AN ELONGATED PROJECTILE, 
579 
Let OX, OF, OZ be three rectangular axes drawn in fixed directions 
in space, meeting a sphere described with centre 0 and unit radius in 
z 
X, Y, Z. (In the figure the eye is supposed to be placed at 0 —the 
centre of the sphere—and to be looking at the concave side of the 
.surface.) 
Let the axes OA, OB, OC of the figure meet the sphere in A, B, C. 
The axes XYZ may be brought into the position ABC by a rotation 
(1) through an angle \J/ suppose, about the axis OZ, bringing them into 
the position FBZ; (2) by a rotation, through an angle 6 suppose, about 
the axis OB, bringing them into the position BBC) and (3) by a rota- 
tion, through an angle suppose, about the axis OC, bringing them 
into the position ABC. 
Let x, y, z denote the component velocities of the point 0 parallel 
to OX, OY, OZ. We must now express u, v, w, p, q, r in terms of 
so, y, z, 0 , (f) } if/, and apply Lagrange’s equations of motion. 
d /st\ 
dt \8® / 
1 
11 ^ 
II 
O 
.(1) 
d (oT\ 
ST 
dt\hi) 
- v- = 0, . 
by 
.(2) 
d /8T\ 
dt \ 8z ) 
\ 
8*1 § 
II 
o 
.(3) 
d /ST\ 
ST 
dt\S6/ 
II 
|^> 
1 
d /8J'\ 
ST „ 
&/> “ ° .. 
.(5) 
