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.  The general plan of campaign is frequently determined on previous to beginning operations, but the choice of lines and positions must ordinarily result from the ulterior events of the war, and be made by the general as these events occur.

As a general rule, a line of operations should be directed upon the centre, or one of the extremities of the enemy’s line of defence; unless our forces be infinitely superior in number, it would be absurd to act against the front and extremities at the same time.

If the configuration of the theatre of operations be favorable to a movement against the extremity of the enemy’s line of defence, this direction maybe best calculated to lead to important results. (Fig. 4.)

In 1800 the army of the Rhine was directed against the extreme left of the line of the Black Forest; the army of reserve was directed by the St. Bernard and Milan on the extreme right and rear of Melas’s line of defence:  both operations were most eminently successful. (Fig. 5.)

It may be well to remark that it is not enough merely to gain the extremity and rear of the enemy, for in that case it may be possible for him to throw himself on our communications and place us in the very dilemma in which we had hoped to involve him.  To avoid this danger it is necessary to give such a direction to the line of operations that our army shall preserve its communications and be able to reach its base.

Thus, if Napoleon, in 1800, after crossing the Alps, had marched by Turin on Alexandria and received battle at Marengo, without having first secured Lombardy and the left of the Po, his own line of retreat would have been completely cut off by Melas; whereas, by the direction which he gave to his line of operations he had, in case of reverse, every means for reaching either the Var or the Valois. (Fig. 6.) Again, in 1806, if he had marched directly from Gera to Leipsic, he would have been cut off from his base on the Rhine; whereas, by turning from Gera towards Weimar, he not only cut off the Prussians from the Elbe, but at the same time secured to himself the roads of Saalfield, Schleitz, and Hoff, thus rendering perfectly safe his communications in his rear.  (Fig. 7.)