The three principal search curves are the meeting curve, the departure curve, and the arrival curve.
The meeting curve is a circle so constructed that the distance from two points to any point on the curve contains the same ratio; thus, in Fig. 1 the distance between A and B is 60 miles. In the first curve the vessel at A is supposed to move at the rate of 20 miles, and the vessel at B at the rate of 10 miles. If they both move toward each other they meet at a point 40 and 20 miles from A and B respectively. The distance of the center of the circle from the point of first meeting is found by the following formula:
MEETING CURVE.
V = speed of faster vessel.
V'= speed of slower vessel.
r = V’/V
D = distance between vessels.
a = distance run by faster vessel to point of meeting.
b = distance run by slower vessel to point of meeting.
c = distance of center of meeting curve from point of meeting.
a = D/(r+1) b = rD/(r+1) c = rD/(1-r2)
At every point on this circle the distance from B divided by the distance from A is equal to r.
By means of this circle, two vessels, knowing the distance apart and their relative speeds, can form a rendezvous, choosing the shortest, longest, or intermediate distances; and either vessel can select a point where he will meet the other if he knows its course. In Fig. 1 the curves are made for speeds for the vessel at B varying from 10 to 15 miles, with a uniform speed of 20 miles for the vessel at A.
The departure Curve furnishes the means for finding a vessel at a known distance away, the relative speeds of the vessels being known, and the course unknown, or only known within certain limits. Fig. 2 shows the departure curve, the distance between the vessels being 6o miles, the vessel at A having a speed of 20 miles, and the vessel at B having a speed varying from 10 to 15 miles. The first point in the curve is on the line joining AB and at the point where they would meet if steaming toward each other (a = D/(r+1). Each succeeding point (at intervals of 1 hour) is found by means of drawing a circle from B with a radius equal to the number of miles the vessel could steam in that time, and taking the point of intersection on this circle with a circle of a radius of 20 miles struck from the previous point.
It is seen from Fig. 2 that A starts to seek for B at a distance of 60 miles, with the relative speeds of 20 and 10 miles, if B moves toward A, A will meet B on a line joining AB, and at the end of the second hour. B taking any other course will be met by A at some point on the upper curve marked 3, 4, 5, 6, etc. (at the end of the third, fourth, etc., hours), and A can make a complete search of 180° in 12 hours. The nearer B's speed approaches that of A, the smaller the sector becomes that A can search in 12 hours; and with B going 15 miles, A can search but little over a 90° sector. With such a method of search there are too many chances of accidents to make it wise to continue the calculations over 12 hours under ordinary circumstances. The following will be seen in examining Fig. 2: With the distance equal to 6o miles and with the speed of the slower vessel of 10 miles, the searching vessel must have a speed of 20 miles or over unless the course of the vessel sought is known to be within less than 1800; and with the same distance, if the speed of the chase is 15 miles, the speed of the chaser must be 20 miles or over, unless the course of the chase is known within 900; and if the speed of the chase is known to be between 10 and 15 miles and the course is known within 90°, three vessels should be sufficient to conduct the search.
Fig. 3 shows the departure curve with the same relative speeds between the chase and chaser, but the distance apart is 90 miles. The other sectors covered by the chaser are much smaller, and it would require many more vessels to cover the same sector with the same uncertainties as to the speeds of the chase as mentioned before—that is, 10 to 15 miles. . Fig. 4 shows the departure curve with a distance of 120 miles between the two vessels. It shows the increased difficulties for the chaser. Fig. 5 shows the departure curve, the vessels being 120 miles apart and the relative speeds 10 and 20 miles. The middle curve is the one to be followed, the vessels both leaving their points of departure at the same time. The three upper curves are the ones to be followed if the chase leaves B 1, 2, or 3 hours later than the chaser leaves A, and the three lower curves are the ones to be followed if the chase leaves B 1, 2, or 3 hours earlier than the chaser leaves A. Should there be uncertainty as to the time when the chase takes its departure, the number of vessels necessary to conduct the search could be found from Fig. 5.
