lta]A/[Delta]S - I(v)],

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

(91) sin 2[phi] = 2C [I(V) - [Delta]A/[Delta]S],
(92) sin 2[beta] = 2C [[Delta]A/[Delta]S - I(v)].

To simplify the work, so as to look out the value of sin 2[phi] without the
intermediate calculation of the remaining velocity v, a double-entry table
has been devised by Captain Braccialini Scipione [v.03 p.0275] (_Problemi
del Tiro_, Roma, 1883), and adapted to yd., ft., in. and lb units by A. G.
Hadcock, late R.A., and published in the _Proc. R.A. Institution_, 1898,
and in _Gunnery Tables_, 1898.

In this table

(93) sin 2[phi] = Ca,

where a is a function tabulated for the two arguments, V the initial
velocity, and R/C the reduced range in yards.

The table is too long for insertion here. The results for [phi] and [beta],
as calculated for the range tables above, are also given there for
comparison.

_Drift_.--An elongated shot fired from a rifled gun does not move in a
vertical plane, but as if the mean plane of the trajectory was inclined to
the true vertical at a small angle, 2deg or 3deg; so that the shot will hit
the mark aimed at if the back sight is tilted to the vertical at this angle
[delta], called the permanent angle of deflection (see SIGHTS).

This effect is called _drift_ and the reason of it is not yet understood
very clearly.

It is evidently a gyroscopic effect, being reversed in direction by a
change from a right to a left-handed twist of rifling, and being increased
by an increase of rotation of the shot.

The axis of an elongated shot would move parallel to itself only if fired
in a vacuum; but in air the couple due to a sidelong motion tends to place
the axis at right angles to the tangent of the trajectory, and acting on a
rotating body causes the axis to precess about the tangent. At the same
time the frictional drag damps the nutation and causes the axis of the shot
to follow the tangent of the trajectory very closely, the point of the shot
being seen to be slightly above and to the right of the tangent, with a
right-handed twist. The effect is as if there was a mean sidelong thrust w
tan [delta] on the shot from left to right in order to deflect the plane of
the trajectory at angle [delta] to the vertical. But no formula has yet
been invented, derived on theoretical principles from the physical data,
which will assign by calculation a definite magnitude to [delta].

An effect simila