On a Jacobian identity associated with real hyperplane arrangements

For x is an element of (a(j-1), a(j)) (j = 1,..., p + 1; a(0) := -infinity, a(p+1) := infinity) the mapping T-j : w = x- Sigma(l=1)(p) lambda(l)/(x - a(l)) (lambda(l) > 0, a(l) is an element of R) is onto R. It was shown by G . Boole in the 1850's that Sigma(j=1)(p+1)[(partial derivative w/partial de rivative x)(-1) ](x=Tj-1(w)) = 1. We give an n-dimensional analogue of this result. The proof makes use of the Griffiths-Harris residue theorem from a lgebraic geometry.