4. Find the number, places, and limits of the real

3

roots of the equation, 2+2x-5 = 0.

Ans. There is 1 real root, and it lies between the limits 1 and 2.

Each variation is lost when a passes from the preceding value to the root value of V; hence, if the greater number substituted for x is a root of the equation, it is to be counted amongst the roots sought.