the system of rectangular coordinates with the x axis horizontal and
the y axis vertical at the gun, to take account of the effect of curva-
ture, both as it affects the direction lines of gravity and the height
of site. Corrections due to curvature become important only at long
ranges and then the most important is that due to height of site or
the curvature of the earth away from the x axis. The correction
angle at the gun due to curvature is one-half the angle at the centre
of the earth subtended by the range.
Correction for Rotation o/ the Earth. If a projectile is fired due east or west at the equator, it has, at the muzzle of the gun, not only the initial velocity with respect to the earth but also the velocity of the earth at that point. If the motion of the earth were one of translation alone, this fact would have no effect on the trajectory: but since the earth rotates around its axis and the rising projectile gets further and further away from this axis, an effect is produced upon the trajectory. This may be made more plainly evident if we assume the projectile to be fired vertically at the equator. With a motionless earth or one moving in right lines the projectile would fall back to the point from which it was fired. With a rotating earth the projectile has, at the muzzle, the vertical velocity given it by the gun, as well as the horizontal velocity of the earth's surface; as it rises it retains the latter velocity at all heights which it reaches. If we now consider points on that radius of the earth which passes through but above the muzzle of the gun we readily see that they have horizontal velocities due to the rotation of the earth in excess of those of the projectile at corresponding heights. It is evident, therefore, that the projectile will lag behind this radius and will fall to the earth west of the gun. A similar range correction will exist if the gun is fired east or west at the equator at any angle of elevation. The value of the correction is proportional to the angular velocity of the earth and the diameter of the equatorial section and depends besides upon the elements of the particular trajectory.
If the gun is fired along a parallel of latitude we have a somewhat similar condition, differing principally in that the velocity of the earth's surface is now less in the proportion cos /, where / is the latitude. Now if the gun at the equator were fired along a meridian, the correction just referred to would no longer exist as a range correction but would become a deflection correction. For a gun fired at any point of the earth it may be shown mathematically that the corrections due to the causes above may be represented by functions of the form
For range, A cos / sin a
For deflection, D cos I cos a
where / is the latitude, a is the azimuth of the plane of fire measured from the south through the west, and A and D are functions whose values depend upon the trajectories.
The above corrections for both range and deflection arise from the lag of the projectile behind the surface of the earth due to its greater distance from the centre of the earth. Another consideration which gives rise to an additional deflection is the change in the velocity of the earth's surface with latitude. A projectile fired from the pole will be displaced by an angular amount depending upon the time of flight and hence by a lateral amount approximately proportional to the product of the time of flight and the range. For a latitude / the deflection due to this cause is equal to that at the pole multiplied by sin / and may be represented by
B sin I.
A rough value of B is J2XT, where JJ is the angular velocity of the earth. The total displacements due to rotation of the earth are, therefore, given by the following equations:
(37) Range displacement, AX= A cos / sin a
(38) Lateral displacement, AZ = B sin / + D cos I cos a, where A, B, and D are computed by integration from the data of each trajectory. It is in this way that range and deflection cor- rections for the rotation of the earth are worked for incorporation in ballistic tables. If air resistance is neglected the values of A, B, and D may be worked out without integration and are:
(40) A flXT (cot u \ tan <#>)
(41) B-fiXT
(42) D - J QXT tan 0.
Since A changes sign at cot co cot (in vacuum) or at 60", it follows that for this angle of departure in vacuum there is no range correction for rotation at any latitude or azimuth.
However, the resistance of the air markedly affects both range and deflection corrections due to the earth's rotation, and the approxi- mate equations (40) to (42) applicable to trajectories in vacuum are not adequate for these corrections with long trajectories.
Variations from Standard Air Density. In case the observed air density does not follow closely enough the assumed law of equation (2) we may divide the air into zones of height, as for variable wind, and determine air density weighting factors and a ballistic air density. The range correction for a variation in air density of, say, 10% from the normal is first worked out, it being assumed that the law of air density given by equation (2) holds through- out the trajectory. The partial corrections due to the same per- centage variation in each zone are then worked out. The ratios of these partial corrections to the total correction are the weighting factors. When the weighting factors are multiplied by the observed densities in corresponding zones, corrected to their value at the
Temperature
ground following the normal law, and the sum of the products for all the zones is taken, we have the ballistic air density.
