Encyclopaedia Britannica, 11th Edition, "Baconthorpe" to "Bankruptcy" / Volume 3, Part 1, Slice 2

by Various

(84) _φt_θ = C[T(u_φ) - T(u_θ)],

(85) _φx_θ = C cos η [S(u_φ) - S(u_θ)],

(86)

φ

(

y

)

θ

= tan φ - C sec η

[

I(uφ) -

ΔA

]

;

x

ΔS

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) v_θ = u_θ sec θ cos η.

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.