Fig. 6 shows the arrival curve. The distances between A and B are taken as before at 66, 90, and 120 miles, and the speeds of the chase as before at 10, 11, 12, 13, 14, and 15 miles, while the chaser is supposed to have the uniform speed of 20 miles. In this curve the speed of the chase is known, also the position of the point of arrival, with the time of intended arrival. In the curve constructed with V' = 10 and V = 20 and distance of 120 miles, the chaser knows that the chase intends to arrive at B at 6 P. M. and that he can steam at 10 miles. The chaser steaming at 20 miles desires to intercept him as early as possible and before he arrives at B. The vessel steaming for B may start for that point from any direction, but must leave some point 120 miles distant from B 12 hours before the time of his arrival, or at 6 A. M. The chaser leaving at 6 A. M. steers such courses as to intersect the 10-mile curves from B at each hour, and must therefore sight the chase at some point on his curve unless that vessel comes from a direction not covered by the curve. The chase having a speed of 11, 12, etc., miles, and starting from the same distance, must arrive in 120/11, 122/12, etc. hours.
The use of the departure and arrival curves is set forth by Commandants Z. and H. Montichant in their "Essai de Strategie Navale," and they use their theory most skillfully, if fallaciously, to prove the advantages of cruisers and torpedo-boats over battle-ships. They endeavor to show that the chance of one fleet keeping touch with another is so small that a weak fleet of cruisers would be as useful, if not better, than a weak fleet of battle-ships. They look upon a naval war as a great game of hide-and-seek, the stronger fleet being unable to accomplish anything as long as the weaker fleet is able to avoid combat. This is the "fleet in being" theory with a vengeance, and shows why that theory has so many opponents. The French naval writers have generally failed to understand the true theory of the "fleet in being," and have, therefore, been led to adopt the heresies of commerce-destroying and flotillas of torpedoboats. They would lock a battle fleet up in a fortress or force it to run away if the enemy's fleet were the stronger, failing utterly to grasp the real power of an active "fleet in being." A short study of the curves is sufficient to show that if the stronger fleet has nothing to do but to chase the weaker, the war may never end. In the same way one might suppose a strong well-found army rendered useless because unable to capture or destroy a small body of the enemy's cavalry; both suppositions are equally absurd. It is a sound maxim of strategy, that the objective of the fleet is the enemy's fleet; but this must not be carried to the absurd length of allowing the stronger fleet to remain idle because the weaker fleet has long legs. A naval war will be conducted with definite objects—primarily to obtain command of the sea; this insures safety to coast commerce, and renders the enemy's coast and commerce liable to attack, not by mere raids or by solitary commerce-destroyers, but by invasions in force and by extensive blockades. To be sure, command of the sea cannot be definitely secured if the enemy's fleet run away too fast to be brought to action, and the enemy may hold local command of the sea at some distance from your fleet; but if you have undertaken your plans with sufficient force, and they are well conceived, you can provide against any real damage from his fleet and proceed with the objects you have in view, forcing him to give up his game of hide-and-seek and to attack you or remain idle, or to attempt some raid that you can make most dangerous to him and which, if successful, would have no great effect upon the final results of the war. This is the full question of the "fleet in being." It forces you to make the attack with a force sufficiently large to neutralize his and to accomplish your main objects. To do this it must be active, it must meet you on chosen lines, forcing you to destroy it or to relinquish your object. The "fleet in being," while it is the weaker fleet, gains all the advantage of conducting the defensive, of secure bases, short lines of communication, prompt repairs, and the aid of the guns of the advance bases. In the end the war must be settled by the result of the appeal to the guns of the fleet, and not to their heels. While France cannot maintain a battle fleet sufficient to obtain command of the sea from England, it can maintain one that is sufficient to hold the local command of her own coasts, only it must be a real battle fleet, held to active work and not locked up in fortresses. She will throw away her best chances if she acknowledges England's superiority without fighting and ' relies upon coast-defense vessels, cruisers, and torpedo-boats.
While a naval war will not be a game of hide-and-seek, there will be many occasions when the knowledge of the enemy's movements will be most important. The junction of fleets may be prevented or facilitated, and all the questions of chase, scouts, and videttes are likely to arise. It is here that the search curves have their value; and while it is impossible to gain mathematical accuracy in chasing, owing to the many uncertainties that enter into the question, yet by properly utilizing these curves the route can be followed that gives the most chances of success, and the probabilities can be estimated for given conditions. A more complete search can be carried on with fewer chasers by using these curves than by proceeding according to guesswork. There are many times when it would be impossible to cover a large sector in the search, owing to great distance or small advantage in speed, and yet a complete search can be made because the probable sector including the enemy's course is small, and is known to be small owing to strategical conditions.
