In recent years there has been much discussion on the subject of the accuracy of our naval guns at long ranges, brought about not only by the natural increase in practicable ranges, due to the development of guns of larger caliber and greater velocity, but also to the laudable desire to push the effective battle range to the extreme limit possible.

Undoubtedly greater accuracy of weapons and superior skill of personnel have their most telling effects at long ranges, for at these ranges superior accuracy and skill affect most decidedly the relative number of hits obtained.

Some 17 years ago, I undertook a mathematical investigation to ascertain the value of the advantage accruing to a naval force engaging an adversary numerically equal but so placed that a portion of his force was masked or beyond effective range. The result of this investigation showed that the advantage was measured, not by the ratio of the number of the opposing ships actually engaged, but by the square of this ratio. Here was the first statement of the so-called “n square law,” a law discovered independently some years later by an English writer.

The application of this law to long range fighting is this: Suppose our greater accuracy of weapons and superior skill makes us feel confident that at a certain long range we can make 4 hits to our adversary’s 3. Then if we engage a numerically equal force at that range our superior hitting ability makes this numerically equal force equivalent to one only 3/4 as large possessing, ship for ship, a hitting power equal to our own. Therefore, according to the n square law our chances of victory are not as 4 is to 3, but as 16 is to 9. Moreover this law shows that in this case the superior force after annihilating the inferior would have left the equivalent of √7/16 of its original force.

The 11 square law is based upon the material damage inflicted and sustained and does not take into account the moral effect due to the knowledge of the inequality between the damage inflicted and that sustained, a knowledge giving rise to a depression on one side and to a corresponding elation on the other.

Courage and grim determination may serve to minimize this moral effect, but in the inanimate vessels themselves is found no counterbalancing force to offset the physical advantages conferred by superior accuracy of weapons coupled with greater skill of personnel.

The foregoing brief discussion suffices to indicate the importance of attaining the maximum effective battle range. It may be well, here, to determine generally what may logically be called the maximum effective battle range. At short ranges the target presented by an adversary is wholly a vertical one and, in estimating the accuracy of our weapons, we are wholly concerned with vertical and lateral errors. As we increase the range we reach a point where the tangent of the angle of fall is approximately equal to the enemy’s freeboard divided by his beam. At this point the target may be regarded either as a vertical or horizontal one. Beyond this point, the target is largely a horizontal one and we are concerned with range and lateral errors only in estimating the accuracy of our weapons. These errors increase as some function of the range, whereas the dimensions of the target remain fixed. It is obvious that with a given accuracy of weapons and skill of personnel a range may be established beyond which the probable number of hits obtainable by the expenditure of a ship’s allowance of ammunition is insufficient to inflict serious damage upon an adversary.

The range so determined may be regarded as the extreme effective battle range. The accepted maximum effective battle range should be as little inside this extreme range as our judgment indicates to be reasonable. As we cannot always count upon a full allowance of ammunition being on board we might define the maximum effective battle range as that at which the expenditure of 75 per cent of the ammunition allowance will probably produce a number of hits sufficient to cause serious damage to our adversary.

Nothing has been said, so far, regarding rapidity of fire. The time element, of course, must enter into our calculations, but a rapidity of fire purchased at the sacrifice of accuracy of fire is undoubtedly a disadvantage and not an advantage. If we fight at a maximum effective battle range justified by our superior accuracy of fire, maintaining a rapidity of fire consistent with such superior accuracy, it is obvious that a superior rapidity of fire on the part of our adversary avails him nothing. It simply hastens his useless expenditure of ammunition. We must strive to shorten the time between hits without increasing the number of misses. The longer the time between misses the better; it lessens the waste of ammunition.

It is difficult to foresee what will be the ultimate limit of the maximum effective battle-range as defined in the foregoing paragraphs. Before the development of aircraft there appeared to be a limit established by the limit of vision from the most elevated observation position practicable on the firing ship. The possibility of using aircraft for observation seems to indicate that the ultimate limit of the maximum battle range will be established solely by the ultimate accuracy of the weapons employed.

It behooves us, therefore, to increase this accuracy by every practicable means available.

The problem of hitting a target, stated in its simplest form, is (1) to determine what position the target will occupy at a given instant and (2) to direct our guns in such a manner that their projectiles will fall upon that spot at the given instant.

