3. The atmospheric density varies with the altitude, according to
the H function given by equation (2) and is standard at the muzzle.
4. The action of gravity is uniform in intensity, is directed toward the earth's centre and is independent of the geographical position of the gun. Its value = 9-80 metres per second.
5. The G function is a function of the velocity alone and has the values given in Table I.
6. The ballistic coefficient C is constant and known. Differential Variations. Range tables for artillery must give the
data required to lay the gun to strike a target at any desired range, not only for certain conditions fixed as standard for that gun, but also for conditions varying considerably from the standard. The variation may be in the initial conditions as in muzzle velocity, ballistic coefficient, or angle of departure. The initial variations in the ballistic coefficient may be due to variations in the density of the air at the gun, variations in the weight of the projectile or variations in the coefficient of form. Again, the variations may be in subsequent conditions, as in the existence of a range or cross wind, in H(y) or the air density curve for the day, in the rotation and curvature of the earth. The latter is introduced as a variation since it is not considered in the calculation of basic trajectories for the construction of ballistic tables, but its effects are material at long ranges.
Ballistic tables having been constructed we may obtain by mere interpolation the important variations such as that in range caused by variations in the initial conditions. Such variations may then be tabulated in convenient form in the range table.
For variations due to abnormal subsequent conditions, it is neces- sary to make special calculations, whether these are to be incor- porated in the ballistic tables, or merely in the range tables. It would be quite possible to calculate a sufficient number of trajectories under assumed abnormal subsequent conditions to enable one to tabulate in ballistic and range tables the variations due to changes in these conditions, but this procedure would require a tremendous amount of work.
Furthermore, a variation in, say the range, due to variations in conditions is the difference between the range under the normal con- ditions and the range under the abnormal conditions. If we deter- mine the variations by determining each range separately and taking the difference we are introducing the errors in two large quantities into a small quantity. The percentage error in the latter will, there- fore, be large.
In view of these considerations, it has been found desirable to consider variations in the elements of the trajectory due to variations from the normal conditions as functions of the variations from the normal conditions and to solve the differential equations of the variations, using the same principles of numerical integration as are used in the solution of the differential equations of the trajectory.
Equations of the Variations. Taking x and y as the coordinates at the time t of the original trajectory of which the differential equations are (9) and (10), let us assume that the coordinates of the modified trajectory corresponding to the same time are #+ and y+1, and ri representing the variations, due to some cause other than wind, in the conditions under which the original trajectory was constructed.
Variations due to wind affect the relative velocity between the projectile and the air and, therefore, the value of E, independently of the variations in x' and y' and will be considered in a later section.
Under this assumption the coordinates of the modified trajectory should satisfy the equations:
(13) *"+{"- -(E+AE) (*'+')
(14) y"+V'=-(E+AE) (/+,') -g.
If we combine these with equations (9) and (ip), and neglect all terms consisting of products of the small quantities , i\ and their derivatives, and AE, we obtain upon solution for " and ij"
(15) "=-E'-*'AE
(16) V' = -Ei?'-y'AE.
On substitution of the value of E obtained from equation (9), trans- position and division by x',
(17) (18)
-AE
x'n"-x"r,'__y'
7 AE.
x'- 1 x'
Since the first members are the derivatives of '/*' and respectively, we may express the integrals as follows:
or,
(19) (20)
(21)
(22)
17 17
X 1 Xo'
- dt
, *v' , n y'
ij' = -. -- x ^-,
X .1 o X
j,
dt.
Here o' and 170' represent the amounts by which the initial com- ponents of velocity differ from x a ' and y<>' respectively.
This set of equations like (9) and (10) may be integrated by the method of numerical integration, but we must first obtain an explicit relation between AE and ', i\' and t\.
C J-T
Effect of the Variations on E. Since by equation (u) E = -- we
may write approximately, / % AE
(23)
AG ' G
Again
and
A/- <* G
AG= -T
dv
dy
f H h H AG =
G AH_
H =
we may write the equation, (24) -g- = ^
Now, (25)
Au=AVa
and Ay = 17.
Consequently we may write,
(26 ) AE^C-i^&G
AC "C"
d logeG
dv d logol
d7
Az;
'A a. d Io R- H A AC -AH J Ay .
dy C
yV
V2
+E
AC
dv dy ) ' "C
The first term of the second member of this equation gives the part of AE due to variations in the components of the velocity and of the height. The last term gives the part due to variations in the ballistic coefficient including variations in the air density. Equa- tion (26) is based on the assumed law of retardation as given by Table I. and the assumed law of air density as given in Equation (2). There is no trouble, however, in making differential corrections for variations from these assumed laws.
The term
d logeG
dv
i dG . , , f.r- ls found vG dv
from the G function
Table I. and tabulated with as an argument in Table III. below.
100
If we assume for H the exponential formula given by equation (2)
d logcH = -0001036 a constant.
we have
TABLE III.
 | A table should appear at this position in the text. See Help:Table for formatting instructions. |
Values of -^r- vG dv
For use in making differential corrections. Argument t^/ioo (v in metres). The expressions -05423, etc. mean -000423, -00000378, etc.
oo 200 400 600 800 1,000 1,200 1,400 1,600 i, 800 2,000 2,200 2,400 2,600 2,800 3,000 3,200 3,400 3,600 3,800 4,000 4,200 4,400 4,600 4,800 5,000 5,200 5,400 5,600 5,8oo 6,000 6,200 6,400 6,600 6,800 7,000
-7T vG dv
-0,412 -04276 -04314 -04400 -04434 -04305 -04175 -04110 -Os774 -05585 -05462 -05378 -05317 -05271 -05237 -05209 -O 6 i88 -05171 -05157 -05145 -05136 -O 6 I28 -O 6 i2i -05114 -05108 -05104 -05102 -0.996 -06975 -05955
-05937 -05921
7,200
7,400
7,600
7,800
8 >o
8,200
8,400
8,600
8,800
9,000
9,200
9,400
9,6oo
9,800
10,000
12,000
14,000
16,000
18,000
20,000
22,000
24,000
26,000
28,000
30,000
32,000
05378,
j__dG vG dv
05875 05865 05855 05845 06836 05827 05818 05810 05802 06794
05785 06779 0,771 0.764 0.757 0,693 0.637 0,586
' Oa5 ^ 06488
0,446 05411 05380 06354 06332 05311
-05897 -05886
