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yl

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).

00ï¿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 .

00zl, z2,...zn) yl, y2,.

00x 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.

00We 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.

00yl, y 2,...yn) (zl, z2,...zn z1, z 2, ï¿½ï¿½ï¿½zn xi, 'X' 2,...

00y m (xly2 - x2y1) 2 (x0,3 - x 3 yl) 2...

00Moreover, instead of having one pair of variables x i, x2 we may have several pairs yl, y2; z i, z2;...

00y1 = x 15+f2n; fï¿½ y2 =x2-f?n, f .a b = ax+ (a f) n, l; n u 2 " 2 22 2 +` n) u3 n-3n3+...+U 2jnï¿½ 3 n Now a covariant of ax =f is obtained from the similar covariant of ab by writing therein x i, x 2, for yl, y2, and, since y?, Y2 have been linearly transformed to and n, it is merely necessary to form the covariants in respect of the form (u1E+u2n) n, and then division, by the proper power of f, gives the covariant in question as a function of f, u0 = I, u2, u3,...un.

00If yl is known this gives 72.

00lh), which in the present orthography is written ii as in Castilian, but formerly used to be represented by iy or yl (Iletra, Ii t e r a iiengua, 1 i n g u a).

00Chem., 1832, 3, p. 2 49) is to be regarded as a most important contribution to the radical theory, for it was shown that a radical containing the elements carbon, hydrogen and oxygen, which they named benzoyl (the termination yl coming from the Gr.

00Suppose n dependent variables yl, y2,Ã¯¿½Ã¯¿½Ã¯¿½yn, each of which is a function of n independent variables x1, x2 i Ã¯¿½Ã¯¿½Ã¯¿½xn, so that y s = f s (x i, x 2, ...x n).

00Ã¯¿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,Ã¯¿½Ã¯¿½Ã¯¿½xn), and we may consider the three determinants which i s 7xk, the partial differential coefficient of z i, with regard to k .

00zl, z2,...zn) yl, y2,.

00x n / I l yl, y2,Ã¯¿½Ã¯¿½Ã¯¿½yn Theorem.-If the functions y 1, y2,Ã¯¿½Ã¯¿½Ã¯¿½ y n be not independent of one another the functional determinant vanishes, and conversely if the determinant vanishes, yl, Y2, ...y.

00We can solve these, assuming them independent, for the - i ratios yl, y2,...yn-iÃ¯¿½ Now a21A11 +a22Al2 Ã¯¿½ Ã¯¿½ Ã¯¿½ = 0 a31A11+a32Al2 +Ã¯¿½ Ã¯¿½Ã¯¿½ +a3nAln = 0 an1Al1+an2Al2 +Ã¯¿½Ã¯¿½Ã¯¿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.

00yl, y 2,...yn) (zl, z2,...zn z1, z 2, Ã¯¿½Ã¯¿½Ã¯¿½zn xi, 'X' 2,...

00x n/ yl, Y2,...y n j ' x 1, Ã¯¿½ Forming the product of the first two by the product theorem, we obtain for the element in the ith row and kth column aZ, ayl az i ayz azi ayn ayl + e +...+ where or as a21 a22 Ã¯¿½Ã¯¿½Ã¯¿½a2,i -1 a2,i +1 .Ã¯¿½Ã¯¿½a2n a31 Ã¯¿½Ã¯¿½Ã¯¿½a3,i -1 a3,ti+ 1 Ã¯¿½Ã¯¿½Ã¯¿½a3n Ã¯¿½Ã¯¿½Ã¯¿½yi -)tin,and a7,2 Ã¯¿½Ã¯¿½Ã¯¿½a,,,i -1 an,i+1 ...anÃ¯¿½' a21 a22 Ã¯¿½Ã¯¿½Ã¯¿½a2, -1 a32 -1 I.

00y m (xly2 - x2y1) 2 (x0,3 - x 3 yl) 2...

00Moreover, instead of having one pair of variables x i, x2 we may have several pairs yl, y2; z i, z2;...

00y1 = x 15+f2n; fÃ¯¿½ y2 =x2-f?n, f .a b = ax+ (a f) n, l; n u 2 " 2 22 2 +` n) u3 n-3n3+...+U 2jnÃ¯¿½ 3 n Now a covariant of ax =f is obtained from the similar covariant of ab by writing therein x i, x 2, for yl, y2, and, since y?, Y2 have been linearly transformed to and n, it is merely necessary to form the covariants in respect of the form (u1E+u2n) n, and then division, by the proper power of f, gives the covariant in question as a function of f, u0 = I, u2, u3,...un.

00If yl is known this gives 72.

00lh), which in the present orthography is written ii as in Castilian, but formerly used to be represented by iy or yl (Iletra, Ii t e r a iiengua, 1 i n g u a).

00

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