The two parts are largely independent, but closely coordinated. The first part is essentially the fire control problem, the second part is chiefly a matter of ordnance and gunnery and demands accuracy of weapons and their accessories. It is needless to remark that skill of personnel is demanded in both parts.

In the solution of the fire-control problem there must appear some form of tracking. Some perform this tracking by means of rangefinder distances, and bearings; others measure the distance by means of the guns themselves for, in reality, that is what we do when we attempt to “spot on.” No matter which method or combination or variation of methods we adopt we must have confidence in our “yard stick.” Rangefinder and gun alike must give readings on a fixed range differing little from the mean of all the readings taken by the same instrument; in other words, the mean error must be small. The French have a rangefinder mount carrying three rangefinders of the same size and type, and presumably having the same mean error and by an ingenious device utilize a continuous mean of the readings of the three instruments.

If D be the mean error of each instrument the mean error of the mean of their three readings is D_{0} = D/√3 = .57735D. In this way they get the same accuracy that could be obtained from a single instrument having a mean error equal to 57.735 per cent of the mean error of each one of the three instruments actually employed. The same principle can be applied to the gun when it is used as a rangefinder. By firing a 3-gun salvo, instead of a single gun for ranging shots, there would be obtained a mean range corresponding in accuracy to that obtainable by a gun having a mean error 57.735 per cent of that of each gun used, provided we could estimate accurately the position of the mean point of impact of the salvo.

No matter what method be employed we obtain, at certain instants, more or less accurate determinations of the enemy’s position and from these positions a more or less accurate estimate of the enemy’s course and speed. The more accurate the instruments employed, the more nearly correct will be our determination of the enemy’s position, course and speed. It must be borne in mind, however, that however accurate these successive positions of the enemy may be, there will always remain the error due to the uncertainty of the enemy’s movements during the interval of time from the last observation before our shots are fired until they land. In order to reduce this possible error to a minimum, we must cut this time interval down to the smallest possible. This means reducing as much as possible the time from the last observation to the firing of our shots and using the highest muzzle velocity practicable, so as to reduce the time of flight as much as possible.

Several articles have been written seeking to demonstrate that the mean errors of the gun should not be below a certain percentage of the fire control error in order to obtain the maximum percentage of hits. The arguments advanced appear reasonable, provided we regard the fire control error as being wholly independent of the gun error. The two errors are, however, closely related and it appears to me that so long as we use the gun in any way to measure the range we must have it as accurate as possible, and the two errors will approach the same value.

Let us consider the simplest case possible—-a stationary ship firing at a fixed target whose distance is only approximately known. Suppose two ships fire 4-gun salvos—one with guns having a mean error of 100 yards and the other with guns having a mean error of 200 yards. Now the mean error of a salvo range—that is, the mean variation of the center of impact of a salvo from the true range—is in one case 100/√4 = 50 yards and in the other 200/√4 = 100 yards. This means that the chances are 9 to 1 that the salvo of one ship will not fall more than 100 yards from the point where it was predicted to fall. In the other case, the chances are the same that this distance will not exceed 200 yards.

Suppose in both cases the best estimate of the distance of the target is 18,000 yards and that the first salvo from each ship falls 600 yards short of the target. One ship feels reasonably certain that the true range of the target lies between 18,700 and 18,500. The other ship with the same degree of certainty knows the true range of the target lies between 18,800 and 18,400 yards. Let each ship increase the range 600 yards for the second salvo and let us consider the following cases: (1) the second salvo falls 200 yards short of the target; (2) 100 yards short; (3) at the target; (4) 100 yards over and (5) 200 yards over. Below are tabulated the limits of range as established by first salvo, by second salvo and both salvos:

An examination of the table shows clearly how reduction of the mean error of the gun reduces the fire control error. Moreover, a small mean error means generally well bunched salvos which facilitates the estimation of the distance of the center of impact to the target by observers in aircraft or captive balloons.

Even after the true range has been established and the fire is continued with a constant sight bar range, the centers of impact of salvos, at best, can be confined to an area whose center is the target and whose dimensions are directly proportional to the mean errors of the gun in range and deflection. It is obvious in this case that the salvos, whose centers of impact are confined to the smaller area, will score the greater number of hits.

Where target and firing ship are both moving the problem is more complicated but the same principles apply.

It is believed that the mean errors of rangefinders can be established for various ranges and then the rangefinders be used much the same as guns in measuring distances, the average of several simultaneous readings corresponding to a salvo of guns.

