_Interior lines of operations_ are those adopted by one or two armies to oppose several hostile bodies, and having such a direction that the general can concentrate the masses and maneuver with his whole force in a shorter period of time than it would require for the enemy to oppose to them a greater force.[12] _Exterior lines_ lead to the opposite result, and are those formed by an army which operates at the same time on both flanks of the enemy, or against several of his masses. _Concentric lines of operations_ are those which depart from widely-separated points and meet at the same point, either in advance of or behind the base. _Divergent lines_ are those by which an army would leave a given point to move upon several distinct points. These lines, of course, necessitate a subdivision of the army. There are also _deep lines_, which are simply _long lines_. The term _maneuver-lines_ I apply to momentary strategic lines, often adopted for a single temporary maneuver, and which are by no means to be confounded with the real _lines of operations_. _Secondary lines_ are those of two armies acting so as to afford each other mutual support,--as, in 1796, the army of the Sambre and Meuse was secondary to the army of the Rhine, and, in 1812, the army of Bagration was secondary to that of Barclay. _Accidental lines_ are those brought about by events which change the original plan and give a new direction to operations. These are of the highest importance. The proper occasions for their use are fully recognized only by a great and active mind. There may be, in addition, _provisional_ and _definitive lines of operations_. The first designate the line adopted by an army in a preliminary, decisive enterprise, after which it is at liberty to select a more advantageous or direct line. They seem to belong as much to the class of temporary or eventual strategic lines as to the class of lines of operations. These definitions show how I differ from those authors who have preceded me. Lloyd and Bulow attribute to these lines no other importance than that arising from their relations to the depots of the army: the latter has even asserted that when an army is encamped near its depots it has no lines of operations. The following example will disprove this paradox. Let us suppose two armies, the first on the Upper Rhine, the second in advance of Dusseldorf or any other point of this frontier, and that their large depots are immediately behind the river,--certainly the safest, nearest, and most advantageous position for them which could possibly be adopted. These armies will have an offensive or defensive object: hence they will certainly have lines of operations, arising from the different proposed enterprises.