188 
ON THE MOTION OF ELONGATED PROJECTILES. 
Hence by the formula on the middle of p. 126, is initially 
finite where z is the coordinate giving the distance from the vertical 
plane initially containing the tangent: this is obviously untrue, and 
as a matter of fact, neglecting as Greenhill does any displacement 
of the centre of gravity from the centre of figure, the fifth differential 
coefficient with respect to the time is the first that does not vanish. 
Again, the angle that Greenhill neglects in identifying the axis 
and tangent is ft on page 129, and for it he gets an expression 
where 
• 2c e n da 
sm 2/3 = — 6 -j- 
Ci — Co at 
sin2„'=8 
TV k 2 2 V 3 
so that /3' is initially finite and only vanishes by accident. 
The method adopted in the following essay is to obtain six first 
integrals of the equations of motion of a solid of revolution moving in a 
perfect liquid under gravity and thence to determine the initial motion. 
This agrees with the facts of Rev. F. Bashforth’s Chapter, p. 128, 
in that by (xvii.) 6"' is negative, so that the axis rises, by (xviii.) i|r IV is 
negative, therefore the axis turns to the right and by (xix.) the 
shot moves to the right. The order of differential coefficients also 
agrees with the succession of the phenomena, the axis first rising 
and then turning to the right. 
I have not attempted to work out a numerical case, knowing 
nothing of the practical methods, but a computer might find the 
values of the velocities of the c. G. at intervals and by substituting 
in the integrals of the equations of motion, get the values of the 
6, r , y at these times and thence the drift. 
1. In the course of the following analysis with regard to the motion 
under gravity of a solid of revolution through a liquid, it will be 
assumed that the liquid is perfect and incompressible and that it 
extends to infinity. 
If we take a system of rectangular axes GA, GB, GG fixed in 
the body, such that G is the centre of gravity and GG the axis of 
revolution, and if the motion of the body be defined by the instan¬ 
taneous angular velocities of p, q, r about, and the translational 
velocities u , v, w of the origin G parallel to, the instantaneous positions 
of these axes, then Kirch off has shown (Gesammelte Abhandlungen , 
p. 391) that the kinetic energy of the solid and fluid together may 
be taken as given by:— 
2 T= P {u 2 + v 2 ) + Rw 2 
+ A ( p 2 + q 2 ) + Cr 2 
+ 2 N(uq-vp). 
[It may, perhaps, here be remarked that although this expression 
is correctly given by Kirchoff and Lamb, § 116 (c.), yet Basset, Vol. I. 