The sketch opposite shows a method of ascertaining the mean errors of rangefinders at various ranges.

It is evident that comparisons of the rangefinder readings with the true distances obtained from the plotting by the shore stations will give the rangefinder errors for the various ranges.

If a suitable place such as a strait can be found, where ships may pass on approximately opposite courses about 15,000 yards apart, and stations can be established on each side of the strait, then two vessels can steam on approximately opposite courses and take simultaneous rangefinder readings on each other. Each vessel’s course will be plotted by its own set of stations and a comparison of the two plottings will give the actual ranges at various times. Some form of trophy might be awarded to the vessel obtaining the most nearly correct set of ranges. In both of these tests, excellent practice in taking long varying ranges under way will be obtained and considerable knowledge of the working accuracy of the rangefinders will be accumulated.

Compared with the gun as a range measuring instrument, the present rangefinder has this disadvantage—its mean error increases at least as the square of the range, whereas that of the gun should increase approximately as the first power of the range. It would, therefore, appear that at some range their mean errors would be numerically equal—within this range the rangefinder would be the most accurate range measuring instrument but beyond this range the gun would give the more accurate measurements.

It seems to me that as the maximum effective battle range is increased we must depend more and more upon the accuracy of our guns, not only to hit the enemy after we have established the true range but also to determine the true range. Our elevated observers, especially if we place them in aircraft are in a better position to utilize the range measuring ability of the gun than that of the rangefinder. Looking down upon the range, an observer can align his gun “yard stick” and estimate very accurately the amount it overlaps or falls short of the true range. One may regard the ship as a storehouse of a large number of such “yard sticks,” each labelled with its supposed true length but each varying from its tagged length by an amount not exceeding a known quantity depending upon the mean error of the gun. The observer calls for a certain length “yard stick,” lays it down t>n the range and observes how much too short or too long it appears to be. He immediately knows the true range lies between certain limits, he calls for another “yard stick” and establishes new limits for his true range and by comparison with the first limits established narrows the limits between which the true range must lie. For example, suppose an observer is observing the fire of a ship firing 12-gun salvos, with guns having a mean error of 200 yards at 20,000 yards. The observer knows that if he calls for a 20,000 yard 12-gun salvo “yard stick” that the chances are over 9 to 1 that its true length will lie between 20,120 and 19,880 yards, and that the chances are about 3 to 2 that its true length will lie between 20,060 and 19,940 yards. If, therefore, he applies the “yard stick” he receives from the ship, and finds it 200 yards too short he feels reasonably certain that the true range lies between 20,320 and 20,080 yards and that it is more than probable that its true length lies between 20,260 and 20,140 yards. If he decides to call for a 20,100-yard “yard stick” and finds it 100 yards too long, he feels reasonably certain that the true range lies between 20,120 and 19,880 yards. Combined with the first results, this makes him feel reasonably certain that the true range lies between 20120 and 20,080 yards. He can, therefore, call the true range 20,100 yards and feel reasonably certain that the range thus established is not more than 20 yards in error.

So far, I have dealt with that part of the problem of hitting a target that has to do with the accurate determination of the position of the target at a given instant, and have shown how important is the accuracy of our guns, regarded merely as range-finding instruments. There remains that part of the problem which relates to placing our projectiles at the given instant on the position occupied by the target. It is needless to remark that in this part of the problem the accuracy of our guns is all important, for it is obvious that accurate determination of the position of the target and rapidity of fire avail us nothing if we be unable to place our projectiles where we desire them to fall.

Now human experience shows that the perfect instrument never has and probably never will be made. Even when we seek to solve the simplest physical problem, such as ascertaining the true distance between two fixed points, we find that no matter how carefully and accurately made our measuring instruments may be, and no matter how carefully we use it under conditions that appear so far as we can judge, to be identical, we do not obtain identical values.

Experience has taught mankind to make a number of measurements under conditions which appear to be identical and to accept the arithmetical mean of these measurements as being the nearest approximation to the true value attainable with the instruments available. This faith in the arithmetical mean is based upon the assumption that the difference between any particular reading and the true value is made up of the algebraic sum of a number of elemental errors, each one of which is of such a nature that equal positive and negative values are equally probable. It follows that the positive and negative limits of value of the elemental errors must be equal. Therefore, in an infinite number of measurements each elemental error would occur in pairs of equal magnitude but opposite sign. In the arithmetical mean, therefore, all elemental errors would cancel out leaving the arithmetical mean equal to the true value. It is obvious that if there be an elemental error constant in size and sign in each measurement, this error will appear undiminished in the arithmetical mean. Moreover, if there be an elemental error whose positive and negative limits are unequal, the arithmetical mean will contain the mean value of this error no matter how great may be the number of measurements made.