Rear Admiral C. Marchese, of the Italian Navy, made an elaborate attack in the Rivista Marittima on the search curves as set forth in the "Essai de Strategie Navale." The weight of his argument is that there will not be sufficient difference between the speeds of the two fleets to permit the use of the curves with advantage. No one would more quickly agree with this than the writers whose essay he attacks. They appreciate, and he appears to forget, that the chasers will be fast cruisers, destroyers, etc., when the chase may be a battle fleet, limited in speed by its slowest vessel and by the necessity of a long steaming radius. The chaser may have secure coaling bases close at hand, while the chase may have to fight before securing a base, and, therefore, forced to economize coal.
Admiral Marchese proposes another method of search, and does some little mathematical computations to analyze the meeting curve and illustrate his use of it. The meeting curve was analyzed by Lieutenant H. 0. Rittenhouse, U. S. Navy, in the Proceedings of the U. S. Naval Institute for 1885, twelve years before the appearance of Admiral Marchese's paper. The gist of the Admiral's proposition is that the enemy's course being known within certain limits, the chaser can set a course at an angle to the direction of the line joining the chase and the chaser, and that while he will pass through the possible positions of the chase only twice, the points where his course intersects the meeting curve, the average distance from the chase will be less than in the departure curve, and that under favorable conditions this average distance will be within the space covered by the outposts of the fleet. He fails to notice that the limitations are similar to those in the case of the departure curve, and unless the distance apart when the search commences is small and the superiority of the speed of the chaser is marked, the average distance may be too great for observation. Again, there are only two points on the meeting curve where the two vessels can meet, whereas every point on the departure curve is a point of possible meeting. Where there are so many possibilities of error it would seem better to steer for some point or points where, if no error is made, the vessels will actually meet, so that the circle of probable error surrounding this point can be estimated and be within the vision of the chaser, than to steer on a line where they will probably pass each other at a distance that may be greatly increased by the probable error. In other words, the probable error is great enough without adding to it an average distance.
Naturally a chasing fleet would steer at an angle to the direction joining its position with the chase, if the course of the chase is known within 180’, so that it would probably be nearer the chase when it (the chase) was discovered by the scouts than if it steered directly toward the first position. The course taken, the speed, and the time of departure of the chasing fleet will be known to its scouts, so that, if they find the chase, they can plot the position of their fleet, and then, by using the meeting curve, can carry the information to the fleet at the earliest practicable moment.
The search curve was attempted in the last English maneuvers and failed. The failure is generally ascribed to foggy weather. The information as to the maneuvers is not sufficiently exact to permit the departure curve, as used, being traced. The Blenheim was at Blacksod Bay, and left there for the rendezvous of her fleet about midnight of the 7th, steaming at the rate of 12 miles. The Powerful and Terrible were north and south of the bay, at distances stated of from 30 to 50 miles, and left in chase of the Blenheim two to four hours later at the rate of 17 miles. If they started just two hours after the Blenheim they would have searched 90° each and met at a point 168 miles to the westward of the bay, having steamed 204 miles each (see Fig. 7, upper curve). If they started four hours afterward they could not have completed the 90° curve in 12 hours. It would have taken about 15 hours, and they would have met at a point 228 miles to the westward of the bay, having steamed about 255 miles. From the various reports it appears probable that the Powerful, being 50 miles north of Blacksod Bay, started on the search at 4 A. M., four hours after the Blenheim left the bay, and continued the search until 7 P. M., and having then completed the search of a 90° sector, without sighting the Blenheim or Terrible, steamed to the rendezvous of her own fleet, which she rejoined about 7 A. M. the following morning. The Terrible, from the south, was unable to keep up her speed of 17 knots and joined her fleet some time after the Powerful. If such were the case, the search for the Blenheim might have failed even in clear weather, as there were too many chances of error during so long a run as that of the Powerful to admit of certainty, and without error the Blenheim might have passed out of the sector searched by the Terrible, ahead of her and out of sight of the Powerful.
