THE AERODYNAMICS OF A SPINNING SHELL. 
325 
magnitude of a single impulse, at not too great a value of the yaw, probably has the 
values 
14-3 foot-poundals at 2470 f.s., 
8*9 foot-poundals at 1140 f.s. ; 
the values at other velocities may be roughly obtained by linear interpolation. 
The effect on the observed motion of the axis due to an impulsive couple was 
calculated, and it was found that rough values could be assigned for the magnitude 
of the impulsive couple acting at any card. On calculating the total effect on the 
observed value of s it was found that the probable correction required varied from 
2|- to 4|- per cent, in the various groups. This correction was applied before 
constructing Table I. and figs. 4 and 5. The figures of Table II. have not been 
corrected for this effect as their accuracy is not great enough to make it worth while 
to do so. 
Q 
Part III.— Methods of Obtaining and Solving the Equations of 
Motion of a Spinning Shell. 
§ 3.0. Introductory. 
On the assumptions discussed in Part I. the equations of motion of a spinning 
shell can be written down at once by the rules of rigid dynamics. Three different 
types of these equations will be found of use in practice, all of which may be obtained 
most simply as special cases of the vector equations of motion of the shell, referred to 
axes rotating in the most general manner. The use of the vector notation, in the 
initial stages of the discussion, has the further advantage of showing most clearly 
the meaning of the various terms, and of presenting the results in a symmetrical 
form. 
In order to simplify the general equations, the only components of the force system 
impressed by the air, retained in the initial discussion, are It, L, M, and the spin- 
retarding couple I (= ANT). The remaining components are of less importance and 
will be inserted later on in § 3.5. 
After obtaining the general equations the three special types are deduced. They 
may be described as follows :— 
Type a .—Equations in terms of direction cosines, referred to axes moving with 
the tangent to the corresponding plane trajectory. 
Type —Equations in terms of direction cosines or spherical polar co-ordinates, 
referred to' axes moving with the tangent to the actual twisted trajectory. 
Type y .—Equations similar to the equations of energy and angular momentum 
of a top (spherical polar co-ordinates), referred to the axes used for type f3. 
In each case the equations obtained are simplified by certain approximations, and 