Based upon the accuracy of the arithmetical mean and the nature of the elemental errors, as set forth above, mathematicians have evolved a theory of errors and deduced the “ error function ” which is a continuous function between the limits of +∞ and —∞ and, therefore, covers all errors between those limits.

Inasmuch as it is unlikely that in any given measurement, all or a great majority of the elemental errors will have the same sign, the conclusion is reached that small errors are more frequent than larger ones.

As has been said, the theory of errors cover all errors between +∞ and —∞ and so tacitly assumes that infinite errors in either direction are possible. Now, as a matter of fact, in practically all physical problems an infinite error in either direction is an impossibility. This discrepancy between theory and practice, however, is easily reconciled as the probability of having an error greater than a certain moderate size is very small and in general can be neglected. Practically we do not consider the possibility of committing an error greater than 4 times the mean error. According to the theory of errors in every 2000 measurements we should expect to find 3 errors greater than 4 times the mean error.

Mankind has not been satisfied to accept the arithmetical mean as being the best value obtainable, but has sought some means of ascertaining how near the true value this mean really is. In the size of certain defined errors is found the best indication of the accuracy of the results obtained. These defined errors are (1) the probable error, (2) the mean absolute error, and (3) the mean error.

The probable error is that error which is just as likely to be exceeded as not. This error is the one generally adopted by scientists as the criterion of the accuracy of their measurements. The mean absolute error is thus defined—if we find the variation of each measurement from the true value and, considering only the absolute value of these variations (disregarding signs), obtain the mean value of these variations, this is the mean absolute error. For reasons to be explained later, this is the criterion generally adopted for ascertaining the accuracy of fire. Generally, the word “absolute” is omitted, as the true “ mean error ” is seldom employed and is only of academic interest. The true mean error is obtained by a process similar to that for obtaining the mean absolute error except that each variation is squared and the square root of the mean of the squared variations gives the value of the mean error.

All these errors may be expressed in terms of a quantity known as the measure of precision. If their values be obtained from an infinite number of measurements, such values are called observational values. If obtained from a finite number of measurements by assuming the arithmetical mean value to be the true value, the values of the errors thus obtained are called apparent values. The errors as defined are in fixed numerical ratio, thus: if unity represent the probable error the mean absolute error is 1.1829 and the mean error is 1.4826.

If D be the true value of any one of these errors, D' be the apparent value of the same error derived from n measurements,

and D_{0} be the corresponding error of the arithmetical mean, the following equations are easily deduced:

D’ = D√(n-1/n) D_{0} = D/√n = D’/√(n-1)

For example, suppose the apparent mean error of a gun deduced from 100 shots is 200 yards, then

200 = D√99/100 = .99499d or D = 200/.99499 = 201 yards

D_{0} = 201/√100 = 200/√99 = 20.1 yards

This means that the value of the mean error which we derive from ioo shots is within one yard of the value we would obtain from an infinite number of shots. Moreover, the mean error of the center of impact of the ioo shots is 20.r yards. This means that if we fired an infinite number of groups, of 100 shots each, and measured the absolute variation of the mean point of impact each group of 100 shots from the mean point of impact of all the shots and obtained the mean value of these variations, this mean value would be 20.1 yards. If we take 4 times the mean error to be equal to the extreme error and fire a number of groups of 100 shots each, we would expect all the mean points of impacts of the groups to be confined in a distance equal to 8x20.1 = 160.8 yards. Similarly all the individual shots should be confined to a distance equal to 8x201 = 1608 yards.

With the value D = 201 yards (derived from 100 shots) let us see what we may expect in 12-gun salvos.

D’ = 201√11/12 = 192.4 yards

D_{0} = 201/√12 = 58 yards

We should, therefore, expect that the spread of a salvo not to exceed 8 X 192.4=1539.5 yards and the mean points of impact to be confined in a band not exceeding 8 X 58=464 yards in width.

