JOURNAL

OF THE

UNITED STATES ARTILLERY.

VOL. VIII. No. 2.

SEPTEMBER-OCTOBER, 1897. WHOLE NO. 28.

THE PROBABILITY OF HIT WHEN THE PROBABLE ERROR IN AIM IS KNOWN: WITH A COMPARISON OF THE PROBABILITIES OF HIT BY THE METHODS OF INDEPENDENT AND PARALLEL FIRE FROM MORTAR BATTERIES.

BY MANSFIELD MERRIMAN, PROFESSOR OF CIVIL ENGINEERING IN LEHIGH UNIVERSITY.

The term "probability of hit" means the probability of hitting a target when one shot is fired, or if several shots be fired the probability of hit multiplied by the number of shots gives the probable number of hits. In general discussions of the probability of hit the inaccuracy in aim is regarded merely as one of the accidental errors which cause the dispersion of the shots. It is here proposed to investigate the case where the probable error of shot dispersion under accurate aim is known, and where also the probable error of the aim under ordinary conditions has been determined. Thus for a mortar the former may be found by firing a number of shots with the same range and azimuth, while the latter may be ascertained by tests on the instruments which locate a floating target in range and azimuth.

The target is supposed to be a rectangular object floating on the water, and in the case of a ship it will be taken as a rectangle slightly shorter than the total length and about the same area as the deck. One of the sides of the target will be taken parallel to the plane of fire, and its length in range be designated by 2A, while the width of the target in azimuth is denoted by 2a.

Let ri be the probable error of shot dispersion in azimuth and R, that in range, as determined under circumstances of accurate aim. Let r, be the probable error of the designated position in azimuth and R, that in range, as determined from tests on the

azimuth and range finders. Let a shot be fired with the intention of hitting the center of the target; it is required to find the probability of hit on the target or deck whose area is 4a4.

Α SMALL TARGET.

When a and A are very small a simple expression for the probability of hit may be established. Considering first the case of errors in azimuth, or those at right angles to the plane of fire, let h, and h2 be the measures of precision in shot dispersion and in azimuth aim, these being connected with the probable errors by the relations h, r, p, hr, p, where p 0.4769363. Let the center of the small target be indicated by the azimuth finder as at a position whose error is x, then

is the probability of hit on the strip 2a when the error x exists. But the probability of this error x in azimuth aim is

Y2 = π h2 e-h2x2 dx.

2

(2)

The product of these two probabilities gives the probability of hit on the strip whose width in azimuth is 2a and whose length in range is infinite, under the combined error of shot dispersion and ordinary aim, or

is the probability of hit in azimuth for any error x in azimuth aim. Then the probability of hit when nothing is known about the error x is the sum of the probabilities of hit which can occur in the different independent ways, that is, the sum of all the possible values of (3). This sum is

2

Now let h, and h, be replaced by p/r, and p/r,, and let ri+be represented by, where may be called the resultant probable. error in azimuth. Then (4) becomes

which is the probability of hit in azimuth for the small target. By applying the above reasoning to errors in range similar equations result, a, r,, r, being replaced by A, R1, R; and

is the probability of hit in range for the small target.

(6)

Formula (5) gives the probability of hit for the small strip whose width in azimuth is 2a and whose length in range is infinity; and formula (6) gives the probability of hit for the small strip whose width in range is 2A and whose length in azimuth is infinity. To find the probability of hit for the area 2ax 24 these expressions are to be multiplied together, and

is the probability of hit for the small target.

(7)

If the aim be always perfectly accurate, there being no errors in the azimuth and range finders, these formulas become

which are the expressions given by the common theory where the probable errors in aim are not separately considered.

Although these formulas are strictly correct only when a and A are very small it may be shown that values computed from (5) and (6) are in error only one-half of one per cent when a = { r and AR, and only two per cent when ar and AR. Hence in many cases of target practice they can be safely used for numerical computations.

A LARGE TARGET.

Following a similar line of reasoning, formulas for the probability of hit for a target of any size may be deduced. Taking first the investigation for azimuth, let the center of the target be designated by the azimuth finder at a position whose error is x. The probability of this error is given by (2). Let z be any distance in azimuth from the middle of the target; then the probability of hit on the strip dz for this error x is

and the product of (2) and (9) gives the probability of hit under the two combined causes of error. The sum of all possible values of this product is the probability of hit in azimuth for the strip dz. Then z must be varied so as to cover the whole width 2a. Thus, the expression to be integrated is

where -∞ and ∞ are the limits for x, and -a anda are the