362 MESSRS. R. H. FOWLER, E. G. GALLOP, C. N. H. LOCK AND H. W. RICHMOND : 
4.23. The Exact Motion in a High Angle Trajectory.— It will be shown in the 
next section that, for a trajectory of much higher elevation than 50 degrees, the 
approximations for the particular integral break down, and the equations of type a, 
are not applicable to the later stages of the trajectory when the velocity has fallen 
much below 500 f.s. These later stages occur after the initial oscillations have been 
damped out, and are suitable for the use of equations of type ft. These equations 
can be integrated step-by-step on the basis of the wind channel values of R, L, and 
M (fig. 2), which apply to velocities up to 700 f.s. The process is analogous to the 
usual method of calculating a plane trajectory, but rather more laborious, and has 
been carried out in one case only for a 3-inch 12^-lb. shell fired at 70 degrees with a 
muzzle velocity of 2450 f.s. At 40 seconds the yaw has reached the large value of 
60 degrees and is still increasing. This is partly due to the large initial value of the 
stability factor (about 4*0) indicating that the spin is unnecessarily large for this 
shell. The results of comparing the drift with observation were again fairly 
satisfactory in this case; but details of these results cannot be given here. 
§ 4.3. Estimate of the Errors in the Various Solutions. 
In the development of the various solutions of the equations of motion in Part III., 
it was found necessary to neglect certain terms. We shall now proceed to examine 
these terms in succession, and to determine, as far as possible, their numerical values, 
using the values of the various force components obtained from our experiments. By 
so doing we shall justify the use of the*solutions by showing that the terms neglected 
are all very small over the range covered by the jump card experiments. In the 
applications to the later parts of a trajectory, the solutions break down in certain 
cases, and an examination of the error terms enables us to define the circumstances 
under which the solutions are valid. We proceed to examine the various terms. It 
is necessary as a rule to distinguish the terms neglected in obtaining the comple¬ 
mentary function from the terms neglected in obtaining the particular integral. 
In the complementary function, m, n, y, z are periodic functions of the time with 
periods comparable with £3. For the solution to be applicable we have to assume 
that S is always small (say d<0*l radian). Then m, n, y , z are all small quantities 
comparable with S, and m'/Q, m"/Q 2 , &c., are also comparable with S, while (l—l), I'/Q, 
l"/Q 2 , &c., are of the order of <f. In neglecting terms independent of 6\ from the 
equations (3.202), (3.203), we are guided by the condition that all terms neglected 
should be of the order of S 2 compared with those retained. As regards the terms 
containing 6\ or 0" 1 , it appears that the maximum value of 6\/Q in the 50 degrees 
trajectory (for rifling 1 in 30) is 30 x 10 -5 , its initial value being 5 x 10 -5 . Hence all 
terms such as nm'6\, n6\ 2 are completely negligible in obtaining the complementary 
function. 
If all terms in 6\, 6" 1 are removed from equations (3.202), (3.203), they become 
