332 MESSRS. R. H. FOWLER, E. G. GALLOP, C. N. H. LOCK AND H. W. RICHMOND : 
plane trajectory can no longer be regarded as a valid first approximation, and the only 
possible method is to obtain equations of motion which are suitable for direct step-by- 
step integration. For this purpose the following set of moving axes are most suitable, 
as they reduce the equations of motion of the centre of gravity to its simplest form. 
We take the true direction of motion OP for the axis 1 and a horizontal line at right 
angles to OP for the axis 3. We define the position of OP by spherical polar 
co-ordinates 6, \fs with respect to axes fixed in direction at O, see fig. 10. Then X has 
components (l, 0, 0), 0 has components (— \Js' sin 0, — \p~'cos6, O'), Gr has components 
( — g sin 0, —g cos 6 , 0) and A components (l, m, n) as before. 
Fig. 10. OX, Y, Z are fixed axes, OY being the upwards vertical; the plane XOY contains the 
line of fire. 
Equation (3.105), when written out in full, becomes very complicated. To simplify 
it, we can, under certain circumstances, neglect the angular momentum about a 
transverse axis compared to the angular momentum about the axis of the shell. The 
legitimacy of this approximation, which is equivalent to putting B = 0 in (3.105), is 
discussed in § 4.33. It should be stated that this type of approximation also is 
classical,* but that the equations we obtain are apparently new and of a wide range 
of validity. 
As before, we have 
( 3 . 301 ) N' - -Nr. 
The second and third components of (3.105) reduce to . 
ANm'—AN {—nyj/ sin 6 — 16') — —^n, 
ANA—AN (—Zi// cos 6+m\Js' sin 6) = iu.m, 
* See, e.g., Charbonnier, ‘Traits de Balistique ExtArieure,’ Livre V., Chap. IV. 
