is there, for example, about eight times as great as when the distance to the target is 2125 or 2475 m. We conclude from this, inversely, that if these shots are observed as above mentioned, the probability that the distance to the target is 2300 m. is greater than the probability for any other distance; and that it is about eight times more probable that the target distance is 2300 m, than that it is 2125 m. (or 2475 m.) TABLE IV. Log. Num. Log. Num. Log. Num. Log. Num. Log. Num. The probability that the fork of 200 m. is correctly constructed, or that the target stands within the limiting distances-2200 and 2400 m. remains equal to unity (1), since the area of the crosshatched surface is equal to the entire surface bounded by the curve (Plate II, fig. 1). The probability is, as already mentioned, theoretically infinitely great, on occount of the supposition that of all shots fired will be correctly and incorrectly observed; therefore the two branches of the curve will run parallel to the axis of abscissas. However, it appears that the probability of the incorrect observation of a shot falling 200 m. in front or rear of the target is very small, and indeed a shot falling 200 m. in front or in rear of the target will be observed with a much greater degree of certainty than that indicated by the ratio of one incorrect observation in every ten. We will therefore consider the observation of all shots falling more than 300 m. from the target as absolutely reliable; that is to say, we will take into consideration only about 300 m. (to scale) of the curve on each side of its highest point, which point corresponds to the most probable target distance. This will approximately correct the error of our | ရာ 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9.00 9 01 9.07 9.23 9.48 9.70 9.85 9.92 9.95 9.95 9.95 9.95 9.95 9.95 9.95 9.95 9.95 9.95 From Table III 8.95 8.95 9.95 8.95 8.96 9.02 9.18 9.43 9.65 9.80 9.87 9.90 9.90 9.90 9.87 9.80 9.65 9.43 9.18 9.02 8.96 8.95 8.95 8.95 8.95 Sums. 7.95 7.95 7.95 7.95 7.96 8.02 8.18 8.43 8.66 8.87 9.10 9.38 9.60 9.75 9.79 9.75 9.06 9.38 9.13 8.97 8.91 8.90 8.90 8.90 8.90 0.009 0.009 0.009 0.009 0.009 0.010 0.015 0.027 0.046 0.074 0.130 0.240 0.400 0.560 0.620 0.560 0.400 0.240 0.130 0.093 0.081 0.079 0.079 0.079 0.079 supposition, that all shots will be correctly observed. In our example also, the probability that the target stands between 2200 and 2400 m. (that the 200 m. fork is correctly constructed) is equal to 1, as in fig. 1, the area of the crosshatched surface is equal to that lying between the ordinates for 2000 and 2600 m. It is easily understood that the areas are always proportional to the equidistant ordinates (the sums that are given in the bottom line of Table III), by which we notice that the shaded surface has for its limit the ordinates for 2200 and 2400 m. ; therefore the ordinate for 2000 and 2600 m. should be accepted as only half as great. These sums are for the shaded surface 5.56, and for the entire surface 7.86. Let w denote the desired probability for the correct construction of the fork ; then we have w:1:: 5.56:7.86; hence w 5.56 7.86 = = .707; in other words, under the supposition made (mean spread 50 m., of all shots incorrectly and correctly observed), of every hundred forks of 200 m., the probability is that 70.7 will be correctly and 29.3 incorrectly constructed. For the construction of the fork of room. a shot must be fired with the elevation for 2300 m., and the observation taken. To ascertain the probability for a correctly constructed fork a strip for short shots with the mark () at 2300 m. will be placed under the line of Table III giving the sums of the logarithms; the sums of the logarithms in the table and on the strip will be taken and the corresponding numbers set down (see Table V). The bottom line of this table shows that the greatest probable target distance is not more than 2350 m. We calculate from this the area of the surface bounded by the curve from 2050 to 2650 m. (the value of the ordinates for distances over 2600 m. remains at 0.079). The probability that the target lies between 2300 and 2400 m. (that the 100 m. fork is correctly constructed) is found by the foregoing practical method to be 2140 4083 0.524; that is, for every 100 forks of 100 m., there will probably be 52.4 correctly, and 47.6 incorrectly constructed. Let us extend our investigation to a fork of 50 m. For this purpose a shot is fired at 2350 m. If this gives a + observation (over shot) the probability that the target stands between 2300 and 2350 m. is 0.387. If it gives, on the other hand, a minus observation (short shot) the probability of the correct construction of the fork of 50 m. (target between 2350 and 2400 m.) is reduced to 0.314 or in other words, nearly of all forks of 50 m. will be incorrectly constructed (see fig. 3 and 4 of the Plate).* We have previously investigated two cases, namely: In case I, the observations of the four shots furnish a very good guarantee that the target stands between 2200 and 2400 m., the probability of this amounting to 0.707, after the delivery of the first two shots; it is increased, after the observation of the last two shots, to 0.878 (see fig. 3). In the second case, on the other hand, the probability for the correctness of the 200 m. fork is diminished by the observation of the last two shots to o 503; that is, in this case the probability that the target lies at a greater distance than 240c m. is proportionally greater (see fig. 4). For this reason the Austrian Firing Regulations contain the provision that when only one of the shots fired for the construction and narrowing of the fork is observed in front of the target, and all the others in rear of