THE AERODYNAMICS OF A SPINNING SHELL. 
329 
The acceleration due to gravity is represented by the vector G, whose modulus 
is g* Under these conditions the vector equation of motion of the centre of 
gravity is 
(3.111) ^{vX}-v[X.Q] = - — X+kv {A-X cos <5} + G. 
dt 777/ 
Taking the scalar product of both sides into X, equation (3.111) reduces to 
(3.112) v' = -R/m+(G. X), 
On substituting this value of v' in (3.111), and dividing by v, we obtain 
(3.113) X'-[X .0] - k {A —X cos <5} + {G-(G. X)X}/v. 
Equations (3.104), (3.105), (3.112), and (3.113) determine the motion completely. 
§ 3.2. Equations of Motion of Type a. 
When a shell is initially sufficiently stable, and leaves the muzzle so that its initial 
disturbance is small, it will be shownf that the axis OA and the direction of motion 
OP deviate, at any time t, by small angles only from the direction of the tangent to 
the corresponding! plane trajectory at the same time. This is true of the early part 
of all trajectories, and for the whole of a trajectory whose initial elevation is less than 
45 degrees—at any rate, when the muzzle velocity is fairly large Under these 
circumstances we may follow the classical§ treatment in regarding the plane trajectory 
as a first approximation to the actual trajectory. It is then convenient to refer the 
motion to axes moving with the tangent to this plane trajectory. The axis Ol is the 
tangent to the plane trajectory drawn in the direction of motion ; axis 02 is the 
upward normal; and axis 03 is horizontal and to the right, as viewed from the gun. 
The components of A and X are (l, m, n) and ( x , y, z ), which are therefore the direction 
cosines of OA and OP respectively. 
It will now be shown that it is possible to express the complete motion 
approximately in terms of the two complex variables, m + in and y + iz, and the 
elements of the plane trajectory. We suppose that the equations of the plane 
trajectory have been numerically solved, so that, e.g., v 1 and 6 U the velocity and 
inclination in the plane trajectory, may be regarded as tabulated functions of t. 
* The vector G may, if desired, be regarded as representing any force which acts through the centre of 
gravity and is a function of position only. 
t See §4.21. 
\ The corresponding plane trajectory is the trajectory which would be described by the same shell, with 
the same initial velocity and initial direction of motion, if its yaw remained always zero. 
See, e.g., Cranz, ‘ Zeitschrift fur Math. u. Phys.’ The equations we obtain, however, appear to 
be new. 
2 z 2 
