THE AERODYNAMICS OF A SPINNING SHELL. 
3G7 
any stage is less in numerical value than the last term retained.* Hence the 
numerical estimates of the third term, obtained above, justify the use of the first 
term only to obtain an approximate value of the drift at all elevations up to 50 degrees 
and for the initial part of a trajectory at any elevation. 
Part V.— Summary and Conclusion. 
§ 5.0. Summary of Preceding Results. 
In the earlier parts of this paper we have suggested a tentative set of components 
for the complete force system acting on a shell moving through air (or other medium), 
in which this complete system may be assumed to depend at any moment only on the 
position and velocities of the shell. We have submitted these suggestions to the test 
of experiment, and found that, so far as we have carried the analysis in this paper, 
the experiments confirm our suggestions, and provide, when the yaw is small, 
numerical values for two of the force coefficients (jf M with a probable error of 2 per cent, 
and f L with a probable error of 10 per cent.) for velocities up to double the velocity 
of sound. Plough values for a third coefficient f n are also provided. It appears 
probable that the other components (except of course the drag) are much less 
important, and that values of the yaw up to perhaps 10 degrees may be regarded as 
small in this connection. 
It is convenient to summarize here what we do and do not know about the 
components of the force system on the shells used in this trial. The values of the 
drag coefficient f R may be regarded as known for all velocities at zero yaw. The 
values of f M and f h are roughly known for velocities up to vja = 2*0, and values of 
the yaw less than 10 degrees. From wind channel experiments f R , / M and f L are all 
known for all values of the yaw when v/a is small, and these determinations probably 
apply so long as v/a <0-7. The damping effects are only known roughly, but 
sufficient is known to estimate how long a shell will take effectively to settle down to 
a steady state of motion. 
On the other hand the variation of f n with yaw is entirely unknown except from 
wind channel experiments, and so is the variation of f M and f L at values of the yaw 
greater than 10 degrees. The rate of diminution of the axial spin is unknown and so 
is the size of the swerve effect, though this latter is not likely to be important. 
The variation of f R with yaw could be studied experimentally by a suitable 
combination of jump card observations, with the use of the solenoid chronograph to 
determine as exactly as possible the deceleration of the shell at every point. The 
values of jf M and f L for larger values of the yaw could be obtained by a detailed 
analysis of unstable rounds in which large values of the yaw are realized. A start 
* The equation is now of the first order in -q only, so that the exact solution may be written down in 
the form of an integral. By successive integration by parts we obtain the expansion (3.632) together 
with an integral representing the error after n terms. 
3 E 
YOL. CG’XXI.-A. 
