ON THE MOTION OF ELONGATED PROJECTILES. 
189 
§ 181, (5) by making use of wrong considerations, gets instead of 
the above term in N, 2Xw (g > 1 + « 2 ):' this would obviously be altered 
on turning the GA, GB axes about GG through an arbitrary angle and 
is hence impossible.] 
/ jy\ 
If the origin of coordinates be now moved to the point (0, 0, -p), 
then 2 T assumes the form 
P ( u 2 + v 2 ) + Rw 2 + A {p 2 + q 2 ) + On 2 
where the letters P, R, G are unaltered and the new A is less than 
2 i\T 2 
the old A by —p-. The form of 2 T is now the same as for an 
ellipsoid of revolution and if F be the resultant component of 
momentum to which the fixed axis OZ is taken parallel and the 
directions of the axes are given by the ordinary Lagrangian co¬ 
ordinates 6, t/t, <fi, then Greenhill has shown (Quarterly Journal, Yol. 
xvi., p. 256) that under no forces a prolate spheroid has a steady 
motion given by 
where 
z 
6 = a, \Js — /x, T—n 
Ft 
'l 1\ . 
%=- 
D — -j. Sill a cos a Sill Lit 
U ' 
k r pj 
Ft 
a i\ . 
y=-\ 
„ - n sin a COS a COS lit 
u 
\ R PJ 
n 1 
'cos 2 a sin 2 a\ 
z=F[ 
, R + R ) t 
A cos a n 2 -Cn fi + F 2 
cos a = 0. 
2. When gravity is taken into account we refer the motion of the 
origin fixed in the body to a set of rectangular axes fixed in space of 
which OZ is drawn vertically upwards and OX, OT such that the 
initial linear momentum lies in the plane XOZ. We shall have in 
terms of the Lagrangian cords x, y, z, 0, </>, yjr. 
