lead to a result which is known to be wrong in this case. The method therefore cannot be a correct one.

So far as the Report shows, the average case cited on page 344 in the Summary may be a case where the advantage with reference to probability of "at least one hit" is on the side of the independent system. The statement there made, however, that the parallel would be superior in 672 cases out of 772 would still be true in the sense in which the Board must have meant it, that is, in the sense of 672 positions of the ship out of 772, although the independent method might give the greater general probability of at least one hit." This apparant inconsistency is explained by the fact that the superiority of the independent system in the 100 positions of the ship may be great enough to more than offset the superiority of the system in the other 672 positions of the ship. The statement made in the Summary of the Report is in the sense explained, always true; but at the same time the general superiority may be on the side of the independent method. The language of the Summary is therefore very misleading to the casual reader. To one who has carefully analyzed and understands it, it has no pertinent meaning; and in fact means nothing more than that the parallel method scatters the shot more widely than the independent method; an assumption with which the Report starts out.

Since, according to the Report, our inaccuracy of fire is to result so largely from the inaccuracy of the officially adopted range-finder, that is a subject requiring very serious consideration. It is a serious question as to whether we should rely on a system which at the long ranges is so inaccurate as represented in the Report.

If we are to use so inaccurate a system we should not rely on one range-finder to fire all our mortars by, but should have several. By a well known and simple principle of probabilities the chances of at least one range-finder being correct where several are used independently is much greater than the probability that any one will be correct. If therefore we fired our four groups of mortars each with a range determined by its own rangefinder operated independently of the others, we would increase our chances of firing at least one group with a correct range and azimuth, and therefore increase our chances of hitting the ship.

If the columns of figures in the Report be compared, it will be found that in no case, except that of the first group, is the number of shot "wasted" (by more than one shot striking in a rectangle the size of the ship's deck) much greater for the inde

pendent than for the parallel system; and by comparing the different tables it will be seen that the greater the probable error the less this difference becomes. In no case is this waste' so great as in the case of the experimental firing at Sandy Hook, as shown in the plates accompanying the article by the author of the typical battery in the May-June 1896 number of the Artillery Journal.

At the 3000 yards range 7 shot fell within a space less than the deck of the Royal Sovereign, located with center about 40 yards from the center of impact and having her longer axis nearly parallel to the shorter axis of the battery. Four fell in another equal space at about the same distance in an opposite direction from the center.

At the 6000 yards range 6 shot struck within a space equal to the ship's deck for one position, and five for another position, both positions at a considerable distance from the center of impact.

For both ranges it will be seen from the diagrams that the ship could pass clear across the field in a direction nearly at right angles to the larger axis of the battery and close by the center, and have room to spare, without being touched by a shot.

Economy of shot and regularity of dispersion is not therefore shown in this, the only experimental firing we have had from which to draw conclusions in this connection.

In no case do the columns of figures in the Report make it evident that there is a greater probability of "at least one hit" by the parallel method than by the independent, unless it be in the case of the first group. On the contrary these figures in the table for the second group indicate clearly that for longitudinal position of ship this probability would be greater for the independent system, since the numbers in columns 16 are greater than the corresponding numbers in 17, and those in column 18 greater than those in 19, down as far as opposite 110 in column 1, or for a distance of 110 yards from the center of impact. This 110 yards from center of impact includes as much as 88 per cent of the shots of the independent system and 85 per cent of those of the typical battery. It is then, I think, evident that in this case also, the independent system has the advantage both with reference to "probable number” and “probability of at least one."

We know that when the probable shot error is large, and also when the inaccuracy of aim is not great, we have too great scattering without resorting to artificial means by the form of the battery and parallel fire.

On the whole I think we are justified in concluding that at the effective mortar ranges we have sufficient scattering unavoidable, and that we would get the better effect by aiming the mortars independently at the target. This is equivalent to saying that if parallel fire is to be used it should be from mortars brought as near together as practicable. When this is the case the two systems become practically one, and for the sake of convenience. it would be wise to aim them parallel.

Certainly there has been nothing approximating to proof, either experimental or by mathematical investigation, that at the long ranges the effect of mortar fire would be increased by the form of the typical battery, using with it the parallel method of aiming.

