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 ½(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,