Let f 1 , f 2 , … , f n f_1, f_2, \ldots, f_n f 1 , f 2 , … , f n g 1 , g 2 , … , g n g_1, g_2, \ldots, g_n g 1 , g 2 , … , g n 2 n 2n 2 n n − 1 n - 1 n − 1 ( x i ) 1 ≤ n (x_i)_1 \leq n ( x i ) 1 ≤ n
∣ f 1 ( x 1 ) f 1 ( x 2 ) ⋯ f 1 ( x n ) f 2 ( x 1 ) f 2 ( x 2 ) ⋯ f 2 ( x n ) ⋮ ⋮ ⋱ ⋮ f n ( x 1 ) f n ( x 2 ) ⋯ f n ( x n ) ∣
\begin{vmatrix}
f_1(x_1) & f_1(x_2) & \cdots & f_1(x_n) \\
f_2(x_1) & f_2(x_2) & \cdots & f_2(x_n) \\
\vdots & \vdots & \ddots & \vdots \\
f_n(x_1) & f_n(x_2) & \cdots & f_n(x_n)
\end{vmatrix}
f 1 ( x 1 ) f 2 ( x 1 ) ⋮ f n ( x 1 ) f 1 ( x 2 ) f 2 ( x 2 ) ⋮ f n ( x 2 ) ⋯ ⋯ ⋱ ⋯ f 1 ( x n ) f 2 ( x n ) ⋮ f n ( x n ) is equal to the ratio of the two determinants
∣ g 1 ( x 1 ) g 1 ( x 2 ) ⋯ g 1 ( x n ) g 2 ( x 1 ) g 2 ( x 2 ) ⋯ g 2 ( x n ) ⋮ ⋮ ⋱ ⋮ g n ( x 1 ) g n ( x 2 ) ⋯ g n ( x n ) ∣
\begin{vmatrix}
g_1(x_1) & g_1(x_2) & \cdots & g_1(x_n) \\
g_2(x_1) & g_2(x_2) & \cdots & g_2(x_n) \\
\vdots & \vdots & \ddots & \vdots \\
g_n(x_1) & g_n(x_2) & \cdots & g_n(x_n)
\end{vmatrix}
g 1 ( x 1 ) g 2 ( x 1 ) ⋮ g n ( x 1 ) g 1 ( x 2 ) g 2 ( x 2 ) ⋮ g n ( x 2 ) ⋯ ⋯ ⋱ ⋯ g 1 ( x n ) g 2 ( x n ) ⋮ g n ( x n ) ∣ f 1 ′ ( ξ 1 ) f 1 ′ ( ξ 2 ) ⋯ f 1 ( n − 1 ) ( ξ n ) f 2 ′ ( ξ 1 ) f 2 ′ ( ξ 2 ) ⋯ f 2 ( n − 1 ) ( ξ n ) ⋮ ⋮ ⋱ ⋮ f n ′ ( ξ 1 ) f n ′ ( ξ 2 ) ⋯ f n ( n − 1 ) ( ξ n ) ∣
\begin{vmatrix}
f_1'(\xi_1) & f_1'(\xi_2) & \cdots & f_1^{(n-1)}(\xi_n) \\
f_2'(\xi_1) & f_2'(\xi_2) & \cdots & f_2^{(n-1)}(\xi_n) \\
\vdots & \vdots & \ddots & \vdots \\
f_n'(\xi_1) & f_n'(\xi_2) & \cdots & f_n^{(n-1)}(\xi_n)
\end{vmatrix}
f 1 ′ ( ξ 1 ) f 2 ′ ( ξ 1 ) ⋮ f n ′ ( ξ 1 ) f 1 ′ ( ξ 2 ) f 2 ′ ( ξ 2 ) ⋮ f n ′ ( ξ 2 ) ⋯ ⋯ ⋱ ⋯ f 1 ( n − 1 ) ( ξ n ) f 2 ( n − 1 ) ( ξ n ) ⋮ f n ( n − 1 ) ( ξ n ) ∣ g 1 ′ ( ξ 1 ) g 1 ′ ( ξ 2 ) ⋯ g 1 ( n − 1 ) ( ξ n ) g 2 ′ ( ξ 1 ) g 2 ′ ( ξ 2 ) ⋯ g 2 ( n − 1 ) ( ξ n ) ⋮ ⋮ ⋱ ⋮ g n ′ ( ξ 1 ) g n ′ ( ξ 2 ) ⋯ g n ( n − 1 ) ( ξ n ) ∣
\begin{vmatrix}
g_1'(\xi_1) & g_1'(\xi_2) & \cdots & g_1^{(n-1)}(\xi_n) \\
g_2'(\xi_1) & g_2'(\xi_2) & \cdots & g_2^{(n-1)}(\xi_n) \\
\vdots & \vdots & \ddots & \vdots \\
g_n'(\xi_1) & g_n'(\xi_2) & \cdots & g_n^{(n-1)}(\xi_n)
\end{vmatrix}
g 1 ′ ( ξ 1 ) g 2 ′ ( ξ 1 ) ⋮ g n ′ ( ξ 1 ) g 1 ′ ( ξ 2 ) g 2 ′ ( ξ 2 ) ⋮ g n ′ ( ξ 2 ) ⋯ ⋯ ⋱ ⋯ g 1 ( n − 1 ) ( ξ n ) g 2 ( n − 1 ) ( ξ n ) ⋮ g n ( n − 1 ) ( ξ n ) where
ξ 1 = x 1 , ξ 1 < ξ 2 < x 2 , ξ 2 < ξ 3 < x 3 , … , ξ n − 1 < ξ n < x n
\xi_1 = x_1,\quad \xi_1 < \xi_2 < x_2,\quad \xi_2 < \xi_3 < x_3,\quad \ldots,\quad \xi_{n-1} < \xi_n < x_n
ξ 1 = x 1 , ξ 1 < ξ 2 < x 2 , ξ 2 < ξ 3 < x 3 , … , ξ n − 1 < ξ n < x n (apply exerc. 9 of I, p. 38).
Particular case where g 1 ( x ) = 1 , g 2 ( x ) = x , … , g n ( x ) = x n − 1 g_1(x) = 1, g_2(x) = x, \ldots, g_n(x) = x^{n-1} g 1 ( x ) = 1 , g 2 ( x ) = x , … , g n ( x ) = x n − 1