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§ 20 men meet a "Red" force of 1,000 men in a single pitched battle, or whether the whole " Blue " force concentrates on 500 of the "Red" force, and, having annihilated them, turns its attention to the other half; there will, presuming the "Reds" stand their ground to the last, be half the "Blue" force wiped out in the annihilation of the "Red" force in the first battle, and the second battle will start on terms of equality—i.e., 500 "Blue" against 500 "Red."

§ 21. Modern Conditions Investigated. Now let us take the modern conditions. If, again, we assume equal individual fighting value, and the combatants otherwise (as to "cover," etc.) on terms of equality, each man will in a given time score, on an average, a certain number of hits that are effective; consequently, the number of men knocked out per unit time will be directly proportional to the numerical strength of the opposing force. Putting this in mathematical language, and employing symbol b to represent the numerical strength of the "Blue" force, and r for the "Red," we have:—


 * $db/dt$ = -r × c (1)

and


 * $dr/dt$ = -b × k (2)

in which t is time and c and k are constants {c = k if the fighting values of the individual units of the force are equal). The reduction of strength of the two forces may be represented by two conjugate curves following the above equations. In Fig. 2 {a) graphs are given representing the case of the "Blue" force 1,000 strong encountering a section of the "Red" force 500 strong, and it will be seen that the "Red" force is wiped out of existence with