Range.  s.  s/C.  S(v).  v.  T(v).  t/C.  t.  T(v_{0}).  v_{0}.  D(v_{0})  φ/C.  φ.  β/C.  β. 

0  0  0  20700.53  2150  28.6891  0.0000  0.000  28.6891  2150  50.9219  0.0000  0.000  0.0000  0.000 
500  1500  518  20182.53  1999  28.4399  0.2492  0.720  28.5645  2071  50.8132  0.1087  0.315  0.1135  0.328 
1000  3000  1036  19664.53  1862  28.1711  0.5180  1.497  28.4301  1994  50.6913  0.2306  0.666  0.2486  0.718 
1500  4500  1554  19146.53  1732  27.8815  0.8076  2.330  28.2853  1918  50.5542  0.3677  1.062  0.4085  1.181 
2000  6000  2072  10628.53  1610  27.5728  1.1163  3.225  28.1310  1843  50.4029  0.5190  1.500  0.5989  1.734 



Remaining Velocity  To strike an object 10 ft. high range must be known to  Slope of Descent.  5' elevation or depression alters point of impact.  Elevation.  Range.  Fuse scale for T. and P. middle No. 54 Marks I., II., or III.  50% of rounds should fall in.  Time of Flight.  Penetration into Wrought Iron.  

Range.  Laterally or Vertically.  Length.  Breadth.  Height.  
f/s  yds.  1 in.  yds.  yds.  °  '  yds.  yds.  yds.  yds.  secs.  in.  
2154  ..  ..  ..  0.00  0  0  0  ..  ..  ..  ..  0.00  13.6 
2122  1145  687  125  0.14  0  4  100  1/4  ..  0.4  ..  0.16  13.4 
2091  635  381  125  0.29  0  9  200  3/4  ..  0.4  ..  0.31  13.2 
2061  408  245  125  0.43  0  13  300  1  ..  0.4  ..  0.47  13.0 
2032  316  190  125  0.58  0  17  400  1 1/4  ..  0.4  ..  0.62  12.8 
2003  260  156  125  0.72  0  21  500  1 3/4  ..  0.5  0.2  0.78  12.6 
1974  211  127  125  0.87  0  26  600  2  ..  0.5  0.2  0.95  12.4 
1946  183  110  125  1.01  0  30  700  2 1/4  ..  0.5  0.2  1.11  12.2 
1909  163  98  125  1.16  0  34  800  2 3/4  ..  0.5  0.2  1.28  12.0 
1883  143  85  125  1.31  0  39  900  3  ..  0.6  0.3  1.44  11.8 
1857  130  78  125  1.45  0  43  1000  3 1/4  ..  0.6  0.3  1.61  11.6 
1830  118  71  125  1.60  0  47  1100  3 3/4  ..  0.6  0.3  1.78  11.4 
1803  110  66  125  1.74  0  51  1200  4  ..  0.6  0.3  1.95  11.2 
1776  101  61  125  1.89  0  55  1300  4 1/2  ..  0.7  0.4  2.12  11.0 
1749  93  56  125  2.03  0  59  1400  4 3/4  ..  0.7  0.4  2.30  10.8 
1722  86  52  125  2.18  1  3  1500  5  ..  0.7  0.4  2.47  10.6 
1695  80  48  125  2.32  1  7  1600  5 1/2  25  0.8  0.5  2.65  10.5 
1669  71  43  125  2.47  1  11  1700  5 3/4  25  0.9  0.5  2.84  10.3 
1642  67  40  100  2.61  1  16  1800  6 1/4  25  1.0  0.5  3.03  10.1 
1616  61  37  100  2.76  1  22  1900  6 1/2  25  1.1  0.6  3.23  9.9 
1591  57  34  100  2.91  1  27  2000  7  25  1.2  0.6  3.41  9.7 
The last column in the Range Table giving the inches of penetration into wrought iron is calculated from the remaining velocity by an empirical formula, as explained in the article Armour Plates. 
High Angle and Curved Fire.— "High angle fire," as defined officially, "is fire at elevations greater than 15°," and "curved fire is fire from howitzers at all angles of elevation not exceeding 15°". In these cases the curvature of the trajectory becomes considerable, and the formulae employed in direct fire must be modified; the method generally employed is due to Colonel Siacci of the Italian artillery.
Starting with the exact equations of motion in a resisting medium,
 (43)
 (44)
and eliminating r,
 (45)
and this, in conjunction with
 (46)
 (47)
reduces to
 (48)
the equation obtained, as in (18), by resolving normally in the trajectory, but di now denoting the increment of i in the increment of time dt.
Denoting dx/dt, the horizontal component of the velocity, by q, so that
 (49)
equation (43) becomes
 (50)
and therefore by (48)
 (51)
It is convenient to express r as a function of v in the previous notation
 (52)
and now
 (53)
an equation connecting q and i.
Now, since v=q sec i
 (54)
and multiplying by dx/dt or q,
 (55)
and multiplying by dx/dx or tan i,
 (56)
also
 (57)
 (58)
from which the values of t, x, y, i, and tan i are given by integration with respect to q, when sec i is given as a function of q by means of (51).
Now these integrations are quite intractable, even for a very simple mathematical assumption of the function f(v), say the quadratic or cubic law, f(v)=v^{2}/k or v^{3}/k.
But, as originally pointed out by Euler, the difficulty can be turned if we notice that in the ordinary trajectory of practice the quantities i, cos i, and sec i vary so slowly that they may be replaced by their mean values, η, cos η, and sec η, especially if the trajectory, when considerable, is divided up in the calculation into arcs of small curvature, the curvature of an arc being defined as the angle between the tangents or normals at the ends of the arc.
Replacing then the angle i on the righthand side of equations (54)(56) by some mean value η we introduce Siacci's pseudovelocity u defined by
 (59)
so that u is a quasicomponent parallel to the mean direction of the tangent, say the direction of the chord of the arc.