Range.	s.	s/C.	S(v).	v.	T(v).	t/C.	t.	T(v0).	v0.	D(v0)	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

RANGE TABLE FOR 6-INCH GUN.

Charge ${\displaystyle {\Bigg \{}}$ weight, 13 lb 4 oz. ${\displaystyle {\frac {55.01}{0.504}}.}$ gravimetric density, nature, cordite, size 30.

${\displaystyle {\Bigg |}}$

Projectile ${\displaystyle {\bigg \{}}$ Palliser shot, Shrapnel shell. Weight, 100 lb.

${\displaystyle {\Bigg |}}$

Muzzle velocity, 2154 f/s. Nature of mounting, pedestal. Jump, nil.

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) ${\displaystyle {\frac {d^{2}x}{dt^{2}}}=-r\cos i=-r{\frac {dx}{ds}},}$

(44) ${\displaystyle {\frac {d^{2}y}{dt^{2}}}=-r\sin i-g=-r{\frac {dy}{ds}}-g,}$

and eliminating r,

(45) ${\displaystyle {\frac {dx}{dt}}{\frac {d^{2}y}{dt^{2}}}-{\frac {dy}{dt}}{\frac {d^{2}x}{dt^{2}}}=-g{\frac {dx}{dt}},}$

and this, in conjunction with

(46) ${\displaystyle \tan i={\frac {dy}{dx}}={\frac {dy}{dt}}{\Bigg /}{\frac {dx}{dt}},}$

(47) ${\displaystyle \sec ^{2}i{\frac {di}{dt}}=\left({\frac {dx}{dt}}{\frac {d^{2}y}{dt^{2}}}-{\frac {dy}{dt}}{\frac {d^{2}x}{dt^{2}}}\right){\Bigg /}\left({\frac {dx}{dt}}\right)^{2},}$

reduces to

(48) ${\displaystyle {\frac {di}{dt}}=-{\frac {g}{v}}\cos i,{\mbox{ or }}{\frac {d\tan i}{dt}}=-g{\frac {g}{v\cos i}},}$

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) ${\displaystyle v\cos i=q,\,}$

equation (43) becomes

(50) ${\displaystyle dq/dt=-r\cos i,\,}$

and therefore by (48)

(51) ${\displaystyle {\frac {dq}{di}}={\frac {dq}{dt}}{\frac {dt}{di}}={\frac {rv}{g}}}$

It is convenient to express r as a function of v in the previous notation

(52) ${\displaystyle Cr=f(v)\,}$

and now

(53) ${\displaystyle {\frac {dq}{di}}={\frac {vf(v)}{Cg}},}$

an equation connecting q and i.

Now, since v=q sec i

(54) ${\displaystyle {\frac {dt}{dq}}=-C{\frac {\sec i}{f(q\sec i)}},}$

and multiplying by dx/dt or q,

(55) ${\displaystyle {\frac {dx}{dq}}=-{\frac {Cq\sec i}{f(q\sec i)}},}$

and multiplying by dx/dx or tan i,

(56) ${\displaystyle {\frac {dy}{dq}}=-{\frac {Cq\sec i\tan i}{f(q\sec i)}};}$

also

(57) ${\displaystyle {\frac {di}{dq}}={\frac {Cg}{q\sec i.f(q\sec i)}},}$

(58) ${\displaystyle {\frac {d\tan i}{dq}}={\frac {Cg\sec i}{q.f(q\sec i)}},}$

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)=v²/k or v³/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 right-hand side of equations (54)-(56) by some mean value η we introduce Siacci's pseudo-velocity u defined by

(59) ${\displaystyle u=q\sec \eta ,\,}$

so that u is a quasi-component parallel to the mean direction of the tangent, say the direction of the chord of the arc.