192 
ON THE MOTION OF ELONGATED PROJECTILES. 
A better form for calculation is obtained by multiplying (iv), (v), (vi) 
by x y z and adding:— 
P (£ 2 +y 2 + $) + (R-P)w 2 =Mx + (K- Mgt) z 
. •. Hoc + ( K- Mgt) z + A (\p sin 2 6+6 2 ) + Cn 2 + 2 Mg {^~p cos = ^ A..(xii) 
3. If an elongated shot be fired from a rifled gun at an elevation f3 
in the plane XOZ with initial velocity U and spin n we shall have as 
equations of motion, 
[whence 
| Px+[R - P)w sin 6 cos \J/ = RU cos /3 
' ~ P) w sin 6 sin \|^ = 0 
Pz + (R-P) w cos 6 — RU sin /3- Mgt. 
Rw — RU cos/3sin 6 cos y\r + (RU sin/3 — Mgt) cos 6] 
( Ayjs sin 2 6 + Cn cos 6—yRU cos/3 = Cn sin /3 
RU i?cos/3+(I2i[7 smfi-Mgt)z+A (0 2 + \p sin 2 0) 
+ 2 Mg (z - p cos = R U 2 - g sin /3, 
for initially 
0=|-/3, yj, = 0, 
x= U cos /3, y = 0, £= £7sin/3, w— U, 0 = 0, \p- = 0. 
4. It is of interest to know at any rate the initial motion: to do 
this we expand x y z 6 \Jr in powers of r the time that has elapsed 
since the beginning of the motion : thus:— 
t "‘2 
X = U COS p . T+ '~2J- + 3 T t3# ” 
y = 
T* 1 "1” ~ -4- 
2! ^ 3:1 ■ 
z=U sin /3. r + r 2 +|yr 3 +... 
Hr* 3 + H’ J+ 5T t,+ - 
'\' = 
r T >+^ + 
2! ^ 3! ^ 
Substituting in the three momentum and the first angular equation 
to the first power of t we get 
P[U cos /3 + x" r] + (R - P) [( U cos (3 + x" r) cos 2 /3 + ( U sin /3 + z" r) sin /3 cos /3] = H 
Pu"t= 0 , 
