hence we may take A' equal to some power of 2; and it is easily seen that the third power will do, so that we may make x = y. Hence the given equation becomes y441 y3574 y2-3144 y 5760 = 0. The commensurable roots of which are found, as in art. 211, to be y= 4, 6, 15, and 16; so that the roots of the given equation are x = 1, 2, 1, and 2. x= 2. Find the commensurable roots of the equation 8x3+34 x2 - 79 x + 30 = 0. Incommensurable Roots whose difference is greater than E. There is a mistake yo 6. Find all the roots of the equations. One of the t 6 x3 +7 x2 + 39 x + 63 = 0 Leg 7. Find the commensurable roots of the equation Ans. and - 2. SECTION IV. Incommensurable Roots. 215. A real root, which cannot be exactly expressed in numbers, is called an incommensurable root. 216. Problem. To find the incommensurable roots of an equation, the difference between every two of whose roots is known to exceed a given quantity E. Solution. Form the arithmetical progression continued to the superior limit of the positive roots; also the progression 0,- E, -2 E, — 3 E, &c. continued to the superior limit of the negative roots. Substitute, successively, each term of each of these progressions for the unknown quantity in the first member of the equation, reduced to the form of art. 160. Decimal places to be retained. When two successive values of the first member thus obtained are affected by opposite signs, a root must, by art. 190, be contained between the corresponding terms of the series, and not more than one root, because the difference between two roots is greater than that between two terms of the series. Either of these two terms of the series must, therefore, differ from the root by a quantity less than E, less, therefore, than unity, when E is equal to or less than unity; and may be assumed as the first approximation to this root. But if one or more integers are contained between the two terms of the series, which must be the case when E is greater than unity, the successive substitution of these integers for the unknown quantity of the equation will give an integer, which differs from the root by a quantity less than unity, and which is to be preferred as the first approximation to its value. Representing this first approximation by w, and the quantity by which it differs from the true root by h, substitute w + h for the unknown quantity in the equation; and since h is less than unity, its powers must be still smaller, and the terms containing them may be omitted, so that the result may be reduced to the form Ah B 0. The value of h, obtained from this equation, substituted in w+h, Decimal places to be retained. gives a second approximation to the value of the root, from which a third approximation may be obtained in the same way that this was obtained from the first approximation, and so on to higher approximations. 217. Scholium. It is advisable to reduce all the calculations of this solution to decimal fractions; and retain in each multiplication or division only one or two decimal places beyond the probable accuracy of the approximation. The third approximation is rarely exact beyond three places of decimals; but every further approximation may usually be relied upon to twice as many places of decimals as its preceding approximation. 218. Scholium. The real roots of most equations which are met with in practice differ from each other by more than unity, or, at least, only one real root is usually included between two successive integers, so that the substitution of all the integers contained between the extreme limits of the roots will usually lead to their discovery. Even if this substitution of the integers is unsuccessful, it is rarely useless, for the progressive increase or diminution of the values of the first member will usually indicate the integers which are nearest to the roots; and the substitution of fractional values which differ but little from these integers can hardly fail of success. 219. Scholium. Before applying this solution to an equation, it is almost always advisable to transform it, as in art. 170, to a form in which the second term is wanting. Examples of Incommensurable Roots. EXAMPLES. 1. Find the real roots of the equation x3 15 x2 + 72 x 109 = 0. Solution. Transform it, as in art. 170, by substituting for x, and the limits of u are 1+3, and · 13. Now the substitution for u of gives for the corresponding values of the first member 19, 3, 1, 1, 3, — 1, — 17; gives, by neglecting all the powers of h, but the first, |