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§ 28 of the red. Alternatively q represents numerically a second "red" army of the strength necessary in a separate action to place the red forces on terms of equality, as in Fig. 5b.

§ 28. A Numerical Example. As an example of the above, let us assume an army of 50,000 giving battle in turn to two armies of 40,000 and 30,000 respectively, equally well armed; then the strengths are equal, since (50,000)2 = (40,000)2 + (30,000)2. If, on the other hand, the two smaller armies are given time to effect a junction, then the army of 50,000 will be overwhelmed, for the fighting strength of the opposing force, 70,000 is no longer equal, but is in fact nearly twice as great—namely, in the relation ofnb49 to 25. Superior morale or better tactics or a hundred and one other extraneous causes may intervene in practice to modify the issue, but this does not invalidate the mathematical statement.

§ 29. Example Involving Weapons of Different Effective Value. Let us now take an example in which a difference in the fighting value of the unit is a factor. We will assume that, as a matter of experiment, one man employing a machine-gun can punish a target to the same extent in a given time as sixteen riflemen. What is the number of men armed with the machine gun necessary to replace a battalion a thousand strong in the field? Taking the fighting value of a rifleman as unity, let n = the number required. The fighting strength of the battalion is, (1,000)2 or,

or one quarter the number of the opposing force.

This example is instructive; it exhibits at once the utility and weakness of the method. The basic assumption is that the fire of each force is definitely concentrated