By consulting the several departure curves in the various figures it will be seen that uncertainty as to time of the departure of the chase will greatly decrease the chances of success of the chaser; and if the sector to be searched is large, more than one vessel must conduct the search and over different curves, the number depending on the amount of the uncertainty. But it would seem as if the time of the departure of the chase would be known within a reasonable degree of accuracy if the fact of her departure was known at all. Of course it must be assumed that the chase proceeds over a straight course, without dodging, otherwise the theory of search falls to the ground and the whole search becomes a matter of guesswork. But the speed of the chase may well be a subject of uncertainty, although it would ordinarily be known within moderate limits. There are two ways by which the search can be made more complete with an additional number of cruisers. One way is to send each chaser over a different curve, assuming 'different speeds for the chase, and the other to send them all over the same curve at certain intervals apart. The latter method has the advantage of keeping your cruisers in hand, as they can readily reassemble, can be informed if the chase is discovered, and can keep touch as well as send information to the fleet of the discovery; and the accuracy of the search, owing to the intervals being preserved, is fairly uniform, whereas in the first method the intervals between the vessels increase with the length of the search. In chasing over the same curve it must be the curve required by the fastest chase. It is easy to find the number of chasers necessary to make the search reasonably thorough. If the search is to continue 12 hours and the speeds of the chase are to be considered between 12 and 15 knots, the number of vessels required would be four, for the chase can cover 18o miles in 12 hours at 15 knots, and it would take 15 hours at 12 knots, so that to sight the 12-knot vessel and all intermediate speeds a chaser must pass the same spot at 13, 14, and 15 hours after starting. If the search is to continue only 6 hours, four chasers will be sufficient to search for all speeds between 15 and 10 knots, for the chaser will cover 90 miles in 6 hours at 15 knots and it will take 9 hours at 10 knots.
Radial lines have been used more frequently than any other plan of search, except the haphazard method, which is endeared to some because it cannot be accused of being founded upon geometry or any other means by which past human experience is utilized for future benefit.
Radial lines can be used to advantage when the distance between chase and chaser is small and the difference in speeds is large. This can be readily seen by reference to Fig. 1, with D = 60, V = 20, V’ = 10 or less. By radial lines three cruisers could make a close search during daytime and in good weather and complete the search in about 6 hours, gaining 6 hours over the method by departure curve. This is similar to Admiral Marchese's method. But the numbers required to conduct the search by radial lines and the time necessary to complete it increase rapidly with an increase in distance or decrease in difference in speeds. So that, except for extreme cases, radial lines require more chasers and give less chances of success than departure curves.
Many examples of the use of the departure curve could be shown. Fig. 8 illustrates the departure curves drawn for various conditions of time for departure and speed of the enemy with our scouts' speed at 20 knots. In one case information is received by our scouts off Cape Henry that the enemy has left Delaware Bay and is just passing Five Fathom Bank Lightship; in the other our scouts receive the information off Chincoteague that the enemy has left the Chesapeake and is 17 miles to the southward of Cape Henry. In place of scouts our chasers might be a fleet of destroyers, which, if they sighted the enemy in the daytime, might retain touch and strike him at night.
The arrival curve is more limited in its application, and it would be used most frequently by scouts wishing to join the main fleet as early as possible and knowing the general direction from which the fleet is steering at a given speed toward a certain rendezvous. Commandants Z. and H. Montichant, in La Marine Francaise for December, 1895, give a further example of the use of the search curve, in which they employ four cruisers that proceed to the point of meeting in line, with intervals of 10 miles between cruisers, when the cruiser farthest from the diameter strikes the meeting curve they form column and follow the departure curve. They appear to have failed to account for the time required to deploy 40 miles. With a division already deployed in the right direction this method might prove useful.
To understand the theory or the practice of these curves requires little knowledge of mathematics, and they can be drawn on the chart by any one who has sufficient knowledge to enable him to plot his course. While in theory they are accurate, the many conditions that may be uncertain will cause the chaser using them to fail frequently in practice. But they offer the best chances of success with the smallest number of ships when used with judgment. The chances of success can be fairly well estimated, and under many circumstances likely to arise in time of war success can be practically assured.