APPENDIX. No. III. Extermination of unknown Quantities, and I. THE of Radicals. HE extermination of unknown quantities from fimple equations, may always be performed by fubftitutions, but fometimes more easily by the addition, and fubtraction of the equations themselves. This laft method will be beft explained by an example. Suppofe that ax+byc, and dx+fy=g, where a, b, c, &c. are given quantities, that may be either pofitive or negative. To exterminate y, multiply the first equation by f, the coefficient of y in the fecond, and the fecond by b, the coefficient of y in、 the first, and subtract the latter from the former; we have afx+bfy=cf bdx+bfy=bg, And therefore afx-bdx=cf-bg, 2. Let there be now three unknown quantities, and three equations, viz. ax+by+cz+d=0, fx+gy+bz+l=0, mx+ny+px+q=0; multiply the firft by b, and the second by abx+bby+cbx+db=0, and (ab-cf)x+(bb—cg)y+db-clo. In the fame manner, an equation without z, is found from the fecond and third, viz. (fp—bm)x+(gp—hn)y+lp—bq=0. From From the two laft equations, y is to be exterminated, as in the former example, and we will then have, (bh—-cg )( lp—hq )—( gp—hn ) ( dh—cl ) (ah—cf)( gp—bn )—( bh—cg )( fp−−hm) —(ab—cf)( lp—bq)+( fp—hm )( dh—cl); (ab-cf)(gp-bn)—(bb—cg )(fp—bm) (fn―gm.)(dg—bl )—(ag—bf)(In—gg). (ag-bf)(hn-gp)—( cg—bh )( ƒn—gm) 3. By the fame method may the unknown quantities be exterminated from equations of the higher orders; but it will then be convenient to write fingle letters inftead of the coefficients, whether known or unknown, of the different powers of the quantity to be exterminated. For example, let y2-3xy+ay—x2=0, and y2-by+2ax—4x2+b2=0; to exterminate y. As the coefficient of 2 is 1, in both equations, no fubftitution for it is necessary, but that the method may be general, we fhall suppose A=1, B=a-3x, C=—x2; alfo, in the fecond equation, D1, E=—b, F F=2ax-4x2+62; the two equations are then Ay2+By+C÷6, and Dy2+Ey+F=0. To exterminate y2, multiplying the first by D, and the fecond by A, we have ADy2+BDy+CD=0, ADy2+AEy+AF=0; Therefore (BD-AE)y+CD-AF =0, Again, to obtain another value of y, multiply the first equation by F, and the second by C; then AFy2+BFy+CF=0, and CDy2+CEy+CF=0, therefore (AF-CD)y2+(BF—CE)y=0. that is, (AF-CD)y+BF-CE=0, or (AF-CD)2=(BD—AE) (CE—BF). As 1 y does not enter into this equation, if we restore the values of A, B, and C, we have an equation involving only x and known quantities, and have thus obtained a general theorem for the extermination of one of the unknown quantities from any two quadratic equations. In the prefent cafe, the equation becomes, (b2+2ax—3x2)2 = (bx2—(a—3x)(b2+2á x −4x2))(a+b—3x), or, 27x4-30ax3+12a2x2-2ax—a'b'—0 —15bx3+11 abx2+2ab2x—ab3 -2b⋅ x2-2a2bx-b4. - +363x. The fame method, it is evident, may be applied, whatever be the number, or the order of the equations. 4. By a fimilar process, the radical quantities that enter into any equation may be exterminated. This cannot always be effected by involution only; for, on the contrary, by that operation, the number of radicals may fometimes be increased. The method, on the other hand, which we are going to lay down, is univerfal: It |