which is by Art. 423 equivalent to the product of 0 0 Y=ay+bz+cx, and Z =az + bx+cy. Since x y z = x3+y3+z3 - 3xyz, and the other determinants are of the same form, we see that the product of any two expressions of the form a3+y3+z3 − 3xyz can be expressed in the same form. [See Art. 155, Ex. 4.] are the co-factors of a1, b1, &c. in the expansion of the determinant [a1b2c3]. is employed to denote the system of four determinants obtained by omitting any one of the columns. 426. We conclude with the following important applications of determinants. 427. Simultaneous Equations of the First degree. The solution of any number of simultaneous equations of the first degree can be at once obtained by means of the foregoing properties of determinants. First take the case of the three equations Multiply the equations in order by A1, A„, Ag, where A1, А„, Д ̧ are the co-factors of a, a, a, respectively in the 1 2' Then we have by addition (α ̧Ã ̧ + α ̧Ã ̧ + α ̧Ã ̧) x + (b ̧A ̧ + b,A,+b ̧Ã ̧) Y 21 2 2 3 that is [a, b, c,] x = [k, b, c,], 2 2 for from Art. 420 the coefficients of y and z are zero. Now consider n equations of the form α1x12+b1x12+ c1x2+d ̧æ1 +......= k ̧. As before, multiply the equations in order by A,, A., A,, &c. the co-factors respectively of a, a, a,, &c. in the determinant [a, b2 c...]; then we have by addition 2 2 1 2 (α11 + α‚Â ̧+α‚Â ̧ +.....) x = k1Ã ̧ + k„A2+ k ̧Ã ̧+......., the coefficients of y, z, &c. being all zero by Art. 420. and it will be found that each determinant is Ex. 2. Solve the equations -20, so that and the values of y, z and w can be written down from that of x. Multiply the equations in order by C1, C2, C3, the co-factors of c1, C, C respectively in the determinant Then by addition we have Δ = b1 a ba ca which is the required condition. The three homogeneous equations ax+by+c1z=аqx+by+CgZ =α3x+by+C3Z=0 are obviously satisfied by the values x=y=z=0. If however x, y, z are not all zero, it follows from the above that the condition [ab2c3]=0 must hold good. It can be shewn in a similar manner that the condition that n equations of the form ax+by+...+k1 = 0, with (n - 1) unknown quantities, may be simultaneously true is [a,b,c,... k2] = 0. 429. Sylvester's method of Elimination. This is a method by which x can be eliminated from any two rational and integral equations in x. The method will be understood from the following examples. Ex. 1. Eliminate x from the equations and ax2+ bx+c=0 and px2+qx+r=0. From the given equations we have ax3+ bx2+cx =0, ax2+ bx + c = 0, px3+qx2+rx =0, px2+qx+r=0. Now we may consider the different powers of x as so many different unknown quantities; and the result of eliminating x3, x2 and x from the four last equations is by Art. 428 [This result is equivalent to that obtained in Art. 153, Ex. 3.] Ex. 2. Eliminate x from the equations ax3+bx2+cx+d=0 and px2+qx+r=0. From the given equations we have Eliminating x, x3, x2, x from the five last equations as if the different powers of x were so many different unknown quantities, we have the condition |