By combining, as above, Hamilton's linear and vector-function with his celebrated vector square-root of the negative of Laplace's operator, an operator of great use in physical applications of mathematics is obtained. With the notation employed in the author's paper “On Green's and other Allied Theorems,” Trans. R.S.E. 1870, § 17, it is shown to be generally expressible in the form of

where α,β,γ, are any three unit vectors (not necessarily rectangular), and α1,β1,γ1 any three vectors whatever.