expressions which possess the relation of the square and simple power only, we may replace this expression by a new symbol, and proceed to solve the equation with respect to it. Thus, if the equation be 6x − x2 + 3√(x2 − 6x + 16) = 12 (1), we make √√(x2 – 6x + 16) = u, or x2 - 6x + 16 = u2, and therefore 6x-16-u: we thus get the quadratic equation 16-u2 + 3u = 12 (2), where the values of u are 4 and 1: but inasmuch as the form of the proposed equation (1) excludes a negative value of u, we confine our attention to the arithmetical root of equation (2)*: we thus get {(x + 3)2 + (x + 3)}2 − 7 (x + 3)2 = 7 x + 711 is easily reducible to the form {(x + 3)2 + (x + 3)}2 − 7 {(x + 3)2 + (x + 3)} = 690, which becomes, if we make u = (x+3)2+(x+3) u2 - 7 u = 690. The values of u are 30 and -23: those of x are 2, -9, -17±√-91 which are the four roots of the proposed biqua 2 , • If we take the negative value of u or -1, the resulting roots 3±√6, correspond to the equation 6x - x2-3√x2-6x + 16 = 12, and are merely roots of solution of the original equation. The biquadratic equation, if reduced to the ordinary form, is +14x3+66x2119 r - 630 = 0, which may be solved by the extraction of its square root. The num are 4 and 20 of which the second must be rejected, being merely a root of solution: there is, therefore, only one proper root of the proposed equation. 678. The number of proper solutions of an equation is not ber of pro- affected by a common denominator of the index of the unknown tions of an symbol, or its powers: for if the highest power of this symbol per solu equation not affected by a deno- was x P the process of solution would give us the values of x-, minator of and the number of those values would not be increased by the the index of the un known symbol. 1 transition from the values of a to those of a: thus the equation x+6x3 = 891 There are three cube roots of 1, which are 1, -1√√1 (Art. 2 1, 2 − 1 ± √ √−1 (Art. 670) : there are, therefore, six values of x in this equation. † For (1)§ = (1)} and! ( − 1) § = ( − 1)} . would have the same number of proper solutions with the equation x2+6x=891, and the solutions themselves only differ in the roots of one equation being the fifth powers of those of the other. If, however, we should suppose the equation x+6x=891 to admit equally all the forms which are proper to the different symbolical values of a3 (which will be found hereafter to be 5 in number,) we should have 5 equations and 30 roots, of which 6 only are the proper roots of the proposed equation, the rest being roots of solution. It will appear, likewise, that the reduction of the indices of the unknown symbol, when one or more of them are fractional, to a common denominator, will convert roots of solution into proper roots, unless the change from one form to the other is accompanied by a limitation of the roots to be extracted, in consequence of such a change, to their arithmetical values only: thus the proper root of the equation is 4, and 9 is the root of solution: but 4 and 9 are equally proper roots of the equation unless we are equally restricted to the arithmetical value of in the two equations (1) and (2). In a similar manner, there are only three proper roots of the equation which include the roots of the former equation (1). CHAPTER XXI. The bino mial theorem when the index is a whole number. THE BINOMIAL THEOREM AND ITS APPLICATIONS. 679. IN Chap. VIII. (Art. 486...), we have proved, when the index n is a whole number, that (1 + x)" = 1 + n x + n ( n − 1 ) 22 + n (n − 1) (n − 2) — . 2. 3 x2 1.2 1.2.3 and it will be seen, from an examination of this series, and of the law of its formation, that the powers of x and their divisors are independent of n, and that the coefficients of The same series is index is are n, n (n − 1), n (n − 1) (n − 2), .. - being, for the (1+r)th term, the continued product of the descending series of natural numbers from n to n − r + 1. 680. This series for (1 + x)" is perfectly general in its form, equivalent though n is specific in its value, and it will continue therefore, when the by "the principle of the permanence of equivalent forms" (Art. perfectly 631) to be equivalent to (1+x)", when n is general in value as well as in form: and it will consequently admit, in virtue of this equivalence, of being immediately translated into the whole series of propositions respecting indices and their interpretation, which are given in Chapter XVI *. general in value as well as in form. The pro 681. Thus "the general principle of indices" (Art. 635) duct of the shews that series for (1 + x) * and (1 + x)"' (1 + x)" (1 + x)"' = (1 + x)"+n' is the series for all values of n and n', and consequently the product of the will be equivalent to the series for (1 + x)+"', or for the product -n(n-1) may be replaced by n (n + 1): the product-n(n-1) (n-2) by n (n + 1)(n+2), and similarly for the subsequent terms. (1 − x)", 683. The series for (1-x)" is deducible, in virtue of the The series same principle (Art. 631), from that of (1 + x)", by changing for the signs of those terms which involve the odd powers of x: we thus get, as in Art. 491, and in a similar manner, the series for (1 - x)-" will become Series for |