Suppose n dependent variables yl, y2,ï¿½ï¿½ï¿½yn, each of which is a function of n independent variables x1, x2 i ï¿½ï¿½ï¿½xn, so that y s = f s (x i, x 2, ...x n).
ï¿½ Oxl d 2x 77n If we have new variables z such that zs=4s(yl, Y2,...yn), we have also z s =1 Y 8(x1, x2,ï¿½ï¿½ï¿½xn), and we may consider the three determinants which i s 7xk, the partial differential coefficient of z i, with regard to k .
Zl, z2,...zn) yl, y2,.
X n / I l yl, y2,ï¿½ï¿½ï¿½yn Theorem.-If the functions y 1, y2,ï¿½ï¿½ï¿½ y n be not independent of one another the functional determinant vanishes, and conversely if the determinant vanishes, yl, Y2, ...y.
We can solve these, assuming them independent, for the - i ratios yl, y2,...yn-iï¿½ Now a21A11 +a22Al2 ï¿½ ï¿½ ï¿½ = 0 a31A11+a32Al2 +ï¿½ ï¿½ï¿½ +a3nAln = 0 an1Al1+an2Al2 +ï¿½ï¿½ï¿½+annAln =0, and therefore, by comparison with the given equations, x i = pA11, where p is an arbitrary factor which remains constant as i varies.