Matrix Differentiation of the Characteristic Function

The following work is a sequel to three previous communications, and more particularly to the first. The present object is to shew the effect of repeated operation with the matrix differential operator , when it acts upon a scalar matrix formed from an n rowed determinant | x ij |, or sums of principal minors, the n 2 elements x ij being treated as independent variables. Thus when z is a scalar quantity ω z means the matrix [∂z/∂x ij ], whose ij th element is the derivative.