will be engaged all the time. It only works where each unit (soldier, ship, etc.) can kill only one equivalent unit at a time. For this reason, the law does not apply to machine guns, artillery with unguided munitions, or nuclear weapons. The law requires an assumption that casualties accumulate over time: it does not work in situations in which opposing troops kill each other instantly, either by shooting simultaneously or by one side getting off the first shot and inflicting multiple casualties. Note that Lanchester's square law does not apply to technological force, only numerical force; so it requires an N-squared-fold increase in quality to compensate for an N-fold decrease in quantity. Example equations Suppose that two armies, Red and Blue, are engaging each other in combat. Red is shooting a continuous stream of bullets at Blue. Meanwhile, Blue is shooting a continuous stream of bullets at Red. Let symbol A represent the number of soldiers in the Red force. Each one has offensive firepower α, which is the number of enemy soldiers it can incapacitate (e.g., kill or injure) per unit time. Likewise, Blue has B soldiers, each with offensive firepower β. Lanchester's square law calculates the number of soldiers lost on each side using the following pair of equations. Here, dA/dt represents the rate at which the number of Red soldiers is changing at a particular instant. A negative value indicates the loss of soldiers. Similarly, dB/dt represents the rate of change of the number of Blue soldiers. The solution to these equations shows that: If α=β, i.e. the two sides have equal firepower, the side with more soldiers at the beginning of the battle will win; If A=B, i.e. the two sides have equal numbers of soldiers, the side with greater firepower will win; If A>B and α>β, then Red will win, while if A<B and α<β, Blue will win; If A>B but α<β, or A<B but α>β, the winning side will depend on whether the ratio of β/α is greater or less than the square of the ratio of A/B. Thus, if numbers and firepower are unequal in opposite directions, a superiority in firepower equal to the square of the inferiority in numbers is required for victory; or, to put it another way, the effectiveness of the army rises proportionate to the square of the number of people in it, but only linearly with their