Of course the distribution of the fragments will not be regular; the density will be greater near the vertices, and will fall off rapidly from that region. THE MOST FAVORABLE POSITION FOR THE MEAN POINT OF Burst. In Fig. 6 DD represents the covering line and MNPQ the surface to be struck, situated behind it. From the fact that the densest grouping of the fragments is at,,, the point of burst will have the most favorable position when the zone of dispersion covers the target surface in the manner represented in the figure. Larger areas of the zone of dispersion may indeed, be made to cover MNPQ, if the point of burst be put farther back; but the effect will not be increased, because a portion of the surface of dispersion very densely charged with fragments is then replaced by one much more lightly charged. However, the position of S, in the relative positions of zone of dispersion and target surface represented in Fig. 6, gives the mean position of the point of burst, from which we can go horizontally forward or backward only the distance between vertices, And this distance between vertices, as experience shows, and as is evident from Fig. 5, is very small. The most favorable position obtained for the point of burst of a single shot must, for a group of shots, be converted into the mean point of burst. Fig:73 In Fig. 7, let D4 represent the cover and AR the target surface behind it; AD d, ARf; the system of coordinates, XOY, has its origin at the crest of the parapet D. If through D we pass a line making the angle, with the horizontal, then VR will be equal to the depth of the target surface which can be struck, and the point of burst sought S will be on this line. The point S, with the coordinates ʼn and J, is R.S passing through R with the inclination + m m also on the line to the horizon, so that S, is the deepest vertical section through the apexcalled apex section for short-of the cone of explosion. In order to find the coordinates h and J, let the equations for the lines DS and RS exist simultaneously. m 112 The equation of DS is y = x. tan (3, + "), and that of SR, determined by the point R and the inclination +, is y+d= (x +ƒ) tan (3, + »), or y = x . tan (3, + ») +ƒ. tan (3, + »)—d. If the equations of the two lines exist simultaneously we have Example. Making the same assumptions as in the preceding examples, and taking d=1m., and ƒ= 3 m., it is required to compute the values of t, and h. Remembering that, 55°, 45° and 7°, equation 13 gives = P2 = equation 14 gives J = = and equation 15 gives h = 4.72. tan 62° 8.88 m. The relation of J to h is that of cot 62° = 0.532. Finally, to obtain a complete idea of the effectiveness, it may be required to determine the area in which the point of burst must lie, in order that the apex section can strike on the target surface. A simple consideration will show that this area is enclosed by the line DB drawn through D parallel to the highest trajectory SR of the apex section, and the line RC drawn through R, parallel to the lowest trajectory SD of the apex section. m m The position of the point s relative to R is evidently the same as that of the point S with reference to D. m Of course, for shrapnel effect, only those projectiles come into play which explode above the tangent to the trajectory drawn through the point D, i. e. in the space B DEC; the shrapnel in the space SDT strike full on the deepest vertical axial section, those in the space BDS only partially. m m Similar considerations lead to the determination of the space within which ordinary shrapnel must explode in order to be dangerous to a given target. This is shown in Fig. 8. ST is the direction of the tangent to the trajectory, SB and SC the highest and lowest elements, respectively, of the cone of dispersion. Now, if DB, is drawn parallel to SB, and C,R parallel to S C, we have the space B,C,, in which the shrapnel must explode to have effective action. In case of round bodies, like balloons, tangents must be drawn in place of grazing lines. The determination of the point s is a simple matter: the equations of the grazing lines or tangents, as the case may be, are constructed, and are then made to coexist. We will leave that to be done by the reader. [Translated by 1st Lieutenant JOHN P. WIsser, First Artillery.] A NEW GENERAL BALLISTIC TABLE. The new ballistic table is based upon the new resistance formula, viz: r F(v)=0.2002 v48.05 +√(0.1648 v-47.95)2 + 9.6 + retardation, (or resistance per unit of mass) in metres, v velocity, in meters, weight in kilogrammes of a cubic metre of air divided by 1.206, i= coefficient of form, a and p = diameter and weight of the projectile in metres and kilogrammes. This formula, by making i=1, represents the results of the English and Russian experiments (Bashforth and Mayevski); and by making i=0.896, represents the Dutch (Hojel) and those obtained at Meppen (Krupp).* F(u) = - S (u)' were calculated by quadratures. In the first place, 101 values of D() were calculated by Simpson's formula, Placing Au= 2, from u = 1, from u = 120 metres, 180 to u From these were obtained, by simple and quadratic interpolation, values of u corresponding to values of the argument D(u) having a constant difference of 10. By similar interpolations were obtained the values of after having calculated the values of the two functions by Simpson's formula, with a constant difference of 50 in the values of (u).* Finally the values of A(u) = — u du F(u) were all calculated by the trapezium formula without interpolation. Formulas (22), (23), (24), of our second article on the resistance of the air have been used as a check in those portions of the table relating to high velocities. For low velocities the well known formulas of quadratic resistance have served the same purpose. As to medium velocities, we have made numerous applications of the general ballistic formulas, making use not only of the new table, but also of the old one published in our Ballistics, and have found only exceedingly slight differences in the results; the divergence between the numerical values of the resistance for medium velocities, as given by the old and new formulas, being very small. For the initial values of D, J, A, T, we have taken respectively 1000, 0.1, 100, and 1; by adopting which the table may be extended considerably beyond the velocity 1500 metres without encountering negative values, although we are of the opinion |