236 
THE EFFECT OF THE ROTATION OF THE EARTH 
We wish now to transform these equations to a set of axes with 
origin at the gun, z vertical^ upwards x to the north and y to the east. 
To find the vertical direction put u, v, and w, each equal to zero, and 
the components of the initial acceleration will be the components of g. 
Hence if c fi be the latitude of the station (angle between the plumb line 
and the plane of the equator) we have 
— g cos fi=x Q = -fsind + aoa 2 , 
- g sin fi=z 0 = — f cos 6 , 
and the direction cosines of the vertical upwards at the gun are 
proportional to (/sin 6 — aw 2 ) ; o ; and /cos 0. 
Hence, to rotate the axes to the required directions, we write 
(— x sin (fi + z cos fi) for x and (x cos fi + z sin fi) for z, with similar 
transformations for u , w, and a, c. If at the same time we move the 
origin up to the gun the constant terms disappear, and we have 
. , (cos <ot + cousin (ot -1 t 2 } . , 
x=g sm fi cos fi -j-g — - 2 j JrUt ( cosZ ^ + sin ^ 9 cos 
- vt sin cf> sin cot + wt sin (j) cos $ ( 1 - cos oat) 
, (sin oat — oat cos oo£l ...... . , . 
y=g COS ^- 2 - y + ut sm (p sm oat + vt COS oat 
- wt cos (fa sin oat I 
( . . COS oat + oat sin oat- 1 , . 9 , 
% — g |cos 2 (fa - - 2 - +sm 2 0-j | 
+ ut sin cos (j) (1 - cos oat) +vt cos 0 sin oat + wt (cos 2 (fa cos oat + sin 2 cfa) 
( 8 ), 
•( 9 ), 
.( 10 ). 
The equations so far are exact, on the assumption that the attrac¬ 
tion of the earth is constant in magnitude and relative direction. As 
however co is very small compared with the other quantities in the 
equations, we may expand the circular functions, retaining only the 
first few terms. Keeping as far as w 2 we have 
x—ut— vt sin <p . oat — (ut sin (fa — wt cos cfa + \g cos (fit 2 ) — sin <fi 
oa 2 t 2 
y = Vt J r(ut sin (fi-wt cos (fi+^g cos (fit 2 ) oat—vt. 
oa 2 t 2 
z—wt- \gt 2 + vt cos cfi . oat + (ut sin cfi - wt cos (fi + £g cos fit 2 ) — cos fi 
■( 11 ), 
( 12 ), 
(13). 
To determine the deflection produced in any given case, let V 
be the muzzle velocity, a the azimuth of the direction in which the 
gun is laid, a' that of the direction to the shot at a time t, ft the angle 
of departure (elevation of gun plus jump) and S the deflection of the 
shot to the right (a - a). 
