"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) d2x = -r cos i = -r dx , dt2 ds
 (44) d2y = -r sin i - g = -r dy - g, dt2 ds

and eliminating r,

 (45) dx d2y - dy d2x = -g dx ; dt dt2 dt dt2 dt

and this, in conjunction with

 (46) tan i = dy = dy / dx , dx dt dt
 (47) sec2 i di = ( dx d2y - dy d2x ) / ( dx ) 2, dt dt dt2 dt dt2 dt

reduces to

 (48) di = -g cos i, or d tan i = -g , dt v dt 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) v cos i = q,

equation (43) becomes

(50) dq/dt = -r cos i,

and therefore by (48)

 (51) dq = dq dt = rv . di dt di g

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

(52) Cr = f(v),

and now

 (53) dq = vf(v) , di Cg

an equation connecting q and i.

Now, since v = g sec i

 (54) dt = -C sec i , dq f(q sec i)

and multiplying by dx/dt or q,

 (55) dx = - C q sec i , dq f(q sec i)

and multiplying by dy/dx or tan i,

 (56) dy = - C q sec i tan i ; dq f(q sec i)

also

 (57) di = Cg , dq q sec i . f(q sec i)
 (58) d tan i = C g sec i , dq 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) u = q sec η,

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

Integrating from any initial pseudo-velocity U,

 (60) t = C ∫ Uu du , f(u)
 (61) x = C cos η ∫ u du , f(u)
 (62) y = C sin η ∫ u du ; f(u)

and supposing the inclination i to change from φ to θ radians over the arc,

 (63) φ - θ = Cg cos η ∫ du , u f(u)
 (64) tan φ - tan θ = Cg sec η ∫ du . u f(u)

But according to the definition of the functions T, S, I and D of the ballistic table, employed for direct fire, with u written for v,

 (65) ∫ Uu du = ∫ du = T(U) - T(u), f(u) gp
 (66) ∫ u du = S(U) - S(u), f(u)
 (67) ∫ g du = I(U) - I(u); u f(u)

and therefore

(68) t = C[T(U) - T(u)],

(69) x = C cos η [S(U) - S(u)],

(70) y = C sin η [S(U) - S(u)],

(71) φ - θ = C cos η [I(U) - I(u)],

(72) tan φ - tan θ = C sec η [I(U) - I(u)],

while, expressed in degrees,

(73) φ° - θ° = C cos η [D(U) - D(u)],

The equations (66)-(71) are Siacci's, slightly modified by General Mayevski; and now in the numerical applications to high angle fire we can still employ the ballistic table for direct fire.

It will be noticed that η cannot be exactly the same mean angle in all these equations; but if η is the same in (69) and (70),

(74) y/x = tan η.

so that η is the inclination of the chord of the arc of the trajectory, as in Niven's method of calculating trajectories (Proc. R.S., 1877): but this method requires η to be known with accuracy, as 1% variation in η causes more than 1% variation in tan η.

The difficulty is avoided by the use of Siacci's altitude-function A or A(u), by which y/x can be calculated without introducing sin η or tan η, but in which η occurs only in the form cos η or sec η, which varies very slowly for moderate values of η, so that η need not be calculated with any great regard for accuracy, the arithmetic mean &FRAC12;(φ + θ) of φ and θ being near enough for η over any arc φ - θ of moderate extent.

Now taking equation (72), and replacing tan θ, as a variable final tangent of an angle, by tan i or dy/dx,

 (75) tan φ - dy = C sec η [ I(U) - I(u) ] , dx

and integrating with respect to x over the arc considered,

 (76) x tan φ - y = C sec η [ xI(U) - ∫ x0 I(u)dx ] ,

But

 (77) ∫ x0 I(u)dx = ∫ uU I(u) dx du du
 = C cos η ∫ Ux I(u) u du g f(u)
 = C cos η [A(U) - A(u)]

in Siacci's notation; so that the altitude-function A must be calculated by summation from the finite difference δA, where

 (78) δA = I(u) uδu = I(u)δS, gp

or else by an integration when it is legitimate to assume that f(v)=vm/k in an interval of velocity in which m may be supposed constant.

Dividing again by x, as given in (76),

 (79) tan φ - y = C sec η [ I(U) - A(U) - A(u) ] x S(U) - S(u)

from which y/x can be calculated, and thence y.

In the application of Siacci's method to the calculation of a trajectory in high angle fire by successive arcs of small curvature, starting at the beginning of an arc at an angle φ with velocity v, the curvature of the arc φ - θ is first settled upon, and now

(80) η = &FRAC12;(φ + θ)

is a good first approximation for η.

Now calculate the pseudo-velocity u from

(81) u = v cos φ sec η,

and then, from the given values of φ and θ, calculate u from either of the formulae of (72) or (73): -

 (82) I(u) = I(u) - tan φ - tan θ , C sec η
 (83) D(u) = D(u) - φ° - θ° . C cos η

Then with the suffix notation to denote the beginning and end of the arc φ - θ,

(84) t = C[T(u) - T(u)],

(85) x = C cos η [S(u) - S(u)],

 (86) ( y ) = tan φ - C sec η [ I(u) - δA ] ; x δS

δ now denoting any finite tabular difference of the function between the initial and final (pseudo-) velocity.

Fig. 2.

Also the velocity v at the end of the arc is given by

(87) v = u sec θ cos η.

Treating this final velocity v and angle θ as the initial velocity v and angle φ of the next arc, the calculation proceeds as before (fig. 2).

In the long range high angle fire the shot ascends to such a height that the correction for the tenuity of the air becomes important, and the curvature φ - θ of an arc should be so chosen that y the height ascended, should be limited to about 1000 ft., equivalent to a fall of 1 inch in the barometer or 3% diminution in the tenuity factor τ.