"such equation cannot have a single root of the form √±b; "✓ being an irreducible surd." For ifbbe substituted 66 p=a+B+r+√±6 q=aß+ay+By+(a + B+ r) √±b where p, q, r, s are evidently quantities involving the irreducible surd, or its multiples, (a+B+r) √±b, (aß+ay+By) √±b, (aßy) √±b. In the same manner, it might be shewn that an equation with integral coefficients cannot possibly contain a single root of the form a+ √±b, or a-√±b. 14. But the case is different when an equation contains two roots, one of the form a+b, and the other of the form ab; for such is the symmetrical arrangement of the roots in the combinations by which the coefficients p, q, r, 5, &c. are formed (in Art. 2.), that in the sum of these combinations the radical parts destroy each other; as will appear from the following investigation. I. Let a quadratic equation be formed of the two roots a+b and a-b; then p=(a+√±b)+(a−√ ±b)=2a and consequently the equation itself is x2-2ax+(a'Fb)=0, an equation whose coefficients are integral.* II. Let * That a quadratic equation whose coefficients are integral may have its two roots in a radical form, appears also from the ordinary solution of it (as in Art. 81. EL); for the two roots of the equation x2-px+q=0 are II. Let a cubic equation be formed of the three roots a+√±b, a−√±b,c; then p=(a+b)+(a−√±b)+c=2a+c q=(a+√±b) (a−√±b)+(a+√±b)c + (a−√±6) c=(a*Fb)+2ac r=(a+b) (a−√±b)c=(a+b)c, and the equation itself is x3− (2a+c) x2 + ((a* Fb)+2ac) x—(a'Fb) c=0. III. In the same manner it might be shewn that the biquadratic equation whose roots are a +√√± b, a −√± b, c, d, is x*— (2 a + c + d) x3 + ((a*Fb)+2ac+c+d)) x* −((a2+b)(c+d)+2acd)x+(a2+b) cd=0.* These examples are sufficient to shew the manner in which roots of the form ab lie hid among the integral coefficients (p, q, r, s, &c.) of the equation (A)=0. And as it has already been shewn (Art. 13.) that an equation so constituted can contain no single root of that form, it follows that roots of this form enter every equation by pairs. They are indeed the corresponding roots of the several quadratic equations of which the equation (A)=0 is composed. 15. Let the sign of b be negative, then all the roots of the form ab contained in the equation (A)=0 are impossible. Now by Art. 3, every equation of n dimensions may be considered as composed of the several combinations of equations inferior to it, the sum of whose indices amounts to n. Hence, if n be even, the equation may be considered as • The operation for finding the coefficients of this equation will be much facilitated by arranging the values of q, r, (Art. 2.) in the following manner; viz. q=aß+a+By+ad + ßd+yd=aß +(x+ß) y +(a+ß+y) d and then substituting a++b for a, ab for ß, e for y, and d ford. Λ as consisting entirely of quadratic equations, whose number amounts ton, and consequently must have either 0, 2, 4, 6, &c. or n impossible roots, according as the roots of these the roots of these if any several quadratic equations are possible or impossible; but if be n be an odd number, the equation may be considered as composed of (n-1) quadratic equations and one simple equation, and therefore may contain either (), 2, 4, 6, &c. or n−1 impossible roots, and must contain at least one possible root. 16. In demonstrating the foregoing properties of the equation (A)=0, we have gone altogether upon the supposition that this equation is composed of n simple equations, x-α=0, x-8=0, x-y=0, x-8=0, &c. To infer from hence, however, that every equation of n dimensions may be resolved into n simple equations, would be to argue in a circle. The truth of the proposition "that every equation has as many roots as it has dimensions" must depend entirely upon the means we possess for solving equations; what those means are, will be shewn in the Fourth Chapter; but it may not be improper to explain how far the proof of this proposition may be deduced from the properties of the equation (A)=0, as constructed in Art. 2. Let us take for example the equation x 1 — px 3 + q x2 —r ̧x+s=0; and let us consider its coefficients p, q, r, s as known quantities, a+B+r+d=p aß +ay+ad + By + ßd +rd=g aßy+aßd+and+Byd=r αβγδας we have as many equations as unknown quantities for deter- By+Bd+rd=q−(αß+ay+ad)=q—pa+a2 =ra(By + ßd+nd) ^ of But 2 or a* — pa3 + q¤2¬7a+s=0; an equation which is identical with the given equation. * In the same, manner it might be shewn that the equations for determining the values of 6,, or d are s-pß3+98 2 — rß + s = (), y•—py3+qy3—ry+s=0, or d1—pd2+qd1Hrds0 respectively, so that to find the values of a, ß, y, d, by this method, requires the solution of an equation of the same dimensions as the given equation; and the same is true for equations of all dimensions. Hence the direct proof of the proposition extends no further than to those orders of equations, of which the general method of solution is known. Our remaining observations upon this subject must, therefore, be reserved for the end of the Fourth Chapter. initesne sdi mon cr brent da. SZ. ~ -1= 17 T CHAP. II. ON THE TRANSFORMATION OF EQUATIONS. By the transformation of equations, is meant the changing of them into others, whose roots shall bear some given relation to the roots of the original equations. 17. If it be required to transform the equation (A) =x-px+9x - &c.... ± vxFw=0, whose roots are a, ß, y, d, &c., into one whose roots shall be m times the roots of this equation, let y=mx, or x= y m 2; then for x and it's powers in the equation (4)=0 substitute/ and its powers, and the resulting equation will be m y"—mpy"—'+m2qy*~2—m3ryTM3+ &c...±m*¬vyFm"w=0, an equation whose roots are ma, mß, my, mè, &c., where m may be any quantity whatever, either integral or fractional. Hence this rule for transforming an equation into one whose roots shall be any multiple or part of the roots of the given equation; "Multiply each term of the given equation, "beginning with the second, by the successive powers of the "whole number or fraction expressing the value of such mul |