so that [eta] need
not be calculated with any great regard for accuracy, the arithmetic mean
1/2([phi] + [theta]) of [phi] and [theta] being near enough for [eta] over
any arc [phi] - [theta] of moderate extent.

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

(75) tan [phi] - dy/dx = C sec [eta] [I(U) - I(u)],

and integrating with respect to x over the arc considered,

(76) x tan [phi] - y = C sec [eta] [xI(U) - [Integral,0:x] I(u)dx],

But

(77) [Integral,0:x] I(u)dx = [Integral,U:u] I(u) dx/du du
= C cos [eta] [Integral,x:U] I(u) {u du}/{g f(u)}
= C cos [eta] [A(U) - A(u)]

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

(78) [Delta]A = I(u) 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) tan [phi] - y/x =
C sec [eta] [I(U) - {A(U) - A(u)}/{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 [phi] with velocity v_[phi], the curvature
of the arc [phi] - [theta] is first settled upon, and now

(80) [eta] = 1/2([phi] + [theta])

is a good first approximation for [eta].

Now calculate the pseudo-velocity u_[phi] from

(81) u_[phi] = v_[phi] cos [phi] sec [eta],

and then, from the given values of [phi] and [theta], calculate u_[theta]
from either of the formulae of (72) or (73):--

(82) I(u_[theta]) =
I(u_[phi]) - {tan [phi] - tan [theta]}/{C sec [eta]},
(83) D(u_[theta]) =
D(u_[phi]) - {[phi]deg - [theta]deg}/{C cos [eta]}.

Then with the suffix notation to denote the beginning and end of the arc
[phi] - [theta],

(84) _[phi]t_[theta] = C[T(u_[phi]) - T(u_[theta])],
(85) _[phi]x_[theta] = C cos [eta] [S(u_[phi]) - S(u_[theta])],
(86) _[phi](y/x)_[theta] =
tan [phi] - C sec [eta] [I(u_[phi]) - [Delta]A/[Delta]S];

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

[Illustration: FIG. 2.]