Hadamard original notation for global inversion theorem

Question

I'm reading the original article of J. Hadamard: "Sur les transformations ponctuelles" (1906). He considers "la plus petite valeur du rapport" (the minimum value of the ratio): $\sqrt{\frac{dX_1^2+dX_2^2+\cdots+dX_n^2}{dx_1^2+dx_2^2+\cdots dx_n^2}}$. Here $X_i=f_i(x_1,x_2,\cdots x_n)$ for $i=1,2,\dots, n$. How can I ``translate'' this in modern notation? I know that this minimum value is $\|Jf(x)^{-1}\|^{-1}$ (modern literature), but I want to know how exactly was this quantity established by Hadamard.