, 



September 30, 1920] 



NATURE 



•53 



on a shell of mass m is usually expressed by the 

 formula : — 



R _F(z/) 



»i = C/0/ 



where v is the velocity of the shell relative to the 

 air, /(y) is the reciprocal of the density of the air 

 at height y, F{r) is an experimentally determined 

 function of the velocity, and C depends on the 

 size and shape of the shell. 



Standard functions ^{v) and /(y) are used in all 

 ordinary calculations ; the quantity C is deter- 

 mined for any particular shell by comparing the 

 results of firing trials with trajectories calculated 

 for the same muzzle velocity and elevations as 

 used in the trial, and two or three values of C. 



Primary Ballistic Problems. 



Before the advent of the anti-aircraft gun the 

 point of fall was the only point of a trajectory 

 of any great practical importance, and this couid 

 be found to a certain degree of accuracy by means 

 of approximate integrals of the equations of 

 motion, for high-velocity guns at small elevations, 

 and for low-velocity howitzers at high elevations, 

 which were the two cases of importance before 

 the war. 



But when guns began to be used at higher 

 elevations, and the muzzle velocities of howitzers 

 were increased, these approximate solutions be- 

 came unsatisfactory ; also, with the development 

 of the anti-aircraft gun, came the necessity 

 for calculating whole trajectories, instead of 

 merely a point on each trajectory. Later, it 

 was found necessary to know the whole tra- 

 jectory, even for guns only intended for use 

 against targets on the ground, in order to solve 

 certain secondary problems, such, for example, as 

 the wind correction to be applied when the wind 

 varied with the height. 



The equations of motion, even of the plane 

 trajectory, are formally insoluble, which is not 

 surprising, considering that the air resistance 

 which enters into them contains two functions, 

 V{v) and /(y), of an empirical nature. The only 

 really satisfactory way of obtaining numerical 

 solutions is to carry out a numerical integration 

 of the equations of motion. 



To perform this integration a step-by-step 

 method is employed. That is to say, the tra- 

 jectory is divided up into a series of fairly short 

 intervals, and the integration through each in- 

 terval in turn performed by means of suitable 

 approximate formulie, the size of the interval 

 being chosen to make the errors negligible. The 

 complicated way in which the different variables 

 ire connected makes it impossible to use directly 

 ;iny of the ordinary integration formulae, such as 

 Simpson's rule. 



Methods of step-by-step integration have, of 



course, long been known in astronomy; they 



seem, however, to have been regarded until 



recently as too laborious for balli.stic work except 



NO, 2657, VOL. 106] 



in special cases. However, during the war those 

 concerned with ballistic calculation were forced to 

 use them, for reasons already mentioned, and 

 gradually with experience methods were evolved 

 which were both simple to carry out and not too 

 lengthy. The use of a series of intervals of the 

 same length, and of the finite differences of 

 various quantities at the ends of successive inter- 

 vals, both simplifies the integration and makes 

 possible a complete check on the numerical work. 



When two or more trajectories of the same gun 

 with different elevations have been calculated by 

 these methods, it is obviously possible to deter- 

 mine intermediate trajectories by interpolation. 

 Theoretically, interpolation methods are of a sub- 

 sidiary nature ; in practice, if simple and accurate, 

 they are often very useful. 



For a range table, or for the graduation of 

 sighting apparatus, either for flat or high-angle 

 fire, interpolation from the data furnished by thte 

 calculation of trajectories is necessary. 



Thus in a flat range table the elevation neces- 

 sary to reach a given range is tabulated as a 

 function of the range, but in calculating a tra- 

 jectory an exact value of the elevation is taken, 

 and the range is found. The interpolation in this 

 case is usually done graphically. 



For a high-angle range table the question is 

 more complicated, for this table is one of double 

 entry, giving the elevation required to reach vari- 

 ous points in a two-dimensional region. A table 

 obtained by graphical interpolation usually needs 

 some smoothing. This process, though fairly 

 simple for a single entry table, is almost pro- 

 hibitive for a table of double entry. A scheme of 

 accurate numerical interpolation was therefore 

 evolved ; this scheme as a whole is rather elab- 

 orate, but the individual calculations are very 

 simple. 



In England the greater part of the numerical 

 work of ballistics is carried out by means of 

 calculating machines. 



Secondary Ballistic Problems. 



The development of the methods of solution of 

 the secondary problems in general arose in the 

 first place from the necessity of finding the effect 

 of a wind, or change of atmospheric density from 

 standard, which varied along the trajectory. These 

 are the most important secondary problems in 

 practice, but the methods can be extended to 

 others with little difficulty. 



To make the problem more manageable, only 

 "first order " effects of applied variations arc con- 

 sidered. That is to say, it is assumed that the 

 effects of such variations are additive, so that, for 

 example, the effect of a given wind and a given 

 phange of atmospheric density acting together is 

 the sum of the effects of each separately. In cases 

 of practical importance the error is probably very 

 small. 



The problem of calculating the effect of a wind 

 variable along the trajectory is generally divided 

 into two parts, the determination of the effect of 

 unit constant wind, and the determination of the 



