AN ELONGATED PROJECTILE. 
583 
and since from equation (14), differentiating with respect to t, 
c 66 r sin 00 = 2c u sin 0 cos 00i{/ + sin 2 
therefore 
cJO + c* 44 sin 0 cos OOif/ 2 + r 44 sin 2 Oi[/if/ — F 2 sin 0 cos 0 
and integrating, 
I) -O, 
k c u ($ 2 + sin 2 Oxj/0 + |c 66 '/ 
n( 
cos 
2 0 
+ 
'33 
.in 2 (9\ _ 
C 11 / 
(17) 
a constant. The kinetic energy is therefore constant—a principle we 
might have assumed. ’ 
Eliminating \j/ between equations (14) and (17), we shall obtain an 
equation the integral of which will give 0 in terms of elliptic functions 
of t. 
Considering, however, only the case of the steady motion of the 
body, when the point 0 describes a uniform helix whose axis is in the 
direction of the axis OZ, the angle 0 between the axis OC of the body 
and OZ remaining constant; putting 6 = a, a constant, and \j/ = /*, a 
constant, and therefore ^ = y-t, we have 
and therefore 
cc — F (——- —^ sin a cos a cos at, 
V33 %/ 
i] — F (~ -sin a cos a sin at, 
V33 c il/ 
,, /cos 2 a sin 2 a\ 
z = F( -+- ; 
\ C 33 C 11 / 
F (1 1 \ . . , 
x sa — (- ) sm a cos a sin [it, 
V- V33 c il' 
F ( \ 1 \ . 
■u sass — — j —■ — } sill a COS a COS at, 
V c 33 *11/ 
T7 /cos 2 a , sin 2 a\ 
F i + ) t ., 
the equations of the helix described by the centre 0 of the body. 
Equation (15) becomes, since 0=0, 
C 44 sin a cos a [i 2 — c 66 r sin a ju + F^ sin a cos a (— -—^ = 0; 
V 33 c n/ 
and dividing out by sin a, 
c 4Jt cos a[F — c 66 r/x + F% (— —* —^ cos a == 0 . 
V 33 c n/ 
