Two related ways to invariantly study higher derivatives, new-tensors and derivative strings, are compared, and the idea of suppression operator, or multiplier, is introduced. It renders the formal expression of the new objects almost as simple as that of ordinary tensors. Jacobians as a subgroup of the full second-order group are identified, and the harmonic subgroup leaving Laplace's equation invariant is determined. The wider principle of general covariance entailed by the full group is considered. In particular, electrogravitic potentials are defined, and the exploration of their field theory begun.