581 
AN ELONGATED PROJECTILE. 
Conversely, if we express x, y, z in terms of u, v, w, 0, if/, we shall 
have 
X cos if/ + y sin if/ =s component velocity in the direction OF 
== (u cos 0 — v sin (j >) cos 0 + w sin 0 
JF F 
=3 — — (sin 0 cos 2 0 + sin 0 sin 3 0) cos 0 + — sin 6 cos 0 
c n C S3 
== F (— — sin 0 cos 0; 
\ c 33 c ll/ 
r— x sin if/ + y cos if/ ss component velocity in the direction OF 
= u sin 0 + v cos 0 =5 0 ; 
and 
therefore 
X 
= F 
(-- 
| sin 6 
cos 
0 
cos 
\ c 33 
c u/ 
y 
= F | 
X- 
| sin 0 
cos 
0 
sin 
\ c 33 
also 
z 
= — 
u sin 0 
cos 
0 + v 
sin 
0 
sin 
0 ■ 
cos 2 0 sin 2 0\ 
, c 33 c n ) 
( 11 ) 
( 12 ) 
(13) 
We have therefore expressed x, y, z —the component velocities of 
the point 0 parallel to the fixed directions OX, OY, OZ —in terms of 
6, if/, and constants, the angle 4> not appearing; and u, v] w —the com¬ 
ponent velocities of the point 0, parallel to the axes OA, OB, OC, 
moving in space and fixed in the body—in terms of 0, 0 >, and constants, 
the angle if/ not appearing. 
We must now express p, q, r in terms of 6 , <0, 0 and 0, <p , if /—their 
rates of increase per unit of time. 
We have 
P — sin 00 — sin 0 cos 00 , 
q == cos 00 + sin 0 sin 00, 
r = <p + cos Oij/; 
and expressing T in terms of 0, 0, if/, 0, 0, 0, 
T= IF* (— + —) + io u + sin* (ty!) + \c m (j, + cos 6$)*; 
\ c 33 C 11 / 
and therefore 
