Page:EB1911 - Volume 03.djvu/292

Integrating from any initial pseudo-velocity U,

(60)  ${\displaystyle t=C\int _{u}^{U}{\frac {du}{f(u)}},}$

(61)  ${\displaystyle x=C\cos \eta \int {\frac {udu}{f(u)}},}$

(62)  ${\displaystyle y=C\sin \eta \int {\frac {udu}{f(u)}},}$

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

(63)  ${\displaystyle \phi -\theta =Cg\cos \eta \int {\frac {du}{uf(u)}},}$

(64)  ${\displaystyle \tan \phi -\tan \theta =Cg\sec \eta \int {\frac {du}{uf(u)}}.}$

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)  ${\displaystyle \int _{u}^{U}{\frac {du}{f(u)}}=\int {\frac {du}{gp}}}$	${\displaystyle =T(U)-T(u),\,}$
(66)  ${\displaystyle \int _{u}^{U}{\frac {udu}{f(u)}}}$	${\displaystyle =S(U)-S(u),\,}$
(67)  ${\displaystyle \int _{u}^{U}{\frac {gdu}{uf(u)}}}$	${\displaystyle =I(U)-I(u),\,}$

and therefore

(68)  ${\displaystyle t=C\left\lbrack T(U)-T(u)\right\rbrack ,}$

(69)  ${\displaystyle x=C\cos \eta \left\lbrack S(U)-S(u)\right\rbrack ,}$

(70)  ${\displaystyle y=C\sin \eta \left\lbrack S(U)-S(u)\right\rbrack ,}$

(71)  ${\displaystyle \phi -\theta =C\cos \eta \left\lbrack I(U)-I(u)\right\rbrack ,}$

(72)  ${\displaystyle \tan \phi -\tan \theta =C\sec \eta \left\lbrack I(U)-I(u)\right\rbrack ,}$

while, expressed in degrees,

(73)  ${\displaystyle \phi ^{\circ }-\theta ^{\circ }=C\cos \eta \left\lbrack D(U)-D(u)\right\rbrack ,}$

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)  ${\displaystyle y/x=\tan \eta ,\,}$

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 1/2(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)  ${\displaystyle \tan \phi -{\frac {dy}{dx}}=C\sec \eta \left\lbrack I(U)-I(u)\right\rbrack ,}$

and integrating with respect to x over the arc considered,

(76)  ${\displaystyle x\tan \phi -y=C\sec \eta \left\lbrack xI(U)-\int _{0}^{\phi }I(u)dx\right\rbrack .}$

But

(77)  ${\displaystyle \int _{0}^{x}I(u)dx}$	${\displaystyle =\int _{u}^{U}I(u){\frac {dx}{du}}du}$
	${\displaystyle =C\cos \eta \int _{u}^{U}I(u){\frac {udu}{gf(u)}}}$
	${\displaystyle =C\cos \eta \left\lbrack A(U)-A(u)\right\rbrack }$

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

(78)  ${\displaystyle \Delta A=I(u){\frac {u\Delta u}{gp}}=I(u)\Delta S,}$

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

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

(79)  ${\displaystyle \tan \phi -{\frac {y}{x}}=C\sec \eta \left\lbrack I(U)-{\frac {A(U)-A(u)}{S(U)-S(u)}}\right\rbrack }$

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 vφ, the curvature of the arc φ—θ is first settled upon, and now

(80)  ${\displaystyle \eta ={\frac {1}{2}}(\phi +\theta )}$

is a good first approximation for η.

Now calculate the pseudo-velocity u_φ from

(81)  ${\displaystyle u_{\phi }=v_{\phi }\cos \phi \sec \eta ,\,}$

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

(82)  ${\displaystyle I(u_{\theta })-I(u_{\phi })-{\frac {\tan \phi -\tan \theta }{C\sec \eta }},}$

(83)  ${\displaystyle D(u_{\theta })=D(u_{\phi })-{\frac {\phi ^{\circ }\theta ^{circ}}{C\cos \eta }}.}$

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

(84)  ${\displaystyle {}_{\phi }t_{\theta }=C\left\lbrack T(u_{\phi })-T(u_{\theta })\right\rbrack ,}$

(85)  ${\displaystyle {}_{\phi }x_{\theta }=C\cos \eta \left\lbrack S(u_{\phi })-S(u_{\theta })\right\rbrack .}$

(86)  ${\displaystyle {}_{\phi }\left({\frac {y}{x}}\right){}_{\theta }=\tan \phi -C\sec \eta \left\lbrack I(u_{\phi })-{\frac {\Delta A}{\Delta S}}\right\rbrack ;}$

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

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

(87)  ${\displaystyle v_{\theta }=u_{\theta }\sec \theta \cos \eta .\,}$

Treating this final velocity v_θ and angle θ as the initial velocity v_φ 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 φy_θ 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.

A convenient rule has been given by Captain James M. Ingalls, U.S.A., for approximating to a high angle trajectory in a single arc, which assumes that the mean density of the air may be taken as the density at two-thirds of the estimated height of the vertex; the rule is founded on the fact that in an unresisted parabolic trajectory the average height of the shot is two-thirds the height of the vertex, as illustrated in a jet of water, or in a stream of bullets from a Maxim gun.

The longest recorded range is that given in 1888 by the 9·2-in. gun to a shot weighing 380 ℔ fired with velocity 2375 f/s at elevation 40°; the range was about 12 m., with a time for flight of about 64 sec., shown in fig. 2.

A calculation of this trajectory is given by Lieutenant A. H. Wolley-Dod, R.A., in the Proceedings R.A. Institution, 1888, employing Siacci’s method and about twenty arcs; and Captain Ingalls, by assuming a mean tenuity-factor τ=0·68, corresponding to a height of about 2 m., on the estimate that the shot would reach a height of 3 m., was able to obtain a very accurate result, working in two arcs over the whole trajectory, up to the vertex and down again (Ingalls, Handbook of Ballistic Problems).

Siacci’s altitude-function is useful in direct fire, for giving immediately the angle of elevation φ, required for a given range of R yds. or X ft., between limits V and v of the velocity, and also the angle of descent β.

In direct fire the pseudo-velocities U and u, and the real velocities V and v, are undistinguishable, and sec η, may be replaced by unity so that, putting y=0 in (79),

(88)  ${\displaystyle \tan \phi =C\left\lbrack I(V)-{\frac {\Delta A}{\Delta S}}\right\rbrack .}$

Also,

(89)  ${\displaystyle \tan \phi -\tan \beta =C\left\lbrack I(V)-L(V)\right\rbrack \,}$

so that

(90)  ${\displaystyle \tan \beta =C\left\lbrack {\frac {\Delta A}{\Delta S}}-I(v)\right\rbrack ,}$

or, as (88) and (90) may be written for small angles,

(91)  ${\displaystyle \sin 2\phi =2C\left\lbrack I(V)-{\frac {\Delta A}{\Delta S}}\right\rbrack ,}$

(92)  ${\displaystyle \sin 2\beta =2C\left\lbrack {\frac {\Delta A}{\Delta S}}-I(v)\right\rbrack .}$

To simplify the work, so as to look out the value of sin 2φ, without the intermediate calculation of the remaining velocity v, a double-entry table has been devised by Captain Braccialini Scipione