From 1915, however, the nature of the fighting on the western
front called for the development of extreme ranges in all artillery,
and the easiest and quickest method of increasing the range of a
given gun was to modify or redesign its mount so as to permit the
piece to be fired at the angle of elevation that would produce the
maximum, or at any rate the necessary, range. The method was
adopted by all the armies for all calibres of land guns. Further-
more, anti-aircraft guns were designed to permit of all angles of
elevation up to 90 degrees. Thus for the first time it became
necessary to have a knowledge of all the elements along the tra-
jectory and not merely of the range, time of flight, etc., of the
horizontal trajectory. Soon after the war started, improvements
in projectiles, which had been developing slowly since 1900,
began to make themselves felt in still further increasing ranges.
Causes which led to New Methods. Siacci's method involves an assumption (see 3.274, Equation 59), which introduces an error, if an attempt is made to complete the whole trajectory in a single arc, when the angle of departure is more than 20 degrees. The method of " successive arcs," based on Siacci (see 3.275), has been used extensively and has the required accuracy, providing the arcs taken are short, but the method is laborious and has other disadvantages arising from the discontinuity of the suc- cessive arcs. To overcome these difficulties and at the'same time simplify calculations on trajectories, England and France and later the United States adopted the method of numerical in- tegration of the differential equations of motion of the projectile as the standard method of solution. In all these countries the best mathematical talent was brought to bear on the solution of this problem, which in peace-time had received the attention only of a limited number of officers and others connected with the military and naval services and of a few civilians.
The outline of the method of numerical integration given below is that first proposed by F. R. Moulton in the United States, and developed to a high degree by the mathematicians and others associated with him in the study of ballistic problems during the World War. Other methods worked out in England and France, while possessing the same advantages over the older methods, are perhaps not so simple in their application.
Preliminary Assumptions. For purposes of small arc computa- tions, the retardation of the projectile with normal air density at the gun is represented by
C where R is the retardation of the projectile,
v, the velocity in metres of the projectile in the direction of its
motion. vG(v), a function of v, experimentally determined; the retardation
due to air resistance of a projectile of ballistic coefficient = i,
moving horizontally at the height of the muzzle of the gun in
air at a temperature of 15 C. and a pressure of 760 mm., 78 %
saturated with water. H(y), a function of the altitude y (above the muzzle of the gun);
the ratio between the density of the air at that altitude and its
density at a zero altitude. C, the ballistic coefficient.
Law of Air Resistance. The results obtained from any mathematical analysis of the motion of a projectile depend for their accuracy upon the care with which the law of air resistance has been experimentally determined. (For a description of the method and calculations by which Bashforth's ballistic tables, including the law of air resistance, were determined, see 3.271, 272.) In later experiments the same essential methods were followed with the use of more accurate instruments and with projectiles more nearly of the modern form. Such are the Krupp experiments (see 3.273), and the Gavre Commission experiments made in 1888. Chief Engineer Garnier has smoothed out the irregularities in the results of the Gavre Commission firings and has thus obtained a law of air resistance which, while not differing essentially in any region from the results of experiments, is of a continuous character. This cannot be said of Zabudski's law based upon various powers of the velocity.
The G Function. The retardation of the standard projectile due to standard air resistance is put in the form v G(v) for convenience in numerical integration. The function G(v) here represents the ratio
between the retardation and the velocity at each instant. G(v) as
smoothed out by Chief Engineer Garnier is tabulated with as
an argument, velocities and retardations being expressed in his tab- ulated form in metres.
On the next page (p. 388), Table I. gives an abridged table of the G Function (G is the retardation divided by the velocity, for C = i and at surface air density), based on the French
vciui.iLy, iui * i cuiu cit Burittus aii uenbicy;, udbtu on me rrenc tables, giving 10 log G with the argument ; v expressed in
metres per second.
The B Function. The retardation function is sometimes written 2 B(i;), and then B() is the ratio between the retardation and the square of the velocity. In those regions and under those conditions where the'" square law " of resistance holds true, B(i>) is a constant.
Figure i shows Mayevski's and Zabudski's values for B(z>) or **
as compared with Garnier's smoothed-out Gavre Commission values. The tremendous change in the law in the neighbourhood of the velocity of sound is to be noted. More recent but uncompleted
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experiments indicate that the disturbance in the vicinity of the velocity of sound may be changed in amount and displaced in position by changes in the form of the projectile.
Density Function. The air density function H(y) is intended to represent the normal change in density of the air with height. The value of the density function here assumed is,
(2) _ H(y) = lo-- 000045 "
where y is in metres. The coefficient of y is subject to seasonal variations. (See Cours de Ballistique G. Sugot, 1918.)
The density function merely expresses the law of change of density with altitude. It is quite possible to calculate trajectories in air that do not follow this or any other continuous law, providing we know the density at each height. It is necessary, however, in the calculation of ballistic tables to follow some definite law in order to make the tables consistent throughout. Seasonal variations and other variations from the assumed law are taken care of in differential corrections as will be explained below.
The Ballistic Coefficient. The ballistic coefficient is represented by the formula,
TV
d\ r -^s where
(3) <- = i d?
w is the weight of the projectile in pounds.
d, the diameter of the projectile in inches.
i, a factor called the coefficient of form which accounts for differ-
