+a2a4 -a2a 3a4 +a3a4 &c., &c. By carefully examining the above results, we discover the following properties : (249.) The coefficient of x in the first term is always 1. The coefficient of the second term is the sum of all the roots with their signs changed. The coefficient of the third term is the sum of all the products of the roots taken two and two. The coefficient of the fourth term is the sum of all the products of the roots with their signs changed, taken three and three. And so on for the succeeding terms, until we reach the last term which is independent of x, and is equal to the continued product of all the roots, with their signs changed. (250.) The general form of an imaginary or impossible root of an equation is a+ √-b. The only factor which will render a+ √-b rational is a-√-b. We have just seen, that the last term of our general equa tion is composed of the continued product of all its roots. Hence, if a+√-b is a root of this equation, then also will a-v-b be a root. In the same way, if a'+ √-b' is a root, then will a' — √ —b' be a root, and so for other imaginary roots. From this we infer the following properties : (251.) Every equation has an even number of impossible roots, or else none at all. An equation of an even degree may have all its roots impossible; but if they are not all impossible, two of them at least are possible. If all the roots of an equation are impossible, then whatever values are substituted for x in that equation, the results will always be affected with the same signs. An equation of an odd degree has at least one real root. (252.) If we divide both members of the identical equa 3 A an (1-2) (1-2) (1-4) ----(1-2) S Taking the logarithms of both members, we find If we actually take the logarithm of the left-hand member of (A), by formula (C), Art. 220, where A, A, A-14 + + + 2 n 2 xn +&c. (B) х 1 Ai 4 By taking the logarithms of the terms of the right-hand member of (A), we get 1 - (a tartaz ambien By equating the coefficients of the like powers of x, in (B) and (C), (Art. 183,) we find the following interesting properties: (253.) These relations make known the sum of the mth powers of all the roots of an equation in terms of its coefficients. (254.) If we suppose the general equation is deprived of its second term, or which amounts to the same thing, if we suppose 41=0, the above results of (D) will become If in this equation we suppose x=u+x', u being a new unknown quantity, and x' an indeterminate quantity, we shall have (u+x')"+A1(u+x')n−1+A2(u+x′)n−2 An-1 (u+x')+A„=0, (2) N-1 which, when expanded by the Binomial Thcorem, becomes "+ng? tx'n +421-1 =0 (3) 11" x 2 un-2 n +4,-1X +4 Now, since a' is wholly arbitrary, we are able to give it such a value as to satisfy this condition nx'+A.=0; which A is done by making a= This value of x' substituted in (3) will give an equation of the following form : u"+Bzun-2+B3u1-3 Br-1u+B=0, (4) which is deprived of its second term. (256.) Hence, to cause the second term of an equation to disappear, we must replace the unknown by a new unknown augmented by the coefficient of the second term with its sign changed, and divided by the number denoting the degree of the equation. EXAMPLES. 1. Transform the quadratic equation x2 + A,x+4y=0, into a new equation wanting its second term. A Assumę x=u and it will become 2 A1 u- +A2= 2 2 this, when reduced, becomes |