| 1 | , |
| (1 - s1X1) (1 - s2X2) ... (1 - snXn) |
which is a function of the products s1x1, s2x2, s3x3, ... snxn only is
| 1 |
| |(1 - a11s1x1) (1 - a22s2x2) (1 - a33s3x3) ... (1 - annsnxn)| |
where the denominator is in a symbolic form and denotes on expansion
1 - Σ |a11|s1x1 + Σ |a11a22|s1s2x1x2 - ... + (-)n |a11a22a33 ... ann| s1s2 ... snx1x2 ... xn,
where |a11|, |a11a22|, ... |a11a22, ... ann| denote the several co-axial minors of the determinant
|a11a22 ... ann|
of the matrix. (For the proof of this theorem see MacMahon, “A certain Class of Generating Functions in the Theory of Numbers,” Phil. Trans. R. S. vol. clxxxv. A, 1894). It follows that the coefficient of
xξ11 xξ22 ... xξnn