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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).

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.

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

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

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

y1 = 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.

If yl is known this gives 72.

lh), 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).

Chem., 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.

Suppose 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).

Ã¯¿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 .

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

x 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.

We 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.

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

x 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.

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

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

y1 = 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.

If yl is known this gives 72.

lh), 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).

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