The value of S is dependent on the least dimension of the grain, called the web thick ness," since when this is burned through, the grain is either completely consumed or else dis integrated. As grains, in the United States service, are of nitrocellulose powder and of a standard pattern, it is clear that the value of S depends on the web thickness alone except that the content of moisture affects E. and also and it will cause considerable variations in these, corresponding to variations in moisture.
This method permits the experimental de termination of the powder characteristics E. and S and the quantity This latter is of considerable importance, as it is a measure of the work done on the gases themselves in giving to them a motion of trans lation and also in disturbing the mass of gas. This latter action may involve a great deal of energy and depends, to a large extent, on the shape of the chamber. Accordingly, K may have a value much greater than unity and its value has a decided bearing on the shape of the chamber, and consequently is important in the design of a gun. The pressures and velocities at any point of the bore may be readily calculated, thus furnishing valuable data for gun design. Muzzle velocities and maximum pressures may be easily calculated for given conditions of loading. The direct problem is thus completely solved.
This method has been used with great suc cess in the Coast Artillery Service of the United States Army, and a similar method is used at the School of Application at Fontaine bleau.
The method is originally due to Captain Leduc, of the French Artillery, though the details of the mcthod as here given have been developed in this country.
Exterior Ballistics.—A projectile in flight describes a curve called its trajectory. This trajectory is a curve of which the vertical height is related to the horizontal distance traveled in a manner dependent on the cir cumstances of its motion. The origin is usually taken as the muzzle of the gun ; the horizontal distance to any point on the trajectory being represented by x and the height above the origin by y, both in feet. The velocity in its path of the projectile at this point is repre sented by v, the inclination of the path to the horizontal by 0, the time of flight from the origin to (x,y) by t. The curve (due to the action of gravity) is convex upwards and has a maximum ordinate or greatest height repre sented by yo, corresponding to a horizontal distance or range At this point the inclination is zero. At the
origin x and y are zero, 0 is represented by 4,, v by V; e is zero. The point at which the projectile again finds itself at the level of the gun is called the "point of fall?) The law of motion already presumed gives the following relations: d2x d (v cos 0) — F (v) cos 0 dP dt C d2y (v sin 0) — F (v) sin 0— g.
dt2 C From these d (v cos 0) v F (v) d gC v d 0 —d-1 =— g cos 0 at sec 0 d 0 --vm dx vcos0 di —0 de dy =v sine dt= =20 taxied 13 e As the trajectory is a curve of relation be tween x and y, so that 3/f(x) it is readily shown from the above relations that • dy = tan @ _d2y —g r— dx2— v2 cos2 d2y —2 r2 F (v) 0 0 yor_ dx 3 g V 1 - C —2; F (v) v4 cos' C It is thus clear that, with a known relation between v and F (v) the trajectory may be, theoretically at least, completely defined in terms of x, y, and constants. On the other hand, if the form of the trajectory is completely de fined, F (v) in its relation to v will be deter mined. There are thus two ways of approach ing the subject, and the problems involved in the two cases are known as the "direct)) and the ((inverse° problems. The direct problem is based on values of F (v) derived from firings through screens, the time of passage being elec trically recorded, and the rate of loss of ve locity thus found. Determinations of F (v) in vanous countries differ considerably; and ex perimental firings have recently been conducted to further define the law. These firings are all made almost horizontally and the effect of the inclination of the projectile on its presenta tion to the air is thus not experimentally known. It is certain, however, that in high angle fire (as with mortars, regarding which many data are available), the air resistance is greatly in creased,— in many cases almost doubled, and that this effect increases with the inclination. The inverse problem, therefore, is one of great importance for several reasons, amonK which is the fact that a great mass of data is avail able, and the additional facts that such data pro ceeded from actual practical firings and that conclusions deduced by proper methods from such data must duplicate the results obtained in practice.