it (or vice versa), the shot in question shall be repeated in order to "control" the fork. We will now investigate the correctness of the fork construction when one cause of incorrect forks, namely, incorrect observation These calculations as well as all that follow have been worked out with three place logarithms, and range intervals of 12.5 m. instead of 25 m. is entirely elininated, and the scattering" is the only remaining source of error. The method of the examination is exactly the same as before, except that we turn to Table I instead of Table III for the application of our "strips." We find then, that the probability for the correct formation of the 200 m. fork increases from 0.707 to 0.851, and that of the 100 m. fork, from 0.524 to 0.707 (see fig. 5 and 6). If the shot fired at 2350 m. falls in rear of the target (case I), the probability that the 50 m. fork is correctly constructed is 0.509 (against formerly 0.387); if, on the contrary, it falls in front of the target (case II), the probability of the correct construction of the fork is 0.485 (against formerly 0.314). In each case the increase in the probability of the correct construction of the 50 m. fork due to correct observation is very small. The most of my honored readers will be astonished at the large number of incorrectly constructed forks, and will perhaps think that these results are in contradiction to those of actual experience.* But the opposite is the case, as I have shown conclusively in the work intitled" Correctness of Fork Construction." I have frequently brought to my notice, during the revision of Shooting Lists, or in discussions, the entirely erroneous view that the fork will be correctly constructed if all the shots fired for the purpose are correctly observed. It is evident that by such a definition we shall receive a greater number of "correctly constructed" forks. But the correctness of the fork construction is as much influenced by a shot having a large deviation from the mean trajectory, as by an incorrect observation. If, in spite of correct observation, a fork be incorrectly constructed on account of the deviation of one of the individual shots which, under certain circumstances, may be but a very small deviation, the battery commander naturally discovers no fault. But the mere fact that the fork is incorrect (e. g. if for a target of 2199 or 2401 meters, a fork of 2200/2400 is constructed) will make no difference. Incidentally notice, that the fork may be correctly constructed by means of incorrect observation; if, for example, a shot which must cause an incorrect construction of the fork in consequence of its deviation, be accidentally incorrectly observed. We may also have a case of correct fork construction if the probable target distance is so near one of the two fork limits that, by a continuation of the shots upon this limit, we find we have determined the range (that is, the least practicable correction-25 m.-would not improve the situation of the mean trajectory, and would perhaps I will later compare the results of calculation and experience. make it worse). According to this definition, there would be among a 100 forks of 200 m. 78.3 instead of 70.7 correctly constructed; and of the forks of 100 m., 65.3 instead of 52.4. Now, every incorrect construction of the fork is not equivalent to a failure of the shooting; the correct construction of the fork only insures a greater success. According to the Firing Regulations for the Field Artillery, a fork of 100 m. will be constructed by firing with shrapnel, alternating with time fuse shrapnel, and fired at the two fork limits. The Target Practice Book regards the bursting point as unfavorable, if among six shots more than one falls in rear of the target; in other words, if the mean “ bursting distance" comes under 20 m.; on the other hand, we read (Par. 41) that mean bursting distances of from 30 to 130 m. give good results. Proceeding on the supposition that mean bursting distances from 20 to 130 m. will probably give good results, we can by time fuse shrapnel firing, alternately upon both limits of the 100 m. fork, expect to get good results at least at one of the limits, and from one-half of all the shots fired, if the target stands no more than about 30 m. nearer than the short fork limit, or about 80 m. farther than the long fork limit, supposing that the fuses burn correctly. If the fork is constructed between 2300 and 2400 m., we may therefore count upon good results if the target stands between 2270 and 2480 m. The probability that this is the case, taking from Table V, is 3242 = 0.794; that is, in 20.6 % of all cases we cannot expect to receive effective results from the alternate firing at the two limits of the 100 m. fork. According to par. 94 of the Target Practice Book, if it is plain. that the range is short, the firing in these circumstances should alternate upon the long fork limit and a distance 100 m. greater. In our example we may therefore, as will appear later, be permitted to count upon effective results if the target stands at a range not greater than 2580 m. The probability that the target stands between 2270 and 2580 m., is calculated to be 0.87, i. e., in 13% of all cases there will be no sufficient effect obtained. It is most likely then, if we get no apparent results, that the target stands nearer than 2270 m., when in changing the time fuse shrapnel firing to the short fork limit, according to our rule, the remainder of the percussion fuse shots are observed in rear of the target, in which case we should proceed to the construction of a new fork. It must here be remarked, that these deductions only hold for the case where the bisecting shot of the 200 m. fork (shot at 2300 |