Results to assist in the application of the ellipsoidal bounds provided by standard state-bounding algorithms are presented. They include derived bounds on scalar state-dependent quantities and the state values which determine them, tests of intersection and inclusion of ellipsoids, measures of how much an ellipsoid may be changed without altering its inclusion in another, and an ellipsoidal inner bound for the set reachable in the worst case from an ellipsoidal set by ellipsoidally bounded forcing, in a linear system. Approximations are suggested for the most computationally demanding result. Ways in which these results might be employed in aerospace interception problems are discussed to illustrate their utility.