former curves, since these are plotted for unit air density. When the retardation for any velocity is found from the first curves, it is multiplied by the air density corresponding to the height at which the projectile is then located.

The air-density curve is plotted from the equation of densities found by Bessel to satisfy refraction data, as shown in Chauvenet's discussion of atmospheric refraction, and corresponds closely to similar data assumed by Alger and Ingalls.

FUNDAMENTAL CONSIDERATIONS

Suppose that at a known point in the air, the direction and amount of motion, or the velocity V, of the projectile are known. The retardation caused by the air friction will be in the direction of V, at that instant, and its amount is given by the curves. The velocity V can be resolved into vertical and horizontal components, V and V, and likewise the retardation r at this point can be similarly resolved into r, and r, these latter being in the same ratio to r as are V, and V to V..

After an indefinitely small time, dt, the velocity

V
V

has been

reduced by air friction by the amount rdt. This can be resolved into r1 =rdt and r1=rdt, these being the horizontal and vertical components, or the reductions in the horizontal and vertical velocities.

In the same interval of time, gravity causes a further reduction in the vertical velocity, considering a velocity upwards as positive, amounting to gdt.

The velocity horizontally has then, in the interval of time dt, been reduced to V1–rdt, and the velocity vertically has been

The combination of these new components of the velocity by means of a table of squares or similar means gives a new velocity V along the tangent, whence a new value of r can be deduced. If we take the mean of the two vertical velocities and multiply by dt, we get the vertical travel during this interval, and, likewise, from the two horizontal velocities, the horizontal travel, whence

the coordinates Y and X at the new point can be determined, since the coordinates at the original point are known.

In this way, step by step, we can find V, r, X, and Y, by intervals of dt. This is the method of analytical geometry.

MECHANICAL INTEGRATION

The interval dt is indefinitely small, and unless some method can be found for determining a soluble equation of motion, the analytical method is impracticable.

If, however, we take n successive intervals, each equal to dt, and if the successive values of r are г, r1, and r2, etc., and of V, V, and Vv, similar quantities with the proper suffixes, then after an interval of time ndt the horizontal velocity would be

The time interval ndt is now a measurable quantity, and can be

I Ꮴ called ▲t. The quantity is the mean value during the ΣΥ n Ꮴ

interval At of the horizontal component of the retardation r, the quantity is likewise the means of the vertical component ΣΥ n V

of r, and

h

Σε

n

V

is the mean value of g during the interval.

During a definite time interval At therefore, the horizontal velocity V has been reduced to V-r▲t, and the vertical velocity from V to Vo-ro' At-g'At, where r', r, and g' are the mean values of the quantities rh, r, and g during the interval At.

v

་

Now, if the interval At is not too large, the mean values rn', ro', and g' are for practical purposes the same as the means of the values of rn, rv, and g at the beginning and end of the interval At. The first set of values we know, since we know V, V, and Vn.

The second set is found as follows, by the well-known method of successive approximations:

Having tabulated Vh, Vv, V, r, Th, rv, and g for the first known position of the projectilę, we estimate new values of r1⁄2, rv, and g

after the interval of time At. The value of g changes very slowly and no great error is involved in assuming the new value the same as the old.

Having assumed the next values, the means of the two values of rh, r, and g are taken, and new values are found for V and V. by subtracting from the first values of V the mean of the two values of rh, multiplied by At, and from the first value of V, the mean of the two values of r,, plus the mean of the two values of g, each multiplied by At.

Having thus found approximate values of V and V, at the end of the interval At, they are combined by a table of squares or similar means to give the new value of V. A new value of r is now taken from the curves, for the new value of V, and it is corrected for height by finding the value of 8 corresponding to the new height, and multiplying r by this value of 8. The new height is easily estimated, differing from the old height approximately by V.At.

With the new value of r, corrected for height, we find new values of r and rv. If the originally assumed values of these quantities had been correct, they would be the same as those now found. If not, we repeat the operation, using the new values found for r and rv.

After one or two of these successive approximations, we find that the values of the quantities for the end of the interval remain unchanged. They are then the correct quantities.

Having now found V and V, for the beginning and end of the interval of time At, the horizontal travel in the interval, or AX, is At times the mean of the two values V, and the vertical travel, or AY, is At times the mean of the two values of Vv.

Knowing the original coordinates X and Y, we now find the new ones by applying AX and Y.

PRACTICAL APPLICATION

Sheet B shows a form used for working out a trajectory. The first values, for t=0, are the muzzle velocity for V, coordinates X and Y each zero (in the usual case), and other values derived from V.

These values being tabulated, new values of r and r, are assumed and tabulated for the next interval. Without experience

or other guide there is not much accuracy to be expected in this
first approximation.

The two values of r are now averaged, and the average, mul-
tiplied by At, subtracted from V. Likewise, the two values of ro
are averaged, and, after the value of g has been added to this
average, the sum is multiplied by At and the product subtracted
from V. Thus new values of V and V, are tabulated.

These are now combined into a new value of V, by using a table
of squares if the angle of departure is large, or by the use of a
slide rule if it is small. In the latter case, V is found by increasing

We now consult the curves, and obtain for the value of V the
corresponding value of r. Estimating the altitude of the pro-
jectile at the end of the interval, we also take the proper value of

& from the curves, and multiply r by 8. The product is the proper new value of r to be tabulated. This is now multiplied, by slide to get rv.

V

rule, by to get r1, and by

V

V

We now usually find a small difference between the assumed and derived new values of r and r. The derived values are substituted, new averages of r1⁄2 and r, found, and V and V, corrected. In most cases it is unnecessary to correct further.

h

The two values of V and V, are then averaged, multiplied by At, and tabulated as AX and AY. These are then added to the previous values (in this case zero) of X and Y to find the succeeding values.

The process is then repeated for the next interval of time, and so on, until the value of Y again becomes zero, or the value that is finally wanted.

The remaining steps are obvious.

The last remaining correction is found by interpolation, the final value of X divided by 3 to change it to yards, and the value of AR, above described (see page 74), is subtracted to get the corrected value of R.

It will be noted that ▲t, the time interval, is I second. This simplifies the calculations considerably, and for ordinary trajectories it is sufficiently small. It is possible, for the longer trajectories, to use a larger interval without great error, and the amount of this error can be estimated. For smaller trajectories, for instance with small-arms, it would be necessary to use a shorter interval, in some cases as small as one-tenth second. This is because the value of r changes rapidly.

The values of all quantities used are determined to the nearest tenth of a unit when At=1, except for V, which is determined to the nearest unit. Closer approximation is not necessary, if the old mathematical rule is followed that a fractional result is given to the nearest last significant figure used, and that if the fraction is one-half, the nearest figure is used which is even, not odd.

The columns of differences are very important, tending to control the accuracy and indicate errors before the work has gone too far. The differences at r and r, are also important to determine the values of the next succeeding quantities.

The 20" slide rule is all that is necessary for this work, except that at angles of the trajectory greater than about 20° it is more