The foregoing numerical examples illustrate the following points: (a) from the results of firing a comparatively small number of well-aimed rounds under conditions as nearly identical as possible a very accurate determination of the true value of the mean error may be made, (b) from the results of such a determination the results to be expected from firing a few rounds from a single gun or a salvo from a limited number of guns may be predicted, (c) the maximum spread of salvos is directly proportional to the value of the mean error, (d) the mean point of impact of successive salvos of 11 guns each may be regarded as single impacts made by a more accurate weapon whose mean error is 1/√n times the mean error of each gun in the salvo.

Sufficient has been said to indicate the great importance of eliminating or at least reducing to a minimum the elemental accidental errors, thereby reducing the mean error of our guns. Only in this way can we increase the accuracy of the determination of the range by use of the gun and increase our ability to place the mean point of impacts of our salvos at or near the target, thereby increasing the density of impacts over the target area and consequently the probable number of hits. It is useless to theorize about the relation to be maintained between the fire control error and the mean error in order to obtain the maximum number of hits. The fact remains that the fire control error, in the last analysis, must be reduced to the smallest possible and this can only be done effectively by reducing the mean error of the gun to the minimum. When this has been accomplished, if a greater spread of salvo be found to be desirable it can be readily accomplished by deliberately elevating the guns to slightly different angles of elevation without in any way decreasing our ability to place the center of impact at or near the target.

Reference has already been made to constant errors and to errors having unequal positive and negative limits which latter errors may be decomposed into two parts, one part being a constant error equal to the mean value of the error and the other part being an accidental error whose maximum absolute value is equal to one half the difference between limits of the original error.

Of the first type of constant error are those due to maladjustment of sighting devices and those entering into the “Ballistic Correction” and therefore allowed for by that correction. It may be observed, however, that owing to the inaccuracies of the formula: by which the ballistic correction is computed, a portion of the constant errors remain unallowed for.

Of the second type of constant error are the personal errors of persons engaged in laying the guns. Probably no one’s personal error is constant but varies between limits which in general are unequal in the positive and negative directions. On a moving platform one pointer generally fires too soon, another too late, but neither one by an absolutely fixed amount.

For the above reason salvo firing by pointer fire must generally be inferior to director fire. In the first case the dispersion of a salvo must be increased by the various personal errors of the various pointers. In the second case the dispersion remains unchanged but the salvo as a whole falls too far or too nearby an amount equal to the directorscope operator’s personal error.

A little experience enables one to estimate and allow for this personal error of the directorscope operator. The same is true of any constant error that exists in all the shots of a salvo.

The process of allowing for such constant errors is much the same as that involved in applying an instrumental index correction to an observed angle.

Where constant errors exist in only a portion of a ship’s battery it is very difficult from the results of firing to determine the nature and magnitude of these errors and often their very existence may remain concealed.

For example, suppose a ship having 12 guns mounted in 4 triple-gun turrets fires 3 salvos from each turret singly. There will result 4 groups of 9 shots each. From the fall of shot in each group there may be computed the mean point of impact of each group and the apparent mean error of each shot in the group, and then the true value of the mean error of each shot in a group. There will result 4 values of the mean error of a single shot. If there be no individual errors in the various guns these 4 values will be substantially the same. Suppose D, the real mean error derived from D', the apparent mean error, by the formula

D' = D √8/9 be equal to 200 yards.

Now the mean error of the centers of impact by turrets will be D_{0} = D/√9 = D/3. Treating the four centers of impacts as the fall of four shots the apparent mean error of a single center of impact may be computed and substituted in the formula

D’_{0} = D_{0} √3/4 = D_{0}√3/2 = D/2√3 = D√3/6

Now since in computing D'_{0} we add the absolute values of the 4 errors and divide by 4, the sum (S) of the errors must be

S = 4D’_{0} = 2D/√3 = D2√3/3 = 231 yards

Therefore, the following errors of the centers of impact by turrets are consistent with the value of D — 200 yards, and the existence of no turret constant errors.

I II III IV

+ 75 yards +40 yards —40 yards 75 yards

Suppose now we introduce in turret I a constant error equal t_{0} —150 yards and in turret IV a constant error of +150'yards. The errors of the centers of impact become:

I II III IV

-75 yards +40 yards -40 yards +75 yards

It is obvious that such a fall of shot is also consistent with the value D — 200 yards and gives no indication of the existence of the constant errors existing in turrets I and IV.

