190 
ON THE MOTION OF ELONGATED PROJECTILES. 
2 T= P [x 2 +y 2 +z 2 ] + (R-P) [(£ cos \|/- + y sin \^) sin 6+z cos <9] 2 
+ A \6 2 + \p sin 2 &\ + (?[</> + \j/ cos 6] 2 
^for z ~ p cos $ is the height of the c. G.^ . 
The Lagrangian equations of motion are 
:\ =0 
/ ’ dt\dy) dt\dz) 
d /dT\ dT MNg . . 
dtW)-Te = - T~ sine ’ 
dt \d}js/ d\js 
d^/dJT' 
dt \0i' 
d/dT\ 
dt \d(frj 
= 0. 
..(i) 
..(ii) 
.(iii) 
The first three equations integrate and give 
Px+(R- P) sin 6 cos \j/. w=H .(iv) 
Pi/ + (R-P) sin 6 sin y/r. w=0 .(v) 
Pz+(R-P) wcos 0 =K-Mgt .(vi) 
These equations will give x ij z when 6 , yjr are known. 
From (ii) we have 
— - [(i2 - P) iv ( - x sin ^ + y cos yjs) sin 6] = 0. 
dt \d\js / 
