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fact, when the value of B is worked out for all parts of any trajectory, it is seen that it is rarely less than one, and nearly always greater, so that its average value could never be strictly considered as equal to one. For instance, at a nominal range of 30,000 yards for the 16" gun of 2600 f. s. I. V., B is 1.09 at the gun, is reduced to 1.00 at the vertex, and increases again to 1.28 at the end of its travel. At 35,600 yards, these figures are 1.16, 1.00, and 1.35.
The means of the values of B for each trajectory, which should be compared with the assumed mean value of 1.000, are, for the same gun, as follows:
Nominal range Mean B
When these values of B are substituted for the assumed value of 1.000 in the Alger or Ingalls ballistic formulae, the nominal ranges, with a given angle of departure, are reduced, in some cases considerably. The amount of reduction of range can also be found approximately by the following method: If k is the percentage error in B, t the time of flight, x the mean horizontal retardation of the projectile, then the reduction in range is \k X f2, which must be divided by 3 to reduce to yards. The value of x is found if we have the initial horizontal velocity Vh, and the range X (feet) traversed, since X — Vht — \ X f2. This has been done for the above gun, and the following errors in range due to the error in B are found approximately:
Nominal range &Rg (yards)
Similar methods are followed in finding the error in /. This constant is found from a table of densities of the atmosphere for different heights above the surface of the earth, and any one value of / is the inverse of the density at the height of the projectile, the density at the surface being unity.
Alger assumes a mean value of / for the whole trajectory, equal to the value corresponding to a height equal to two-thirds the maximum ordinate. This is correct only when the orbit is a parabola, which is never the case at longer ranges.
The values of / have been found for each complete trajectory for the above gun, at a number of points, and the mean values derived. These means differ from the value corresponding to the height equal to two-thirds of the maximum ordinate, found by the Alger method, by amounts as follows:
Nominal range Error in / (per cent)
These errors give approximate errors in range as follows:
Nominal range &R/ (yards)
The above errors are of the same sign as those caused by the error in B, and both reduce the nominal range, when the angle of elevation is constant. The sum of the two errors must therefore be subtracted from the nominal range given by the Alger method, to get a closer approximation to the actual range.
There are two other minor sources of error, that caused by the assumption that g is constant and that caused by the assumption that g always acts in the same direction. As a matter of fact, at longer ranges g is less in the upper parts of the trajectory, and its direction, as the trajectory is traversed, inclines more and more from its original direction, the maximum inclination being the angle at the earth's center subtended by the line joining the two ends of the trajectory; or, roughly, if p is this angle in minutes of arc, R
The former error will increase the range, the latter will reduce it, for a constant angle of elevation, and the errors are nearly equal, so that it is practically safe to neglect these two causes of error, if the Alger method is used.
The amount of the latter error, in yards, is necessary in the method to be described later, and is found as follows: The inclination of g introduces a component equal to g sin p, where p is the inclination of g at any point. In practical cases p is a small angle, and its sine is equal to the angle, also its cosine is practically unity, so that no vertical error results. The error in the horizontal retardation is therefore gp, and the maximum is gp. Half the maximum can be taken as the mean value, and the error in range is therefore £/2 • \gp, or \gpt2, in feet, or -}^gpt- in yards. This error, called £\R, for 35,600 yards nominal range, is 59 yards.
The errors in B and / above found were allowed for in the value of C in Alger's ballistic formulae, for ranges of 15,000, 25,000, and 35,600 yards, and calculation by this method then showed a reduction of the range closely corresponding to the errors found by the approximations described above.
It will thus be seen that the Alger method, as applied to the 16" gun, gives ranges that are too long by about the following amounts, using the adopted value of c, the coefficient of form, and including miscellaneous minor errors:
Nominal range Error in range (yards)
20,000 .' 170
At ranges less than 15,000, the errors are negligible. These errors are somewhat reduced, and that at 15,000 yards eliminated, when we change the value of c, the coefficient of form, from .70 to .69. It will be evident that, having neglected the errors in B and / at 15,000 yards, the range near which c was determined, the value of c adopted must be incorrect by the total of the errors in B, f, etc., and in a direction opposite to that of those errors. By reducing c to .69, the error in range at 15,000 yards disappears, and the errors at the longer ranges are slightly reduced, but not by more than about 250 yards at the maximum range.
By the method of mechanical integration described below, all the above errors are avoided, and the results of the mechanical integration are practically in accordance with the results found by Alger's method, when the corrected values of the constants are used. These corrected values cannot, however, be predicted, so that Alger's method cannot be used for longer ranges. This restriction does not apply to the mechanical integration method, nor is there any restriction whatever to its use, as long as the basic assumption holds, that the axis of the shell remains tangent to the trajectory.
The method is also on theoretical grounds sufficiently accurate, if the time interval chosen is not too large, so that it can with confidence be used for high angle fire, as long as the axis of the shell stays in the trajectory.
The only limit to the application of mechanical integration is where the axis of the shell leaves the tangent to the trajectory to such an extent that the Mayevski friction results can no longer be applied. As long as this is not the case, the method is applicable, and this adherence of the shell axis to the tangent is the only basic assumption. ..3?.'
If the shell leaves it to a small extent, as is always the case, there will be a secondary correction, which, transversely, appears as drift. There is, necessarily, a corresponding correction in the plane of fire. Both these corrections become larger with longer ranges or higher angles of fire.
Mayevski's Friction Results
These have been assumed as they stand, but for use have been plotted as curves of retardation on a velocity base, for each individual projectile. The only changes from Mayevski's values have been made at the points where his formulae change, viz., at 1800, 1370, and 1230 f. s. If plotted exactly according to his. values, there would be cusps in the curves at these points. This is obviously impossible, and the curves have been faired off at these points, not enough, however, to make any appreciable change in results. , .