If we change the signs of the constant errors in turrets I and IV the fall of shots become:

I II III IV

+ 225 yards +40 yards —40 yards —225 yards

The existence of errors in I and IV approaching in value the extreme error 4D„ would make us suspect the existence of constant errors. Moreover, the apparent value of D'_{0}, derived directly from this fall of shot, equals 132.5 yards; the value computed from D = 200 yards, is about 58 yards. Calling D"_{0} the apparent value and D'„ the computed value, we might put D"_{0}^{2} = DV +C- and compute C which will be found to be about ±118 yards. Applying this to the fall we might make a second computation and find (7 =±46 yards. Applying this second correction we would over compensate by 14 yards.

Such a procedure, however, cannot be adopted except in special cases where other reasons may indicate such a course.

The case cited is simplified by the symmetry of the fall of shot assumed. When no such symmetry exists the situation becomes more complicated.

Suppose in the case assumed we had introduced into I a constant error of —75 yards and into IV one of +75 yards thus reducing- to o the apparent error of these two turrets. Now the apparent mean error is 20 yards—much less than 58 yards, its computed value. Yet no one would have the temerity to try to increase its apparent value to its computed value and certainly would not meddle with turrets I and IV where the chances are equal that he would apply the correction in the wrong direction.

From the foregoing one might infer that the detection of constant errors from the analysis of the fall of a limited number of shots would be a hopeless task. In general this is more or less true as regards constant errors of the same order of magnitude as the mean error caused by the accidental errors. *This emphasizes the importance of reducing the magnitude of the accidental errors, for it must be apparent that could these accidental errors be wholly eliminated, thus making the mean error, due to their existence, nil, every constant error would be made to appear in its true sense and magnitude.*

Of course the real remedy for errors, constant as well as accidental, lies in the direction of prevention rather than of cure.

Still, much can be done by searching analysis. In the case just cited it is probable that analysis of the individual fall of shot by various combinations of groups would reveal the fact that only the shots from turrets II and III would give entirely consistent results and lead us to infer the existence of constant errors in turrets I and IV. Such analysis might also give us some clue as to their sign and magnitude.

Throughout this discussion I have purposely adopted the mean absolute error as the measure of accuracy of fire. This seems to me to be the only logical measure.

For some time many people have used the size of the so- called “ pattern as the criterion of accuracy of fire and articles have been written with the object of establishing a numerical relation between the size of the “ pattern ” and the “ mean dispersion,” dependent upon the number of shots contained in the pattern. So far as I have been able to determine the term mean dispersion ” is absolutely synonymous with “ apparent mean absolute error ” as determined from the fall of the number of shots under consideration. The size of the “pattern” is determined by the fall of two shots no matter how many shots may be in the pattern—one shot has the greatest positive error, the other the greatest negative error of all the shots under consideration.

If there be n shots in a salvo and it is stated that the pattern is d yards in size, the only information really given is that the distance between the two extreme shots is d yards. This means nothing more than that the algebraic difference of the largest apparent positive error and the largest apparent negative error is d yards. We gain no information as to the fall of the remaining (n-2) shots.

If 10 salvos of 10 shots each have been fired and from the fall of shot the apparent mean error has been computed and we are told that its value is 100 yards, we can immediately calculate that the real mean error is very approximately equal to 100.5 yards and we could make a sketch showing very closely how the shots must have fallen and where the mean points of impact of the various salvos were located.

If, concerning the same firing, we are told that the mean pattern is 500 yards we gain only incomplete knowledge regarding 20 shots—of the remaining 80 we can only infer that in each of the 10 salvos 8 shots were bunched in a space less than 500 yards wide.

Everyone knows that a large error of a certain type of gun means a large pattern when salvos are fired by a number of such guns, but a large salvo pattern does not necessarily mean a large mean error. A mean error so small as to be practically negligible combined with a large individual Constant error varying from gun to gun in magnitude and sign will give a consistently large pattern. Moreover, the pattern size will give no indication of the existence of these constant errors and might lead to the erroneous assumption of the existence of a large mean error and consequently large elemental accidental errors.

Consideration of pattern size, only, undoubtedly may force upon us the conclusion that we have a white elephant on our hands; it can never indicate the way to rid ourselves of the undesirable animal.

