It is usual to give in Range Tables the values of the heights and breadths of the 50 per cent bands as determined at the proving ground, the probable vertical and lateral deviation being deduced by the ordinary methods of the theory of errors from an experimental group of rounds. The chance of hitting a target-band of given height or breadth can then be obtained from the Table of Probability Factors by means of the ratio (probability factor) between the dimensions of the target-band and those of the 50 per cent zone. It is usual, moreover, to adopt as the value of the width of the band that contains all hits, four times that of the 50 per cent zone; because from the Table of Probability Factors it appears, that to the factor 4 corresponds, for the chance, the value 0.99, very nearly unity. Now the above procedure gives results which do not generally tally with those obtained experimentally, and the difference increases with the reduction of the number of rounds of the experimental series considered. I deem it therefore useful limiting myself to the above-mentioned case, a very important one for practical purposes, of finding the ratio between the greatest deviation in a group of hits and the probable deviation of the same group to expound the results of many observations made, together with some remarks dealing with the same.
The average value experimentally obtained of the ratio between the width of the zone that contains all hits, and that of the 50 per cent zone, in a considerable number of series and for a different number of rounds in each series, is given in the following table deduced from the experimental results obtained from a large quantity of groups of rounds fired at the proving ground of Viareggio.
TABLE I.
Number of rounds in each series. (a) | Number of the experimental series considered. (b) | Value of the ratio between the width of the zone that contains all hits and those of the 50% zone. (c) |
3 | 116 | 1.50 |
4 | 65 | 1.73 |
5 | 40 | 1.95 |
6 | 40 | 2.03 |
8 | 98 | 2.24 |
10 | 25 | 2.45 |
12 | 20 | 2.58 |
14 | 20 | 2.59 |
16 | 20 | 2.63 |
19 | 20 | 2.78 |
In spite of the fact that the above values (column c) have been deduced from a considerable number of trials (column b), it would be natural to hesitate before definitely accepting as reliable the empirical law they show forth, and applying it to practical purposes.
It is therefore specially interesting and important to seek some possible relationship between the said values and theoretical deductions. The following remarks can be made on the subject:
Calling E the 50 per cent zone and pk the chance of hitting, with a single shot, the band kE, then 1 - pk is the chance of a single shot missing; and, therefore, in n rounds, the probable number of hits outside the kE band will be n (1 - pk), an expression which increases with the increase of n; which is the same as to say that the width of the zone, that probably contains all hits, increases together with the number of hits.
Now it appears right to hold that, in a series of n rounds, the dimension of the band that contains them all, should be approximately determined by the value of k, given by the following formula:
n(1 - pk) < 1.
Calculating, according to the aforesaid criterion, the values of k for the values of n of column (a) of the preceding table, one obtains:
TABLE II.
For n= | k= |
3 | 1.46 |
4 | 1.73 |
5 | 1.91 |
6 | 2.05 |
8 | 2.28 |
10 | 2.44 |
12 | 2.58 |
14 | 2.68 |
16 | 2.77 |
19 | 2.88 |
The comparison between these values of k and those given in column (c) of Table I shows how the criteria above expounded has a really noteworthy practical confirmation, the differences being very small or non-existent in the greater number of cases, and only somewhat noticeable for the greater values of n, for which, however, the experimental values of k would require a further confirmation; because the greater the number of rounds in a series the more likely it is that the conditions have changed, with consequent perturbation of the results as compared with the deductions to which the theory of casual error leads. Consequently, there is need for a very considerable number of tests, so that such perturbation also may enter within the scope of the laws which regulate the casual errors, viz., should on the average compensate one another.
It seems, therefore, plausible to conclude that, at least while the number of rounds of a series is contained within the limits considered in this paper, the values of k given in Table II may be legitimately held as the most probable values of the relative width of the band that contains all the hits of the series.
An important practical application of the above presents itself when one wishes to establish the order of merit of gun-layers in a firing competition, in relation to the number of hits on the target; in which case it is necessary to consider also, so as to assure the greatest possible accuracy, the number of rounds each gun-layer is allowed to fire. For instance; in firing with a 6-inch gun, full-charge, A.P. shell, at 3000 meters, the 50 per cent zone being mt. 2.65 in height, the height of the target containing all hits should be assumed at mt. 3.87, if the number of rounds to be fired be 3, at 5.43, if the said number be 6, and at 6.84, if it be 12. It is clear that the difference between these values is anything but insignificant.
It should be noted that in the above example the width of the band containing all the n hits of the series has been referred to the similar dimension of the 50 per cent zone given by the Range Table and therefore obtained from a number of hits often different from n. This, however, makes no difference theoretically, whilst also the comparison of the mean values of many experimental data shows that the values of the probable error obtained from series differing in number of hits may, within the limits of approximation necessary to us, be considered equal.
NOTE. In confirmation of the results above set forth, the Superintendent of Experiments at Shoeburyness, Major A.C. Currie, writes to the author that he has tried his factors in many ways and found a wonderful agreement with the experimental data. Taking longitudinal errors, with new guns and with worn out guns, having errors ranging from 10 yards to 750 yards, his results were as follows (using same headings as in Table I):
(a) | (b) | (c) |
5 | 33 | 1.92 |
10 | 61 | 2.39 |
Similarly, with lateral errors and with variations of muzzle velocity, he found experiment closely agreeing with the above deductions.