328 MESSRS. R. H. FOWLER, E. G. GALLOP, C. N. H. LOCK AND H. W. RICHMOND: 
The force components that we propose to include at this stage are It, L, M, and 
ANT. To simplify the algebra we write* 
L = Kmv sin S, M = p. sin S. 
The various components can then be represented by the following vectors :— 
(i.) The drag It, by the vector — It A ; 
(ii.) The cross-wind force L, by the vector! kuiv {A —X cos d} ; 
(iii.) The couple M, by the vector! jx [X . A] ; 
(iv.) The couple ANT, by the vector —ANTA. 
The complete equation for the angular motion is therefore 
(3.103) H' — [H . 0] = /x [X . A] — ANTA, 
where H is given by (3.102). Taking the scalar product of both sides ol (3.103) into 
A, we obtain, with the help of (3.011)—(3.013), 
(3.104) N'=-NI\ 
After substituting for N', equation (3.103), written in full, reduces to 
(3.105) ANA' + B [A . A"] — 2B ( A . 0) A'-B (A . 0') A +BO' 
-AN [A. 0] + B ( A . 0) [A . 0] = fi [X. A]. 
3.11. The Equations of Motion of the Centre of Gravity .—The velocity of the 
centre of gravity is represented by the vector vX, and its acceleration is therefore 
represented by the vector 
In addition to the drag and cross-wind force impressed by the air, we shall suppose 
that gravity is acting on the shell. 
* The mass and velocity of the shell are m and v respectively. For the rest of the notation see § 1.31, 
t If a perpendicular AD be drawn from A to OP, DA is parallel to the direction of the cross-wind 
force L, and its length is sin 8, if OA is of unity length. The vector DA is equal to the difference of the 
vectors OA and OD, so that it is equal to A - X cos 8. Hence {A- X cos 8} /sin 8 is the unit vector 
parallel to the cross-wind force. Similarly [ X . A]/sin 8 is the unit vector normal to the plane AOP 
i.e., parallel to the axis of the couple M. It is easy to verify, with the help of (3.012), that 
A - X cos 8 = ([X. A]. X]. 