If in the 10 salvos just considered we have means of identifying the shots from each of the 10 guns then we may group the shots by guns and compute the mean error by guns.

if the 10 values thus obtained are substantially the same we may rightly infer that all the guns have the same mean error and consequently have no appreciable individual accidental errors. Now if we compute the mean error by salvos we will get 10 .other values. These may or may not be substantially the same. In either case if these values are not consistent with the values obtained by individual gun groups, we may rightly conclude that some or all the guns have constant errors. An inspection of the two groupings followed by a trial and error process will probably indicate the size and magnitude of these constant errors whose elimination will bring the various values of the mean error into agreement. The foregoing reasoning presupposes that all salvos are fired under as nearly identical conditions as possible, such variations as may unavoidably exist should be allowed for by the best available means of correction.

If a successful means of identifying the various shots can be developed, the foregoing method of firing as many all-gun salvos as there are guns in a salvo will constitute the simplest and best method of calibrating a ship’s battery.

It has the advantage over methods involving the firing of the guns singly in that all shots are fired under practically battle conditions so far as the actual firing of the guns is concerned.

I have dwelt at some length upon the importance of ascertaining the mean error of our guns at long ranges and then taking such steps as may be necessary to reduce these mean errors to a minimum. Only in this way can we hope to increase the accuracy of our guns to the extent necessary to insure victory in a long range battle.

As has been pointed out, the mean error is simply the algebraic sum of a number of elemental accidental errors. In the rare case in which all these elemental errors have a maximum size and the same sign we have an extreme error which experience shows to be equal to 4 times the mean error. Now, if we can enumerate all these elemental errors and estimate the maximum value of each, it is evident that we may by addition obtain synthetically a value of the extreme error. In general, if we compare such synthetic values with those derived from observational values of the mean error we find the observational value much greater than the synthetic value. We are, therefore, forced to the conclusion that either we have underestimated the size of the elemental errors enumerated or, more likely, have failed to include in our list of elemental errors one or more whose existence we probably have not even suspected.

So far as the gun and its mount is concerned, they may be considered as having performed their full duty if they deliver the projectile at the muzzle of the gun at predetermined angles of elevation and train and endowed with predetermined translational and rotational velocities.

Recent accurate observations indicate that the angles of elevation and train remain unchanged until after the projectile has left the gun. Such errors, therefore, that arise from variations in these angles must be due to errors in setting.

Proving ground records of velocity firing do not indicate serious variations in muzzle velocities with the same weight and index of charge but persistent effort is being made to reduce existing variations. Recent large caliber experimental firings seem to have brought to light the possibility of large variations in muzzle velocity by the location of the charge in cases where the total length of charge is materially less than the distance between mushroom face and projectile base. When all sections of the charge were placed together at the ignition end of the chamber 3 per cent more than service velocity and 18 per cent more than service pressure were obtained, whereas normal velocity and pressure were obtained when one section was placed at the ignition end of the chamber and the remaining sections were placed at the other end. That such variations may obtain in service seems to be indicated by the result of recent carefully conducted experimental firing of the main battery of one of the capital ships. The firing was at a fixed angle of elevation and included both single gun firing and firing of various groups in salvo including all-gun salvos. Great care was taken to make all loading conditions the same. The range was about 18,000 yards. In every case, without exception, the range of the mean point of impact of each group was, very uniformly about 1000 yards greater than the best predicted range that could be made allowing for every known ballistic variation. An increase of 3 per cent over service velocity would account for 700 of the 1000 yards increase found by observation.

Up to the present, little attempt has been made to measure the rotational velocity actually attained by the projectile. In the near future, it is hoped to obtain accurate data upon the rotational velocity actually attained and also data upon the reduction of rotational velocity that takes place along the trajectory.

After the projectile leaves the muzzle any additional errors that may be introduced is a matter of exterior ballistics and much depends upon efficient projectile design to insure stability and accuracy of flight.

Undoubtedly, the most critical portion of the trajectory is that part where the projectile is still influenced by the powder gases emerging from the muzzle. These rapidly expanding gases are in turbulent motion and rapidly attain a very high velocity, exerting upon the projectile sufficient force to accelerate it for a considerable distance beyond the muzzle of the gun. It seems to me to be highly improbable that the center of pressure for these gases will lie in the longitudinal axis of the projectile. If the center of pressure be eccentric, a couple of considerable magnitude may be brought into play creating a tendency to angular deviation that must be counteracted by the stability of the projectile. It is obvious that an angular variation at the origin of the trajectory produces an increasing lineal error as the range is increased. It is, therefore, highly important to reduce such angular deviation to a minimum.

