In the same way the other two roots may be found to be Solution. Since two successive terms are wanting, this Examples of Incommensurable Roots. equation must, by art. 202, have at least two imaginary roots; and also since the substitution of which evidently cannot be satisfied by a positive value of y, the given equation cannot have negative roots. The limits 3 of its roots are therefore 0, and 1+√10, that is, 1 and 3,2. The substitution then for x, of 0, 1, 2, 3, 4, gives for the corresponding values of the first member 19, 1, 11, 121, 451; so that if there is a real root, it is probably nearly 1. Now the substitution of 0,9 and of 1,1 for x, gives for the values of this first member: 2,3122 and - 0,0718; so that there must be a real root between 1 and 1,1, and another between 1,1 and 2; and these roots are found as above to be nearly 1,0922 and 1, 5914. 3. Find the approximate real root of the equation x3 12x+132 = 0. Ans. - 5,872052. 4. Find the approximate real roots of the equation x4 +8 x2 + 16 x 440= 0. Ans. 3,976, and -4,3504. 5. Find the approximate real roots of the equation 5x36x+2=0. Ans. 0,856, 0,379, and 1,234. - 220. Problem. To find an inferior limit of the dif ference between two roots of an equation. Inferior Limit of the Difference of the Root. Solution. Let the equation be A x2 + B xn−1 + &c. = 0; represent two of its roots by x' and x", and their difference by D, so that we have The elimination of x and x" between these equations gives an equation containing only D, whence the inferior limit of D is determined as in art. 209. 221. Scholium. It is plain, from the remarks of art. 218, that the solution of this problem can rarely be of any practical use, and yet it is important to complete this chapter. EXAMPLES. 1. Find the inferior limit of the difference of the roots of the equation 7x3 6 x + 2 = Solution. The elimination of x and x" between the equa Inferior Limit of the Difference of the Root. 2. Find the inferior limit of the difference between two 3. Find the inferior limit of the difference between two roots of the equation 100 x 3 27 x + 5 = 0. Ans. 0,3. 4. Find the equation for determining the difference between two roots of the equation Ax3Bx+ C = 0. Ans. A3 D6-3AB2 D2-27 A C2 +2 B30, in which D denotes the required difference. Value of Continued Fractions. CHAPTER IX. Continued Fractions. 222. A continued fraction is one whose numerator is unity, and its denominator an integer increased by a fraction, whose numerator is likewise unity, and which may be a continued fraction. 223. Problem. To find the value of a continued fraction which is composed of a finite number of fractions. Solution. Let the given fraction be |