2.) AJ being equal to BJ, A will still cover every thing in rear of DC.
(Fig.

3.) If the army A is obliged to cover the point _a_, the army B will cover all the space without the circle whose radius is _a_ B; and of course A continues to cover the point _a_ so long as it remains within this circle _a_ B.
_A line of operations_ embraces that portion of the theatre of war which an army or _corps d'armee_ passes over in attaining its object; _the front of operations_ is the front formed by the army as it advances on this line.
When an army acts as a single mass, without forming independent corps, the line it follows is denominated a _simple line of operations_.
If two or more corps act in an isolated manner, but against the same opposing force, they are said to follow _double_ or _multiple lines_.
The lines by which Moreau and Jourdan entered Germany in 1796, were double lines; but Napoleon's advance by Bamberg and Gera, in 1806, although moving in seven distinct _corps d'armee,_ formed but a single line of operations.
_Interior lines of operations_ are those followed by an army which operates between the enemy's lines in such a way as to be able to concentrate his forces on one of these lines before the other can be brought to its assistance.

For example, Napoleon's line of operations in 1814, between the Marne and the Seine, where he manoeuvred with so much skill and success against the immensely superior forces of the allies.
_Exterior lines_ present the opposite results; they are those which an army will form in moving on the extremities of the opposing masses.

For example, the lines of the Marne and the Seine, followed by the army of Silesia and the grand Austro-Russian army, in the campaign of 1814.
Burgoyne's line of operations, in 1777, was double and exterior.
_Concentric lines_ are such as start from distant points, and are directed towards the same object, either in the rear or in advance of their base.
If a mass leaves a single point and separates into several distinct corps, taking divergent directions, it is said to pursue _eccentric lines_.
Lines are said to be _deep_, when the end to be attained is very distant from the base.
The lines followed by a secondary or auxiliary force are denominated _secondary lines_.
The lines pursued by the army of the Sombre-et-Meuse in 1796, and by Bagration in 1812, were _secondary lines_, as the former were merely secondary to the army of the Rhine, and the latter to that of Barclay.
_Accidental lines_ are those which result from a change in the primitive plan of campaign, which give a new direction to the operations.

These are of rare occurrence, but they sometimes lead to important results.
The direction given to a line of operations depends not only on the geographical situation of the country, but also on the positions occupied by the enemy.