The laws of probability and chance, as expounded in a treatise on mathematics, are based upon conditions which never actually obtain in connection with gun fire on board ship but, nevertheless, these laws have a definite bearing on salvo dispersion and it is essential that the officer controlling a battery firing at long range have a detailed knowledge of these laws and appreciate the effect of natural dispersion oh the spread of salvos.
If it were possible to construct a perfectly designed gun, firing perfect ammunition, controlled by perfect equipment, and to eliminate the atmospheric and other outside influences on the flight of projectiles, the laws of probability and chance would play little part in practical gunnery. Successive shots from such equipment would fall in the same place and gunnery would be reduced to an exact science.
This is not the case, however. Perfect equipment cannot be made, and successive shots fired from the same gun at the Naval Proving Grounds under the most nearly similar conditions that can be produced do not fall in the same place but are seen to move, both in range and deflection, forming a pattern, the size of which depends upon the design of gun and projectile, the consistency of the propellant charge and the variation in atmospheric effect upon the flight of the projectile. If a great number of shots are fired, under as nearly as possible the same circumstances, it will be found that the impacts are grouped closely together around one point and are more and more widely separated as the distance from that point increases.
The point about which the impacts are grouped is called the mean point of impact and its co-ordinates are the average co-ordinates of all the impacts. The deviations of projectiles from their mean point of impact are closely analogous to what are called "accidental errors" and they obey the same laws. Small deviations are more frequent than large ones; positive and negative deviations are equally probable, and are therefore equally frequent if the number of shots is great; very large deviations are not to be expected at all and when one occurs it is frequently due to some avoidable mistake. The average of the deviations from the M. P. I. of a large number of impacts is called the true mean error of the gun.
In practical gunnery, we are interested only in the effect of these deviations on the dispersion of shots in a salvo and in the variations which occur in the mean points of impact of successive salvos. When we consider the deviation of projectiles we are not dealing with definite numbers (as in dice throwing), but with values which may be anything whatever between certain limits and we cannot assign any finite measure to the probability that a deviation shall have a definite value because the number of values it may have is unlimited. We can, however, measure the probability that a deviation will fall between certain limits or that it will be greater, or less, than an assigned quantity.
If a large number of shots (y) are fired under the same circumstances and (m) shots are observed to fall 100 yards from the M. P. I., while the remainder (y—m) fall more than 100 yards from that point, we say that the probability that a shot will fall within 100 yards of the M. P. I. is m/y and the probability that a shot will be more than 100 yards away is Y—m/y. The probability that a shot will be less than 100 yards or more than 1000 yards from the M. P. I. is m/y + y—m/y =1; that is, it is a certainty.
The probability curve is plotted with distances from the M. P. I. as abscissæ and the numbers of shots within those distances as ordinates, and, when the total number of shots is infinite, may be considered to consist of an infinite number of rectangles whose width is reduced to the infinitesimal dz. The width of each rectangle becomes the elementary area ydz and the whole area
thus obviating the necessity of dividing a partial area by the whole area whenever a probability is to be computed.
The plotted form wh.ch this curve takes is illustrated in the accompanying pattern analysis diagram, in which a true mean error of 200 yards has been assumed in order to bring out clearly the points illustrated. The curve is built up of a series of rectangles whose bases are the distances every 50 yards (one-fourth the error assumed) on each side of the M. P. I. and whose heights are the percentages of shots to be expected within those limits. The figures written in the corners of the rectangles represent the additional percentages for the additional 50 yards distant from the M. P. I. and are additive as the distance from the M. P. I. increases. This diagram also shows the maximum spread and maximum movement of the M. P. I. to be expected for 12-gun, 10-gun and 6-gun salvos, which will be discussed later. It also shows the 50 per cent zone for an infinite number of shots.
The probability that a single shot will fall more than four times the mean error from the M. P. I. is very small, and such rounds are very likely to have been fired under circumstances dissimilar to those of other rounds. The pattern, therefore, may be considered as eight times the mean dispersion. The probability that the deviation of any shot will be numerically less than any given quantity is
where a is the limiting distance from the M. P. I. Values of P,. with the ratio a/D as the argument, have been calculated and tabulated in Alger as follows, so that it is only necessary to know the mean dispersion to pick out the probability of a single shot falling within certain limits.
As an illustration of the use of this table, taking a=200 yards and D=100 yards, we find that for a/D = 200/100 =2, P= .889; that is, for a gun whose mean error is 100 yards the probability of a single shot falling within200 yards of the M. P. I. is .889. If P is the probability that the deviation of a single shot will not be greater than a, then evidently 100 P will be the probable number of shots out of 100 which will fall within the limits ±a; in other words, 100 P is the percentage of hits to be expected in an area 2a wide with its center at the M. P. I. Thus, we see that the area in which 25 per cent of all shots may be expected to fall is ±.4D; the 50 per cent zone is ±.846D; the 75 per cent zone is ±1.445D. The half-width of the 50 per cent zone, .846D, is the probable deviation since it is the deviation which is just as likely to be exceeded as it is not to be exceeded.
