Lanchester-Type Equations of Combat which Account for Multiple Killing Hits

Differential equations are derived for “aimed-fire” combat losses with multiple killing hits taken into account. The multiple killing hits arise because more than one combatant may aim at the same target. The exchanged fire may occur (1) in salvos and (2) random in time. Effects of supporting fires are included. Conditions for equally balanced forces are obtained. The equations in general differ from the usual Lanchester equations, but they do reduce to the Lanchester equations if the single shot probability of kill is small. The force balance equation for random fire does not depend on the time the weapons spend in transit to the target, but it does depend on the fraction of time during which an unfired weapon is committed to a particular target so that the aim cannot be shifted even if the target is killed, and the weapon is launched against the killed target anyway.