Effects of Temperature Variations. The temperature of the air affects both its density and its elasticity. In so far as it affects density, corrections in ballistic results, due to changes in temperature, are accounted for by the density correction, and when once the density is known no further reference need be made to temperature. The effect of temperature on the elasticity of the air is in addition to and almost independent of its effect on density. Elasticity of the air may be measured by the velocity of sound therein. This is known to increase as the square root of the absolute temperature and is only slightly affected by density.
In fig. I of the B curve above, note was made of the disturbance in the neighbourhood of the velocity of sound. If the velocity of sound is moved to the right or left on the V-axis by a change of temperature, the B curve will be similarly displaced and hence the E function used in equations (9) and (10) will be changed. With the quadratic resistance law, the B curve would be a right line parallel to the V-axis, and no change would be caused in B. G or E by a change of temperature.
Trajectories used for ballistic table data are worked out for normal temperature
I5C. = 59F.=288A., and are so tabulated.
Standard Temperature. It would be more logical if trajectories were worked out under some law of temperature gradient, similar to that assumed for the density gradient, equation (2). Taking account of the " gas law " derived from Boyle's and Charles' laws, the density law given by equation (2) and the theorem of static equilibrium which requires the difference in pressure at altitude y and sea level to be due only to the weight of the intervening layer, A. A. Bennett has arrived at the following formula for temperature aloft:
Cf . _ TCVin -000045y
.' =122-63X10 -^ This represents fairly mean midsummer temperature in the United States. For mean midwinter temperatures subtract 12-5 C. =22-5 F. throughout. Corrections may be worked out to enable one to pass from the tabular data, based on constant temperature, to data based on the temperature gradient given by these equations.
Rotation of Projectiles. Thus far we have considered the motion as merely that of a material point, or at any rate we have taken no account explicitly of the change in the air resistance which may result from the oblique presentation of an elongated projectile. If pro- jectiles were spherical, as formerly, there could be but one presenta- tion or one section exposed to air pressure, though the projectile might rotate in any direction.
Modern projectiles are given a motion of rotation by the rifling to prevent them from tumbling end over end and thus meeting with vastly increased and irregular air resistance. A projectile so designed as to place the " centre " of pressure in rear of the centre of mass, would doubtless travel head-on without having rotation. Attempts have been made to design such projectiles with some success as far as the ability to travel head-on is concerned ; but it has always been necessary to increase the total head-on resistance, by the addition of a tail or similar device, to such an extent as to make them inferior to projectiles stabilized by rotation. Projectiles of this type are now used as bombs to be dropped from aircraft but are not fired from guns.
Yaw of the Projectile. It has been determined by experiments that elongated projectiles do not always make round holes through cardboard screens called " jump cards " placed at short distances in front of the gun. By placing a sufficient number of these cards, it is found that the holes change in regular cycles, from greater to less and again to greater elongation. For a given round, certain positions of the jump cards, if thickly spaced, give holes of the greatest elongation, corresponding to the maximum yaws of the projectile, and certain other positions give holes of the least elongation corre- sponding to the minimum yaws. If jump cards are placed from near the gun up to 500 or 600 yd. from it, it will be found that the maximum yaws will diminish in value, the first one that appears in front of the gun being the greatest. By yaw is understood the angle between the direction of motion of the centre of gravity and the axis of the projectile.
In the Aerodynamics of a Spinning Shell by R. H. Fowler, E. G. Gallop, C. N. H. Lock and H. W. Richmond, F.R.S. (Phil. Trans., series A, vol. 221), the authors present a very complete analysis of British jump-card experiments conducted by them. While the existence of initial instability of projectiles had long been known, knowledge of its laws and its effects on drift range and accuracy were vague, prior to these British experiments made in 1918.
Causes of Yaw. A projectile fitting perfectly in the gun and hav- ing the centre of gravity of every cross section on the axis of figure, will move in the direction of that axis after leaving the muzzle, unless some force should start an angular motion of the longer axis. For the short distance with which we are here concerned the effect of gravity in curving the trajectory away from the axis is neglected. A projectile not fitting perfectly, or having its axis of figure not coin- cident with its dynamic axis, will yaw slightly in the gun. On leaving the muzzle it may receive an additional yaw from the powder