If the M. P. I. falls a distance d from the center of the danger space of a target whose danger space is equal to S, then the percentage of hits to be expected from an infinite number of shots is:
on a target whose danger space is 100 yards, if the M. P. I. falls 200 yards from the center of the danger space, the mean error of the gun being considered to be 100 yards. The same percentage of hits would be made whether the M. P. I. fell over or short.
The following table shows the percentage of hits to be expected, on a target whose danger space is 100 yards, from a gun whose true mean error is 100 yards for different locations of the M. P. I.:
It will be noted that the percentage of hits decreases rapidly as the M. P. I. is moved away from the center of the danger space. This reduction takes place more rapidly in the case of a pattern resulting from a small mean error than in the case of a pattern due to a large mean error. The following table shows the percentages of hits to be expected on a 100-yard danger space from an infinite number of shots fired from guns having mean errors of from 50 yards to 250 yards where the M. P. I.'s are various distances from the center of the danger space. If the fire cannot be regulated to bring the M. P. I. on the center of the danger space, it may be detrimental to have too small a mean error. From examination of the table we find that, if the M. P. I. is 50 yards from the center of the danger space, a gun with a 50-yard mean error should get about twice as many hits as a gun with a mean error of 150 yards. If, however, the regulation of fire brings the M. P. I. within only 200 yards of the center of the danger space, the gun with a 50-yard error gets practically no hits, while the gun having a mean error of 150 yards should get about 12 per cent hits. The most efficient mean error, and therefore pattern, is directly dependent upon the ability to control the fire; that is, upon the relative positions of the M. P. I. and the center of the danger space of the target.
Heretofore, we have considered only shots fired from a single gun. If several shots could be fired at the same time from one gun or if a salvo is fired from several exactly similar guns, the spread of the salvo, or the pattern, will have a certain relation to the number of shots in the salvo and to the mean error of the gun. The probability that a 12-gun salvo will make as large a pattern as an infinite number of shots is very small. The probable salvo limits are the limits outside of which no shot will probably fall. If the total probability outside of such limits is therefore unity, or less, it must be I/n for each of n shots in a salvo. The probability of falling within these limits is therefore
If only a few shots are fired in a salvo, the M. P. I. of those shots does not necessarily coincide with the M. P. I. of an infinite number of shots and the M. P. I.'s of successive salvos would shift back and forth. The greater the number of shots in a salvo, the more nearly we may expect the M. P. I. of the salvo to coincide with the M. P. I. of an infinite number of shots. The error of the M. P. I. of a salvo varies inversely as the square root of the number of shots in the salvo and we have
The following table shows the patterns and movements of M. P. I. to be expected when salvos containing various numbers of shots are fired:
The application of the foregoing theory to the firing of a battleship at sea is rather difficult and we are reduced to considering probabilities. The main battery of a dreadnought consists of from 8 to 12 different guns, in as many different mounts, in from 4 to 6 double or triple turrets with as many different foundations, and the whole structure of the ship is subject to hogging, and bending, so that the shots are not fired under "similar circumstances." Also errors are introduced due to the oscillation of the guns about the center of gravity of the ship in rolling and pitching; sight setting, gun laying, ramming of projectiles, different temperatures of powder, different resistances in firing circuits and differences in bore-sighting all produce errors; and, finally, the errors of spotting and fire control take the M. P. I. off of the target. In addition, we should not forget that the number of shots fired is comparatively small and our reasoning must be based on the expected performance for only a few shots, not on the exact figures of the laws of probability and chance which are based on an infinite number of exactly similar rounds.
When a 12-gun salvo is fired from a ship, we know (assuming a mean error of 100 yards for the gun) that the maximum pattern to be expected from gun errors alone is 766 yards and the average 12-gun pattern should be 435 yards if no preventable errors (as distinguished from accidental errors) are present. If the spread of the salvo is 1000 yards, at least 234 yards (1000-766) and probably more of this spread is due to what we have called preventable errors and, if a large number of such salvos were considered, we would expect that 565 yards of this error was due to that cause, only 435 yards being due to accidental errors. We further find from the table that accidental errors may cause the M. P. I. of successive 12-gun salvos to move as much as 232 yards (± 116 yards). Any movement greater than this is surely due to preventable errors, while a portion of this movement, the exact amount of which is indeterminate, may also be due to that cause.
A thorough understanding of the above probabilities and possibilities will be of great assistance to an officer controlling the fire of a battery. He should know the true mean error of his guns and from this can calculate the spread and movement of M. P. I. of salvos to be expected from accidental errors. He will have, from previous firings, data which will enable him to make a comparison between the theoretical spread and movement of M. P. I. and the spread and movement of M. P. I. actually obtained in firing. It is believed that such an examination will reveal the fact that preventable errors cause as much, or slightly more, spread and movement of M. P. I. as is caused by accidental errors. The information gathered from such an analysis, combined with observation of the fall of shots, will enable the control officer to correct the range in a systematic manner which is in accord with the laws governing the probability of gun fire.