I propose the following method for getting the relative probabilities of "at least one hit" by the two systems: Let Q, be successively the probability that a shot from one mortar of the independent battery will fall in range zones the width of target in range direction; q, the same in azimuth. Then Q, q, is the probability that the shot will fall in retangle the size of target; and 1-(1-Q,,) is the probability of at least one hit in the rectangle, and when multiplied by the ordinates of the two accuracy (or range-finder) curves gives probability that ship will be be in that rectangle.

For typical battery we have for probability that "at least one shot" will fall in above retangle the expression

1

in which P, is range probability for the two front groups and P, the same for the two rear groups; 1 is azimuth probability for the two right groups and ⁄ the same for the two left groups. We must then as before multiply by the ordinates of range-finder (or accuracy) curve.

By applying these methods to a sufficiently large number of rectangles, we may from the respective sums of the results obtain the relative probabilities desired.

CONCLUSIONS.

I. The Report of the Board of Officers contains nothing that shows in any case either the "probable number of hits" or the "probability of at least one hit in a volley," for either the typical or the independent battery; and therefore there is nothing in the mathematical work of the Board to justify any conclusions as to the relative merits of the two systems.

2. The Summary" of the Report given on page 344 is true from the point of view of the Report; but the method there employed would show that, from that point of view, the typical battery would in all cases be the superior even in cases where it has been shown that the independent battery would give both the greater,, probable number of hits" and the greater "probability of at least one hit" in a volley. This "Summary" is therefore very misleading to the casual reader; and to one who has analyzed it and fully understands it, it has no pertinent meaning whatever. 3. My original proposition, given in italics at the bottom of page 289, is mathematically true, and the demonstration given established that proposition; namely, that the independent method gives the greater probable number of hits, which is the same as saying that in the long run it will give the greater number of hits.

4. For reasons of simplicity, or convenience, it is not objectionable, but advantageous, to use parallel fire from groups of mortars where these mortars are in close proximity to each other, as for example in the small groups of four in the typical battery. 5. No method of range-finding has yet been officially adopted which is sufficiently accurate for long ranges, such as are required for effective fire.

6. With such inaccuracy of range-finding as that represented in the Report, it would be unwise (if it could be avoided), to fire a large number of mortars depending for range on a single rangefinder. The probability of hitting the ship would be very greatly increased by having a range-finder for each four groups of mortars, and firing each group with the range determined by its own range-finder; or we might use for all the mortars the mean of the several range-finder readings.

7. From the point of view of the artillerist no justification of the typical mortar battery has ever been established either by means of experimental firing or by mathematical investigation.

SEPTEMBER 7, 1897.

THE THEORETICAL AND PRACTICAL TRAINING

OF THE LIGHT ARTILLERY GUNNER.

[Essay read before the Officers' (Artillery) Lyceum, Presidio, California, March 5, 1897].

The part to be played by Light Artillery in battle is universally acknowledged to be a most important one. The day was, and not so far distant, when field artillery held but an auxiliary status. The relative superiority of this arm over infantry, under modern conditions and within the first zone of fire action, has made it the principal arm throughout that zone; it is therein. accorded first consideration in the choice of position. It there performs, independently, a most responsible task. Only when the inner limit of this outer field is traversed does it subordinate its movements to the infantry, while at the same time rendering indispensable assistance.

Considering the vital function to be performed by the field artillery, in view of its stride forward of late years in material development, it is somewhat surprising that corresponding attention, comparatively speaking, has not been given to its comprehensive and systematic training, particularly along subordinate lines. The thoughts and study of artillery officers seem to have been devoted chiefly to the seacoast or heavy artillery, with its more numerous technicalities, and perhaps greater complexities attending its growth and service. Keeping pace with the development of harbor defense and the consideration of the general duty of this branch of the line, the training of its personnel, the value and necessity of the same, have received a proportionate share of recognition,-throughout the arm and by the highest authorities of the army. Special reference is made to the personnel, as it is with the individual soldier, or a class of individual soldiers, that this discussion has to deal.

Carbaugh's book, Cronkhite's work, Ingall's Hand-book, Chester's Ballistics, and some other publications, while intended for non-commissioned officers, find their more natural application in the hands of the heavy artilleryman; they do not satisfactorily, if at all, cover or touch the ground for the light artillery gunner.

There is no book specially prepared for the latter that I know of.