306 MESSRS. R. H. FOWLER, E. G. GALLOP, C. N. H. LOCK AND H. W. RICHMOND: 
values of f h from the values of fu for two different positions of the centre of 
gravity. It is found that f L cannot conveniently be directly observed. 
It will be found convenient in the practical use of (1.132) to introduce the force 
component normal to the axis of the shell. If is the corresponding coefficient, it is 
easily seen that, when the yaw is small, 
(1.133) /n=/r+/l 
and that it is that is directly determined by the variations of / M . 
No other relations between the various coefficients are available. Previous to the present experiments, 
when no definite information existed as to the form of f h and / M as functions of v / a , special arbitrary 
assumptions have been made, in order to carry out calculations of the drift of a shell, or of the twisted 
curve described by its centre of gravity. The authors have made considerable use of the assumptions 
that the fractions 
f n ( vla , 8 ) , /l ( v / a , 5) f m ( v / a , 5) 
/r ( v / a , 0) ’ / R ( v / a , 0) ’ / R ( v / a , 0) 
are independent of v / a , and have determined their values by wind-channel observations. Cranz,* using 
essentially the same assumptions, has calculated the values of these fractions by an empirical law due to 
Kummer. It must be emphasised that the use of any assumption of this type is of very dubious validity, 
and that, so far as experiments have yet gone, they have not confirmed any such assumptions. When 
the values of the coefficients / R , / M and / L are required for a shell of any given external shape they can and 
must be determined by direct experiment. 
1.14. In the preceding sections, we have built up, by synthetic arguments, what 
appears to be the most probable complete force system. It will be seen that in so 
doing we have actually introduced what can be regarded as a complete system of 
three forces and three couples referred to three axes at right angles. Owing, how¬ 
ever, to the complex nature of the reactions, it appears to us to be essential to 
construct our force system in this manner, instead of attempting to analyse a complete 
system of three forces and three couples, and assign each component to its proper 
causes. In this construction, we have been guided by considerations of symmetry, 
the theory of dimensions, the analogy with the theory of the aeroplane, and also, of 
course, by the all-important fact that the results of this construction are in 
agreement with experiments, so far as these have yet been carried. Of our seven 
components by far the most important are It, L and M; then, some way behind, H. 
Our experiments were designed to determine L and M, and if possible to throw some 
light on the size of H, and in these objects a successful start has been made.' As a 
result, it seems reasonable to expect that the preceding specification of the complete 
force system will prove to be adequate; but much more work on these and other 
linest is still required. With the numerical knowledge already obtained, which is 
* 1 Zeitschrift fur Math. u. Phys.,’ vol. XLIII., p. 184. 
t For instance, the determination of the coupie I that destroys the axial spin and the behaviour of 
f R as a function of 8. 