A possible way of reducing the suddenness of change of conditions as the projectile emerges from the muzzle is to remove a number of the lands for some distance from the muzzle retaining a sufficient number of lands to maintain the longitudinal axis of the projectile coincident with the axis of the bore so long as the projectile is within the gun. This procedure is along the lines which, it is understood, were adopted by the Germans in their long-range gun. It is understood that they left a considerable length of the bore at the muzzle unrifled. Their projectile, being rifled on its cylindrical surface, there was no need to retain any lands in the bore for centering purposes.

The problem of satisfactory projectile design is a complicated one, the solution of which, in my opinion, will, necessitate a considerable amount of experimentation. For armor piercing ability, the projectile proper must have a stout blunt point but for ranging qualities the outward form must have a long slender point. This necessitates fixing a long pointed wind shield over the blunt pointed projectile.

In the determination of the stability of flight, the location of the center of gravity, the point where the line of action of the resistance intersects the longitudinal axis, the moments of inertia about the longitudinal axis and about a transverse axis through the center of gravity and the rotational velocity about the longitudinal axis must be considered. The interrelation of these factors is probably very complex and experimental data bearing upon this interrelation is very meagre. Consequently we are confronted at times with results that are difficult to explain. In some recent 6"/53 firing, standard projectiles at 3000 f. s. velocity gave evidence of considerably more accurate flight in a gun rifled 1 in 50 turns than in a gun rifled 1 in 25 turns, in which gun, however, the accuracy was markedly better than in a gun rifled I in 37.5 turns. On the other hand, in the 4"/50 with standard projectiles, similar in form to the 6" standard projectiles, steady flight could not be obtained in a gun rifled 1 in 50 turns when fired at 2900 f. s. velocity but in a gun rifled 1 in 25 turns normal flight was obtained.

These results are difficult to reconcile. The rotational velocity of the projectile fired at 2900 f. s. velocity in the 4" gun rifled 1 in 50 turns is the same as that which would have been obtained at 3000 f. s. velocity in a 6" gun rifled 1 in 34.5 turns. It may be more than a mere coincidence that the rifling of the least accurate 6" gun was not far from this value.

On the other hand, although the shells wobbled considerably in the 1 in 50 4-inch gun, resulting in a mean range considerably smaller than that obtained in the 1 in 25 4-inch gun, the mean dispersion of the low pitched rifled gun was fully as good if not better than that obtained in the gun with the higher pitched rifling.

One may wonder why I have rather harped upon the subject of the mean error and have stated and restated facts that should be obvious to everyone.

I have been impelled to adopt such a course because I have observed a too general tendency to adopt an illogical measure of accuracy—a measure that has filled our literature with wailings about the shortcomings of our weapons; spent our energies in more or less futile efforts and has advanced us but little toward the solution of our difficulties.

I remember reading, during the progress of the late war, a lengthy dissertation upon dispersion emanating from a large group of officers, in which was given at length and in detail a list of possible causes of the dispersion of our guns.

These possible causes, if I remember correctly, were as numerous as are the products of the Heinz factories, and so far as I could make out most of these possible causes and all of those popular food products were about equally closely connected with the subject of dispersion.

In my opinion, the first step towards accuracy of fire at long ranges is the determination of the mean errors at these ranges. The second step is to study and analyze these errors with a view to decomposing them into their elemental parts. Lastly comes the elimination or reduction of these elemental errors and the determination of their residual values.

These steps may and probably will proceed concurrently; all involve much time and labor but our achievements in the last step must be the measure of the success of our efforts.

The elimination or reduction of elemental errors, so far as I can foresee, means, for the gun and mount, the greatest accuracy and uniformity of workmanship with minimum tolerances. It means for the foundations an accuracy and rigidity permitting minimum distortions permanent or temporary.

It means for the powder, a purity of materials and accuracy of manufacture, insuring uniformity of finished product so that when made up in charges of fixed weight and length to insure uniform loading conditions it will give uniform results.

Lastly, it means for the projectile a form and disposition of metal that will insure accurate flight throughout its maximum trajectory.

Only by proceeding along these lines we can achieve at long ranges an accuracy of fire to which we can, as politicians say, “point with pride” and in which we can have a confidence, amounting to assurance, of victory in long-range battle.