358 MESSRS. R. H. FOWLER, E. G. GALLOP, C. N. H. LOCK AND H. W. RICHMOND : 
below 700 feet per second. Hence, when the yaw exceeds 0-1 radian, the wind 
channel values for the various force components as shown in fig. 2 can be used; it 
will, however, be necessary to abandon the above method of solution and proceed by 
the step-by-step integration of the equations of type /3. 
Throughout the following work all numerical results will be based on a set of plane 
trajectories of a 16-lb. shell, of external form A, fired at a muzzle velocity of 2000 f.s., 
calculated by the ordinary ballistic methods.* The various elements of the trajec¬ 
tories at elevations of 30 degrees and 50 degrees and a list of constants for the 
service shell, to which the calculations apply, are given in Table VIIIa. 
From the value of £ for the general solution, as given in § 3.65, we can deduce the 
true path of the centre of gravity in terms of the tabulated elements of the plane 
trajectory. Let (X x , Yj, 0) be the co-ordinates of the shell in the plane trajectory at 
time t, and (X, Y, Z) the corresponding co-ordinates in the true (twisted) trajectory. 
The direction cosines of the tangents to the two trajectories are Y \fv 1 , 0), or 
(cos 0], sin 0 1? 0) and Xl/v, Y'/v, Z'/v, so that, to the usual order of approximation, 
(4.201) c£ = (Y'-Y'O (cos 0 a )M“ (X'-X' a ) (sin Oj/v. + iZ'/v, 
= (H'-kZOM, 
say, while the condition v = v x gives 
(4.202) (X'-X'O cos 0!+ (Y'-Y\) sin 0 X = 0. 
It is convenient to separate the parts of the solution arising from the comple¬ 
mentary function and the particular integral. To determine the latter, we use 
equations (3.632), (3.633), and (4.201), obtaining 
(4.203) 
71 = ^ p — 4&vc0/ dt 
Vi Jo cQ 
= cVo 
say, neglecting the terms i9"jQ in (see § 3.20). 
Therefore 
(4.204) 
where \p- may be written (since —O'Jc = g/v j) 
This equation defines \Js. 
A = 
< 7 *AN dt _ A g p N f L (v/a) dt 
ixv x mr Jo / M {via) v 2 
To the same approximation (X' — X'j)/^ and (Y' — Y'^/vi are 0(l/Q 2 ), so that 
(Xj —X) and (Yj—Y) are small compared to Z, so long as the approximations hold. 
The above result is identical in form with the “ classical ” formula of Mayevski, 
* Trajectories were calculated with the ballistic coefficient 1 ’75. 
